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abstract: 'I will review the latest results for the presence of diffuse light in the nearby universe and at intermediate redshift, and then discuss the latest results from hydrodynamical cosmological simulations of cluster formation on the expected properties of diffuse light in clusters. I shall present how intracluster planetary nebulae (ICPNe) can be used as excellent tracers of the diffuse stellar population in nearby clusters, and how their number density profile and radial velocity distribution can provide an observational test for models of cluster formation. The preliminary comparison of available ICPN samples with predictions from cosmological simulations support late infall as the most likely mechanism for the origin of diffuse stellar light in clusters.'
author:
- Magda Arnaboldi
title: Intracluster Planetary Nebulae as dynamical probes of the diffuse light in galaxy clusters
---
[ address=[INAF, Osservatorio Astronomico di Torino, Strada Osservatorio, 20, I-10025 Pino Torinese]{} ]{}
Observations of Diffuse Light
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The study of the intracluster light (ICL) began with Zwicky’s (1951) claimed discovery of an excess of light between galaxies in the Coma Cluster. Its low surface brightness ($\mu_B > 28$ mag arcsec$^{-2}$) makes it difficult to study the ICL systematically (Oemler 1973; Thuan & Kormendy 1977; Bernstein et al. 1995; Gregg & West 1998; Gonzalez et al. 2000).
Presence of diffuse light can be revealed by tails, arcs or plumes which are narrow (about $\sim 2$ kpc) and extended ($\sim 50 - 100$ kpc), or as a halo of light at the cluster scales, which is present in the 200 -700 kpc radial range. In the Coma cluster, Adami et al. (2005) have searched for ICL small features using a wavelet analysis and reconstruction technique. They identified 4 extended sources, with 50 to 100 kpc diameter and V band magnitudes in the 14.5 - 16.0 range. Quantitative, large scale measurements of the diffuse light in the Virgo cluster were recently attempted by Mihos et al. (2005): these deep observations show the intricate and complex structure of the ICL in Virgo.
On the cluster scales, the presence of diffuse light can be revealed when the whole distribution of stars in clusters is analysed in a way similar to Schombert’s (1986) photometry of brightest cluster galaxies (BCGs). When this component is present, the surface-brightness profiles centred on the BCG turn strongly upward in a $(\mu,R^{1/\alpha})$ plot for radii from 200 to 700 kpc. This approach to ICL low surface brightness measurements was taken by Zibetti et al. (2005), who studied the spatial distribution and colors of the ICL in 683 clusters of galaxies at $z\simeq 0.25$ by stacking their images, after rescaling them to the same metric size and masking out resolved sources.
In nearby galaxy clusters, intracluster planetary nebulae (ICPNe) can be used as tracers of the ICL; this has the advantages that detection of ICPNe are possible with deep narrow band images and that the ICPN radial velocities can be measured to investigate the dynamics of the ICL component. ICPN candidates have been identified in Virgo (Arnaboldi et al. 1996, 2002, 2003; Feldmeier et al. 2003, 2004a) and Fornax (Theuns & Warren 1997), with significant numbers of ICPN velocities beginning to become available (Arnaboldi et al. 2004).
The overall amount of the ICL in galaxy clusters is still a matter of debate. However, there is now observational evidence that it may depend on the physical parameters of clusters, with rich galaxy clusters containing 20% or more of their stars in the intracluster component (Gonzalez et al. 2000; Gal-Yam et al. 2003), while the Virgo Cluster has a fraction of $\sim 10$% in the ICL (Ferguson et al. 1998; Durrell et al. 2002; Arnaboldi et al. 2002, 2003; Feldmeier et al. 2004a), and the fraction of detected intragroup light (IGL) is 1.3% in the M81 group (Feldmeier et al. 2004b) and less than 1.6% in the Leo I group (Castro-Rodríguez et al. 2003). Recent hydrodynamical simulations of galaxy cluster formation in a $\Lambda$CDM cosmology have corroborated this observational evidence: in these simulated clusters, the fraction of the ICL increases from $\sim$ 10% $-$ 20% in clusters with $10^{14} M_\odot$ to up to 50% for very massive clusters with $10^{15} M_\odot$ (Murante et al. 2004). Strong correlation between ICL fraction and cluster mass is also predicted from semi-analytical models of structure formation (Lin & Mohr 2004).
The mass fraction and physical properties of the ICL and their dependence on cluster mass will be related with the mechanisms by which the ICL is formed. Theoretical studies predict that if most of the ICL is removed from galaxies because of their interaction with the galaxy cluster potential or in fast encounters with other galaxies, the amount of the ICL should be a function of the galaxy number density (Richstone & Malumuth 1983; Moore et al. 1996). The early theoretical studies about the origin and evolution of the ICL suggested that it might account for between 10% and 70% of the total cluster luminosity (Richstone & Malumuth 1983; Malumuth & Richstone 1984; Miller 1983; Merritt 1983, 1984). These studies were based on analytic estimates of tidal stripping or simulations of individual galaxies orbiting in a smooth gravitational potential. Nowadays, cosmological simulations allow us to study in detail the evolution of galaxies in cluster environments (see, e.g., Moore et al. 1996; Dubinski 1998; Murante et al. 2004; Willman et al. 2004; Sommer-Larsen et al. 2005). Napolitano et al. (2003) investigated the ICL for a Virgo-like cluster in one of these hierarchical simulations, predicting that the ICL in such clusters should be unrelaxed in velocity space and show significant substructures; spatial substructures have been observed in one field in the ICPNe identified with \[O III\] and H$\alpha$ (Okamura et al. 2002).
Diffuse light in clusters from cosmological simulations
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Cosmological simulations of structure formation facilitate studies of the diffuse light and its expected properties. Dubinski (1998) constructed compound models of disk galaxies and placed them into a partially evolved simulation of cluster formation, allowing an evolutionary study of the dark matter and stellar components independently. Using an empirical method to identify stellar tracer particles in high-resolution cold dark matter (CDM) simulations, Napolitano et al. (2003) studied a Virgo-like cluster, finding evidence of a young dynamical age of the intracluster component. The main limitations in these approaches is the restriction to collisionless dynamics.
Murante et al. (2004) analyzed for the first time the ICL formed in a cosmological hydrodynamical simulation including a self-consistent model for star formation. In this method, no assumptions about the structural properties of the forming galaxies need to be made, and the gradual formation process of the stars, as well as their subsequent dynamical evolution in the non-linearly evolving gravitational potential can be seen as a direct consequence of the $\Lambda$CDM initial conditions.
Murante et al. (2004) identified 117 clusters in a large volume of $192^3\, h^{-3}{\rm Mpc}^3$, and analyze the correlations of properties of diffuse light with, e.g., cluster mass and X-ray temperatures. Galaxies at the centers of these clusters have surface-brightness profiles which turn strongly upward in a $(\mu,R^{1/\alpha})$ plot. This light excess can be explained as IC stars orbiting in the cluster potential. Integrating its density distribution along the line-of-sight (LOS), the slopes from Murante et al. (2004) simulations are in agreement with those observed for the surface brightness profiles of the diffuse light in nearby clusters.
At large cluster radii, the surface brightness profile of the ICL appears more centrally concentrated than the surface brightness profile of cluster galaxies (see Figure \[fig1\] ). The prediction of ICL being more centrally concentrated than the galaxy cluster light has been tested observationally. Zibetti et al. (2005) have presented surface photometry from the stacking of 683 clusters of galaxies imaged in the g- , r-, and i-bands in the SDSS. They have been able to measure surface brightness as deep as $\mu_r \sim 32$ mag arcsec$^{-2}$ for the ICL light and $\mu_r \sim 29.0$ for the total light, out to 700 kpc from the BCG. They finds that the ICL is significantly more concentrated than the total light.
From the simulations carried out by Murante et al. (2004), they also obtained the redshifts $z_{form}$ at which the stars formed: those in the IC component have a $z_{form}$ distribution which differs from that in cluster galaxies, see Figure \[fig2\]. The “unbound” stars are formed earlier than the stars in galaxies. The prediction for an old stars’ age in the diffuse component agrees with the HST observation of the IRGB stars in the Virgo IC field, e.g. $t> 2$Gyr (Durrell et al. 2002), and points toward the early tidal interactions as the preferred formation process for the ICL. The different age and spatial distribution of the stars in the diffuse component indicate that it is a stellar population that is not a random sampling of the stellar populations in cluster galaxies.
![Schombert–like analysis on the [*stacked*]{} 2D radial density profile (BCG + ICL) of clusters in the Murante et al (2004) simulation (triangles). The light excess is evident at large cluster radii. The solid line shows the function $
\log \Sigma(r) = \log \Sigma_e -3.33 [(r/r_e)^{1/\alpha} -1]$, with best–fit parameters $\log \Sigma_e = 20.80$, $r_e = 0.005$, $\alpha =
3.66$ to the BCG inner stellar light. Also shown are the averaged 2D density profile of stars in galaxies (dotted line) and in the field (dashed line). In the inserts, the results are shown from the same analysis for the most luminous clusters with $T>4$ keV (left panel), and for less luminous ones with $0<T<2$ keV (right panel). The resulting best–fit parameters are respectively $\log \Sigma_e = 16.47$, $r_e =
0.11$, $\alpha = 1.24$ and $\log \Sigma_e = 23.11$, $r_e = 0.00076$, $\alpha = 4.37$. In the main plot and in the inserts the unit $(R/R_{200})^{1/\alpha}$ refers to the $\alpha$ values given by each Sersic profile. From Murante et al. (2004).[]{data-label="fig1"}](arnaboldi.fig1.eps){height=".3\textheight"}
Murante et al. (2004) studied the correlation between the fraction of stellar mass in the diffuse component and the clusters’ total mass in stars, based on their statistical sample of 117 clusters. This fraction is $\sim 0.1$ for cluster masses $M > 10^{14}h^{-1}M_\odot$ and it increases with cluster mass: the more massive clusters have the largest fraction of diffuse light, see Figure \[fig2\]). For $M \sim
10^{15}h^{-1}M_\odot$, simulations predict as many stars in the diffuse component as in cluster galaxies.
![Left: Fraction of stellar mass in diffuse light vs. cluster mass. Dots are for clusters in the simulated volume; asterisks show the average values of this fraction in 9 mass bins with errorbars. Right panel: histograms of clusters over mean formation redshift, of their respective bound (dashed) and IC star particles (solid line). Mean formation redshifts are evaluated for each cluster as the average on the formation redshift of each star particle. From Murante et al. (2004).[]{data-label="fig2"}](arnaboldi.fig2.eps){height=".3\textheight"}
Predicted dynamics of the ICL
-----------------------------
In the currently favored hierarchical clustering scenario, fast encounters and tidal interactions within the cluster potential are the main players of the morphological evolution of galaxies in clusters. Fast encounters and tidal stirring cause a significant fraction of the stellar component in individual galaxies to be stripped and dispersed within the cluster in a few dynamic times. If the timescale for significant phase-mixing is on the order of few cluster internal dynamical times, then a fraction of the ICL should still be located in long streams along the orbits of the parent galaxies. Detections of substructures in phase space would be a clear sign of late infall and harassment as the origin of the ICL.
A high resolution simulation of a Virgo-like cluster in a $\Lambda
CDM$ cosmology was used to predict the velocity and the clustering properties of the diffuse stellar component in the intracluster region at the present epoch (Napolitano et al. 2003). The simulated cluster builds up hierarchically and tidal interactions between member galaxies and the cluster potential produce a diffuse stellar component free-flying in the intracluster medium. The simulations are able to predict the radial velocity distribution expected in spectroscopic follow-up surveys: they find that at $z=0$ the intracluster stellar light is mostly dynamically unmixed and clustered in structures on scales of about 50 kpc at a radius of $400-500$ kpc from the cluster center.
![Projected phase-space diagram for a simulated ICPN sample in a N-body simulation of a Virgo-like cluster. From Napolitano et al. (2003).[]{data-label="fig3"}](arnaboldi.fig3.eps){height=".3\textheight"}
Willman et al. (2004) and Sommer-Larsen et al. (2005) have studied the dynamics of the ICL in cosmological hydrodynamical simulations for the formation of a rich galaxy cluster. In a Coma-like rich cluster, Willman et al. (2004) finds that the ICL show significant substructure in velocity space, tracing separate streams of stripped IC stars. Evidence is given that despite an un-relaxed distribution, IC stars are useful mass tracers, when several fields at a range of radii have measured LOS velocities. According to Sommer-Larsen et al. (2005), IC stars are colder than cluster galaxies. This is to be expected because diffuse light is more centrally concentrated than cluster galaxies, as found in cosmological simulations (see Murante et al. 2004) and confirmed from observations of intermediate redshift clusters (Zibetti et al. 2005), and both the ICL and galaxies are in equilibrium with the same cluster potential.
Intracluster planetary nebulae in the Virgo cluster: the projected phase space distribution
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Intracluster planetary nebulae (ICPNe) have several unique features that make them ideal for probing the ICL. The diffuse envelope of a PN re-emits 15% of the UV light of the central star in one bright optical emission line, the green \[OIII\]$\lambda 5007$ Åline. PNe can therefore readily be detected in external galaxies out to distances of 25 Mpc and their velocities can be determined from moderate resolution $(\lambda /\Delta \lambda \sim 5000)$ spectra: this enables kinematical studies of the IC stellar population.
PNe trace stellar luminosity and therefore provide an estimate of the total IC light. Also, through the \[OIII\] $\lambda 5007$ Åplanetary nebulae luminosity function (PNLF), PNe are good distance indicators, and the observed shape of the PNLF provides information on the LOS distribution of the IC starlight. Therefore ICPNe are useful tracers to study the spatial distribution, kinematics, and metallicity of the diffuse stellar population in nearby clusters.
Current narrow band imaging surveys
-----------------------------------
Several groups (Arnaboldi et al. 2002, 2003; Aguerri et al. 2005; Feldmeier et al. 2003, 2004a) have embarked on narrow-band \[OIII\] imaging surveys in the Virgo cluster, with the aim of determining the radial density profile of the diffuse light, and gaining information on the velocity distribution via subsequent spectroscopic observations of the obtained samples. Given the use of the PNLF as distance indicators, one also obtain valuable information on the 3D shape of the Virgo cluster from these ICPN samples.
![Aguerri et al. (2005) surveyed fields in the Virgo cluster core. The CORE field was obtained at the ESO MPI 2.2m telescope, and the SUB field with the Suprime Cam at the 8.2m Subaru telescope. The FCJ field is from prime focus camera of the Kitt Peak 4m telescope and the lower right field, LPC, was observed at the La Palma INT. From Aguerri et al. (2005).[]{data-label="fig4"}](arnaboldi.fig4.eps){width=".6\textwidth"}
Wide-field mosaic cameras, such as the WFI on the ESO MPI 2.2m telescope and the Suprime Cam on the Subaru 8.2m, allow us to identify the ICPNe associated with the extended ICL (Arnaboldi et al. 2002, 2003; Okamura et al. 2002; Aguerri et al. 2005). These surveys require the use of data reduction techniques suited for mosaic images, and also the development and refining of selection criteria based on color-magnitude diagrams from photometric catalogs, produced with SExtractor (Bertin & Arnout 1996).
The data analysed by Aguerri et al. (2005) constitute a sizable sample of ICPNe in the Virgo core region, constructed homogeneously and according to rigorous selection criteria; a layout of the pointings is shown in Figure \[fig4\]. From the study of five wide-fields they conclude that the number density plot in Figure \[fig6\] shows no clear trend with distance from the cluster center at M87, except that the value in the innermost FCJ field is high. However, the spectroscopic results of Arnaboldi et al. (2004) have shown that 12/15 PNe in this field have a low velocity dispersion of 250 kms$^{-1}$, i.e. in fact they belong to the outer halo of M87, which thus extends to at least 65 kpc radius. In the SUB field, 8/13 PNe belong to the similarly cold, extended halo of M84, while the remaining PNe are observed at velocities that are close to the systemic velocities of M86 and NGC 4388, the two other large galaxies in or near this field. It is possible that in a cluster as young and unrelaxed as Virgo, a substantial fraction of the ICL is still bound to the extended halos of galaxies, whereas in denser and older clusters these halos might already have been stripped. If so, it is not inappropriate to already count the luminosity in these halos as part of the ICL. However, in Figure \[fig6\] the plot of the PN number density with radius is also shown for the case in which the PNe in the outer halos of M87 and M84 are removed from the FCJ and SUB samples. In this case, the resulting number density is even more nearly flat with radius, but there are still significant field-to-field variations; in particular, the remaining number densities in SUB and LPC are low.
When one wishes to compare the luminosity of the ICL at the positions of Aguerri et al. (2005) fields with the luminosity from the Virgo galaxies, one adds in further uncertainties, because the luminosities of nearby Virgo galaxies depend very much on the location and field size surveyed in the Virgo Cluster. Aguerri et al. (2005) consider therefore their reported intervals in surface brightness to be their primary result, while the relative fractions of the ICL with respect to the Virgo galaxy light are evaluated for comparison with previous ICPN works, and considered them to be more uncertain.
From the study of four wide fields in the Virgo core, Aguerri et al. (2005) obtain a mean surface luminosity density of $2.7 \times
10^6$ L$_{B\odot}$ arcmin$^{-2}$, rms = $2.1 \times 10^6$ L$_{B\odot}$ arcmin$^{-2}$, and a mean surface brightness of $\mu_B$ = 29.0 mag arcsec$^{-2}$. Their best estimate of the ICL fractions with respect to light in galaxies in the Virgo core is $\sim
5\%$. However, there are significant field-to-field variations. The fraction of the ICL versus total light ranges from $\sim 8\%$ in the CORE and FCJ fields, to less than 1% in the LPC field, which in its low ICL fraction is similar to low-density environments (Castro-Rodríguez et al. 2003). This latter field corresponds to the lowest luminosity density in the mosaic image of the Virgo core region from Mihos et al. (2005).
![Number density of PNe (top) and surface brightness (bottom) in our surveyed fields. In the top panel, circles show the measured number densities from Table 3 of Aguerri et al. (2005), and error bars denote the Poisson errors. For the LPC field our upper limit is given. For the RCN1 field at the largest distance from M87, the uncertainty from the correction for Ly$\alpha$ emitters is substantial and is included in the error bar. The large stars with Poisson error bars show the number densities of PNe in FCJ and SUB fields not including PNe bound to the halos of M87 and M84. In the lower panel, circles show the surface brightness inferred with the average value of $\alpha$ in Table 4 of Aguerri et al. (2005), and error bars show the range of values implied by the Poisson errors and the range of adopted $\alpha$ values. Triangles represent the measurements of the ICL from RGB stars; error bars indicate uncertainties in the metallicity, age, and distance of the parent population as discussed in Durrell et al. (2002). The stars indicate the surface brightness associated with the ICPNe in the FCJ and SUB fields that are not associated with the M87 or M84 halos but are free flying in the Virgo Cluster potential, (Arnaboldi et al. 2004). The dashed line and diamonds show the B-band luminosity of Virgo galaxies averaged in rings (Binggeli et al. 1987). Distances are relative to M87. The ICL shows no trend with cluster radius out to 150 arcmin. From Aguerri et al. (2005).[]{data-label="fig6"}](arnaboldi.fig7.eps){height=".3\textheight"}
Spectroscopic follow-up
=======================
ICPNe are the only component of the ICL whose kinematics can be measured at this time. This is important since the high-resolution N-body and hydrodynamical simulations predict that the ICL is un-relaxed, showing significant substructure in its spatial and velocity distributions in clusters similar to Virgo.
The spectroscopic follow-up with FLAMES of the ICPN candidates selected from three survey fields in the Virgo cluster core was carried out by Arnaboldi et al. (2004). Radial velocities of 40 ICPNe in the Virgo cluster were obtained with the new multi-fiber FLAMES spectrograph on UT2 at VLT. The spectra were taken for a homogeneously selected sample of ICPNe, previously identified in three $\sim 0.25$ deg$^2$ fields in the Virgo cluster core. For the first time, the $\lambda$ 4959 Å line of the \[OIII\] doublet is seen in a large fraction (40%) of ICPNe spectra, and a large fraction of the photometric candidates with m(5007) $ < 27.2$ is spectroscopically confirmed.
The LOS velocity distributions of ICPNe in the Virgo cluster core.
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With these data, Arnaboldi et al. (2004) were able for the first time to determine radial velocity distributions of ICPNe and use these to investigate the dynamical state of the Virgo cluster. Figure \[fig5\] shows an image of the Virgo cluster core with the positions of the imaged fields. The radial velocity distributions obtained from the FLAMES spectra in three of these fields are also displayed in Figure \[fig5\]. Clearly the velocity distribution histograms for the three pointings are very different.
In the FCJ field, the ICPNe distribution is dominated by the halo of M87. There are 3 additional outliers, 2 at low velocity, which are also in the brightest PNLF bin, and therefore may be in front of the cluster. The surface brightness of the ICL associated with the 3 outliers, e.g. the ICPNe in the FCJ field, amounts to $\mu_B \simeq
30.63$ mag arcsec$^{-2}$, in agreement with the surface brightness measurements of Ferguson et al. (1998) and Durrell et al. (2002) of the intracluster red giant stars.
The M87 peak of the FCJ velocity distribution contains 12 velocities with $\bar{v}_{p} = 1276\pm 71$ km s$^{-1}$ and $\sigma_{p} = 247\pm
52$ km s$^{-1}$. The average velocity is consistent with that of M87, $v_{sys} = 1258$ km s$^{-1}$. The distance of the center of the FCJ field from the center of M87 is $15.'0\simeq\,65$ kpc for an assumed M87 distance of $15$ Mpc. The value of $\sigma_p$ is very consistent with the stellar velocity dispersion profile extrapolated outwards from $\simeq 150''$ in Figure 5 of Romanowsky & Kochanek (2001) and falls in the range spanned by their dynamical models for the M87 stars. The main result from our measurement of $\sigma_p$ is that M87 has a stellar halo in approximate dynamical equilibrium out to at least $65$ kpc.
In the CORE field, the distribution of ICPN LOS velocities is clearly broader than in the FCJ field. It has $\bar{v}_{C} = 1491\pm 290$ km s$^{-1}$ and $\sigma_{C} = 1000\pm 210$ km s$^{-1}$. The CORE field is in a region of Virgo devoid of bright galaxies, but contains 7 dwarfs, and 3 low luminosity E/S near its S/W borders. None of the confirmed ICPNe lies within a circle of three times half the major axis diameter of any of these galaxies, and there are no correlations of their velocities with the velocities of the nearest galaxies where these are known. Thus in this field there is a true IC stellar component.
The mean velocity of the ICPN in this field is consistent with that of 25 Virgo dE and dS0 within 2$^\circ$ of M87, $<v_{\rm dE,M87}> =
1436\pm108$ km s$^{-1}$ (Binggeli et al. 1987), and with that of 93 dE and dS0 Virgo members, $<v_{\rm dE,Virgo}> = 1139\pm67$ km s$^{-1}$ (Binggeli et al. 1993). However, the velocity dispersion of these galaxies is smaller, $\sigma_{\rm dE,M87}=538\pm 77$ km s$^{-1}$ and $\sigma_{\rm dE,Virgo}=649\pm 48$ km s$^{-1}$.
The inferred luminosity from the ICPNe in the CORE field is $1.8\times
10^9 L_{B,\odot}$. This is about three times the luminosity of all dwarf galaxies in this field, $5.3\times 10^8 L_{B,\odot}$, but an order of magnitude less than the luminosities of the three low-luminosity E/S galaxies near the field borders. Using the results of Nulsen & Böhringer (1995) and Matsushita et al. (2002), Arnaboldi et al. (2004) estimate the mass of the M87 subcluster inside 310 kpc (the projected distance $D$ of the CORE field from M87) as $4.2\times 10^{13} M_\odot$, and compute a tidal parameter $T$ for all these galaxies as the ratio of the mean density within the CORE field to the mean density of the galaxy. They find $T=0.01-0.06$, independent of galaxy luminosity. Since $T\sim D^{-2}$, any of these galaxies whose orbit [*now*]{} comes closer to M87 than $\sim 60$ kpc would be subject to severe tidal mass loss. Based on the evidence so far, a tantalizing possibility is that the ICPN population in the CORE field could be debris from the tidal disruption of small galaxies on nearby orbits in the M87 halo.
In the SUB field the velocity distribution from FLAMES spectra is again different from CORE and FCJ. The histogram of the LOS velocities shows substructures related to M86, M84 and NGC 4388, respectively, and in Figure \[fig5\] the projected phase space is shown. The association with the three galaxies is strengthened when we plot the LOS velocities of 4 HII regions (see Gerhard et al. 2002) detected with FLAMES in this pointing. The substructures in this distribution are highly correlated with the galaxy systemic velocities. The highest peak in the distribution coincides with M84, and even more so when we add the LOS velocities obtained previously at the TNG (Arnaboldi et al. 2003). The 10 TNG velocities give $\bar{v}_{\rm
M84} = 1079\pm 103$ km s$^{-1}$ and $\sigma_{\rm M84} = 325\pm75$ km s$^{-1}$ within a square of $4 R_e \times 4 R_e$ of the M84 center. The 8 FLAMES velocities give $\bar{v}_{\rm M84} = 891\pm 74$ km s$^{-1}$ and $\sigma_{\rm M84} = 208\pm54$ km s$^{-1}$, going out to larger radii. Note that this includes the over-luminous PNe not attributed to M84 previously. The combined sample of 18 velocities gives $\bar{v}_{\rm M84} = 996\pm 69$ km s$^{-1}$ and $\sigma_{\rm
M84} = 293\pm50$ km s$^{-1}$. Most likely, all these PNe belong to a very extended halo around M84 (see the deep image in Arnaboldi et al.1996). It is possible that the somewhat low velocity with respect to M84 may be a sign of tidal stripping by M86.
Future prospects and Conclusions
================================
The observations indicate that the diffuse light is important in understanding cluster evolution, the star formation history and the enrichment of the Intracluster Medium. Measuring the projected phase space distribution of the IC stars constrains how and when this light originates, and the ICPNe are the only abundant stellar component of the ICL whose kinematics can be measured at this time.
These measurements are not restricted only to clusters within 25 Mpc distance: by using a technique similar to those adopted for studies of Ly$\alpha$ emitting galaxies at very high redshift, Gerhard et al. (2005) were able to detect PNe associated with the diffuse light in the Coma cluster, at 100 Mpc distance, in a field which was previously studied by Bernstein et al. (1995); see also O. Gerhard’s contribution, this conference. Now it has become possible to study ICL kinematics also in denser environments like the Coma cluster, and we can explore the effect of environments with different densities on galaxy evolution.
M.A. would like to thank the organizing committee of the Conference on Planetary Nebulae as Astronomical Tools (Gdansk Poland, 28 June-2 July 2005) for the invitation to give this review. This work has been done in collaboration with Ortwin E. Gerhard, Kenneth C. Freeman, and J. Alfonso Aguerri, Massimo Capaccioli, Nieves Castro-Rodriguez, John Feldmeier, Fabio Governato, Rolf-P. Kudritzki, Roberto Mendez, Giuseppe Murante, Nicola R. Napolitano, Sadanori Okamura, Maurilio Pannella, Naoki Yasuda. M.A. wishes to thank ESO for the support of this project and the observing time allocated both at La Silla and Paranal Telescopes. M.A. wishes to thank the National Astronomical Observatory of Japan, for the observing time allocated at the Subaru Telescope. This work has been supported by INAF and the Swiss National Foundation.
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abstract: 'We consider the signal detection problem in the Gaussian design trace regression model with low rank alternative hypotheses. We derive the precise (Ingster-type) detection boundary for the Frobenius and the nuclear norm. We then apply these results to show that honest confidence sets for the unknown matrix parameter that adapt to all low rank sub-models in nuclear norm do not exist. This shows that recently obtained positive results in [@CGEN15] for confidence sets in low rank recovery problems are essentially optimal.'
address:
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Institut für Mathematik\
Universität Potsdam\
Am Neuen Palais 10\
14469 Potsdam\
Germany\
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Statistical Laboratory\
Center for Mathematical Sciences\
University of Cambridge\
Wilberforce Road\
CB3 0WB Cambridge\
United Kingdom\
author:
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title: On signal detection and confidence sets for low rank inference problems
---
Introduction
============
Consider the Gaussian design trace regression model $$\label{model}
Y_i=\mathrm{tr}(X^i\theta) + \epsilon_i,~~~~i=1, \dots, n,$$ where $\epsilon \sim N(0, I_n)$ is an i.i.d. vector of Gaussian noise. Here the matrices $X^i$ are $d\times d$ square matrices with i.i.d. entries $X^i_{mk} \sim N(0,1)$, and $\theta$ is the unknown $d \times d$ matrix we want to make inference on. We are interested in the case where the model dimension $d^2$ is possibly large compared to sample size $n$, but where $\theta$ has low rank $k$, in which case we write $\theta \in R(k), 1 \le k \le d$. This setting serves as a prototype for various matrix inference problems such as those occurring in compressed sensing [@CP11] or in quantum tomography [@GLFBE10]. We consider here a high-dimensional regime where $\min(d,n) \to \infty$, reflecting contemporary statistical challenges.
The first problem we study in this paper is the *signal detection problem* with low-rank alternatives: We want to test the hypothesis $$H_0: \theta = 0 ~~vs.~~H_1: \theta \neq 0, \theta \in R(k), \|\theta\| \ge \rho,$$ where $\|\cdot\|$ equals either the Frobenius norm $\|\cdot\|_F$ or the nuclear norm $\|\cdot\|_*$ (defined in detail below), and where $\rho$ should be the minimal ‘signal strength’ condition for the above hypothesis testing problem to have a consistent solution (in the sense of Ingster, see [@I03]). We will show that the minimax optimal detection boundary in Frobenius norm is of the form $$\rho \approx \min \left(\sqrt{\frac{d}{n}}, n^{-1/4}\right)$$ whereas in nuclear norm it is $$\rho \approx \min \left(\sqrt{\frac{kd}{n}}, \sqrt {\frac{k}{n^{1/2}}}\right).$$ A remarkable feature is that for the Frobenius norm the detection rate *does not depend at all* on the complexity of the alternative hypothesis (the rank $k$), whereas for the nuclear norm it does. The phase transition between the two regimes in these rates depends precisely on whether the sample size $n$ exceeds the dimension $d^2$ of the maximal parameter space $R(d)$ or not. The upper bounds in our proofs are related to the papers [@ITV10; @ACCP11] about the detection boundary in the sparse regression setting, and our main contribution consists in deriving the matching lower bounds for low rank alternatives.
Our interest in the detection boundary is triggered by the second problem we investigate here: the question of existence and non-existence of adaptive confidence sets for low rank parameters. It follows from general decision-theoretic principles (see Chapter 8.3 in [@GN15] and also [@HN11; @BN13]) that the answer to this question is closely related to a ‘composite version’ of the detection problem (see (\[compo\]) below). This approach was employed in [@NvdG13] to prove that adaptive and honest confidence sets for the parameter $\theta$ do not exist in *sparse* regression models if an $\ell_2$-risk performance beyond $O(n^{-1/4})$ is desired. In contrast in the recent paper [@CGEN15] it was shown that if sparsity constraints are replaced by low rank conditions, then adaptive and fully honest confidence sets exist over the entire parameter space $R(d)$. Adaptation means here that the expected Frobenius norm diameter of the confidence set reflects the minimax risk over arbitrary low rank sub-models $R(k), 1 \le k \le d$. The fact that the detection rates obtained here in Frobenius norm are independent of the rank constraint $\theta \in R(k)$ provides another heuristic explanation of the result in [@CGEN15].
Moreover [@CGEN15] constructed another confidence set whose diameter adapts to low rank sub-models in the stronger nuclear norm distance, and that is honest for all $\theta$’s that are *non-negative definite and have trace equal to one*, that is, whenever $\theta$ is the density matrix of a *quantum state*. Such a constraint on $\theta$ is natural in a quantum physics context considered in [@CGEN15], but not in general. The question arises whether it is essentially necessary or not. In the present paper we show that indeed the existence results of [@CGEN15] are specific to the geometry induced by the Frobenius norm or to the quantum state constraint, and that nuclear-norm adaptive and honest confidence sets over general low rank parameter spaces *do not exist* in the model (\[model\]). For example, our results imply that if one requires coverage of a confidence set over all of $R(d)$ then the worst case nuclear norm diameter for rank-one parameters can be off the minimax estimation rate over $R(1)$ by as much as $\sqrt d$. Our results thus further illustrate the subtleties involved in the theory of confidence sets for high-dimensional parameters, and that the positive results in [@CGEN15] are of a rather specific nature.
Our proofs are given in the simplest model where both the design and the noise are Gaussian, and the matrices involved are of square type. As usual, our results extend without major difficulty to sub-Gaussian design and noise, to certain correlated random designs, and also to non-square matrices, at the expense of slightly more technical proofs. Generalisations of our results to the matrix completion problem are currently under investigation.
Main results
============
Notation
--------
We write $\mathbb M_d$ for the set of $d \times d$ matrices with real elements. If $\mathcal X: \mathbb M_d \to \mathbb R^n$ denotes the ‘sampling operator’ $$\theta \mapsto \mathcal X\theta = \big(\mathrm{tr}(X^1\theta), \dots, \mathrm{tr}(X^n\theta) \big)^T,$$ then the model (\[model\]) can be written as $$Y = \mathcal X\theta + \epsilon,$$ where $Y=(Y_1, \dots, Y_n)^T$ and $\epsilon = (\epsilon_1, \ldots, \epsilon_n)^T$. We write $E^X$ for the expectation over the distribution of $\mathcal X$ only, and $E_\theta$ for the expectation conditional on $\mathcal X$. The full expectation is denoted by $\mathbb E_\theta= E^X E_\theta$. The corresponding probability laws are denoted by $P^X, P_\theta, \mathbb P_\theta$ and we employ the usual $o/O/o_P/O_P$-notation with $\min(n,d) \to \infty$.
We denote the standard norm on Euclidean space by $\|\cdot\|_2$, and the associated inner product by $\langle \cdot, \cdot \rangle_2$. Let $\|.\|_F$ be the Frobenius norm over $\mathbb M_d$, i.e. $$\|M\|_F = \sqrt{\mathrm{tr}( M^TM)} = \sqrt{\sum_{j\leq d} \lambda_j^2},$$ where $\lambda_i^2$ are the eigenvalues of $ M^TM$. The associated inner product is $$\langle U, V \rangle_F = \mathrm{tr}(U^TV).$$ We also define the nuclear norm of $M$ as $$\|M\|_{*} = \sum_{j\leq d} |\lambda_j|.$$ These two norms are in fact defined also for matrices that are not of square type. Finally we recall that for any matrix $M \in R(k)$, we have $$\|M\|_{F}\leq \|M\|_{*}\leq \sqrt{k}\|M\|_{F}.$$
Signal detection for low rank alternatives
------------------------------------------
We consider first the following hypothesis testing problem, also known as the signal detection problem: $$\label{maintest}
H_0: \theta = 0~~vs.~~H_1: \theta \in R(k), \|\theta\| \ge \rho.$$ Here the alternative space is restricted to a ‘low rank’ hypothesis $\theta \in R(k)$ for some $1 \le k \le d$. Moreover, for a separation constant $\rho>0$, the detection boundary is described by a ‘signal strength’ condition measured in terms of the size $\|\theta\| \ge \rho$ of the Frobenius-, or of the nuclear norm of $\theta$. In the high-dimensional regime where $\min (n,d) \to \infty$, we want to find the minimal sequence $\rho \equiv \rho_{n,d}$ such that for any $\alpha>0$ a level $\alpha$-test $\Psi=\Psi(Y,\mathcal X, \alpha)$ exists: $$\label{power}
\left[\mathbb E_0 [\Psi] + \sup_{\theta \in H_1} \mathbb E_\theta [1-\Psi] \right] = \mathbb P_0 (\text{reject } H_0) + \sup_{\theta \in H_1} \mathbb P_\theta (\text{accept } H_0) \le \alpha.$$ Recall that a test is simply a random indicator function $\psi=1_A$ where the rejection event $A$ depends only on $Y, \mathcal X, \alpha$, and we require the sum of the type-one and the type-two error of the test to be controlled at any fixed level $\alpha>0$.
\[signalthm\] Consider the testing problem (\[maintest\]) with norm $\|\cdot\|$. Define
$$r_{n,d} = \begin{cases}
\min(\sqrt{d/n}, n^{-1/4})&\text{if $\|\cdot\|=\|\cdot\|_F$}\\
\min(\sqrt{kd/n},\sqrt k/n^{1/4})&\text{if $\|\cdot\|=\|\cdot\|_*$.}
\end{cases}$$
1\) Suppose $\rho \ge Dr_{n,d}$. Then for every $\alpha>0$ there exists a test $\Psi= \Psi(Y,\mathcal X, \alpha)$ and finite constants $D=D_\alpha>0, n_\alpha \in \mathbb N$ such that (\[power\]) holds for every $n \ge n_\alpha$.
2\) Conversely, suppose $\rho = o(r_{n,d})$ and $k=o(d)$ as $\min(n,d) \to \infty$. Then no test satisfying (\[power\]) for every $\alpha>0$ exists. In fact $$\label{ing}
\liminf_{n,d} \inf_{\Psi} \left[\mathbb E_0 [\Psi] + \sup_{\theta \in H_1} \mathbb E_\theta [1-\Psi] \right] \ge 1$$ where the infimum extends over all test functions $\Psi= \Psi(Y,\mathcal X)$.
The tests $\Psi$ constructed in the proof are given in (\[test\]) below and straightforward to implement. Note also that the $\|\cdot\|_*$-separated alternatives are a subset of the $\|\cdot\|_F$-separated alternatives (see (\[nucsep\]) below), and our results imply that an optimal test for the case $\|\cdot\|=\|\cdot\|_F$ is essentially optimal also for $\|\cdot\|_*$.
Confidence sets for low rank recovery
-------------------------------------
Low rank recovery algorithms are well-studied in compressed sensing and high-dimensional statistics, see e.g., [@CP11; @GLFBE10; @K11; @KLT11; @NW11; @CZ15] and the references therein. In the setting of model (\[model\]) they provide minimax optimal estimators $\tilde \theta$ of $\theta \in R(k)$ with (high probability) performance guarantees $$\label{minimax}
\|\tilde \theta - \theta\|^2_F \lesssim \frac{kd}{n},~~~\|\tilde \theta- \theta\|_* \lesssim k \sqrt{\frac{d}{n}}.$$ The question we study here is whether associated uncertainty quantification methodology exists, that is, whether we can find confidence sets $C_n \subset \mathbb M_d$ such that $$\label{coverage}
\inf_{\theta \in \mathbb M_d}\mathbb P_\theta (\theta \in C_n) \ge 1-\alpha,$$ at least for $\min(n,d)$ large enough, and such that the diameter $|C_n|$ of $C_n$ reflects the accuracy of adaptive estimation in the sense that $|C_n|$ shrinks, with high probability, at the optimal rates from (\[minimax\]) whenever $\theta \in R(k)$. We insist here on an *adaptive* confidence set that does not require knowledge of the unknown rank $k$ of $\theta$.
A first result that is proved in the paper [@CGEN15] is that such adaptive confidence sets do exist in the model (\[model\]) if the diameter is measured in Frobenius distance. The construction of this set is straightforward, see [@CGEN15] for details.
For every $\alpha>0$ there exists a confidence set $C_n=C_n(Y,\mathcal X, \alpha)$ such that for all $n \in \mathbb N$, (\[coverage\]) holds, and such that uniformly in $\theta \in R(k_0)$ for any $1 \le k_0 \le k,$ with high $\mathbb P_\theta$-probability the Frobenius-norm diameter $|C_n|_F$ of $C_n$ satisfies $$|C_n|_{F} \lesssim \sqrt{k_0\frac{d}{n}}.$$
A second result that is proved in the paper [@CGEN15] is that an (asymptotic) adaptive confidence set exists also in nuclear norm provided that the “quantum state constraint" is satisfied, namely, provided it is known a priori that $\theta$ is non-negative definite and has nuclear norm one, and provided the coverage requirement in (\[coverage\]) is relaxed to hold only over a maximal model $R(k)$ in which asymptotically consistent estimation of $\theta$ is possible (i.e., $k\sqrt{d/n} = o(1)$). Define $$R^+(k) = R(k) \cap \{\theta \text{ is non-negative definite},~ \text{tr}(\theta)=1\},$$ the set of quantum state density matrices of rank at most $k$.
\[thm:rich\] Assume $k\sqrt{d/n} = o(1)$ for some $1 \le k \le d,$ and let $\alpha>0$ be given. Then there exists a confidence set $C_n=C_n(Y,\mathcal X, \alpha)$ such that $$\liminf_{\min (n,d) \to \infty} \inf_{\theta \in R^+(k)} \mathbb P_\theta (\theta \in C_n) \ge 1-\alpha,$$ and such that uniformly in $\theta \in R^+(k_0)$ for any $1 \le k_0 \le k,$ with high $\mathbb P_\theta$-probability the nuclear norm diameter $|C_n|_*$ of $C_n$ satisfies $$|C_n|_{*} \lesssim k_0\sqrt{\frac{d}{n}}.$$
In fact it is not difficult to generalise the above theorem to the case where the condition $tr(\theta)=1$ is relaxed to $\|\theta\|_* \le 1$.
The next theorem, which is the main result of this subsection, implies that no analogue of Theorem 2 can hold true if the Frobenius norm there is replaced by the nuclear norm, and it also shows that Theorem 3 cannot hold true if $R^+(k)$ is replaced by $R(k)$, that is, if the ‘quantum state constraint’ is relaxed. More precisely, we show that if a confidence set $C_n$ is required to have coverage over the maximal model $R(k_1)$, then the worst case expected nuclear norm diameter of $C_n$ over arbitrary sub-models $R(k_0), k_0=o(k_1),$ depends on the maximal model dimension $k_1$ and does not improve as $k_0 \downarrow 1$. The proof of Theorem \[main\] is based on Part 2) of Theorem \[signalthm\] and lower bound techniques for adaptive confidence sets from [@HN11; @BN13].
\[main\] Let $k_1 \to \infty$ such that $k_1=o(d)$ as $\min(n,d)\to \infty$. Suppose that for any $0<\alpha<1/3$ the confidence set $C_n=C_n(Y, \mathcal X, \alpha)$ is asymptotically honest over the maximal model $R(k_1)$, that is, it satisfies $$\label{cov2}
\liminf_{\min (n,d) \to \infty} \inf_{\theta \in R(k_1)}\mathbb P_\theta (\theta \in C_n) \ge 1-\alpha.$$ Then for every $k_0 =o(k_1)$ and some constant $c>0$ depending on $\alpha$, we have $$\label{adap}
\sup_{\theta \in R(k_0)} \mathbb E_\theta |C_n|_* \ge c \sqrt{\frac{k_1 d}{n}}$$ for every $\min(n,d)$ large enough. In particular no confidence set exists that is honest over all of $\mathbb M_d$ and that adapts in nuclear norm to any model $R(k_0), k_0 =o(\sqrt d)$.
For notational simplicity we have lower bounded the *expected diameter* $|C_n|_*$ in (\[adap\]), but the proof actually contains a stronger ‘in probability version’ of this lower bound.
A few remarks on Theorem \[main\] are in order:
i\) In the least favourable case where one wants coverage over the entire $R(d) = \mathbb M_d$ while still adapting to rank-one matrices (i.e., $k_0=1$), the performance of any honest confidence set is off the minimax optimal adaptive estimation rate $\sqrt{d/n}$ over $R(1)$ by a diverging factor that can be as close to $\sqrt d$ as desired.
ii\) Even if one restricts coverage to hold only for ‘consistently estimable models’ $R(k_1)$ with $k_1 \sqrt {d/n}\to 0$ (as in Theorem \[thm:rich\]), the diameter $|C_n|_*$ can be off the minimax rate of estimation over $R(1)$ by a factor of $\sqrt k_1$.
iii\) We also note that the above result does *not* disprove the existence of adaptive confidence sets for sub-models $R(k_0)$ of ‘moderate rank’ where $k_0 \ge \sqrt d$. While more of technical interest – note that this rules out $n < d^2$ for consistent recovery to be possible – this regime currently remains open (it is related to the apparently hard problem of finding optimal separation rates in the composite testing problem (\[compo\]) below).
Proofs
======
Proof of Theorem \[signalthm\], upper bounds
--------------------------------------------
When $n<d^2$ then define $$\hat r_n = \frac{1}{n}\|Y\|_2^2 - 1,~~\tau_n = n^{-1/2}$$ but when $n \ge d^2$ set $$\hat r_n = \frac{2}{n(n-1)} \sum_{i<j} \sum_{1 \le m \le d, 1 \le k \le d}Y_i X^i_{mk}Y_jX^j_{mk},~~\tau_n=d/n.$$ The test statistic is $$\label{test}
\Psi_n = 1\left\{\hat r_n \ge z_\alpha \tau_n \right\}$$ where $z_\alpha$ are quantile constants chosen below.
These tests work for Frobenius norm separation, by effectively the same proofs as in [@ITV10], using that we can embed the matrix regression model into a vector regression model with $p=d^2$ parameters, and since the separation rates only depend on the model dimension (and not on low rank or sparsity degrees). However, to provide intuition, we give some details, first for the case $n<d^2$: Under $H_0$ we have $Y=\epsilon$ and so $$\mathbb E_0 \Psi_n = \Pr\left(\frac{1}{\sqrt n} \sum_{i=1}^n (\varepsilon_i^2 -E \varepsilon_i^2) >z_\alpha \right) \le \alpha/2$$ for every $n \in \mathbb N$ and $z_\alpha$ large enough (using either Chebyshev’s inequality and $E\varepsilon_i^4 =3$, or Theorem 4.1.9 in [@GN15] for a more precise non-asymptotic bound). Now for the alternatives $\theta \in H_1$ we use the basic concentration result Lemma 1a) in [@CGEN15] which implies that for any fixed $\theta$ the event $$\mathcal E = \left\{\left|(1/n)\|\mathcal X \theta\|_2^2 - \|\theta\|_F^2 \right|\le \|\theta\|_F^2/2 \right\}$$ has $P^X$-probability at least $1-2\exp(-n/24)$, and so, for $n\ge n_\alpha$ such that $2\exp(-n/24) <\alpha/6$, $$\begin{aligned}
\mathbb E_\theta(1-\Psi_n) &= \mathbb P_\theta \left(\hat r_n < z_\alpha \tau_n \right) \\
&= \Pr \left(\frac{1}{n}\|\mathcal X \theta + \epsilon\|_2^2 -1 < \frac{z_\alpha}{\sqrt n} \right) \\
&= \Pr \left(\frac{1}{n}\|\mathcal X\theta\|_2^2 - \frac{z_\alpha}{\sqrt n} < -\frac{2}{n} \epsilon^T\mathcal X\theta - \frac{1}{n}\sum_{i=1}^n( \varepsilon_i^2-1) \right) \\
&\le \Pr \left(\frac{\|\theta\|_F^2}{2} - \frac{z_\alpha}{\sqrt n} < -\frac{2}{n} \epsilon^T\mathcal X\theta - \frac{1}{n}\sum_{i=1}^n( \varepsilon_i^2-1) , \mathcal E\right) + 2 \exp(-n/24) \\
&\le \Pr \left(\left|\frac{2}{n} \epsilon^T\mathcal X\theta\right| > \|\theta\|_F^2/8, \mathcal E\right) + \Pr\left(\frac{1}{\sqrt n} \sum_{i=1}^n (\varepsilon_i^2 -E \varepsilon_i^2) >z_{\alpha/3} \right) +\alpha/6\end{aligned}$$ since, by the hypothesis on $\rho$, we have for $D$ large enough that $$\frac{\|\theta\|_F^2}{2} - \frac{z_\alpha}{\sqrt n} \ge \frac{\|\theta\|_F^2}{4} \ge \frac{2z_{\alpha/3}}{ n^{1/2}}.$$ The last probability is bounded by $\alpha/6$ as under $H_0$ and the last but one probability is also bounded by $\alpha/6$ by a direct (conditional on $\mathcal X$) Gaussian tail inequality (restricting to the event $\mathcal E$: just as in term II of the proof of Theorem [@CGEN15] with $\tilde \theta=0$ there), so that in total we have bounded the testing errors in (\[power\]) by $\alpha/2 + (3/6)\alpha =\alpha$, as desired. The case $n \ge d^2$ follows from similar but slightly more technical arguments, adapting the arguments from proof of Theorem 3 in [@CGEN15], or arguing directly as in Theorem 4.3 in [@ITV10] with $p=d^2$.
The test (\[test\]) also works for nuclear-norm separation since $$H_1^* = \theta \in R(k): \|\theta\|_* \ge c\sqrt{k} \rho$$ is a subset of $$H_1^F = \theta \in R(k) : \|\theta\|_F \ge c \rho$$ in view of the inequality $$\label{nucsep}
\|\theta\|_F \ge (1/\sqrt k) \|\theta\|_* ~~\forall \theta \in R(k),$$ so that $$\mathbb E_0\Psi_n + \sup_{\theta \in H_1^*} \mathbb E_\theta(1-\Psi_n) \le \mathbb E_0\Psi_n + \sup_{\theta \in H_1^F} \mathbb E_\theta(1-\Psi_n) \le \alpha.$$ We now turn to the more difficult lower bounds.
Proof of Theorem \[signalthm\], lower bounds
--------------------------------------------
Let $\Psi$ be any test – any measurable function of $Y, \mathcal X$ that takes values in $\{0,1\}$. Assume $\rho=o(r_{n,d})$ as $\min(n,d) \to \infty$ and let $H_1=H_1(\rho)$ be the corresponding alternative hypothesis.
*Step I: Reduction to averaged likelihood ratios*: Let $\pi=\pi_{n,d}$ be a sequence of finitely supported probability distributions on $\mathbb M_d$ such that $\pi_{n,d}(H_1) \to 1$, and denote by $\pi|H_1$ that measure restricted to $H_1$ and re-normalised to unit mass. Define $$Z= \mathbb E_{\theta \sim \pi} \prod_{i\leq n} \frac{dP_{i}^{(\theta)}}{dP_{i}^{(0)}} \equiv \int \prod_{i\leq n} \frac{dP_{i}^{(\theta)}}{dP_{i}^{(0)}} d\pi(\theta) ,$$ where $dP_{i}^{(\theta)}$ is the distribution of $Y_i|\mathcal X$ when the parameter generating the data is $\theta$, and $dP_{i}^{(0)}$ is the distribution of $Y_i|\mathcal X$ when the parameter generating the data is $0$. Then, by a standard testing lower bound (e.g., (6.23) in [@GN15]), for any $\eta>0$, $$\begin{aligned}
\mathbb E_0 \Psi + \sup_{\theta \in H_1} \mathbb E_\theta(1-\Psi) &\ge \mathbb E_0 \Psi + \mathbb E_{\theta \sim \pi|H_1} \mathbb E_\theta(1-\Psi)\\
& \ge \mathbb E_0 \Psi + \mathbb E_{\theta \sim \pi}\mathbb E_\theta(1-\Psi) - o(1) \\
&= E^X\left[E_0 \Psi + \mathbb E_{\theta \sim \pi} E_\theta(1-\Psi) \right] - o(1) \\
&\geq (1-\eta)\left[1 - \left[\frac{\sqrt{\mathbb E_0(Z-1)^2}}{\eta}\right]\right] - o(1). \end{aligned}$$ Now since $$\mathbb E_0[Z-1]^2 = \mathbb E_0[Z^2] -1,$$ if we show that $\mathbb E_0[Z^2]\le 1+o(1)$ as $\min(n,d) \to\infty$ for a suitable choice of $\pi$, then the lower bound (\[ing\]) will follow by letting $\eta \to 0$. Recall the notation $\mathbb E_\theta = E^X E_\theta$.
*Step II: Computation of $E_0[Z^2]$:* The $(Y_i)$ are independent with distribution $\mathcal N((\mathcal X\theta)_i,1)$ conditional on the design $\mathcal X$, hence $$\begin{aligned}
Z &=\mathbb E_{\theta \sim \pi} \Bigg[ \prod_{i \leq n} \frac{\exp(-\frac{1}{2}(y_{i} - (\mathcal X\theta)_i)^2)}{\exp(-\frac{1}{2}y_{i}^2)} \Bigg]\\
&=\mathbb E_{\theta \sim \pi}\Bigg[ \prod_{i \leq n} \exp(y_{i} (\mathcal X\theta)_i) \exp(- \frac{1}{2} ((\mathcal X\theta)_i)^2) \Bigg]\end{aligned}$$ and can hence write
$$\begin{aligned}
E_0 \big[Z^2 \big] &= \int_{\mathbb R^n} \Bigg(\mathbb E_{\theta \sim \pi} \Big[\prod_{i \leq n} \exp(y_{i} (\mathcal X\theta)_i) \exp(- \frac{1}{2} ((\mathcal X\theta)_i)^2)\Big] \Bigg)^2 \prod_{i\leq n} \frac{1}{\sqrt{2\pi}}\exp(- \frac{y_i^2}{2} ) dy_1...dy_{n} \nonumber\\
&= \int_{\mathbb R^n} \Bigg(\mathbb E_{\theta \sim \pi} \Big[ \exp(- \frac{1}{2} \|\mathcal X \theta\|_2^2) \prod_{i \leq n} \exp(y_{i} (\mathcal X \theta)_i \Big] \Bigg)^2 \prod_{i\leq n} \frac{1}{\sqrt{2\pi}}\exp(- \frac{y_i^2}{2} ) dy_1...dy_{n} .\nonumber\end{aligned}$$
Thus, if $\theta, \theta'$ are independent copies of joint law $\pi^2$, then we have $$\begin{aligned}
E_0 \big[Z^2 \big] &= \int_{\mathbb R^n} \mathbb E_{\pi^2} \Big[ \exp(- \frac{1}{2} (\|\mathcal X \theta\|_2^2 - \frac{1}{2} (\|\mathcal X \theta'\|_2^2)\prod_{i \leq n} \frac{1}{\sqrt{2\pi}}\exp\big(y_{i} (\mathcal X(\theta+\theta'))_i - \frac{y_i^2}{2}\big) \Big] dy_1...dy_{n}\\
&= \mathbb E_{\pi^2} \Bigg[ \exp(- \frac{1}{2} \|\mathcal X \theta\|_2^2 - \frac{1}{2} \|\mathcal X\theta'\|_2^2)\nonumber\\
&~~~~\times \prod_{i \leq n}\int_{y_i} \Big(\frac{1}{\sqrt{2\pi}} \exp\Big(-\frac{1}{2} \big(y_{i} - (\mathcal X(\theta+ \theta'))_i)^2\Big)dy_i \exp\Big(\frac{1}{2} (\mathcal X(\theta+\theta'))_i^2\Big)\Bigg]\nonumber\\
&= \mathbb E_{\pi^2}\left[ \exp\left(\frac{1}{2}\|\mathcal X(\theta + \theta')\|_2^2 - \frac{1}{2} \|\mathcal X \theta\|_2^2 - \frac{1}{2} \|\mathcal X \theta'\|_2^2 \right)\right]\nonumber
$$
*Step III: Integrating over $\mathcal X$:* The $E^X$-expectation of the last expression can be bounded by $$\mathbb E_{\pi^2} \left[\exp \left(\frac{n}{2} (\|\theta+\theta'\|_F^2 - \|\theta\|_F^2 - \|\theta'\|_F^2) \right) E^X \exp \left(\frac{1}{2}(Z_1-Z_2-Z_3) \right)\right]$$ where $$Z_\ell = \|\mathcal X\vartheta_\ell\|_2^2 - n \|\vartheta_\ell\|_F^2,~~\text{with}~~\vartheta_1=\theta+\theta', \vartheta_2=\theta, \vartheta_3=\theta'.$$ The last factor can be bounded, by applying the Cauchy-Schwarz inequality twice, by $$\label{product}
(E^X\exp(Z_1))^{1/2} (E^X \exp(2Z_2))^{1/4} (E^X \exp(2Z_3))^{1/4}.$$ Since $\mathcal X \vartheta_\ell \sim N(0, \|\vartheta_\ell\|_F^2 I_n)$ the distribution of $Z_\ell$ is the one of $\|\vartheta_\ell\|_F^2 \sum_{i=1}^n (g_i^2-1)$ where the $g_i$ are i.i.d. $N(0,1)$. Applying Theorem 3.1.9 in [@GN15] with $\tau_i \equiv 1$ and $\lambda = \|\vartheta_1\|_F^2$ or $\lambda = 2\|\vartheta_\ell\|_F^2, \ell=2,3,$ (and hence setting $\|A\|=1, \|A\|_{HS}=n$ in that theorem) we see that if $\max_\ell \|\vartheta_\ell\|_F^2 \le 1/4$ then $$E^X \exp(Z_1) \le \exp\left(\frac{n \|\vartheta_1\|_F^4}{1- 2 \|\vartheta_1\|_F^2} \right),~~\text{and}~~E^X \exp(2Z_\ell) \le \exp\left(\frac{2n \|\vartheta_\ell\|_F^4}{1- 4 \|\vartheta_\ell\|_F^2} \right),~\ell=2,3.$$ As a consequence if $$\label{viertel}
\max_{\ell=1,2,3} \|\vartheta_\ell\|_F = o(n^{-1/4})$$ then the the product (\[product\]) is bounded above by $1+o(1)$. We conclude that if the prior $\pi$ satisfies (\[viertel\]) almost surely then $$\begin{aligned}
\mathbb E_0[Z^2] = E^XE_0[Z^2] &\le \left(1+o(1) \right) \times E_{\pi^2} \exp \left(\frac{n}{2} (\|\theta+\theta'\|_F^2 - \|\theta\|_F^2 - \|\theta'\|_F^2) \right) \\
& = \left(1+o(1) \right) \times E_{\pi^2} \exp \left(n \langle \theta, \theta' \rangle_F \right) .\end{aligned}$$
*Step IV: Construction of $\pi$ and bounds for $\mathbb E_0[Z^2]$:* Assume for notational simplicity that $d$ is an integer multiple of $k$, the general case needs only minor notational adjustment. Pick independent random $d \times 1$ vectors $v_\ell: \ell=1, \dots, k$ each of which consists of i.i.d. Rademacher entries (i.e., taking values $\pm 1$ with probability $1/2$). Create a matrix $W$ as follows: In the first $d/k$ columns insert $v_1$ times a random sign $B_{1,j}, j=1, \dots, d/k$. Then, in the $\ell$-th block repeat the same with $v_1$ replaced by $v_\ell$, and random signs $B_{\ell,j}, j=1, \dots, d/k$. If $\|\cdot\|=\|\cdot\|_F$ let $\gamma_n=\rho_n/d $ and if $\|\cdot\|=\|\cdot\|_*$ set $\gamma_n = 2\rho_n /(\sqrt k d)$, so that in either case $$\gamma_n= o\left(\min(\sqrt{1/dn}, d^{-1}n^{-1/4}) \right).$$ Define the random matrix $\theta = \gamma_n W$ and let $\theta'$ be an independent copy of it. Thus $$n\langle \theta, \theta' \rangle_F = n\gamma_n^2 \sum_{\ell =1}^k \sum_{m=1}^d \sum_{j=1}^{d/k} v_{\ell,m} B_{\ell,j} v'_{\ell,m} B_{\ell,j}' = n\gamma_n^2 \sum_\ell \sum_m v_{\ell,m} v_{\ell,m}' \sum_j B_{\ell,j} B'_{\ell,j}.$$ As products of Rademacher variables are again Rademacher variables we have, for $\epsilon_{\ell, m},\tilde \epsilon_{\ell,j}$ i.i.d. Rademacher variables (all defined on a suitable product probability space), $$\begin{aligned}
\label{tb}
E_{\pi^2} \exp \left(n \langle \theta, \theta' \rangle_F \right) &= E_\epsilon E_{\tilde \epsilon} \exp \left(n\gamma_n^2 \sum_{\ell} \sum_{m} \epsilon_{\ell, m} \sum_j \tilde \epsilon_{\ell, j} \right) \notag \\
&= \left(E_\epsilon E_{\tilde \epsilon} \exp \left(n\gamma_n^2 \sum_{m} \epsilon_{\ell, m} \sum_j \tilde \epsilon_{\ell, j} \right)\right)^k.\end{aligned}$$ Conditional on the values of $\epsilon$ we set $\lambda = n \gamma_n^2 \sum_{m=1}^d \epsilon_{\ell,m}$ and note that $$|\lambda| \le nd \gamma_n^2 = o(1).$$ By Taylor expansion or standard properties of the hyperbolic cosine (as, e.g., in the proof of Theorem 6.2.9 in [@GN15]) $$E_{\tilde \epsilon} \exp\left(\lambda \sum_{j=1}^{d/k} \tilde \epsilon_{\ell, j}\right) = \cosh (\lambda^2)^{d/k} \le \exp\left(\lambda^2 d/k \right)$$ and thus, since $[EU]^k \le E[U^k]$ for any non-negative random variable $U$, the right hand side in (\[tb\]) is bounded above by $$\left(E_\epsilon \exp\left(\lambda^2 d/k \right) \right)^k \le E_\epsilon \exp\left(\lambda^2 d\right) =E_\epsilon \exp\left(n^2 \gamma_n^4 d \left(\sum_{m=1}^d \epsilon_m\right)^2 \right) \equiv E \exp \left(Z^2/c^2 \right)$$ where the Rademacher sum $Z=\sum_{m=1} ^d \epsilon_m$ is a sub-Gaussian random variable with variance proxy $\sigma^2=d$ (cf. Section 2.3 in [@GN15]). Thus by (2.24) in [@GN15] we have $$E \exp \left(Z^2/c^2 \right) \le 1 + \frac{2}{c^2/2\sigma^2 -1} = 1+o(1)$$ since $$\frac{c^2}{\sigma^2} = \frac{1}{d^2 n^2 \gamma_n^4} \to \infty$$ as $n,d \to \infty$. Summarising all steps so far we conclude $$0 \le E^XE_0[Z-1]^2 = \mathbb E[Z^2] -1 \le 1-1+o(1)=o(1)$$ noting that (\[viertel\]) holds $\pi$-almost surely in view of $$\|\theta\|_F^2 = \gamma_n^2 \|W\|_F^2 = \gamma_n^2 d^2 = o(n^{-1/2}).$$
*Step V: Asymptotic concentration of $\pi$ on $H_1$*: Finally we show that for the above prior we have indeed $\Pi(H_1)\to 1$. First since $\theta$ consists of columns that are linear combinations of at most $k$ distinct vectors $v_\ell$ we immediately have $\theta \in R(k)$ almost surely. Moreover, for the case $\|\cdot\|=\|\cdot\|_F$ we have from the last display and by definition of $\gamma_n$ that $\|\theta\|_F^2 = \rho^2_n$, so $\Pi(H_1)=1$ follows.
For the case $\|\cdot\|=\|\cdot\|_*$ we have to show that $$\pi_{n,d}(\|\theta\|_* \ge \rho_n) \to 1$$ as $\min(n,d) \to \infty$. We can transform $\theta$ into the $d \times k$ matrix $\theta U$ consisting of $k$ column vectors $\gamma_n \sqrt{d/k} v_\ell, \ell=1, \dots, k$. The corresponding $d \times k$ matrix $U$ consists of $k$ column vectors, the $\ell$-th of which has zero entries except for the indices $m \in [\ell d/k, \dots, -1+(\ell+1)d/k]$, where it equals $\sqrt {k/d} B_{\ell, m}$. Thus, $U$ is an orthonormal projection matrix and we deduce that $$\|\theta\|_* \ge \|\theta U\|_*.$$ We can renormalise the column vectors of $\theta U$ so that $$\theta U = \gamma_n \frac{d}{\sqrt k} \left(\dots \frac{1}{\sqrt d}v_\ell \dots \right) \equiv \gamma_n \frac{d}{\sqrt k} V.$$ The $d \times k$ matrix $V$ consists of scaled i.i.d. Rademacher entries, and hence the proof of Lemma 1 in [@NvdG13] (with $n=d, k=k_1=p$ in the first display on p.2868 there) implies that, if $k/d \to 0$, then with probability as close to one as desired, the smallest singular value of $V$ is bounded below by $1/2$ for $d$ large enough. As a consequence $\|V\|_* \ge k/2$ and so, with probability approaching one, $$\|\theta\|_* \ge \gamma_n d\sqrt{k}/2 =\rho_n.$$ Note that the same lower bound holds for $$\label{compdist}
\|\theta-R(k_0)\|_* =\inf_{\theta' \in R(k_0)} \|\theta-\theta'\|_* \ge \sum_{j=k_0+1}^k|\lambda_j| \ge (k-k_0)/2$$ for any $k_0<k$, if the absolute eigenvalues in the last display are assumed to be in decreasing order.
Proof of Theorem \[main\]
-------------------------
Consider the composite testing problem $$\label{compo}
H_0: \theta \in R(k_0) ~vs ~H^c_1: \theta \in R(k_1), \|\theta - R(k_0)\|_*=\inf_{\theta' \in R(k_0)} \|\theta-\theta'\|_* \ge \rho.$$ From (\[compdist\]) with $k=k_1$ and $k_0 = o(k_1)$ we see that for $\min(n,d)$ large enough such that $(k_1-k_0)/2 \ge k_1/4$, the prior $\pi$ from the previous proof with $\gamma_n=4\rho_n /(\sqrt k d)$ asymptotically concentrates on $H_1^c$. As a consequence testing (\[compo\]) is no easier than when $H_0=\{0\}$, so that when $\rho=o(\sqrt{k_1 d/n})$ then the proof of Part 2 of Theorem \[signalthm\] implies $$\label{errors}
\liminf_{n,d} \inf_{\Psi} \left[\sup_{\theta \in H_0} \mathbb E_\theta\Psi + \sup_{\theta \in H_1^c} \mathbb E_\theta(1-\Psi)\right] \ge 1.$$ Now assume by way of contradiction that there exists $C_n$ that satisfies (\[cov2\]) with $\alpha<1/3$ and such that for every $c>0$ there exist infinitely many $n,d$ such that $$\sup_{\theta \in H_0}\mathbb E_\theta |C_n|_*<c \sqrt{k_1d/n}.$$ Passing to the infinite subsequence $\min(n,d) \to \infty$ along which the last inequalities hold, we deduce from Markov’s inequality that $$\sup_{\theta \in R(k_0)}\mathbb P_\theta(|C_n|_* \ge \alpha \sqrt{k_1d/n}) \le c/\alpha<\alpha$$ for $c$ small enough depending only on $\alpha$. Then, by Proposition 8.6.3 in [@GN15] we can construct a test for (\[compo\]) for which the testing errors in (\[errors\]) are no more than $3 \alpha<1$ along the chosen subsequence, a contradiction that completes the proof.
**Acknowledgement.** RN’s research was supported by the European Research Council (ERC) under grant agreement No.647812. This paper was written while AC was a research associate in the University of Cambridge. The authors are grateful to two referees whose comments improved the exposition of this article.
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---
abstract: 'The existence of at least three weak solutions for a kind of nonlinear time-dependent equation is studied. In fact, we consider the case that the source function has singularity at origin. To this aim, the variational methods and the well-known critical points theorem are main tools.'
author:
- 'F. Abdolrazaghi[^1]'
- 'A. Razani[^2][^3]'
- 'R. Mirzaei[^4]'
bibliography:
- 'myref.bib'
title: 'Multiple weak solutions for a kind of time-dependent equation involving singularity'
---
**2010 Mathematics Subject Classification:**[35J20, 34B15]{}
**Keywords**: Sobolev equation, Weak solution, Critical point theory, Variational method, Singularity.
Introduction
============
The linear Sobolev equations have a real physical background ([@barenblatt1960basic; @shi1990initial; @ting1974cooling]) and are studied in [@davis1972quasilinear; @ewing1977coupled]. Because of their complexity, they haven’t exact solutions (except some very especial cases [@aristov2017exact]). There are different methods to study the solution of these problems. One of the standard methods is the fixed point theory that investigate the existence of solutions of nonlinear boundary value problems [@agarwal2010survey; @benchohra2009boundary; @zhang2010positive; @Mokhtarzadeh12; @EhsaniAA; @PournakiAML; @GoodarziAAA; @Razani2014; @dinmohammadi2017analytical; @dinmohammadi2017existence]. The calculus of variation is another impressive technique and for using this technique, one needs to show that the given boundary value problem should possess a variational structure on some convenient spaces [@AbdolrazaghiMMN; @BehboudiFilmat; @chu2017weak; @corvellec2010doubly; @heydari2018efficient; @JeanMawhin; @Khalkhali2013; @Khalkhali2012; @MakvandFilomat; @MahdaviFilomat; @MakvandMJM; @MakvandCKMS; @MakvandTJM; @MakvandGMJ; @li2005existence; @nieto2009variational; @rabinowitz1986minimax; @ragusa2004cauchy; @tang2010some; @goudarzi2019weak].
In the present paper, we study the weak solutions of $$\label{In1}
\left\{
\begin{array}{l}
\frac{\partial u}{\partial t}-\frac{\partial(\triangle u)}{\partial t}=\mu f(x,t,u) \ \ in\ \Omega,\\ u=0\ \ on\ \partial\Omega,\\u(x,0)=g(x) \ \ x\in\Omega,\\\end{array}
\right.$$ where $\Omega$ is a non-empty bounded open subset of $\mathbb{R}^N$ with $\partial \Omega\in C^1$, $\mu$ is a positive parameter, $f:\Omega\times\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R} $ is a Carathéodory function and has a singularity at the origin with respect to the time variable and $g:\Omega\rightarrow \mathbb{R}$ vanishes on $\partial \Omega$.\
The aim of this paper is to find an interval for $\mu$ for which the problem admits at least three distinct weak solutions.
By integrating the first equation of we get $$\label{Co1}
\int_{0}^{t}\frac{\partial u(x,s)}{\partial s}ds-\int_{0}^{t}\frac{\partial\Delta u(x,s)}{\partial s}ds=\int_{0}^{t}\mu f(x,s,u)ds,$$ or $$\label{Co001}
-\Delta u(x,t)=\mu F(x,t,u)-u(x,t)+g(x)-\Delta g(x),$$ where $$\label{Co3}
F(x,t,u)=\int_{0}^{t}f(x,s,u)ds.$$ The equation is a time-dependent elliptic equation.
A function $u:\Omega\rightarrow \mathbb{R}$ is called a weak solution of the problem if $u\in H_{0}^{1}$ and $$\label{p2}
\begin{array}{rl}
\int_{\Omega}\nabla u(x,t)\cdot\nabla v(x)dx&-\mu\int_{\Omega}F(x,t,u(x))v(x)dx+\int_{\Omega}u(x,t)v(x)dx\\
&-\int_{\Omega}g(x)v(x)dx+\int_{\Omega}\Delta g(x)v(x)dx=0,
\end{array}$$ for all $v\in H_{0}^{1}$ and $t\geq0$.
\[functional\] Define the functionals $\varphi,\vartheta:H_{0}^{1}\rightarrow \mathbb{R}$ by $\varphi(u):=\frac{1}{2}{\|u\|}^{2}$ and $$\begin{array}{rl}
\vartheta(u):=&\int_{\Omega}\widetilde{F}(x,t,u)dx-\frac{1}{2\mu}\int_{\Omega}\left(u(x,t)
\right)^2dx+\frac{1}{\mu}\int_{\Omega}g(x)u(x,t)dx\\ & \\
& \quad \quad -\frac{1}{\mu}\int_{\Omega}\Delta g(x)u(x,t)dx,
\end{array}$$ respective, where $\widetilde{F}(x,t,\eta):=\int_{0}^{\eta}F(x,t,s)ds$.
Notice that $\varphi$ and $\vartheta$ are well-defined and $ C^1$, $\varphi^\prime,\vartheta^\prime\in X^{*}$, $\varphi^\prime(u)(v)=\int_{\Omega}\nabla u(x)\cdot\nabla v(x)dx$ and $$\begin{array}{rl}
\vartheta^\prime(u)(v)=&\int_{\Omega}F(x,t,u(x))v(x)dx-\frac{1}{\mu}
\int_{\Omega}u(x,t)v(x)dx\\
& \\
&\quad \quad +\frac{1}{\mu}\int_{\Omega}g(x)v(x)dx-\frac{1}{\mu}\int_{\Omega}\Delta g(x)v(x)dx.
\end{array}$$
A critical point of $I_\mu:=\varphi-\mu\vartheta$ is exactly a weak solution of .
Fix $q\in [1,2^*[$, Embedding Theorem [@bonanno2011three] shows $H_0^1(\Omega)\overset{c}\hookrightarrow L^{q}(\Omega)$, i.e. there exists $c_q>0$ such that for all $u \in H_0^1(\Omega)$ $$\label{p6}
\|u\|_{ L^{q}(\Omega)}\leq c_q\|u\|,$$ where $$\label{p7}
c_q\leq\frac{meas(\Omega)^{\frac{2^*-q}{2^*q}}}{\sqrt{N(N-2)\pi}}\left(\frac{N!}{2\Gamma(N/2+1)}\right)^{\frac{1}{N}},$$ $\Gamma$ is the Gamma function, $2^*=2N/(N-2)$ and $meas(\Omega)$ denotes the Lebesgue measure of $\Omega$.
Three weak solutions
====================
In this section the existence of at least three weak solutions for the problem is proved. Due to do this, we apply [@bonanno2010structure Theorem 3.6] which is given below
\[p1\](see [@bonanno2010structure], Theorem 3.6). let $X$ be a reflexive real Banach space, $\Phi:X\rightarrow \mathbb{R}$ be a coercive, continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on $X^{*}$, $\Psi:X\rightarrow \mathbb{R}$ be a continuously Gateaux differentiable functional whose Gateaux derivative is compact such that $\Phi(0)=\Psi(0)=0.$ Assume that there exist $r>0$ and $\overline{x}\in X$, with $r<\Phi(\overline{x})$, such that:
1. $\frac{\sup_{\Phi(x)\leq r}\Psi(x)}{r}<\frac{\Psi(\overline{x})}{\Phi(\overline{x})};$
2. for each $\lambda\in\Lambda_{r}:=]\frac{\Phi(\overline{x})}{\Psi(\overline{x})},\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}[$ the functional $\Phi-\lambda\Psi$ is coercive.
Then, for each $\lambda \in \Lambda_{r}$, the functional $\Phi-\lambda\Psi$ has at least three distinct critical points in $X$.
Set $$\label{p8}
\begin{array}{ll}
D:=\underset{x\in\Omega}\sup \ dist(x,\partial\Omega),&
\kappa:=\frac{D\sqrt{2}}{2{\pi}^{N/4}}\left(\frac{\Gamma(N/2+1)}{D^N-(D/2)^N}\right)^{\frac{1}{2}},\\
K_1:=\frac{2\sqrt{2}c_1(2^N-1)}{D^2},&
K_2:=\frac{2^{\frac{q+2}{2}}c_q^q(2^N-1)}{qD^2}.
\end{array}$$ Now, we can state the main result.
\[Maint\] Let $f:\Omega\times\mathbb{R}^{+}\times\mathbb{R}\rightarrow\mathbb{R}$ be a Carathéodory function and $g:\Omega\rightarrow \mathbb{R}$ vanishes on $\partial \Omega$. Assume
- There exist non-negative constants $m_1$,$m_2$ and $q \in ]1,\frac{2N}{N-2}[$ such that $$F(x,t,s)\leq m_1+m_2\mid s\mid^{q-1}+\frac{1}{\mu}\left(s-g(x)+\Delta g(x)\right)$$ for all $(x,t,s)\in\Omega\times\mathbb{R}^{+}\times\mathbb{R}$.
- $\widetilde{F}(x,t,\eta)\geq \frac{1}{\mu}\left(\frac{1}{2}\eta^2-\eta g(x)+\eta\Delta g(x)\right)$ for every $(x,t,\eta) \in \Omega\times\mathbb{R}^{+}\times\mathbb{R}$.
- There exist positive constants $a$ and $b<2$ such that $$\widetilde{F}(x,t,\eta)\leq a(1+|\eta|^{b})+\frac{1}{\mu}\left(\frac{1}{2}\eta^2-\eta g(x)+\eta\Delta g(x)\right).$$
- There exist positive constants $\alpha$, $\beta$ with $\beta>\alpha\kappa$ such that $$\frac{\inf_{x \in \Omega}\left(\widetilde{F}(x,t,\beta)-\frac{1}{\mu}\left(\frac{1}{2}\beta^2-\beta g(x)+\beta\Delta g(x)\right)\right)}{{\beta}^{2}}>m_1\frac{K_1}{\alpha}+m_2K_2{\alpha}^{q-2},$$ where $\kappa,K_1,K_2$ are given by .
Then the problem has at least three weak solutions in $H_{0}^{1}(\Omega)$, for each parameter $\mu$ belonging to $\Lambda(\alpha,\beta):= \frac{2(2^N-1)}{D^2}\times \left(\delta_1,\delta_2\right)$, where\
$\delta_1:= \frac{{\beta}^{2}}{\inf_{x \in \Omega}\left(\widetilde{F}(x,t,\beta)-\frac{1}{\mu}\left(\frac{1}{2}\beta^2-\beta g(x)+\beta\Delta g(x)\right)\right)}$ and $\delta_2:=\frac{1}{m_1\frac{K_1}{\alpha}+m_2K_2\alpha^{q-2}}$.
Set $X:=H_{0}^{1}(\Omega)$ and define the functionals $\varphi(u)$ and $\vartheta(u)$ by Definition \[functional\]. Clearly, $\vartheta$ and $\varphi$ satisfy the assumptions of [@bonanno2010structure Theorem 3.6]. By (1) $$\label{m1}
\widetilde{F}(x,t,\eta)\leq \frac{1}{\mu}\left(\frac{1}{2}\eta^{2}-\eta g(x)+\eta\Delta g(x)\right)+m_1|\eta|+m_2\frac{|\eta|^{q}}{q}$$ for every $(x,t,\eta)\in\Omega\times\mathbb{R}^{+}\times\mathbb{R}$. Thus $$\begin{array}{rl}
\vartheta(u):=& \int_{\Omega}\widetilde{F}(x,t,u)dx-\frac{1}{2\mu}\int_{\Omega}\left(u(x,t)\right)^2dx+\frac{1}{\mu}\int_{\Omega}g(x)u(x,t)dx\\& \\
&\quad -\frac{1}{\mu}\int_{\Omega}\Delta g(x)u(x,t)dx\\ & \\
\leq& \frac{1}{\mu}\int_{\Omega}\left(\frac{1}{2}(u(x,t))^{2}-u(x,t) g(x)+u(x,t)\Delta g(x)\right)dx \\ & \\
& \quad +\int_{\Omega}\left(m_1|u(x,t)|+m_2\frac{|u(x,t)|^{q}}{q}\right)dx-\frac{1}{2\mu}\int_{\Omega}\left(u(x,t)\right)^2dx
\\ & \\
&\quad +\frac{1}{\mu}\int_{\Omega}g(x)u(x,t)dx-\frac{1}{\mu}\int_{\Omega}\Delta g(x)u(x,t)dx\\ & \\
\leq& m_1\parallel u\parallel_{L^1(\Omega)}+\frac{m_2}{q}\parallel u\parallel_{L^q(\Omega)}^q.
\end{array}$$ Let $r \in ]0,+\infty[$ such that $ \varphi(u)\leq r$. By , $$\vartheta(u)\leq\left(\sqrt{2r}c_1m_1+\frac{2^{\frac{q}{2}}c_q^qm_2}{q}r^{\frac{q}{2}}\right).$$ Set $\chi(r):=\frac{\sup_{u\in\varphi^{-1}]-\infty,r[}\vartheta(u)}{r}$. Consequently $$\label{m3}
\chi(r)\leq\left(\sqrt{\frac{2}{r}}c_1m_1+\frac{2^{\frac{q}{2}}c_q^qm_2}{q}r^{\frac{q}{2}-1}\right),$$ for every $r>0$.
By , there is $x_0\in\Omega$ such that $B(x_0,D)\subseteq\Omega$. Set $$\begin{aligned}
u_\beta(x,t):=\left\{
\begin{array}{ll}
0 \ \ \ \ \ x \in\Omega\backslash B(x_0,D), \\
\frac{2\beta}{D}(D-|x-x_0|) \ \ x\in\ B(x_0,D)\backslash B(x_0,D/2),\\
\beta \ \ \ \ \ x\in B(x_0,D/2).
\end{array}
\right.\end{aligned}$$ Thus $u_{\beta}\in H_0^1(\Omega)$. So $$\begin{aligned}
\label{m5}
\nonumber \varphi(u_\beta)&=&\frac{1}{2}\int_{\Omega}|\nabla u_\beta(x,t) |^2dx \\
\nonumber&=&\frac{1}{2}\int_{B(x_0,D)\backslash B(x_0,D/2)}\frac{(2\beta)^2}{D^2}dx \\
\nonumber&=&\frac{1}{2}\frac{(2\beta)^2}{D^2}(meas(B(x_0,D))-meas(B(x_0,D/2)))\\
&=&\frac{1}{2}\frac{(2\beta)^2}{D^2}\frac{\pi^{N/2}}{\Gamma(N/2+1)}\left(D^N-(D/2)^N\right).\end{aligned}$$ If we force $\beta>\alpha\kappa$, by (4), $\alpha^2<\varphi(u_{\beta})$ because $\alpha^2<\frac{\beta^2}{\kappa^2}$. Also by assumption (2), $$\label{m6}
\begin{array}{rl}
\vartheta(u_\beta):=&\int_{\Omega}\widetilde{F}(x,t,u_\beta)dx-
\frac{1}{2\mu}\int_{\Omega}\left(u_\beta(x,t)\right)^2dx+\frac{1}{\mu}\int_{\Omega}g(x)u_\beta(x,t)dx\\ & \\
&\quad \quad -\frac{1}{\mu}\int_{\Omega}\Delta g(x)u_\beta(x,t)dx\\ & \\ =&\int_{\Omega}\left[\widetilde{F}(x,t,u_\beta)-\frac{1}{\mu}\left(\frac{1}{2}u_\beta(x,t)^2-g(x)u_\beta(x,t)+\Delta g(x)u_\beta(x,t)\right)\right]dx\\ & \\
\geq&\int_{B(x_0,D/2)}\left[\widetilde{F}(x,t,u_\beta)-\frac{1}{\mu}\left(\frac{1}{2}u_\beta(x,t)^2-g(x)u_\beta(x,t)+\Delta g(x)u_\beta(x,t)\right)\right]dx\\ & \\
\geq &\inf_{x\in\Omega}\left(\widetilde{F}(x,t,\beta)-\frac{1}{\mu}\left(\frac{1}{2}\beta^2-\beta g(x)+\beta\Delta g(x)\right)\right)\frac{\pi^{N/2}}{\Gamma(N/2+1)}\frac{D^N}{2^N}.
\end{array}$$ Next by dividing on , we have $$\label{m7}
\frac{\vartheta(u_\beta)}{\varphi(u_\beta)}\geq \frac{D^2}{2(2^N-1)}\frac{\inf_{x\in\Omega}\left(\widetilde{F}(x,t,\beta)-\frac{1}{\mu}\left(\frac{1}{2}\beta^2-\beta g(x)+\beta\Delta g(x)\right)\right)}{\beta^2}.$$ Using , assumption (4) implies $$\begin{aligned}
\label{m8}
\nonumber\chi(\alpha^2)&\leq&(\frac{\sqrt{2}c_1m_1}{\alpha}+\frac{2^{\frac{q}{2}}c_q^q m_2\alpha^{q-2}}{q})\\
\nonumber&=&\frac{D^2}{2(2^N-1)}(m_1\frac{K_1}{\alpha}+m_2K_2{\alpha}^{q-2})\\
\nonumber&<&\frac{D^2}{2(2^N-1)}\frac{\inf_{x\in\Omega}(\widetilde{F}(x,t,\beta)-U(x,t)-G(x)-\overset{\Delta}G(x))}{\beta^2}\\
\nonumber&\leq&\frac{\vartheta(u_\beta)}{\varphi(u_\beta)}.\end{aligned}$$ Assuming $b<2$ and considering $|u|^b\in L^{\frac{2}{s}}(\Omega)$ for all $u \in X$, Hölder’s inequality for $ u \in X$ implies $\int_{\Omega}|u(x,t)|^b dx\leq\parallel u\parallel_{L^2(\Omega)}^b (meas(\Omega))^{\frac{2-b}{2}}$. Therefore equation shows for all $u \in X$ $$\int_{\Omega}|u(x,t)|^b dx\leq c_2^b\parallel u\parallel^b (meas(\Omega))^{\frac{2-b}{2}},$$ and by assumption (3), $$\begin{array}{rl}
I_\mu(u)=& \varphi(u)-\mu\vartheta(u)\\ & \\
=& \frac{\parallel u\parallel^2}{2}-\mu\int_{\Omega}\widetilde{F}(x,t,u)dx+
\frac{1}{2}\int_{\Omega}\left(u(x,t)\right)^2dx\\&\\
&\quad -\int_{\Omega}g(x)u(x,t)dx+\int_{\Omega}\Delta g(x)u(x,t)dx\\&\\
\geq & \frac{\parallel u\parallel^2}{2}-\mu\int_{\Omega}a\left(1+|u(x,t)|^{b}\right)dx\\ &\\
\geq &\frac{\parallel u\parallel^2}{2}-\mu a c_2^b (meas(\Omega))^{\frac{2-b}{2}}\parallel u\parallel^b-a\mu meas(\Omega).
\end{array}$$ This means for every $
\mu \in \Lambda(\alpha,\beta)\subseteq \left] \frac{\vartheta(u_\beta)}{\varphi(u_\beta)},\frac{\alpha^2}{\sup_{\varphi(u)\leq \alpha^2}\vartheta(u)}\right[$, $I_\mu$ is coercive. Therefore by Theorem \[p1\] for each $\mu \in \Lambda(\alpha,\beta)$ the functional $I_\mu$ has at least three distinct critical points that they are weak solutions of the problem .
Numerical Experiment
====================
Now, we present an example.
$$\label{N1}
\left\{
\begin{array}{l}
\frac{\partial u}{\partial t}-\frac{\partial(\Delta u)}{\partial t}=\frac{1}{100}\frac{99}{100t}\left(1+\frac{\exp(-t)}{99}\right)(8+100u+u^2) \ \in \Omega , u\mid_{\partial\Omega}=0,\\ u(x,0)=\frac{1}{1000}\left(\frac{1}{100}-\left(x_1^2+x_2^2+x_3^2\right)\right) \ \ x\in\Omega,
\end{array}
\right.$$
where $\Omega :=\left\{(x_1,x_2,x_3)\in \mathbb{R}^3, x_1^2+x_2^2+x_3^2\leq0.1\right\}$, then $\mu=0.01, N=3, D=r=0.1, 2^*=6,$ $g(x)=0.001\left(0.01-\left(x_1^2+x_2^2+x_3^2\right)\right),$ $\Delta g(x)=-0.006$ and $f(x,t,u)=\frac{99}{100t}\left(1+\frac{\exp(-t)}{99}\right)(8+100u+u^2)$. Now, setting $q=3$, then $$\begin{array}{l}
c_1\leq0.00445759,\quad c_q\leq0.171543,\\
\kappa=1.16798,\quad K_1\leq8.82557,\quad K_2\leq6.66307.
\end{array}$$ Clearly $F(x,t,s)=\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)\left(8+100s+s^2\right)$, suppose $m_1=9$ and $m_2=1$, then the assumption (1) of the Theorem \[Maint\] is satisfied, i.e. $$\begin{array}{l}
\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)\left(8+100s+s^2\right)\leq \\
9+ s^{2}+\frac{1}{0.01}\left(s-0.001\left(0.01-\left(x_1^2+x_2^2+x_3^2\right)\right)-0.006\right),
\end{array}$$ for all $(x,t,s)\in\Omega\times\mathbb{R}^{+}\times\mathbb{R}$.\
Obviously $\widetilde{F}(x,t,\eta)=\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)\left(8\eta+50\eta^2+\frac{\eta^3}{3}\right)$, then it can be easily verified that the assumption (2) of the Theorem \[Maint\] holds, i.e. for all $(x,t,s)\in\Omega\times\mathbb{R}^{+}\times\mathbb{R}$ $$\begin{array}{l}
\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)
\left(8\eta+50\eta^2+\frac{\eta^3}{3}\right)\geq \\
\frac{1}{0.01}\left(\frac{1}{2}\eta^2- 0.001\eta\left(0.01-\left(x_1^2+x_2^2+x_3^2\right)\right)-0.006\eta\right).
\end{array}$$ Also, by choosing $a=b=10$, the assumption (3) of the Theorem \[Maint\] is satisfied, i.e. for all $(x,t,s)\in\Omega\times\mathbb{R}^{+}\times\mathbb{R}$ $$\begin{array}{l}
\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)\left(8\eta+50\eta^2+\frac{\eta^3}{3}\right)\leq\\
10(1+\eta^{10})+\frac{1}{0.01}\left(\frac{1}{2}\eta^2- 0.001\eta\left(0.01-\left(x_1^2+x_2^2+x_3^2\right)\right)-0.006\eta\right).
\end{array}$$ More, set $\alpha=1$ and $\beta=500>\alpha\kappa$ hence, for all $t\geq 0$, it is not difficult to see that $$\begin{aligned}
\nonumber
162.872&=&\frac{\inf_{x \in \Omega}\left\{\left(
\begin{array}{c}
\frac{99}{100}\left(1+\frac{\exp(-t)}{99}\right)\left(8\eta+50\eta^2+\frac{\eta^3}{3}\right)- \\
\frac{1}{0.01}\left(\frac{1}{2}\eta^2- 0.001\eta\left(0.01-\left(x_1^2+x_2^2+x_3^2\right)\right)-0.006\eta\right) \\
\end{array}
\right)
\right\}}{{\beta}^{2}}\\ \nonumber&>&m_1K_1+m_2K_2=86.0932.\end{aligned}$$ Furthermore, it is observed that $\mu=0.01\in\left]\frac{1}{162.872},\frac{1}{86.0932}\right[$, therefore the problem admits at least three week solutions in according to the Theorem \[Maint\]. **Acknowledgements**\
The authors are very grateful to anonymous reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper very much.
[^1]: f.abdolrazaghi@edu.ikiu.ac.ir
[^2]: Corresponding author
[^3]: razani@sci.ikiu.ac.ir
[^4]: r.mirzaei@sci.ikiu.ac.ir
|
---
abstract: 'Measurement techniques such as Magnetic Resonance Velocimety (MRV) and Magnetic Resonance Concentration (MRC) are useful for obtaining 3D time-averaged flow quantities in complex turbulent flows, but cannot measure turbulent correlations or near-wall data. In this work, we use highly resolved Large Eddy Simulations (LES) to complement the experiments and bypass those limitations. Coupling LES and magnetic resonance experimental techniques is especially advantageous in complex non-homogeneous flows because the 3D data allow for extensive validation, creating confidence that the simulation results portray a physically realistic flow. As such we can treat the simulation as data, which “enrich” the original MRI mean flow results. This approach is demonstrated using a cylindrical and inclined jet in crossflow with three distinct velocity ratios, $r=1$, $r=1.5$, and $r=2$. The numerical mesh is highly refined in order for the subgrid scale models to have negligible contribution, and a systematic, iterative procedure is described to set inlet conditions. The validation of the mean flow data shows excellent agreement between simulation and experiments, which creates confidence that the LES data can be used to enrich the experiments with near-wall results and turbulent statistics. We also discuss some mean flow features and how they vary with velocity ratio, including wall concentration, the counter rotating vortex pair, and the in-hole velocity.'
address: |
Mechanical Engineering Department, Stanford University\
488 Escondido Mall, Stanford CA - 94305
author:
- 'Pedro M. Milani'
- 'Ian E. Gunady, David S. Ching, Andrew J. Banko'
- |
\
Christopher J. Elkins, John K. Eaton
bibliography:
- 'Milani\_IJHFF.bib'
title: Enriching MRI mean flow data of inclined jets in crossflow with Large Eddy Simulations
---
Magnetic Resonance Velocimetry, film cooling, jet in crossflow, validation, inlet conditions
Introduction
============
Magnetic Resonance Imaging (MRI) is a class of experimental techniques used across many scientific fields. In fluid mechanics, MRI-based methods can be used to measure 3D fields in complex turbulent engineering flows. The two main such methods are Magnetic Resonance Velocimetry (MRV), which is adapted for turbulent flows and measures three component mean velocity @elkins20034dmagnetic; and Magnetic Resonance Concentration (MRC), developed by @benson2010three, which is used to measure mean concentration of a passive scalar contaminant.
MRV and MRC are versatile techniques, with many advantages over more conventional diagnostics. First, they measure 3D fields in a Cartesian grid everywhere in the specified region of interest. Second, these techniques can be used in virtually any geometry, since they do not require optical access or the placement of probes. The working fluid must be water, and the test channel is limited in size (it must fit within an MRI scanner) and has to be made of non-magnetic materials. In practice, if a geometry can be 3D printed using stereolithography, the flow through it can be measured. Third, the relatively simple setup from conception to data acquisition allows for quick turnaround of experiments.
Several researchers have leveraged these capabilities to measure different complex turbulent flows. @freudenhammer2014volumetric used MRV to measure mean velocity in an internal combustion engine cylinder and presented detailed flow patterns around realistic intake valves. @siegel2019design measured the velocity field around a spinning projectile to study the Magnus force. @shim20193d applied MRV and MRC to a model urban canopy to understand how contaminants spread in a city under different wind conditions. @ching2018investigation used MRV measurements to study separated flows, and found significant geometric sensitivity of the size and secondary flows in the wake. Due to their flexibility, we expect that the number of groups employing these techniques will grow in the near future since MRI scanners are widely available in medical research facilities.
However, these MRI-based techniques have important limitations in engineering turbulent flows. Firstly, they measure mean velocity and concentration, but cannot directly measure other quantities such as pressure or turbulent statistics like the Reynolds stresses. Also, their resolution might be insufficient to capture some relevant small scale features of the mean turbulent flow. Finally, they do not provide reliable data close to solid boundaries due to inexact alignment and partial volume effects. These drawbacks can be quite significant in certain applications; for example, when designing improved turbulence models, the near-wall behavior is usually of particular interest and it would be crucial to know the turbulent statistics.
The main contribution of the present paper is to demonstrate an approach for obtaining a detailed and reliable dataset whereby MRI data are acquired and subsequently enriched by highly resolved Large Eddy Simulations (LES). The idea is that the simulation geometry and inlet conditions should match the experiment as closely as possible, thus allowing the 3D MRI data to be used to thoroughly validate the numerical results. The simulation serves to complement the experiment and overcome the limitations inherent to MRI experiments. The advantage of performing MRI experiments alongside the Large Eddy Simulations instead of just running the LES is that the validation in 3D of the mean quantities provides confidence that the simulation results actually represent a physical flow and can be treated as data. In case of a mismatch, the 3D data helps to diagnose the root of the problem, which could be inlet conditions, averaging time, etc. Validation could be performed against point or plane data acquired with more traditional techniques such as thermocouple, hot-wire, or particle image velocimetry, but in complex turbulent flows those instruments could be more difficult to set up than the non-intrusive MRI. Besides, if low-dimensional experiments and high-fidelity simulation agree at sparse locations there is no guarantee that the simulation is actually capturing all the relevant 3-dimensional flow physics; if they do not agree, it is very difficult to uncover the causes. This is why the present approach is useful for generating trustworthy datasets in non-homogeneous turbulent flows. In the current work, this is demonstrated in inclined jets in crossflow.
Inclined Jet in Crossflow
-------------------------
The jet in crossflow is an important geometry in which fluid is ejected from an orifice and interacts with the flow passing above the orifice. The jet considered has a circular cross-section of diameter $D$ and is inclined with respect to the main flow by an angle $\beta$ as shown in Fig. \[fig-1-ijicf\]. For a review on the jet in crossflow literature, consult @mahesh_review2013. This geometry has several engineering applications. For example, gas turbine blades are cooled using film cooling, which roughly consists of inclined jets injecting cooler fluid into a hot crossflow to protect the solid walls from the high temperatures [@bogard_review]. Among the parameters that control the flow in a jet in crossflow in the incompressible regime is the velocity ratio $r$, defined as: $$\label{eq-r}
r=U_j/U_c$$ where $U_j$ is the bulk velocity of the jet and $U_c$ is the bulk velocity of the crossflow. In general, low velocity ratio jets stay close to the wall, while high velocity ratio jets penetrate deep into the crossflow.
Previous studies have examined the jet in crossflow. @fric1994vortical used smoke visualization to describe different types of vortical structures present in a turbulent jet in crossflow. @su2004simultaneous and @muppidi2008direct studied scalar transport in a transverse ($\beta=90^\circ$) jet in crossflow with $r=5.7$, the former experimentally and the latter computationally. Among other things, they described decay rates and fluid entrainment. @kohli2005turbulent and @schreivogel2016simultaneous experimentally studied inclined jets at lower values of velocity ratio (which are more relevant for film cooling applications), and reported several turbulent mixing statistics. Some of their main conclusions pertain to the inadequacies of widely used turbulence models in this flow.
In the present paper, the same geometry is studied with three different values of velocity ratio, $r=1.0, 1.5, 2.0$. The MRV and MRC experiments are described in section 2 and the three LES’s that are run to enrich the data are described in section 3. Section 4 discusses mean velocity and concentration results and validates the simulation, and then briefly presents LES turbulence data. Section 5 offers conclusions and paths for future work.
Experimental setup and methods
==============================
Magnetic Resonance Imaging
--------------------------
Experiments are performed on an inclined jet in crossflow. The test section is shown in Fig. \[fig-2-schematicTest\], and consists of a circular hole with diameter $D = 5.8$ mm, a non-dimensional length $L/D = 4.1$, and is inclined at $\beta = 30^\circ$. The main channel has a 50 mm by 50 mm square cross-section, and is large enough for the effects of confinement by the top and side walls to be negligible.
A closed-loop flow circuit is used to provide the mainstream and injected flow, with water as the working fluid. The main channel flow is supplied by a 3/4 hp Little Giant pump (model TE-7-MD-SC) and the injected flow is supplied by a 1/8 hp Little Giant pump (model 5-MD-SC). Before reaching the 50 mm by 50 mm square test section, the main flow fluid is pumped from a reservoir and passes through a flow conditioning section consisting of a diffuser with screens to slow down the flow, a honeycomb section to minimize secondary flows, and a 4:1 contraction to minimize the boundary layer thickness and improve flow uniformity. A trip located on all four channel walls 210 mm upstream of the jet location initiates a new turbulent boundary layer. This geometry is shown in Fig. \[fig-3-schematicFull\]. The injected flow is fed by a plenum attached to the test section. After injection, flow passes through the outlet of the test section and is returned to the reservoir, completing the closed-loop system.
During the experiment, the channel bulk velocity $U_c$ is maintained at 0.50 m/s and the jet bulk velocities $U_j$ are maintained at 0.5, 0.75, and 1.0 m/s for $r$ = 1.0, 1.5, and 2.0, respectively. Flow rates are monitored using Transonic PXL Flowsensors and are controlled using diaphragm valves. The reservoir temperature is held at approximately 21 C throughout all experiments. The channel Reynolds number based on the test section width, channel bulk velocity, and kinematic viscosity of water at this temperature is $Re_C = 25,000$. The hole Reynolds number based on the hole diameter and jet bulk velocity varies from $Re_D = 2,900$ for $r = 1.0$ to $Re_D = 5,800$ for $r = 2.0$. The incoming boundary layer thickness and momentum thickness at the streamwise position of the hole center are measured using hot-wire anemometry (c.f. section 2.2) to be $\delta_{99} = 1.5D$ and $\theta = 0.15 D$, respectively.
Magnetic Resonance Imaging (MRI) techniques are used to obtain the time-averaged, three-dimensional velocity and scalar concentration fields. Data are obtained on a volumetric Cartesian grid that includes the plenum, hole, and main channel over a streamwise length of about $10D$ upstream and $20D$ downstream of injection. The velocity and concentration data for $r=1.0$ were described in @coletti2013turbulent, with additional detail on methods and analysis in @kevin_thesis. Similar methods were used to collect the $r=1.5$ and $r=2.0$ cases in the present work.
Velocity and concentration data were acquired using a 3 Tesla General Electric MRI scanner located at the Richard M. Lucas Center for Imaging at Stanford University. Magnetic Resonance Velocimetry (MRV) was used to obtain 3-component mean velocity data for both $r=1.5$ and $r=2.0$ on a Cartesian grid with a resolution of 0.66 mm by 0.6 mm by 0.6 mm in the streamwise, spanwise, and wall-normal directions, respectively. The MRV technique has been validated and described in detail in @elkins20034dmagnetic. Briefly, this technique measures the velocity of hydrogen protons in water. Copper sulfate is added at a concentration of 0.06 M to enhance the signal-to-noise ratio. Each experiment consists of 15 scans with the flow on and 6 scans with the flow off. Groups of three ‘on’ scans are bookended by ’off‘ scans. Each such pair of ‘off’ scans are averaged together and subtracted from the ‘on’ scans. This procedure compensates for errors in velocity due to eddy-currents [@elkins20034dmagnetic]. Then, all 15 off-subtracted scans are averaged together. Finally, the mean velocity fields are filtered using the method described by @schiavazzi2014matching to reduce spurious noise and produce a nearly divergence free velocity field. During each individual scan, the MRV technique acquires data in spatial-Fourier space over a period of about 2-3 minutes. The entire procedure for a single blowing ratio takes approximately 50 minutes, which is significantly longer than all relevant flow time scales.
Magnetic Resonance Concentration (MRC) was used to collect the mean scalar concentration field at $r=2.0$ on a Cartesian grid of the same resolution as the MRV measurements. The MRC technique has been previously described and validated in @benson2010three. Additional details of the technique, and modifications to flow reservoirs to mitigate contamination of the background signal by the scalar contaminant in a closed flow loop, are given in @kevin_thesis. Briefly, MRC uses a solution of copper sulfate as a contrast agent, where the measured signal magnitude is linearly proportional to the concentration. In a standard experiment, a reference concentration of 0.02 M is injected through the jet while the incoming mainstream flow is pure water (0 M). In practice, four scan types of 24 scans each were used to determine the concentration field. Reference and Background scans wherein the entire channel is filled with the reference concentration and pure water, respectively, were used to correct for large scale variations in the image magnitude due to the spatial sensitivity of the imaging coil. Then, standard scans with the reference concentration injected through the jet and a main flow of pure water are performed to image the mixing. Finally, inverted scans where pure water is injected through the jet and the main flow is held at the reference concentration were used to improve uncertainty in the near-wall data which can be affected by Gibbs artifacts [@benson2010three]. To construct the mean concentration field, the individual Standard and Inverted scans are background subtracted and normalized using the difference between the Reference and Background scans. The results are then averaged together to produce a concentration field which is unity in the plenum and zero in the freestream flow. Each individual scan acquires the spatial concentration field in Fourier space over a period of 2.5 minutes, and the entire experiment takes approximately 4 hours. Therefore, the mean concentration data are acquired over a time period which is orders of magnitude longer than all relevant flow time scales.
The uncertainty in mean velocity measurements due to thermal noise can be approximated from the following formula (@pelc1994quantitative):
$$\delta_{V} = \frac{\sqrt{2}}{\pi}\frac{V_{enc}}{SNR}
\label{eq_unc}$$
Here $V_{enc}$ is the velocity encoding value which dictates the maximum measurable velocity. $SNR$ is the signal to noise ratio defined as the mean signal magnitude in a region of interest downstream of injection divided by the root-mean-square signal within solid regions. For $r=1.5$ and $r=2.0$, the value of $V_{enc}$ in the streamwise, wall-normal, and spanwise directions were 1.5 m/s, 1.4 m/s, and 1.0 m/s, respectively, and the $SNR$ was approximately 35. This gives the average uncertainty across each velocity component as 5$\%$ of the main flow bulk velocity. In practice, the velocity uncertainty is due to signal loss from turbulent dephasing, thermal noise, and statistical uncertainty. However, it has been found that this formula is in good agreement with statistics-based estimates determined from variations between scans and with the typical deviation of MRV from techniques such as hot-wire and PIV in validation test cases (@pelc1994quantitative).
The uncertainty in the mean concentration measurements was determined from a statistical procedure based on the variations between individual Reference, Background, Standard, and Inverted scans. The uncertainty is then propagated through the equation used to compute the concentration field. The details of this procedure are given in @kevin_thesis. The uncertainty is analyzed in a region of interest in the mixing region of the main channel, defined as the region downstream of injection where the concentration is greater than 5$\%$. The uncertainty in mean concentration is 4$\%$ and 6$\%$ of the injected concentration for $r=1.0$ and $r=2.0$, respectively.
The review of @elkins2007review contains more information on the technique and possible use cases of Magnetic Resonance Velocimetry. One important limitation of the present study when compared to realistic jet in crossflow applications is that the working fluid is a liquid and there is no density difference between jet and crossflow. It is possible to mitigate either of those, though that could significantly increase the complexity of the experimental setup and thus is rarely done. For example, some workers used fluorinated gases as a working fluid in MRI velocimetry instead of water (@elkins2007review). It is also possible to vary the properties of the cooling jet within a small band by mixing different liquids with water, such as alcohol or glycerin. Depending on the physics under consideration, however, those extra steps might not be necessary. For instance, @yapa2014comparison compared MRC measurements in water with temperature measurements in air flow with Mach number of 0.7 in the same plane shear layer apparatus and obtained good agreement, demonstrating that the water channel utilizing a temperature-scalar analogy was sufficient to capture the relevant phenomena present in a subsonic compressible flow.
Hot-wire anemometry
-------------------
A hot-wire experiment was performed to better characterize the inlet profiles of the main channel flow and to inform inlet condition specification for numerical simulations, as described in section 3.4. The same channel that was used in the water-based MRI experiments (Fig. \[fig-3-schematicFull\]) was run in air with the same Reynolds number, corresponding to a bulk velocity of $U_c = 7.5$ m/s at a temperature of $23 ^\circ C$. The outlet section of the channel was removed, flow was only present in the main channel, and the injection hole was blocked off, flush to the bottom wall. A hot-wire probe was placed in the symmetry plane of the channel ($z/D=0$), at the same streamwise position as the center of the injection hole ($x/D=0$), and its distance from the bottom wall was carefully controlled using a traverse driven by a stepper motor. The probe is 3 mm long, and a 5 $\mu$m diameter gold-plated wire is mounted, with an active section that is 1 mm long. The overheat ratio employed is 1.8, and the calibration described in @hultmark2010temperature is used, including the temperature correction. A conservative estimate for the calibration uncertainty puts the uncertainty of the velocity measurements at $1.5\%$ when the velocity is relatively low ($u/U_c = 0.5$), and at $0.5\%$ when the velocity is relatively high ($u/U_c = 1.5$).
Computational setup
===================
Three distinct Large Eddy Simulations were conducted, one for each of the velocity ratios ($r=1.0, 1.5, 2.0$). In the current work, the high-fidelity simulations serve to enrich the 3D mean field from MRV/MRC with turbulent statistics and time-resolved quantities. For that, it is crucial to simulate the same flow conditions used in the experiment and thoroughly validate the LES results. Fine meshes are employed to ensure that the viscous sublayer is adequately resolved and that the subgrid scale model contributions are negligible in interesting regions of the flow. More details on the simulations are presented in the following subsections.
Governing equations
-------------------
The filtered, incompressible continuity and Navier-Stokes equations are solved in 3D as shown in Eq. \[eq-continuity\] and Eq. \[eq-ns\].
$$\label{eq-continuity}
\frac{\partial \tilde{u_k}}{\partial x_k} = 0 \newline$$
$$\label{eq-ns}
\frac{\partial \tilde{u}_i}{\partial t} + \frac{\partial \left( \tilde{u}_j \tilde{u}_i \right)}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \tilde{p}}{\partial x_i} + \nu \frac{\partial^2 \tilde{u}_i}{\partial x_j \partial x_j} - \frac{\partial}{\partial x_j} \tilde{\tau}_{ij}$$
The tilde over a variable is shown explicitly to denote grid-filtered quantities; $u_i$ are the Cartesian components of the velocity, and $p$ is the pressure. The unresolved scales are accounted for with the subgrid scale stress, $\tilde{\tau}_{ij} = \widetilde{u_i u_j} - \tilde{u}_i \tilde{u}_j$. The fluid properties, density $\rho$ and kinematic viscosity $\nu$, are constant.
The filtered equation for a passive scalar $c$ is also solved, as shown in Eq. \[eq-scalar\]. The molecular Schmidt number is given by $Sc$ and the subgrid scale mixing is represented by $\tilde{\sigma}_{j} = \widetilde{c u_j} - \tilde{c} \tilde{u}_j$.
$$\label{eq-scalar}
\frac{\partial \tilde{c}}{\partial t} + \frac{\partial \left( \tilde{u}_j \tilde{c} \right)}{\partial x_j} = \frac{\nu}{Sc} \frac{\partial^2 \tilde{c}}{\partial x_j \partial x_j} - \frac{\partial}{\partial x_j} \tilde{\sigma}_{j}$$
To match the properties of water, used in the experiments, the values of density and viscosity were set to $\rho=998 kg/m^3$ and $\nu=1.0 \times 10^{-6} m^2/s$. Thus, the Reynolds number based on hole diameter $D$ and bulk jet velocity $U_j$ varies between $Re_D = 2900$ and $Re_D=5,800$. The value of the molecular Schmidt number $Sc$ was set to $1.0$ in the simulations to ensure that the smallest length scales of the velocity and scalar fields match and are both resolved adequately. This is much smaller than the true molecular Schmidt number of copper sulfate in water, which is of the order of $10^3$ [@emanuel1963diffusion]. However, this discrepancy does not have any practical effect on the scalar field computed: in both cases ($Sc=1$ or $Sc=10^3$), molecular diffusion is negligible compared to turbulent mixing in this flow. Preliminary RANS simulations confirmed that the mean scalar field is insensitive to $Sc$ in this range except extremely close to the bottom wall, where molecular diffusion might compete with turbulent mixing. This coincides with the region where MRC data are unreliable, so there are no scalar data to be matched there. Furthermore, since zero scalar flux boundary condition is maintained at all walls, diffusion is insignificant in that region.
A subgrid scale model is required to determine $\tilde{\tau}_{ij}$ and $\tilde{\sigma}_{j}$ in equations \[eq-ns\] and \[eq-scalar\]. For the momentum equation, the Vreman model is employed [@vreman2004eddy]. This is a linear eddy viscosity model where the eddy viscosity $\nu_{t,SGS}$ depends on the local velocity gradients and cell spacing. For the scalar equation, gradient diffusion and the Reynolds analogy are employed, with turbulent diffusivity $\alpha_{t,SGS} = \nu_{t,SGS}/Sc_{t,SGS}$. The subgrid scale turbulent Schmidt number was set to a constant, $Sc_{t,SGS}=0.85$.
Henceforth, the tilde will be omitted when referring to filtered variables to simplify notation. For example, when referring to the LES, $u_i$ will denote the result of the simulation (which is the filtered quantity), $\bar{u}_i$ will represent the time average of the filtered quantity, and primes will be used to refer to fluctuating quantities in the filtered sense (i.e. $u'_i \equiv u_i - \bar{u}_i$).
Solver
------
The equations of section 3.1 are solved using the unstructured and incompressible solver Vida, developed by Cascade Technologies. It employs the method developed by @ham2007efficient which consists of second-order accurate spatial discretization and explicit time advancement, based on second-order Adams Bashforth. Parallelization is implemented using MPI and each computation is performed on up to 1024 cores. The total computational cost of each simulation varied between $5.2 \times 10^5$ and $1.2 \times 10^6$ CPU-hours as shown in Tab. \[tab-1-cases\].
Domain and mesh
---------------
The full domain simulated in the LES is shown in Fig. \[fig-4-mesh\](a). The crossflow is aligned with the $+x$ direction and the jet comes from a circular hole inclined $30^\circ$ with respect to the streamwise direction. The main channel inlet is located $30D$ upstream of the center of the hole exit, where the origin is located, and extends up to $x/D = 30$. In the experiment, the coolant flow is fed from a rectangular plenum located under the channel. The simulation domain includes the plenum and part of the tube that feeds it following best practices of the film cooling community [@walters1997systematic]. The plenum feed is $3.4D$ long and has a circular cross-section of diameter $2.4D$. The plenum inlet is marked in red, and the main channel outlet is marked in blue.
The mesh is block-structured and contains only hexahedral elements. It was generated with the software ANSYS ICEM. We aimed to have adequate resolution along the bottom wall and the jet hole wall, so that no wall models would be needed in those critical regions. The spacing of the first cell above the wall was set based on preliminary RANS simulations, and the LES results confirm that in all cases the first cell is located at $y^+ < 1.5$ for the bottom wall and at $y^+ < 3.0$ for the cooling hole (with a large majority of cells located at $y^+ < 1.0$). We also required fine overall spacing inside the hole and in the region where the jet and crossflow interact; a typical mesh element there measures approximately $0.02D$ in each Cartesian direction. Fig. \[fig-4-mesh\](b) shows the mesh, and Tab. \[tab-1-cases\] contains the total number of mesh points utilized in each of the three simulations.
--------------- ----------------- ------------------------- ----------------------------------
$r$ Number of cells Time step \[ $D/U_c$ \] Computational cost \[CPU-hours\]
\[3pt\] $1.0$ $40.1M$ $5.0 \times 10^{-3}$ $5.2 \times 10^5$
$1.5$ $43.4M$ $4.1 \times 10^{-3}$ $9.8 \times 10^5$
$2.0$ $48.3M$ $3.4 \times 10^{-3}$ $1.2 \times 10^6$
--------------- ----------------- ------------------------- ----------------------------------
: Total mesh size, dimensionless time step used, and total computational cost for the LES of each velocity ratio.[]{data-label="tab-1-cases"}
To verify that the mesh convergence is satisfactory, a coarser simulation was run for the $r=1$ case. The mesh in the interesting regions of the flow was made coarser by a factor of approximately $1.6$ in all three directions, yielding a mesh with $16.8M$ cells instead of $40.1M$. The coarse simulation ran with the same time step and for the same total time. Fig. \[fig-5-meshconv\] contains comparisons between the coarse and fine LES. Vertical profiles of $\bar{c}$ and $\overline{v'c'}$ in two different streamwise locations show excellent agreement between coarse and fine meshes, which suggests good mesh convergence. The power spectral density $P_u$ was also calculated for a time sequence of the streamwise velocity $u$ at a probe located in the windward shear layer ($x/D=3$, $y/D=1$, $z/D=0$). The plot of non-dimensional $P_u$ as a function of the frequency $f$ is shown in Fig. \[fig-6-meshspectra\]. As expected, it shows that the coarse simulation captures less of the high-frequency variation. However, it is already able to resolve $93\%$ of the variance resolved by the fine simulation, which again suggests that the fine mesh has an appropriate resolution to capture almost all relevant scales of this flow. All subsequent results reported in this paper pertain to the fine mesh LES.
Another important metric to evaluate the mesh resolution is the subgrid scale viscosity, $\nu_{t,SGS}$. For all three simulations, the mean subgrid scale viscosity is less than half of the molecular viscosity everywhere where jet and crossflow interact. This means that the modeled turbulent fluctuations have a negligible impact on the mean flow. Together with the mesh refinement study, this shows that the fine mesh simulations, despite being formally Large Eddy Simulations, have mesh resolutions that are close to that required of Direct Numerical Simulations (DNS).
Time step and averaging
-----------------------
The time step for the Large Eddy Simulations was chosen based on the Courant-Friedrichs-Lewy (CFL) number calculated with the crossflow bulk velocity, $U_c$, and the smallest mesh element anywhere in the domain, which is that immediately above the hole wall. We set the CFL number to $1.0$, and report the resulting time steps in Tab. \[tab-1-cases\]. Note that higher values of $r$ require smaller time steps (which scale as $r^{-0.5}$), since the mesh spacing must be reduced next to the hole wall by the same factor to maintain similar $y^+$.
All simulations are initialized with $u=U_c$, $v=w=0$ everywhere, and $c=1$ in the plenum and in the cooling hole and $c=0$ in the main channel. They were run for $240 D/U_c$ to achieve a statistically stationary state; thereafter, statistics were collected every time step for a simulation time of at least $440 D/U_c$, as specified in in Tab. \[tab-1-cases\]. These simulation times are deemed sufficient for convergence through a comparison of statistics obtained with increasing simulation time. Furthermore, they are significantly longer than the simulation times used by previous authors who computed similar flows [e.g. @muppidi2007direct; @bodart2013highfidelity].
To estimate uncertainty due to incomplete time averaging, we track mean velocity and second order turbulence statistics at three different locations of the flow, shown in Fig. \[fig-7-convergencelocations\]. At each time step $t$, we calculate the time average if the simulation had only run up until that instant. The discrepancy as a function of simulation time, then, is defined as the absolute percent difference between the time average at that time step $t$ and the time average at the final time step (which is the average ultimately reported). Fig. \[fig-8-convergence\] shows plots of such discrepancy for the $r=2$ case and for two different mean quantities, the mean streamwise velocity $\bar{u}$ and mean vertical turbulent scalar flux, $\overline{v'c'}$. The plots show that the oscillations in the perceived statistics are initially high, but as the simulation progresses the oscillations decay indicating lower uncertainty in the statistics. Towards the end of the simulation, such oscillations indicate that our uncertainty in both quantities is well under $5\%$.
It is important to note that the streamwise velocity averages converge much more rapidly than the higher order correlations. A consequence is that it is important to average Large Eddy Simulations for a relatively long time in such fully non-homogeneous geometries if one desires to use the resulting dataset to quantitatively analyze turbulent correlations such as $\overline{u_i'c'}$. For example, @bodart2013highfidelity averaged their similar simulation for $200 D/U_c$; in Fig. \[fig-8-convergence\](b), it is clear that the values of $\overline{v'c'}$ can vary by more than $10\%$ between $t=200 D/U_c$ and the end of the present simulation, at $t=440 D/U_c$. This suggests that a simulation time of $200 D/U_c$ might lead to insufficiently averaged statistics.
Inlet and boundary conditions
-----------------------------
------------------------------------------------------------------------
1. Guess initial parameters for the inlet condition generator.
2. Run a channel flow LES with given inlet condition until convergence.
3. Compare $\bar{u}(y)$ and $u'_{rms}(y)$ between the hot-wire data and different streamwise stations of the channel flow LES:
1. If none of the profiles at different stations agree with the hot-wire data, modify inlet conditions and repeat (2)-(3).
2. If station located at distance $L_{station}$ from the inlet of the channel LES matches the hot-wire data well, we are done. For the jet in crossflow simulation, we use the current inlet condition and set the main inlet to be located a distance $L_{station}$ upstream from the hole.
------------------------------------------------------------------------
The boundary conditions at all walls are no-slip for the velocity field and zero-flux (adiabatic) for the scalar concentration, which match the physical behavior of the experiment. At the bottom wall and the jet wall, the equations are solved directly without wall functions since the mesh resolution is fine enough. At all other solid boundaries, the wall functions of @cabot2000approximate are employed for the velocity field.
Time-varying inlet conditions for the velocity at the channel and plenum feed inlet are generated using a method similar to @xie2008efficient. The inlet generator requires the mean velocity and Reynolds stress 2D profiles, together with a length scale and a time scale. The goal of the present simulations is to match the experimental setup; thus, these quantities must be carefully selected to recover experimental profiles.
To specify inlet conditions for the full simulation, an iterative procedure was employed which involved running preliminary Large Eddy Simulations of a developing square channel flow until a good inlet condition was achieved. Importantly, the channel flow LES uses the same mesh configuration as used in the equivalent region of the final jet in crossflow LES. Hot-wire measurements of mean streamwise velocity, $\bar{u}$, and root-mean-square of the fluctuating streamwise velocity, $u'_{rms}$ are used for validation. The method is described in Tab. \[tab-2-inletalgorithm\]:
The final inlet condition parameters, ultimately used for all three jet in crossflow LES’s, were obtained after several such iterations. They consist of 2D mean profiles based on the 1D hot-wire data, re-scaled to a thinner boundary layer (by a factor of 0.55). The $u'_{rms}$ profile was magnified by a factor of $1.5$ in the boundary layer and by a factor of $5$ in the freestream, because the synthetic turbulence tends to decay rapidly in the first few hole diameters after the inlet. The turbulent length scale required by the synthetic turbulence generator was set to $0.7D$. Fig. \[fig-9-inlet\](a)-(b) shows the hot-wire data along with profiles of $\bar{u}(y)$ and $u'_{rms}(y)$ from different stations downstream of the channel inlet. The agreement for both mean velocity and rms velocity profiles is best for an inlet plane $30D$ upstream of the hole position. Furthermore, the momentum thickness $30D$ downstream of the inlet plane is $0.149D$, very close to the value of $0.15D$ measured in the experiment, as shown in Fig. \[fig-9-inlet\](c). Therefore, we choose to set the inlet plane $30D$ upstream of the hole location.
Other preliminary tests showed that changing the velocity profile at the plenum inlet had small effect on the jet-crossflow interaction. Therefore, a uniform mean velocity and a uniform, low level of isotropic turbulence fluctuations were used in this inlet.
Results and validation
======================
Mean scalar field
-----------------
Fig. \[fig-10-concz0\] shows mean concentration results in the symmetry plane ($z=0$) for two velocity ratios, $r=1$ and $r=2$. In this view, the jet meets the main channel at $y=0$, between $x/D=-1$ and $x/D=1$. Qualitatively, one notes the different behavior of the two configurations. When $r=1$, the scalar concentration initially separates from the bottom wall, but re-attaches at about $x/D=3$ and stays attached all the way to the outlet. At $r=2$, the vertical momentum of the jet is sufficiently high for it to stay completely detached throughout the whole domain. It is not shown here, but the LES results from the $r=1.5$ jet show that it is also completely detached; this implies that for the present geometry, the jet becomes fully detached starting at a critical velocity ratio somewhere between $r=1$ and $r=1.5$.
This regime, in which the jet detaches from the wall however briefly, is the one in which turbulence models fail to predict adequate mean concentrations in the context of RANS simulations [@bogard_review]. Comparing the experimental data to the present simulations (Figs. \[fig-10-concz0\](a) to (b) and (c) to (d)) yields excellent agreement in the symmetry plane, which shows that LES is a tool much better suited at capturing the scalar mixing than RANS models.
The mean scalar concentration evaluated at the wall is of particular interest in film cooling applications, where it is referred to as adiabatic effectiveness. The solid lines in Fig. \[fig-11-concy0\] show the mean scalar concentration at the wall in the LES, averaged between $z/D=-1$ and $z/D=1$, for each velocity ratio. The main feature of jets that separate from the wall right after injection is the sharp drop in the averaged adiabatic effectiveness after injection, as can be seen for all three velocity ratios. RANS simulations employing typical turbulent mixing models fail to predict this behavior (@milani2018approach). Since the $r=1$ jet re-attaches to the wall, its averaged adiabatic effectiveness recovers to much higher levels, and does so over a short streamwise distance. That does not happen with the other two cases: their adiabatic effectiveness curves are similar, and only increase mildly after the initial drop-off. The experimental results are also shown as symbols for $r=1$ and $r=2$. Since the MRI techniques are not appropriate for wall measurements, the experimental values used in Fig. \[fig-11-concy0\] come from the first measurement point in the fluid above the wall. This biases the results to be slightly higher than the actual adiabatic effectiveness and is prone to higher overall noise levels due to partial volume effects. With those caveats, the agreement between LES and MRC in Fig. \[fig-11-concy0\] is good. This comparison illustrates the value of enriching: the experiments, though comprehensive, cannot directly produce the adiabatic effectiveness, so a detailed and validated simulation is used to fill in the gap for the exact geometry.
Figs. \[fig-12-concx2\] and \[fig-13-concx5\] show mean concentration results in streamwise (y-z) planes, in which the main flow direction is out-of-the-page. They are located, respectively, at $x/D=2$ and $x/D=5$. The dominating feature of such mean concentration fields is their kidney shape, previously documented in the literature (e.g. @smith1998mixing). This characteristic shape is caused by the deformation of the jet due to the counter-rotating vortex pair. As expected, higher velocity ratio causes the concentration profiles to be located farther from the bottom wall. Another effect of the velocity ratio regards the jet structure. Increasing $r$ causes the jet area to increase, because the fluid released by the hole must decelerate more as it exchanges momentum with the freestream, and thus must occupy a larger cross-sectional area by conservation of mass. The contours show that most of that increase in area is due to increase in the vertical extent of the jet cross-section: while the jet width increases only mildly from $r=1$ to $r=2$, the jet height increases significantly. This is particularly evident in Fig. \[fig-13-concx5\]. One possible explanation is that, as the jet trajectory moves further away from the wall, there is more space between the jet core and the bottom wall. The turbulent transport in the vertical direction increases since the eddies are farther away from the wall. The spanwise transport, on the other hand, is not magnified by any of these factors.
Finally, Figs. \[fig-12-concx2\] and \[fig-13-concx5\] suggest that there is mild asymmetry in the jet structure about the $z=0$ plane, as has been discussed by authors such as @smith1998mixing. In the present setup, it would be chiefly caused by asymmetry in the plenum feed (see Fig. \[fig-4-mesh\]), not manufacturing tolerances or the main channel inlet.
For a more quantitative comparison between simulation and experiments and between different velocity ratios, Fig. \[fig-14-profilesc\] presents vertical and horizontal profiles of mean concentration. Fig. \[fig-14-profilesc\](a) shows vertical profiles at the centerline at different streamwise positions. It shows that at $x/D=2$ the jet core is still cohesive, and increasing the value of $r$ seems to move the profile up without changing its shape. Further downstream, turbulent mixing acts to spread the jet, and its effects are more pronounced as $r$ increases. Fig. \[fig-14-profilesc\](b) shows double-peaked profiles due to the kidney shape in Fig. \[fig-13-concx5\], and confirms that the width of the jet, measured by its spanwise profile, does not change much across velocity ratios.
The comparison between experiments and simulation in Figs. \[fig-12-concx2\], \[fig-13-concx5\], and \[fig-14-profilesc\] shows that, in general, the mean concentration predictions agree well with the data. Some discrepancy is observed between Figs. \[fig-12-concx2\](a) and \[fig-12-concx2\](c), where the LES seems to slightly underpredict the maximum concentration and the width of the scalar profile. These particular plots should be the most sensitive to small differences in operating conditions and inlet conditions between simulation and experiment, which probably account for the discrepancies observed. For $r=2$ in Fig. \[fig-14-profilesc\](b), the LES and MRC qualitatively agree on the asymmetry across the double peak (showing higher $\bar{c}$ on the $z>0$ side at the lower profile) despite their slight quantitative mismatch, which suggests that simulation has enough fidelity to capture the effect of the plenum feed on the resulting concentration field in the main channel.
Mean velocity field
-------------------
In this subsection, the mean velocity data from the MRV and LES are presented and discussed. For adequate comparison of the interaction between jet and crossflow across different velocity ratios, it is most convenient to normalize the velocity results by the jet bulk velocity, $rU_c$. Fig. \[fig-15-uz0\] shows mean streamwise velocity in $z=0$ planes, superimposed on isocontour lines of mean concentration. In all cases, one notes that before meeting the jet, the incoming boundary layer on the channel wall has thickness comparable to the hole diameter. It grows immediately upstream of the jet due to the adverse pressure gradient induced by the jet’s blockage. @fric1994vortical argued that this process is important for the resulting downstream vortical structure. An observation from the $r=2$ data is that the point of highest scalar concentration is closer to the wall then the point of highest mean streamwise velocity, which appears as an apparent mismatch between the lines and the color contours. This shows that the jet, as measured by the velocity field, penetrates deeper into the crossflow than the jet measured by the concentration field. This effect is also observed using different metrics by @coletti2013turbulent.
Mean secondary flows are significant in jets in crossflow. Fig. \[fig-16-velx2\] shows streamwise planes at $x/D=2$ with information on mean streamwise velocity, mean secondary flows, and mean concentration. The counter rotating vortex pair (CVP) is the most conspicuous feature of this flow, clearly shown by the vectors of in-plane velocity. Instantaneously, the CVP is present, but highly unsteady and asymmetric as discussed by @smith1998mixing; unlike other vortical structures, it also appears in the mean flow. The CVP direction is common-up, and right after injection the center of the vortices is located at approximately $z/D = \pm 0.5$, i.e. above the side edges of the holes. The superposition of the in-plane velocity with the mean scalar concentration shows clearly that the CVP acts to distort the scalar field, creating the kidney shape shown in Fig. \[fig-12-concx2\]. It also acts to sweep in mainstream flow under the jet (which is responsible for the dip in wall concentration shown in Fig. \[fig-11-concy0\]), and push up low velocity fluid located in the boundary layer and under the jet (a phenomenon which is visible in Figs. \[fig-15-uz0\] and \[fig-16-velx2\]).
To further understand and validate the simulation results, it is useful to look at quantitative scalar metrics derived from the 3D data. In the context of mean secondary flows, an important metric is the circulation $\Gamma$, which consists of the line integral of the mean velocity field along a right-handed closed path. At different axial planes, we can define closed lines that form a square and encompass each vortex of the CVP as shown in Fig. \[fig-17-circulation\](a) and calculate the circulation around them. The square has side $1.5D$ and has a corner at $z/D=0$ and $y/D=0.15$; the latter was chosen because MRV data are unreliable closer to the wall, and we are interested in the total circulation from the vortices (not the boundary layer). Since the vortex is common-up, the circulation will be positive in the side marked by $P$ and negative in the side marked by $N$; if the flow were perfectly symmetric, it would have the same magnitude on both sides.
Fig. \[fig-17-circulation\](b) shows dimensionless circulation $\Gamma / (U_c D)$ calculated over the curve shown in Fig. \[fig-17-circulation\](a) defined at different streamwise locations. The lines indicate LES data and the symbols indicate MRV data. To obtain error bars on the MRV calculation, the uncertainty due to possible misalignment of the data was taken into account. The actual curve over which the circulation is to be calculated was shifted randomly in all three directions by up to a single MRI voxel ($0.1D$) fifty times and the error bars indicate two standard deviations around the mean result. The plot contains the absolute value of the circulation in each side marked in Fig. \[fig-17-circulation\](a) ($N$ and $P$). As we can see from the results, the circulation is almost symmetric, with a clear tendency of higher values on the $N$ side (possibly due to asymmetry of the plenum feed). It also seems to increase superlinearly with the velocity ratio $r$, since values at $r=2$ are more than double those of $r=1$. Regarding the agreement between simulation and experiment, the LES reproduces well the trends of the experiment, including the direction of asymmetry, the streamwise decay, and the velocity ratio dependency. However, the simulations seems to consistently overpredict $\Gamma$, sometimes beyond what is explained by misalignment errors alone. This could be attributed to other experimental uncertainties (such as velocity ratio) or to possible differences in the in-hole flow between experiment and simulation, which will be discussed next.
An important feature of the present flow is the flow inside the hole, before the jet meets the crossflow. In idealized conditions, fully developed pipe flow would prevail at the jet exit, which is the case in the experiment of @su2004simultaneous and the simulation of @muppidi2008direct. However, in practical applications, the jet fluid may be expelled by a short pipe, or the pipe could be fed in such a way as to induce important irregularities in the flow. This is often the case in film cooling, where the coolant jet comes from a plenum at the base of a short hole. In order to study possible effects from that, the present work deals with a short hole, of length $4.1D$, fed from a more realistic plenum.
Fig. \[fig-18-velhole\] shows experimental and numerical results for the mean velocity within the hole in the $r=1$ and $r=2$ cases. The coordinate system used is aligned with the inclined hole, where $\hat{s}$ is the unit vector pointing the the axial direction. The contour colors, in all plots, denote mean fluid velocity in the $\hat{s}$ direction, or $\bar{u}_s$. The axial plane is located in the midpoint along the hole between the plenum and the channel bottom wall, and its location is marked as a black line in the centerplane. Vectors in the axial plane show mean in-plane velocity.
It is clear from these plots that the fluid coming from the plenum must turn along the right corner of the hole, causing flow separation on that side. That phenomenon is responsible for the highly non-uniform axial velocity observed in the axial planes shown: much higher velocity on the top, and lower velocity in the wake of the separation bubble. If the pipe were long enough, the flow would develop towards Poiseuille flow; since the pipe is only $4.1 D$ in length, this non-uniform flow persists all the way to injection. The axial planes also show significant secondary flows, with magnitude of up to about 20% of the bulk velocity in the hole. In both cases, there is a counter-clockwise vortex on the top of the planes, and flow along the walls from the top to the bottom.
At first, it might seem intuitive to think that changing velocity ratio from $r=1$ to $r=2$ would impact the flow structure in the hole. That is because the Reynolds number based on hole bulk velocity increases by a factor of two, and the blockage effect as seen by the hole caused by the jet meeting the crossflow becomes less significant as $r$ increases. However, both of these effects seem to be small because Fig. \[fig-18-velhole\] shows that the velocity field non-dimensionalized by $rU_c$ is mostly insensitive to $r$ in this range. The only discernible difference is that the wake of the separation bubble seems to recover slightly faster for $r=1$ compared to $r=2$, an effect captured both in the LES and the MRV.
It is also important to comment on the agreement between the LES and the MRV within the hole. Qualitatively, the plots agree well, including the trends of the secondary flows and the location and size of the separation and its wake. Quantitatively, however, the agreement is not as good as that shown in other regions of the flow. MRV fails to provide high quality data close to solid walls because of signal loss due to partial volume effects. Signal loss is also observed in regions of excessive turbulence. These effects combined with the low experimental resolution at the hole, which is not enough to capture the sharp velocity gradients shown in the LES, greatly increase uncertainty there. Thus, for the present MRI setup, analysis of the data should be limited to broad qualitative features in the hole. Given that the LES matches the experiments elsewhere, this is a region where the LES results are more trustworthy than the MRI data and can be used to enrich them.
Turbulence data
---------------
Two examples of enrichment of MRI data were shown in the mean data, namely concentration at the wall and velocity in the hole. Another broad category of numerical information that complements the MRI data is turbulence statistics, which can be calculated due to the unsteady nature of the LES but are not available in MRI data. The present subsection briefly exemplifies these results.
Fig. \[fig-19-x2turbulence\] shows two turbulence correlations that play an important role in the Reynolds-averaged equations for velocity and concentration. It contains $y-z$ planes at $x/D=2$ for all three velocity ratios. Figs. \[fig-19-x2turbulence\](a)-(c) show one component of the Reynolds stress, $\overline{u'v'}$, with both velocity factors non-dimensionalized by the jet velocity. Figs. \[fig-19-x2turbulence\](d)-(f) show the vertical turbulent scalar flux, $\overline{v'c'}$, with the velocity scaled by the jet velocity. The non-dimensionalization was chosen in an attempt to represent all velocity ratios in the same contour.
Broadly, the two turbulence quantities behave similarly: they are positive on the top shear layer, negative on the bottom half of the jet, and close to zero at the jet core. However, they seem to scale differently as the velocity ratio changes: $\overline{u'v'}$ changes more drastically (with the positive regions intensifying and the negative regions lessening), while the values of $\overline{v'c'}$ (scaled by $r$) are similar across the three simulations. This might be due to changing mean profiles: as can be seen in Fig. \[fig-15-uz0\], the structure of mean concentration gradients does not change much from $r=1$ to $r=2$, but the mean velocity gradients experience quantitative change because in the former the jet core is slower than the freestream and in the latter it is faster. Note also the counter-intuitive behavior of $\overline{u'v'}$ and $\overline{v'c'}$ under the jet: this is a location where $\partial \bar{u} / \partial y > 0$ and $\partial \bar{c} / \partial y > 0$, but $\overline{u'v'}$ and $\overline{v'c'}$ are weakly positive. This suggests that linear eddy viscosity models would struggle to model the turbulence in this region.
Conclusion
==========
In the present work, 3D MRI experiments and highly resolved Large Eddy Simulations were performed on the same geometry, an inclined jet in crossflow, with three distinct velocity ratios ($r=1$, $r=1.5$, and $r=2$). The flow is incompressible, the Reynolds number based on the jet velocity varies between $Re_D = 2,900$ and $Re_D = 5,800$, and the concentration of a passive scalar is also tracked. An important contribution of this work is to document the effort to carefully match operating conditions between MRI experiments and high fidelity simulations, especially that of setting consistent inlet conditions. When this is done properly, good agreement can be expected across the whole 3D domain; then, since the datasets credibly represent the same flow, we say that the LES enriches the MRI dataset with information that the experiment is unable to provide. Future workers who perform MRI experiments can follow some of the present guidelines to complement their experimental data with Large Eddy Simulations, allowing them access to turbulent statistics and near-wall data.
The experimental and numerical data are used to study the inclined jet in crossflow. Under the present conditions, the jets always separate from the bottom wall after injection, but the $r=1$ jet re-attaches while the other two do not. There is also some slight but persistent asymmetry, which can be attributed to the plenum feeding mechanism. Finally, the hole in the present jet is short, as is common in film cooling applications; this causes a highly complex flow in the jet as it meets the crossflow, which is usually not accounted for in previous literature. The flow structure in the hole does not depend much on the blowing ratio in the range studied, however. The analysis of the mean flow also serves to validate the simulation data. In general, the agreement is excellent.
Using the MRI data to validate the LES is especially valuable in a complex non-homogeneous flow since data are available in 3D and mean flow structures can be compared. Access to the full field creates confidence that the whole simulated flow is physical, instead of the unfortunate scenario in which a single number (such as velocity/temperature at a single point, or a mean drag/lift coefficient) is matched despite the predicted flow being qualitatively incorrect. Also, if any discrepancies are found it is easier to trace back what aspects of the simulation might be causing the differences. In the present paper, we performed not just qualitative comparisons, but also quantitative ones that leverage the full 3D domain to calculate metrics, such as the circulation analysis. For other flows, it is important to design these metrics such that (i) they represent the flow physics that one desires to capture and (ii) they are robust to MRI noise and possible misalignment (MRI data consist solely of a Cartesian mesh of data, without the exact location of the walls). The latter constraint usually precludes an analysis in which simulation data are interpolated into the MRI mesh and the values are compared cell-by-cell, because this is highly sensitive to misalignment in locations where local gradients are high. Instead, we compared circulations that were computed independently in each dataset, and quantified the uncertainty due to potential misalignment in the MRV; this metric is useful since it captures the strength of the mean counter rotating vortex pair.
In the present work, some turbulence data were presented mostly to demonstrate the enrichment potential. Such data will be further analyzed in future work, especially in regards to the turbulent scalar flux. The datasets presented here consist of a high quality and validated set of simulations with parameter variation (in this case, the velocity ratio $r$) that will be used for data-driven turbulence modeling in film cooling flows, as was done by @milani2018approach. They can also be made available to any researcher upon request.
|
---
abstract: 'GUI-based models extracted from Android app execution traces, events, or source code can be extremely useful for challenging tasks such as the generation of scenarios or test cases. However, extracting effective models can be an expensive process. Moreover, existing approaches for automatically deriving GUI-based models are not able to generate scenarios that include events which were not observed in execution (nor event) traces. In this paper, we address these and other major challenges in our novel hybrid approach, coined as [[MonkeyLab]{}]{}. Our approach is based on the [*Record*]{}$\rightarrow$[*Mine*]{}$\rightarrow$[*Generate*]{}$\rightarrow$[*Validate*]{} framework, which relies on recording app usages that yield execution (event) traces, mining those event traces and generating execution scenarios using statistical language modeling, static and dynamic analyses, and validating the resulting scenarios using an interactive execution of the app on a real device. The framework aims at mining models capable of generating feasible and fully replayable (i.e., actionable) scenarios reflecting either natural user behavior or uncommon usages (e.g., corner cases) for a given app. We evaluated [[MonkeyLab ]{}]{}in a case study involving several medium-to-large open-source Android apps. Our results demonstrate that [[MonkeyLab ]{}]{}is able to mine GUI-based models that can be used to generate actionable execution scenarios for both natural and unnatural sequences of events on Google Nexus 7 tablets.'
author:
-
bibliography:
- 'ms.bib'
title: 'Mining Android App Usages for Generating Actionable GUI-based Execution Scenarios'
---
=10000 = 10000000 = 1000000
GUI models, mobile apps, mining execution traces and event logs, statistical language models
Introduction {#sec:intro}
============
Background and Related Work {#sec:related}
===========================
Mining and Generating Actionable Execution Scenarios with [[MonkeyLab]{}]{} {#sec:approach}
===========================================================================
Empirical Study Design {#sec:study}
======================
Results and Discussion {#sec:results}
======================
Conclusion and Future Work {#sec:concl}
==========================
|
---
abstract: 'We compute the electromagnetic response and corresponding forces between two silver nanowires. The wires are illuminated by a plane wave which has the electric field vector perpendicular to the axis of the wires, insuring that plasmonic resonances can be excited. We consider a nontrivial square cross section geometry that has dimensions on the order of $0.1 \lambda$, where $\lambda$ is the wavelength of the incident electromagnetic field. We find that due to the plasmonic resonance, there occurs great enhancement of the direct and mutual electromagnetic forces that are exerted on the nanowires. The Lippman-Schwinger volume integral equation is implemented to obtain solutions to Maxwell’s equations for various $\lambda$ and separation distances between wires. The forces are computed using Maxwell’s stress tensor and numerical results are shown for both on and off resonant conditions.'
author:
- Klaus Halterman
- 'J. Merle Elson'
- Surendra Singh
title: Plasmonic Resonances and Electromagnetic Forces Between Coupled Silver Nanowires
---
|[$$\begin{aligned}
}
\def\ear{\end{aligned}$$]{}
Introduction
============
The phenomenon of surface plasma resonance excitation (surface plasmon) was first predicted [@ritchie] as an explanation of observed energy loss spectra of electron beams penetrating thin metal films. Also, at optical wavelengths the reflectance of metal (Al, Ag, Au) mirrors with surface roughness is significantly affected by surface plasmon excitation [@stanford]. When considering scattering of plane waves at wavelength $\lambda$ by particles of dimension $<< \lambda$, Rayleigh scattering theory can often be applied. In this case the shape of the particle is not important and the scattering cross section is typically quite small. However more recently, the study of electromagnetic enhancement and field localization by nano-sized metallic structures via plasmonic resonance has been the focus of research activity. In this case, when plasmonic excitation is involved, the shape and material details of the structure become important with respect to conditions for resonance and this greatly increases the scattering of electromagnetic waves. When two or more nanosized structures are illuminated at a resonant wavelength, the electromagnetic coupling and associated charge redistribution that occurs can result in a significant mutual coupling force. Plasmon resonances lead to extremely large localized fields at specific wavelengths in the the vicinity of nanoparticles, thereby resulting in a large scattering cross section [@r02a] and mutual forces between nanoparticles. The localized fields also play an important role in the surface enhanced Raman scattering. This enhancement can be large enough to enable the detection of a single molecule. A primary driving force for the recent focus on plasmon resonance of nanoparticles is the application of optical properties of these structures in novel optical devices[@r02; @r03] and biosensors [@r04; @r05]. It has been noted that nanowires with different cross sectional shapes such as, cylindrical, triangular and elliptical exhibit multiple resonances depending on their particular shape. For example, the field enhancement in the vicinity of a 20 nm triangular particle can exceed 400 times the incident field amplitude, while it is only 10 for a cylindrical particle of the same size [@r06]. In the past few years, several researchers have investigated the resonance effects of particle sizes larger than 100 nm. Very recently, attention has been focused on resonance effects of nanoparticles in the 20 to 50 nm range [@r06; @f1].
A primary difficulty in dealing with particle size smaller than 100 nm, especially during resonance conditions, is the increasing computational complexity due to rapid variations of the field over short distances. For structures that deviate from the simpler cylinder and sphere geometries, analytical approaches are not available, and one must rely on an effective numerical approach that correctly incorporates the necessary boundary conditions. An efficient numerical procedure for the computation of fields at the plasmon resonance wavelength of nanoparticles of arbitrary cross section is afforded by the Lippman-Schwinger integral equation [@r02a]. This numerical formalism is a volume integral equation utilizing the Green’s tensor for 2D or 3D geometries. In this paper, we calculate the electromagnetic forces exerted on two nanowires with square cross section. The side dimensions of the nanowires are 30 nm and the wavelength of the incident field varies from 300-516 nm.
method
======
We first outline the volume integral equations, which are appropriate for the investigation of nanosystems, and in this case electromagnetic resonances in 2D coupled nanowire structures. The system is assumed infinite in the $z$ direction, and all spatial variation occurs in the $x$-$y$ plane. The physically relevant quantities that we shall focus on are the electric ($\bf E$) and magnetic ($\bf H$) fields and the nature of net forces exerted on a given wire under conditions of plasmonic resonance. In the absence of magnetic media ($\mu=1$), the basic integral equations are written as [@blackbook],
$$\begin{aligned}
\label{LS.1}
{\bf E}({\bf r}) &= {\bf E}^{\rm inc}({\bf r})+k_0^2 \int_S d^2r^{\prime} {\bf G}_e({\bf r},{\bf r}^{\prime}) \cdot \delta\epsilon({\bf r}^{\prime}){\bf E}({\bf r}^{\prime}), \\
\label{LS.2}
{\bf H}({\bf r}) &= {\bf H}^{\rm inc}({\bf r})-i k_0 \int_S d^2r^{\prime} {\bf G}_m({\bf r},{\bf r}^{\prime}) \cdot \delta\epsilon({\bf r}^{\prime}){\bf E}({\bf r}^{\prime}),\end{aligned}$$
where $k_0=\omega/c$, ${\bf r}=(x,y)$, and the integration is over the cross-sections $S$ of the nanowires, which are embedded in vacuum. The free-space electric dyadic Green’s function, ${\bf G}_e({\bf r},{\bf r}^\prime)$, is a second-rank tensor and its explicit form has been given elsewhere [@blackbook]. The magnetic dyadic Green’s function, ${\bf G}_m({\bf r},{\bf r}^{\prime})$, is calculated using ${\bf G}_m({\bf r},{\bf r}^{\prime})=\nabla\times\bigl[{\bf I}\, g_0 ({\bf r},{\bf r'})\bigr]$, where $g_0 ({\bf r},{\bf r'})=(i/4) H_0(k_0 \rho) \exp(i k_z z)$. Since the incident field propagates solely in the $x-y$ plane ($k_z=0$), this yields the following nonzero components: $G^m_{xz}=-i/(4\rho) k_0 (y-y') H_1(k_0\rho)$ and $G^m_{yz}=i/(4\rho) k_0 (x-x') H_1(k_0\rho)$, where $\rho = |{\bf r} - {\bf r}^{\prime}|$. The other components are found straightforwardly using the antisymmetric relation $G^m_{i j}=-G^m_{j i}$. The scalar permittivity contrast is mapped according to $\delta\epsilon({\bf r}) \equiv \epsilon - 1$, when ${\bf r}$ is entirely within the nanowire with permittivity $\epsilon$, and $0$ otherwise. We then cast the integrand of (\[LS.1\]) into the form of a linear matrix equation system by discretizing the wire surface as a fine grid for numerical integration. Once the electric field is found within the scattering medium, ${\bf E}({\bf r})$ is then inserted into (\[LS.2\]) to calculate ${\bf H}({\bf r})$. As a check, we calculated the ${\bf H}$ field directly from the ${\bf E}$ field using the Maxwell-Faraday law and found consistency with the results obtained using Eq. (\[LS.2\]).
To calculate the net force acting on a particular wire, we begin with Maxwell’s stress energy tensor, $\bf T$, which has components,[@jacko] T\_ = . The net force on an object, enclosed by an area $A$ is then given by ${\bf F} = \langle\oint_A ( {\bf T \cdot n} )\, dA \rangle$ where ${\bf n}$ is the outward normal to the surface $A$ enclosing a given nanowire, and $\langle \rangle$ denotes the time average. For the geometry under consideration, and with ${\bf E}=(E_x,E_y)$, only the $T_{xx}, T_{xy}$, $T_{yx}$ and $T_{yy}$ components of the stress tensor contribute to $\bf F$.
numerical results
=================
The Lippmann-Schwinger equation (Eq. (\[LS.1\])) can be cast into the standard form of a linear equation system $\bf A x=b$, and is solved here numerically by an efficient and stabilized version of the conventional BiConjugate gradient method.[@bcg] The matrix ${\bf A}$ does not have to be stored in computer memory, allowing discretization of the scatterer over a very fine scale without undue limitations on the storage requirements. We model the permittivity of the metal nanowires using the experimental silver data in Ref. . The square nanowires are taken to be $30$ nm per side dimension and the grid count per nanowire is $80 \times 80$. This yields a grid spatial resolution of $0.375 \,{\rm nm} \times 0.375 \,{\rm nm}$. Several numerical tests confirmed that this resolution is quite satisfactory.
In general for silver nanowires with small cross sections, the inverse collision time for electrons can increase from the additional scattering events that take place at the surface, and thus scattering from the boundary becomes relevant. This problem has been treated using alternate techniques. The effect of electron screening is expected to also reduce the inverse collision time, however a careful modeling of the particle surface showed [@apell] an effective scattering rate that is comparable to the classical result.
Significant attraction between two metal nanowires can be generated by Coulombic forces. If the gap distance is such that the plasmonic evanescent near-fields overlap, then the potential for coupled-wire resonant modes exist. These coupled-wire modes can be very complex depending on gap distance, permittivity, wavelength, and geometric parameters. These modes can have a widely varying spatial distribution of the electron plasma. It then follows that if the gap fields are enhanced because of a resonance condition, there should be enhanced Coulombic forces.
![(Color online) The wavelength dependence of the $x$-component of the time-averaged total force on the left wire (wire 1) for five different separations $d$, as indicated in the legend. The incident plane wave propagates downwards in the $-\hat{\bf y}$ direction, while the field is polarized along the $\hat{\bf x}$ direction, which is appropriate to excite surface plasmons. The force is always positive, indicating attraction between the nanowires since the adjacent wire (wire 2) feels an identical force that is opposite in sign (not shown). The inset illustrates the corresponding $x$-component of the dipole moment, with the solid gray curve representing the case of a single wire. []{data-label="fig1"}](fig1.eps){width="4in"}
To address this, we begin by examining the forces exerted on the nanowires due directly to the incident wave and the mutual coupling between the wires. The results naturally depend on the propagation direction of the incident field, given by its wavevector ${\bf k} = -k_0(\cos\theta \hat{\bf x} + \sin\theta \hat{\bf y})$. In Fig. \[fig1\], we illustrate the $x$-component of the time-averaged force, $F_x$, on the left wire (wire 1) as a function of the incident field wavelength, $\lambda$, and for various gap distances $d$. The incident electric field vector is polarized along the $\hat{\bf x}$ direction, and the wavevector is directed along $-\hat{\bf y}$. This particular incident field propagation direction yields oppositely directed net forces on each wire that are equal in magnitude. It is evident that $F_x$ is positive for wire 1 over the whole $\lambda$ range shown, indicating attraction among the nanowires. It is seen that for much of the $\lambda$ range, smaller gaps yield greater maximum values of $F_x$. All curves show a decrease and blueshift in the $F_x$ peak as $d$ increases, except for the $d = 20$ nm curve, which shows very little apparent interaction between the wires. This is consistent with a coupled-wire resonance condition that is supported by the $p_x$ inset showing the $x$-component of the dipole moment of wire 1. This quantity is obtained by integrating the appropriate component of the real part of the polarization, ${\bf P}({\bf r}) \equiv 1/4\pi \delta\epsilon({\bf r}) {\bf E}({\bf r})$, over the cross sectional area of the wire. We show here only $p_x$ since this component of the dipole moment encapsulates the relevant charge distribution involved in interwire coupling. In the inset the solid gray curve shows for reference, $p_x$ for a single isolated wire. Note that the $d = 20$ nm curve for $p_x$ is very similar to the solid gray curve, further indicating very little mutual interaction between the wires. For $330 \,\,{\rm nm}\lesssim \lambda \lesssim 400$ nm, $p_x$ has negative slope and becomes more negative quite rapidly with wavelength. This is approximately the region over which the $F_x$ curves reach their primary peak. It is also interesting that for larger wavelengths, there is another set of $F_x$ peaks that are visible in the $d =
3$, $5$, and $7$ nm curves. These peaks become washed out at larger $d$, and approximately correlate with $p_x$ (inset) where the slopes are positive.
\
The sign change in the dipole moment exhibited in Fig. \[fig1\] is easily visualized by means of vector plots. To this end we show in Fig. \[vec:sub\] the electric field vector distribution within the nanowires. The wires are separated by $5$ nm and the top and bottom panels correspond to incident field wavelengths of $\lambda=340$ nm and $\lambda=385$ nm respectively. In the top $\lambda=340$ nm panel, the corner dipole fields are most intense on the outer left and right sides. There is a subsequent nonuniform, strongly position dependent field orientation within the nanowires. The $\bf E$ field on the sides nearest the gap region is relatively small compared to the corner fields. In the bottom $\lambda=385$ nm panel, the situation is markedly different. Here the gap field is dominant and different mode patterns arise. These results are consistent with Fig. \[fig1\], where $F_x$ at $\lambda=340$ nm is small compared to $\lambda=385$ nm.
\
We now vary the incident electric field so that it is now polarized in the $\hat{\bf y}$ direction, and propagates along $-\hat{\bf x}$. We show in Fig. \[fig2:sub:a\], $F_x$ on wire 1 as a function of $\lambda$. Figure \[fig2:sub:b\] depicts the same quantities for wire 2. In general, the net force on each wire is no longer equal in magnitude due to the uneven radiation effects directly from the incident wave. Figure \[fig2:sub:a\] shows that for sufficiently small wire separations ($d \leq 10$ nm), the force on wire 1 changes sign in a continuous manner as $\lambda$ is varied. The force on wire 2 is mainly negative except for a narrow band of $\lambda$ in the vicinity of $\lambda\approx 336$ nm (see Figure \[fig2:sub:b\]), which correlates to when the force on wire 1 is most negative. Thus the wires experience a [*repulsion*]{} at these wavelengths and separations. For larger separations $F_x$ diminishes and tapers towards zero at higher wavelengths, indicating that electromagnetic coupling between the two wires is negligible. The plasmon resonances within these structures reveal themselves through prominent peaks in the force curves in both figures. The peak positions are strongly dependent on $d$, as they broaden and reduce in magnitude as $d$ increases. Comparing the scales in Figs. \[fig1\] and \[fig2:sub\], we find that the overall resonance effects observed for the forces at small separations is reduced somewhat for this incident field direction. The insets in Fig. \[fig2:sub\] show the $y$-component of the electric dipole moment ($p_y$) versus $\lambda$ for wires 1 and 2. For wire 1 and $d=5$ nm, Fig. \[fig2:sub:a\] shows that $p_y$ undergoes a minimum at $\lambda\approx328$ nm and then sharply crosses zero at $\lambda\approx 332$ nm. This trend is similar for the other separations, with only a slight redshifting for larger $d$. For both wire 1 and wire 2 a clear correlation exists between $p_y$ and the associated force; for small enough $d$, the pronounced peaks or depressions in $p_y$ correspond to when $|F_x|$ is largest. These features therefore become redshifted and spread-out with decreasing separation. As one would expect for $d=20$ nm, the general features of $p_y$ approach that of the limiting case of a single nanowire.
![(Color online) Electric field amplitude contours (normalized by the incident field) for two nanowires illuminated by a plane wave with wavevector direction indicated by the arrows. The separation distance is $d=5$ nm. The electromagnetic response of the nanowire pair is quite different depending on the angle of incidence and wavelength. In the two upper plots, the electromagnetic response and subsequent attractive force is greatest for the shortest wavelength. Conversely, in the lower plots the response is greatest for the longer wavelength. These results are consistent with Figs. \[fig1\] and \[fig2:sub\]. []{data-label="onefield"}](fig4.eps){width="5in"}
To contrast the near-field electric field patterns for differing incident wave directions, we display in Fig. \[onefield\] the electric field amplitudes in the region of the two nanowires. The wires are set at a fixed distance of 5 nm and are illuminated at two different wavelengths. The top two panels correspond to an incident field directed downwards (as shown by the arrows) with $\lambda=385$ nm and $\lambda=470$ nm. The bottom two panels are for an incident electric field traveling towards the left. Beginning with the top left panel, at $\lambda=385$ nm there is a large field enhancement from the corners and within the gap between the wires. This is consistent with Fig. \[fig1\], where the force has a maximum at that wavelength. As the wavelength is increased to $\lambda=470$ nm, the system is no longer at resonance and the interaction between the wires has decreased. This follows from Fig. \[fig1\], where the force is reduced by nearly a factor of three from its peak value. A different situation is observed in the bottom panels, where at $\lambda=385$ nm the electric field is small between the wires, although the outer corners are still illuminated. At $\lambda=470$ nm, the field amplitude is intensified predominantly within the gap as the nanowires are electrically coupled. There is a clear relationship between the force profiles in Fig. \[fig2:sub\] and the field amplitude plots. It is apparent that the incident field “sees" a different geometry based on its orientation relative to the coupled structure; downward propagating waves interact with an effectively larger nanowire in addition to the interwire gap that may support additional modes due to the wire-wire interaction.
\
Next, in Fig. \[fxboth:sub\], the force on each of the two nanowires is shown as a function of the separation distance $d$, and for an incident field directed along $-\hat{\bf x}$. Also shown is the corresponding induced net dipole moment, $p_y$, for each wire. There are six $\lambda$ considered and the variation of $F_x$ with $d$ is strongly wavelength dependent. Consider first the longest wavelength, $\lambda=470$ nm where the separation range $d = 4 - 5$ nm shows a large positive $F_x$ in Fig. \[fxboth:sub\](a) and a correspondingly large negative $F_x$ in Fig. \[fxboth:sub\](b). This indicates a resonant condition resulting in a relatively strong attractive force. This is in contrast to the wavelengths, $\lambda=375$ and 385 nm where the $F_x$ curves remain negative and comparatively featureless over the entire $d$ range. Clearly no resonant condition occurs for this shorter wavelength as is also evidenced in Fig. \[fig2:sub\]. These results are consistent with the two lower plots in Fig. \[onefield\] that show little gap field enhancement at $\lambda=385$ nm and a large gap field for $\lambda=470$ nm. The inset plots in Fig. \[fxboth:sub\] are similar in that the maxima in $|F_x|$ show up as discernible peaks in $|p_y|$, reflecting the charge redistribution. In general there is a redshift in the peak values and a sharp decline of $|F_x|$ as $d$ increases, indicating a corresponding shift in the resonance parameters of the mutually interacting wires.
![The normalized $x$-component of the force on wire 1 versus separation $d$. The force on the adjacent wire (wire 2) is identical except for a sign change. The incident field is directed downward as seen in the Fig. \[fig1\]. The inset is the corresponding (normalized) dipole moment and the curves correlate with the legend and the same set of parameters are used as in the main plot. []{data-label="fig4"}](fig6.eps){width="4in"}
The normalized force, $F_x$, on the left wire (wire 1) as a function of $d$ for an incident field propagating downwards (in the $-\hat{\bf y}$ direction) is shown in Figure \[fig4\]. This figure is analogous to Fig. \[fxboth:sub\]. As in Fig. \[fig1\], the force on each wire is equal in magnitude and oppositely directed, yielding an attractive nature over the relatively broad wavelength range studied. The main resonant peaks are present at small separations and then $F_x$ monotonically declines towards zero with increased $d$. The wavelengths that give the largest $F_x$ occur in the range $385-410$ nm. Outside of that range, increasing $\lambda$ has the effect of reducing the force and narrowing its peaks. For sufficient separation distances, the force becomes weakly dependent on the wavelength. These characteristics reveal that there are optimal separations and wavelengths to utilize the plasmon driven force enhancement. The inset shows the normalized $x$-component of the dipole moment which is identical in each nanowire. The curves each follow a similar trend, with large separations maintaining a negative dipole moment, while for $\lambda=375$ nm the possibility exists to induce a positive net dipole moment for small separations. Thus for a particular value of $d$ it is possible to have a vanishing dipole moment in the wire. This follows from regions within the wire structure that have oppositely oriented polarization vectors that sum to zero.
In conclusion, we have studied the electromagnetic response of two long silver nanowires illuminated by a plane wave with the electric field perpendicular to the axis of the wires. The volume integral approach used here provides accurate solutions to Maxwell’s equations over length scales that are much smaller than the wavelength of the driving field. We found that by varying the wavelength, separation distance, and angle of incidence of the electric field, the collective oscillation of the electrons induced appreciable forces and coupling in the nanowires. The results shown here can play a role in the fabrication and design aspects of optical spectroscopy and waveguiding over nanometer length scales.
This work of K. H. and J.M.E. is funded in part by the Office of Naval Research (ONR) In-House Laboratory Independent Research (ILIR) Program and by a grant of HPC resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks as part of the Department of Defense High Performance Computing Modernization Program. S.S. would like to acknowledge support from the ONR/ASEE Summer Faculty Program.
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address: 'Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England, U.K.'
author:
- 'Mark S. Joshi'
- Antonio Sa Barreto
title: Magnetic Inverse Problem
---
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
\#1 [ \#1]{} \#1[\#1 ]{} \#1[ ]{} \#1\#2 \#1\#2[()[\#1]{}[\#2]{}]{} \#1\#2
Computing the Symbol
====================
In this section, we compute the symbol of the scattering matrix - in particular given two magnetic fields which agree to some order, we compute the principal symbol of the difference of the associated scattering matrices in terms of the lead term of the difference of the magnetic fields. We use the techniques of [@smagai], [@smaginv] and [@explicit] to construct the Poisson operator and compute the symbols. We proceed explicitly where possible but occasionally fall back on results from [@smagai] for brevity.
Let $P$ be the operator, $$P = \sum \limits_{j=1}^{n} \left( i \frac{\p}{\p x_j} + A_j \right)^2
+ V = \Delta + i \sum \limits_{j=1}^{n} A_j \frac{\p}{\p x_j} + q$$ where $(A_1,\dots,A_n)$ and $V$ are real-valued classical symbols of order $-2.$ The Poisson operator is then the map that maps $f \in
C^{\infty}(S^{n-1})$ to the smooth function $u$ such that $(P -
\lambda^2)u=0,$ and $$u = e^{i\lambda |x|}|x|^{-\frac{n-1}{2}} f(x/|x|) +
e^{-i\lambda |x|}|x|^{-\frac{n-1}{2}} g(x/|x|) +
O(|x|^{-\frac{n+1}{2}}).$$ The kernel of the Poisson operator will be a smooth function on $S^{n-1} \times \mrn$ with singular asymptotics. The scattering matrix is the map on $C^{\infty}(S^{n-1}),$ $$S(\lambda) : f \mapsto g.$$
Our first result in this section is
With $(A_1, \dots, A_n)$ and $q$ as above, we have that $a^{*}S(\lambda)$ is a zeroth order classical pseudo-differential operator, where $a(\omega) = -\omega.$
We remark that saying that $a^{*}S(\lambda)$ is a zeroth order classical pseudo-differential operator is equivalent to saying that is a classical Fourier integral operator of order $0$ associated with geodesic flow at time $\pi.$ This proposition is clear from any of [@smagai], [@explicit] and [@andras] by just observing that the arguments are not changed by adding a first order short range self-adjoint perturbation, we therefore only present the parts of the proof which relate to the proof of our main result:
\[symbolcomputation\] Let $(A_1,\dots,A_n)$ and $(A^{'}_{1},\dots,A^{'}_{n})$ be real-valued classical symbols of order $-2$ with $S(\lambda)$ and $S'(\lambda)$ the associated scattering matrices. If $A_j - A^{'}_{j}$ is of order $-k$ with $k \geq 2$ then $S(\lambda) - S'(\lambda)$ is of order $1-k.$ If $\sum \limits_{j=1}^{n} (A_j - A^{'}_{j}) dx_j = B$ with lead term $B^{(k)},$ and $B^{(k)}$ is aradial then the principal symbol of $S(\lambda) -
S'(\lambda)$ determines and is determined by $$\int \limits_{0}^{\pi} \langle B^{(k)}(\gamma(s)), \frac{d\gamma}{ds}(s)
\rangle (\sin s)^{k-1} ds$$ for all geodesics $\gamma,$ where we regard $B^{(k)}= \sum B^{(k)}_{j}
dx_j$ as a one form canonically pairing with the vector $\frac{d\gamma}{ds}(s).$
We say a one-form is [*aradial*]{} if its pairing with the radial vector field, $x\frac{\p}{\p x},$ is zero.
Following the ideas of [@smagai], [@explicit], [@smaginv], we look to construct the Poisson operator for the problem as a sum of oscillatory integrals and then use this to read off the properties of the scattering matrix. In particular, we attempt to construct the Poisson operator as $$e^{i\lambda x.\omega}(1+b(x,\omega)),$$ with $b$ a classical symbol of order $-1$ in $x$ and smooth in $\omega.$ We will see that this ansatz works away from the set $\omega = - x/|x| .$ Applying $P-\lambda^2,$ we obtain $$e^{i\lambda x.\omega} \left( -2i\lambda \omega.\frac{\p b}{\p x} +
\Delta b - \lambda \sum \limits_{j=1}^{n} \omega_j A_j (1+b) + i
\sum \limits_{j=1}^{n} A_j \frac{\p b}{\p x_j} +q +qb
\right).$$ That this can be solved smoothly to infinite order in a neighbourhood of $x/|x|=\omega$ is just a repetition of the argument in the proof of Proposition 18 of [@smagai].
We let $b$ have asymptotic expansion $\sum \limits_{j=1}^{\infty}
b_{-j},$ let $A_{l}$ have expansion $\sum \limits_{j=2}^{\infty} A_{l}^{(-j)}$ and $q$ have expansion $\sum \limits_{j=2}^{\infty} q_{-j}.$ In order to continue $b$ smoothly we observe that the lead term is $$-2i\lambda \omega.\frac{\p b}{\p x} - \lambda \sum \limits_{j=1}^{n}
\omega_j A^{(-2)}_j +q_{-2}.$$ We want this to be zero.
Note as these are homogeneous functions this is really an equation on the sphere. We therefore take coordinates $(r,s,\theta)$ where $r=|x|,$ $s$ is the geodesic distance of $x/|x|$ from $\omega$ and $\theta$ is the angular coordinate about $\omega.$ Note the coordinate system depends on $\omega$ but we shall suppress $\omega$ in our notation most of the time.
Now with out loss of generality, we can take $\omega$ to be the north pole. Then $$\omega . \p_x = \p_{x_n} .$$ Now $\cos(s) = \frac{x_n}{|x|}, $ $r=|x|.$ The $\theta$ coordinate will be purely parametric. We have $$\frac{\p s}{\p x_n} = \frac{-1}{\sqrt{1- \frac{x_{n}^{2}}{|x|^2}}} \left(
\frac{1}{|x|} - \frac{x_{n}^{2}}{|x|^3} \right) = -r^{-1} \sin(s),$$ and $$\frac{\p r}{\p x_{n}} = \cos(s).$$ So applying $\omega.\p_{x}$ to $b_{-j}(s,\theta; \omega)r^{-j},$ we obtain $$r^{-j-1} \left[ -\sin(s) \frac{\p b_{-j}}{\p s} - j \cos(s) \right] b_{-j}.$$ Let $W_{-2} = -\lambda \sum \limits_{j=1}^{n}
\omega_j A^{(-2)}_j - q_{-2}.$ So for $j=1,$ we have taking $r=1$ $$2i\lambda (\sin(s) \p_{s} + \cos(s) ) a_{-1} = W_{-2},$$ which is equivalent to $$2i\lambda \p_s (\sin(s) a_{-1} ) = W_{-2}.$$ We want $b$ to be smooth at $s=0,$ so $$\sin(s) b_{-1} = \frac{1}{2i\lambda} \int \limits_{0}^{s} W_{-2} ds',$$ which implies that $$a_{-1}(s,\theta; \omega) = \frac{i}{2\lambda \sin(s)} \int \limits_{0}^{s}
W_{-2}(s',\theta; \omega) ds'. \label{firsterrorterm}$$ This will be singular as $s \to \pi- ,$ but let’s ignore that for now. Now suppose we have chosen the first $j$ terms so we have an error $d \in
S^{-j-1}_{cl} $ with lead term $d_{-j-1}(s,\theta; \omega)r^{-j-1}.$ We then want to solve the transport equation, $$-2i\lambda \omega . \p_z ( a_j r^{-j}) + d_{-j-1} r^{-j-1} =0,$$ as above we get $$2i \lambda [ \sin(s) \p_s +j \cos(s) ] a_j +d_{-j-1} =0.$$ We solve this to obtain, $$a_j(s,\theta, \omega) = \frac{i}{2\lambda (\sin(s))^{j}} \int \limits_{0}^{s}
(\sin(s'))^{j-1} d_{-j-1}(s', \theta; \omega) ds'. \label{genterm}$$ So away from $s=\pi,$ we can achieve an error in $S^{-\infty}$ by applying Borel’s lemma. ie away from the antipodal point. Note that we have a focussing of the geodesics and as well as the fact the solutions blow-up we also have that they will have different values according to the angle. In particular provided $d_{-j-1}$ does not grow faster than $(\pi
-s)^{1-j}$ we have that $a_j$ grows as $(\pi -s)^{-j}.$
Before introducing a second ansatz to cope with the antipodal point, we compare the Poisson operators associated to two different magnetic potentials. Suppose $(A_1, \dots, A_n)$ and $(A^{'}_1, \dots,
A^{'}_n)$ are both classical symbols of order $-2$ and the difference is $(B_1, \dots, B_n)$ which is a classical symbol of order $-k,$ with lead term $(B^{(-k)}_{1}, \dots, B^{(-k)}_{n}).$ The first $k-2$ forcing terms above are then unchanged and the forcing terms at level $k-1$ will differ by $W_{-k} = -\lambda \sum \limits_{j=1}^{n}
\omega_j B^{(-k)}_{j} .$ (Note that the change in the zeroth order term will be lower order.)
Thus the lead term of the difference of the Poisson operators will be $$\frac{i r^{1-k}}{2\lambda (\sin s)^{k-1}} \int \limits_{0}^{s}
W_{-k}(s',\theta; \omega) (\sin s')^{k-2} ds'.$$ This is the important result in our construction as we will see that the lead term of this as $s \to \pi-$ is essentially the principal symbol of the difference of the scattering matrices. We therefore want an invariant interpretation of $$\int \limits_{0}^{\pi} (\sin s)^{k-2} W_{-k}(s,\theta';\omega) ds.$$ If we take $\omega$ to be the north pole and rotate so that $\theta' =
(1,0,\dots,0),$ the computation lies entirely in the $(x_1,x_n)$ plane and $W_k(s)$ equals $-\lambda B_{n}^{(-k)}(s).$ Now if we assume $B$ is aradial then an elementary computation shows that, $$-\langle B^{(-k)}(\gamma(s)), \frac{d\gamma}{ds}(s) \rangle \sin s=
B_{n}^{(-k)}$$ in this case. So by rotational invariance we deduce that in general the lead term of the difference of the Poisson operators is $$\frac{i r^{1-k}}{2(\sin s)^{k-1}} \int \limits_{0}^{s} \langle
B^{(-k)}(\gamma(s')), \frac{d\gamma}{ds'}(s') \rangle (\sin s')^{k-1}
ds'$$ and that the lead singularity as $s \to \pi-$ is $$\frac{i r^{1-k}}{2(\pi-s)^{k-1}} \int \limits_{0}^{\pi} \langle
B^{(-k)}(\gamma(s')), \frac{d\gamma}{ds'}(s') \rangle (\sin s')^{k-1}
ds'.$$
The remainder of the construction of the Poisson operator and the computation of the symbol is now just a repetition of the arguments in [@smagai] or [@explicit]. We sketch these for completeness.
Taking $\omega$ to be the north pole, close to the south pole we look for the Poisson operator in the form, So close to the south pole, we look for the Poisson operator in the form, $$\int \limits_{0}^{\infty} \int \fracwithdelims(){1}{\s|x|}^{\gamma} \s^{\alpha}
\fracwithdelims(){1}{S|x|}^{\gamma} S^{\alpha} e^{i(Sx'.\mu -
\sqrt{1+S^2}|x|)} a\left(\frac{1}{S|x|},S,\mu \right) dS d\mu,
\label{secansatz}$$ with $a(t,\s,\mu)$ a smooth function compactly supported on $[0,\eps) \times [0,\eps)
\times S^{n-2}$ and $\alpha = \frac{n-3}{2}, \gamma = -\frac{n-1}{2}.$ We assume that $\omega$ has been rotated to the north pole. Note that for $|x|$ in a compact set the integral is supported on a compact set and so we have no problems with convergence - in particular the integral yields a smooth function.
In the lower hemi-sphere, away from the south pole, this ansatz is equivalent to the original one - this follows from an application of stationary phase (see [@explicit]). However the lead term in $|x|$ at order $-k$ is now allowed to be singular of order $-k$ as $s \to \pi-$ (which corresponds to $S=0+$) and there is no constraint on the values for different angles matching. This allows the transport equations to be solved right up to the antipodal point and to all orders. The error is then removed by applying the resolvent which yields a term of the form $e^{-i\lambda|x|}|x|^{-\frac{n-1}{2}}h(x)$ with $h$ a classical zeroth order symbol - this term will not affect the singularities in the distributional asymptotics of the Poisson operator.
We recall Proposition 3.4 of [@explicit], which is a special case of Proposition 16 of [@smagai].
\[disasymp\] If $u(x,\omega)$is of the form \[secansatz\] then $ e^{i\lambda|x|} \int u(|x| \theta,\omega) f(\theta, \omega) d\theta
d\omega$ is a smooth symbolic function in $|x|$ of order $-1-\alpha$ and its lead coefficient is $|x|^{-\alpha-1} \la K, f
\ra$ where $K$ is the pull-back of the Schwartz kernel of a pseudo-differential operator of order $\alpha-\gamma-(n-2)$ by the map $\theta \mapsto
-\theta.$ The principal symbol of $K$ determines and is determined by the lead term of the symbol, $a(t,\s,\mu),$ of $u$ as $\s \to 0+.$
The fact that the scattering matrix is the pull-back of a pseudo-differential operator is now immmediate. To deduce Theorem \[symbolcomputation\], we observe that from our computations above we have that the difference of the second ansatzs for the two Poisson operators will be of the same form but with $\alpha$ increased by $k-1$ and the lead term of the symbol as $S \to 0+$ will be a constant multiple of $$S^{n-k-1} \int \limits_{0}^{\pi} \langle
B^{(-k)}(\gamma(s')), \frac{d\gamma}{ds'}(s') \rangle (\sin s')^{k-1}
ds'.$$ So the theorem then follows from Proposition \[disasymp\].
[aa]{} G. Eskin, J. Ralston, Inverse Scattering Problem for the Schrodinger Equation with Magnetic Potential at a Fixed Energy, Commun. Math. Phys. 173, 199-224 (1995) S. Helgasson, [*Groups and Geometric Analysis,*]{}[ Academic Press 1984.]{} M.S. Joshi, [*Recovering Asymptotics of Coulomb-like Potentials,*]{} to appear in S.I.A.M. Journal of Mathematical Analysis M.S. Joshi, [*Explicitly Recovering Asymptotics of Short Range Potentials,*]{} preprint M.S. Joshi, A. Sá Barreto, [*Recovering Asymptotics of Short Range Potentials,*]{} Comm. Math. Phys. 193, 197-208 (1998) M.S Joshi, A. Sá Barreto, [*Recovering Asymptotics of Metrics from Fixed Energy Scattering Data*]{}, to appear in [*Invent. Math.*]{} R.B. Melrose, [*Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean spaces (M. Ikawa, ed),*]{}[ Marcel Dekker, 1994.]{} R.B. Melrose, M. Zworski, [*Scattering Metrics and Geodesic Flow at Infinity,*]{}[ Invent. Math. 124, 389-436 (1996).]{} A. Vasy, [*Geometric Scattering Theory for Long-Range Potentials and Metrics,*]{} I.M.R.N. 1998 no 6, 285-315
|
---
abstract: 'We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel-Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang-Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.'
address:
- |
Department of Mathematics\
University of Southern California\
Los Angeles, LA
- |
Mathematical Sciences Institute\
Australian National University\
Canberra, Australia
- |
Department of Mathematics and Computer Information Sciences\
Mercy College\
Dobbs Ferry, NY
author:
- Sabin Cautis
- Anthony Licata
- Joshua Sussan
title: Braid group actions via categorified Heisenberg complexes
---
Introduction
============
Let $\D(0)$ and $\D(1)$ be graded, triangulated categories and let $\P: \D(0) \rightarrow \D(1)$ and $\Q: \D(1) \rightarrow \D(0)$ be bi-adjoint functors up to a grading shift (which we take to be equal to $2$ for convenience). If $\Q \circ \P \cong \1_0 \la -1 \ra \oplus \1_0 \la 1 \ra$, where $\1_0$ denotes the identity functor of $\D(0)$, then $\P$ is called a spherical functor. This notion is due (in various levels of generality) to Seidel-Thomas [@ST], Horja [@H], Anno [@A] and Rouquier [@Rou1].
The general theory of spherical twists states that $\T := \Cone(\P \circ \Q \la -1 \ra \rightarrow \1_1)$ is an autoequivalence of $\D(1)$. One important reason to consider equivalences coming from spherical functors is that if $\{\P_i\}$ is a configuration of spherical functors, one for each node $i$ of a simply laced Kac-Moody Dynkin diagram $D$, then the associated auto equivalences $\T_i$ will define an action of the corresponding braid group $\Br_D$ on $\D(1)$.
The notion of a spherical twist was generalized in [@CR; @CK2] to that of categorical $\g$ actions, with $\g$ a symmetric Kac-Moody algebra. Such categorical $\g$ actions can be used to construct further examples of braid group actions. Another generalization of spherical twists, which replaces the role of the Kac-Moody algebra by a Heisenberg algebra, is the subject of the current paper. Namely, fix a simply laced Kac-Moody Dynkin diagram $D$ with vertex set $I$. For each $n \ge 0$, suppose we have a collection of additive categories $\D(n)$ together with bi-adjoint functors $$\P_i : \D(n) \rightarrow \D(n+1) \ \ \text{ and } \ \ \Q_i: \D(n+1) \rightarrow \D(n)$$ for any $i \in I$ which give a 2-representation of a particular Heisenberg algebra (see Section \[sec:2heis\]). Roughly speaking, this means that we have isomorphisms $$\label{eq:heiseq}
\Q_i \circ \P_i \cong \P_i \circ \Q_i \oplus \1_n \la -1 \ra \oplus \1_n \la 1 \ra,$$ along with a precise collection of natural transformations of functors. Such a 2-representation generalizes the notion of a spherical functor since $\P_i$ and $\Q_i$ are spherical functors between $\D(0)$ and $\D(1)$.
However, the data of a Heisenberg 2-representation contains more than just a spherical functor. For instance, the action by natural transformations includes an action of the symmetric group $S_k$ on the composition $\P_i^k$. This splits $\P_i^k$ into a direct sum of indecomposable functors $\P_i^{(\lambda)}$ corresponding to irreducible representations of $S_k$ ($\Q_i^k$ also splits analogously). We may then form a complex $$\label{eq:intro1}
\T_i \1_n := \left[ \dots \rightarrow \bigoplus_{\l \vdash d} \P_i^{(\l)} \Q_i^{(\l^t)} \la -d \ra \1_n \rightarrow \bigoplus_{\l \vdash d-1} \P_i^{(\l)} \Q_i^{(\l^t)} \la -d+1 \ra \1_n \rightarrow \dots \rightarrow \P_i \Q_i \la -1 \ra \1_n\rightarrow \1_n \right]$$ (we have omitted the symbol $\circ$ for composition of functors in the above, as we will do for the remainder of the paper). Theorem \[thm:main1\] of the current paper states that these complexes define a categorical action of the associated braid group on the homotopy category of each $\D(n)$. In particular, each $\T_i$ defines an equivalence of categories.
An important example where the setup above holds is the following. Let $A$ be the skew zig-zag algebra (defined in section \[sec:zigzag\]), which is the quadratic dual of the deformed preprojective algebra of a quiver. For $n \geq 0$, we let $A^{[n]}$ denote the wreath product of $A$ with the group algebra of $S_n$ (by convention, we take $A^{[0]} = \C$). By a formal construction, the braid group action of [@KS; @HK] on the homotopy category $\Kom(A \dmod)$ by spherical twists lifts to a braid group action on $\Kom(A^{[n]} \dmod)$ for each $n$.
On the other hand, from the point of view of representation theory of infinite dimensional algebras, it is natural to consider the categories $\Kom(A^{[n]} \dmod)$ together. In particular, in [@CL1] we define 2-representations of a Heisenberg algebra on $\oplus_n A^{[n]} \dmod$. Thus there are two algebraic objects of interest: the braid group action (which is somewhat complicated) and the Heisenberg action (which is simpler). The constructions of the current paper explain precisely the relationship between these two actions. In particular, we prove that integrable 2-representations of the Heisenberg algebra always induce braid group actions.
There is also a geometric version of this setup, where the algebra $A$ is replaced by a surface $X$, the algebra $A^{[n]}$ is replaced by the Hilbert scheme $X^{[n]}$ of $n$ points on $X$, and the triangulated category $\Kom(A^{[n]} \dmod)$ is replaced by the derived category of coherent sheaves $D(X^{[n]})$. Then, as studied by Ploog [@P], if a group $G$ acts on $D(X)$ then there is an induced action of $G$ on $D(X^{[n]})$. In particular, if $G$ is an affine braid group of simply-laced type one can take the surface $X$ to be the ALE space $\widehat{\C^2/\Gamma}$, where $\Gamma\subset SL_2(\C)$ is the finite subgroup associated to the affine quiver by the McKay correspondence. A 2-representation of the associated Heisenberg algebra on $\oplus_n D(X^{[n]})$ was defined in [@CL1], and Theorem \[thm:geom\] then describes the relationship between this 2-representation and the associated affine braid group action on $D(X^{[n]})$.
In the remainder of the introduction we will give a more detailed exposition of the content in this paper.
Heisenberg actions and braid groups
-----------------------------------
To any simply laced Dynkin diagram $D$ one can associate a quantum Heisenberg algebra algebra $\h$. This algebra has generators $P_i^{(n)}, Q_i^{(n)}$ satisfying relations described in Section \[sec:hei\]. A representation $V$ of this algebra breaks up into weight spaces $V = \oplus_{\ell \in \N} V(\ell)$ with $$P_i^{(n)}: V(\ell) \rightarrow V(\ell+n) \text{ and } Q_i^{(n)}: V(\ell+n) \rightarrow V(\ell).$$
In [@CL1] we define a 2-category $\H$ whose Grothendieck group is isomorphic to $\h$. A 2-representation of $\H$ consists of graded, additive categories $\D(\ell)$ where $\ell \in \N$ and, for any partition $\l$, functors $$\P_i^{(\l)}: \D(\ell) \rightarrow \D(\ell+|\l|) \text{ and } \Q_i^{(\l)}: \D(\ell+|\l|) \rightarrow \D(\ell)$$ satisfying various relations described in sections \[sec:2cat\] and \[sec:2rep\].
Now, in the homotopy category $\Kom(\H)$ of $\H$ one can define complexes as in (\[eq:intro1\]) where the differentials are given by certain explicit 2-morphisms described in section \[sec:cpx\]. Given a 2-representation, each $\T_i \1_n$ defines an endofunctor of $\Kom(\D(n))$. The following is the main result of this paper.
\[thm:main1\] For each $n \in \N$, the map $\sigma_i \mapsto \T_i \1_n$ defines a morphism $\Br(D) \rightarrow \Aut(\Kom(\D(n)))$ where $\Br(D)$ denotes the braid group associated with the Dynkin diagram $D$ and the $\sigma_i$’s are its standard generators.
Now, if we take $A$ to be the zig-zag algebra from [@HK] then, following [@CL1], we can define a 2-representation of $\h$ where $\D(n) = A^{[n]} \dmod$. Theorem \[thm:main1\] above then gives us a morphism $\Br(D) \rightarrow \Aut(\Kom(A^{[n]} \dmod))$. Applying Theorem \[thm:geom\] (see also Remark \[rem:conv\]) this also gives us a morphism $\Br(D) \rightarrow \Aut(D(A^{[n]} \dmod))$.
When $n=1$ this gives the braid group action of Khovanov-Seidel [@KS] via spherical twists. For $n > 1$ we recover the action on $D(A^{[n]} \dmod)$ induced from that on $D(A \dmod)$ (see Theorem \[thm:braids\]).
Another braid group action
--------------------------
The complexes $\T_i$ also act on the 2-category $\Kom(\H)$ by conjugation. It would be interesting to describe this braid group action explicitly.
While we do not address the conjugation action on the entire category $\Kom(\H)$ here, in section \[sec:braidH\] we define another braid group action on $\Kom(\H)$ and conjecture (Conjecture \[conj:intertwiner\]) that it agrees with the conjugation action. This additional action is defined explicitly by describing how each generator $\sigma_i^{\pm 1}$ of the braid group acts on the generating 1 and 2-morphisms.
Although conjecturally related to conjugation by the complexes of Theorem \[thm:main1\], section \[sec:braidH\] is independent of the rest of the paper. The proofs in section \[sec:braidH\] are postponed until the appendix.
Lie algebra actions and braid groups
------------------------------------
The story above closely parallels (and is directly related to) that of quantum groups. Recall that for any Dynkin diagram $D$ on has the associated quantum group $U_q(\g)$. One can consider 2-representations of $\g$, which consist of various additive categories $\D(\l)$ indexed by weights $\l$ and functors $$\E_i^{(r)} \1_\l: \D(\l) \rightarrow \D(\l+r\alpha_i) \text{ and } \1_\l \F_i^{(r)}: \D(\l+r\alpha_i) \rightarrow \D(\l).$$ These functors satisfy certain relations lifting those in $U_q(\g)$. For more details see [@KL1; @KL2; @KL3; @Rou2; @CLa].
In analogy with (\[eq:intro1\]), one can then define the Rickard complexes $$\sT_i \1_\l := \left[ \dots \rightarrow \F_i^{(\la \l, \alpha_i \ra + s)} \E_i^{(s)} \la -s \ra \1_\l \rightarrow \dots \rightarrow \F_i^{(\la \l,\alpha_i \ra + 1)} \E_i \la -1 \ra \1_\l \rightarrow \F_i^{(\la \l,\alpha_i \ra)} \1_\l \right].$$ These complexes define a morphism $\Br(D) \rightarrow \Aut(\oplus_\l \Kom(\D(\l)))$ just like the one in Theorem \[thm:main1\]. See [@CR; @CKL; @CK2] for more details.
The relationship between 2-representations of $\h$ and 2-representations of $\g$ is given by the vertex operator constructions from [@CL2]. Thus we expect to have the diagram $$\begin{tikzpicture}[>=stealth]
\draw (0.1,0) -- (5.9,0)[->] [very thick];
\draw (-.5,-.5) node {2-representations of $\h$};
\draw (6.5,-.5) node {2-representations of $\g$};
\draw (3,4.5) node {categorical braid group actions};
\draw (0.1,0) -- (3,3.9)[->] [very thick];
\draw (3.1,3.9) -- (6,0)[<-] [very thick];
\draw (3,.4) node {vertex operator complexes};
\draw (6.5,2) node {Rickard complexes};
\draw (-2,2) node {Theorem \ref{thm:main1} using complexes $\T_i$ from (\ref{eq:intro1})};
\end{tikzpicture}$$ As the above diagram indicates, we should be able to deduce Theorem \[thm:main1\] as a consequence of the braid group actions arising from 2-representations of $\g$ [@CK2] (this essentially amounts to checking that the diagram above commutes). However, there are several technical details required to give a proof in this way (see section \[sec:remarks\] for more details), so in this paper we choose to give a direct construction of the left arrow.
[**Acknowledgments:**]{} The authors benefited from discussions with Jon Kujawa and Eli Grigsby. S.C. was supported by NSF grant DMS-1101439 and the Alfred P. Sloan foundation. A.L. would like to thank the Institute for Advanced Study for support.
Preliminaries
=============
We will always work over a base field $\k$ of characteristic zero.
Dynkin data {#sec:data}
-----------
Let $D$ be a finite graph without edge loops or multiple edges between vertices. We let $I$ denote the vertex set of $D$, and $E$ the edge set. The graph $D$ is is the Dynkin diagram of a symmetric simply-laced Kac-Moody algebra. We define a pairing $\la \cdot, \cdot \ra: I \times I \rightarrow \Z$ by $\la i,j \ra := C_{i,j}$ where $C_{i,j}$ is the Cartan matrix associated to our Dynkin diagram. More precisely: $$\la i,j \ra =
\begin{cases}
2 & \text{ if } i=j \\
-1 & \text{ if } i \ne j \text{ are joined by an edge } \\
0 & \text{ if } i \ne j \text{ are not joined by an edge.}
\end{cases}$$ Associated to this data there is the braid group $\Br(D)$ which is generated by $\{\sigma_i\}_{i \in I}$ subject to the relations $\sigma_i \sigma_j = \sigma_j \sigma_i$ if $i,j \in I$ are not joined by an edge, and $\sigma_i \sigma_j \sigma_i = \sigma_j \sigma_i \sigma_j$ if they are joined.
Fix an orientation $\epsilon$ of $D$. For $i,j\in I$ with $\la i,j \ra = -1$, we set $\epsilon_{ij} = 1$ if the edge is oriented $i \rightarrow j$ by $\epsilon$ and $\epsilon_{ij} = -1$ if oriented $j \rightarrow i$. If $\la i,j \ra = 0$, then we set $\epsilon_{ij} = 0$. Notice that in both cases we have $\epsilon_{ij} = -\epsilon_{ji}$.
Partitions
----------
Let $\l = (\l_1 \ge \l_2 \ge \dots \ge \l_k\geq 0 )$ be a partition. We denote the size of $\lambda$ by $|\l| := \sum_i \l_i $; we write $\l \vdash n$ if $\l$ is a partition of $n$, and denote the transposed partition by $\l^t$. If the number of $\l_i = k$ is $a_k$, we also write $\l = (1^{a_1},2^{a_2},\dots,s^{a_s}\dots)$. For example, in this notation, $(n)^t = (1^n)$. We write $\l' \subset \l$ if $\l'$ if $\l_i \ge \l'_i$ for all $i$.
We denote by $\k[S_n]$ the group algebra of the symmetric group and $s_k = (k,k+1) \in S_n$ the simple transposition. Since the characteristic of $\k$ is 0, $\k[S_n]$ is isomorphic to a direct sum of matrix algebras, $\k[S_n] = \bigoplus_{\l \vdash n} M_{h_\l}(\k)$. Here $\{h_\l\}_{\l\vdash n}$ are positive integers, and $M_s(\k)$ is the algebra of $s$-by$s$ matrices over $\k$. For any partition $\l$ of $n$, we denote by $e_\l \in \k[S_n]$ a minimal idempotent (a matrix unit) in the matrix algebra $M_{h_\l}$ corresponding to $\l$. We denote by $\tau: \k[S_n] \rightarrow \k[S_n]$ the involution which sends $s_i \mapsto -s_i$ for all $i \in I$. The minimal idempotents $e_{\l}$ may be chosen so as to have have $\tau(e_\l) = e_{\l^t}$.
Zig-zag algebras {#sec:zigzag}
----------------
Let $cD$ denote the doubled quiver, with the same vertex set as $D$ and with two oriented edges (one in each orientation) for each edge of $D$. Let $\k[dD]$ denote the path algebra of $dD$. A path in $dD$ is described as a sequence of vertices $(i_1 | i_2 | \dots | i_m)$ where $i_k$ and $i_{k+1}$ are connected by an edge in $D$. If $ D $ has more than two nodes then we define $B^D_\ep$ to be the quotient of $\C[dD]$ by the two sided ideal generated by
- $(a|b|c)$ if $a \ne c$ and
- $\ep_{ab} (a|b|a) - \ep_{ac} (a|c|a)$ whenever $a$ is connected to both $b$ and $c$.
In the above, $e_i$ denotes the constant path which starts and ends at the vertex $i \in I$. If $D$ consists of the single vertex only, we let $B^D_\ep$ be the algebra generated by $1$ and $X$ with $X^2 = 0$. If $D$ consists of two points joined by a single edge, we deÞne $B^D_\ep$ to be the quotient of $\k[dD]$ by the two-sided ideal spanned by all paths of length greater than two. Notice that $B^D_\ep$ is $\Z$-graded by the length of the path (we denote the degree of a path by $|\cdot|$). The $\Z_2$-grading induced from the $\Z$ grading makes $B^D_\ep$ into a $\Z$-graded superalgebra.
The algebras $B^D_\ep$ first appeared in [@HK] in the context of categorifying the adjoint representation of the Lie algebra assciated to $D$.
For $n \ge 0$, we define $\Z$-graded superalgebras $B^D_\ep(n) := ({B^D_\ep)}^{\otimes n} \rtimes \k[S_n]$. As a vector space, we have $B^D_\ep(n) = ({B^D_\ep)}^{\otimes n} \otimes_\k \k[S_n]$, but for the algebra structure the tensor product is in the category of superalgebras (see [@CL1 Section 9.1]). Thus $$(a \otimes b) \cdot (a' \otimes b') = (-1)^{|b||a'|}(aa' \otimes bb'),$$ while $S_n$ acts by superpermutations $s_k \cdot (b_1 \otimes \dots \otimes b_k \otimes b_{k+1} \otimes \dots \otimes b_n) = (-1)^{|b_k||b_{k+1}|} b_1 \otimes \dots \otimes b_{k+1} \otimes b_k \otimes \dots \otimes b_n.$ By convention, $B^D_\ep(0) = \k$. To shorten notation we will write $$e_{i,m} := (1 \otimes \dots \otimes 1 \otimes e_i \otimes 1 \otimes \dots \otimes 1,1) \in B^D_\ep(n)$$ for the idempotent where $e_i$ occurs in the $m$th tensor factor on the right hand side. The $\Z$-grading on $B^D_\ep(n)$ is induced from that on $B^D_\ep$, with the factor $\k[S_n]$ placed in degree $0$.
\[rem:super\] All the constructions in the remainder of the paper will involve $\Z$-graded superalgebras over $\k$ and graded supermodules or superbimodules over such superalgebras. For simplicitly, we will write “algebra", “module“, and ”bimodule", omitting the understood prefixes “$\Z$-graded" and “super".
The wreath functor $(\cdot)^{[n]}$ {#sec:wreath}
----------------------------------
If $A$ is a algebra then we can define a new algebra $A^{[n]} := A^{\otimes n} \rtimes \k[S_n]$. The grading and superstructure on $A^{[n]}$ are inherited from that on $A$, with the understanding that $S_n$ acts on $A$ by superpermutations and that the subalgebra $\k[S_n]\subset A^{[n]}$ is in degree 0. Similarly, if $A_1$ and $A_2$ are algebras and $M$ is an $(A_2,A_1)$-bimodule, then we can define the $(A_2^{[n]}, A_1^{[n]})$-bimodule $M^{[n]} := M^{\otimes n} \rtimes \k[S_n]$.
To describe $(\cdot)^{[n]}$ as a functor, it is convenient to use the language of 2-categories. Let $\catC_a$ be the 2-category whose objects are algebras, 1-morphisms are bimodules, and 2-morphisms are bimodule maps. Composition of 1-morphisms is tensor product of bimodules, and composition of 2-morphisms is composition of bimodule maps.
\[lem:\[n\]functor\] The map $(\cdot) \mapsto (\cdot)^{[n]}$ is defines a 2-functor $$(\cdot)^{[n]}: \catC_a\longrightarrow \catC_a.$$
For $M_1$ is an $(A_2,A_1)$-bimodule and $M_2$ is an $(A_3,A_2)$ bimodule, we define $M_2^{[n]} \otimes_{A_2^{[n]}} M_1^{[n]} \rightarrow (M_2 \otimes_{A_2} M_1)^{[n]}$ by $$(m_1 \otimes \dots \otimes m_n, \sigma) \otimes (m'_1 \otimes \dots \otimes m'_n, \sigma') \mapsto ((m_1 \otimes m'_{\sigma(1)}) \otimes \dots \otimes (m_n \otimes m'_{\sigma(n)}), \sigma \sigma').$$ It is not difficult to check that this map is an isomorphism. Thus $(\cdot)\mapsto(\cdot)^{[n]}$ respects composition of 1-morphisms. It is also clear that $(\cdot)\mapsto (\cdot)^{[n]}$ intertwines compositions of 2-morphisms, for if $f: M_1 \rightarrow M_2$ is a map of bimodules then $$f^{[n]} := (f \otimes \dots \otimes f, 1): M_1^{\otimes n} \rtimes \k[S_n] \rightarrow M_2^{\otimes n} \rtimes \k[S_n]$$ is a morphism $M_1^{[n]} \rightarrow M_2^{[n]}$ with $(f_2 \circ f_1)^{[n]} = f_2^{[n]} \circ f_1^{[n]}$.
The functor $(\cdot)^{[n]}$ is somewhat subtle. In particular,
- $(\cdot)^{[n]}$ is not linear: if $f,g \in {{\rm Hom}}(M_1,M_2)$, then both $(f+g)^{[n]}$ and $f^{[n]} + g^{[n]}$ are well-defined elements of ${{\rm Hom}}(M_1^{[n]},M_2^{[n]})$ but in general they are not equal to each other.
- $(\cdot)^{[n]}$ is not additive: in general $(M_1 \oplus M_2)^{[n]}$ and $M_1^{[n]}\oplus M_2^{[n]}$ are not isomorphic (this is already clear at the level of vector spaces via a dimension count). Subsequently, $(\cdot)^{[n]}$ is neither left exact nor right exact.
However, $(\cdot)^{[n]}$ behaves well with respect to homotopies of complexes. Suppose $M_\bullet = M_0 \rightarrow \dots \rightarrow M_\ell$ is a complex of $(A_1,A_2)$-bimodules. Then the complex $M_\bullet^{[n]}$ is a complex of $(A_1^{[n]},A_2^{[n]})$-bimodules. The slightly subtle part of this definition is the definition of the boundary map in the complex $M_\bullet^{[n]}$; the easiest way to define it is to consider $M_\bullet$ as a supermodule over the superalgebra $A \otimes_\k \k[d]/d^2$, where $d$ has superdegree one. Then $M_\bullet^{[n]}$ is naturally an $(A \otimes_\k \k[d]/d^2)^{[n]} \cong A^{[n]}\otimes_{\k[S_n]} (\k[d]/d^2)^{[n]}$ supermodule. Now the coproduct $$\Delta: \k[d]/d^2 \rightarrow (\k[d]/d^2)^{\otimes n}\subset (\k[d]/d^2)^{[n]}$$ given by $$\Delta(d) = (1\otimes1 \otimes \hdots \otimes d) + (1 \otimes d \otimes 1 \otimes\hdots \otimes 1) + \hdots + (d\otimes 1\otimes \hdots \otimes 1)$$ embeds $A^{[n]} \otimes_\k \k[d]/d^2$ as a subalgebra of $(A\otimes_\k \k[d]/d^2)^{[n]}$. Thus the $(A\otimes_\k \k[d]/d^2)^{[n]}$ supermodule $M_\bullet^{[n]}$ can be restricted to $A^{[n]}\otimes_\k \k[d]/d^2$, and thus $M_\bullet^{[n]}$ may be considered as a complex of $A^{[n]}$-modules.
An important point to keep in mind is that, because all constructions take place in the category of supermodules, the action of $\k[S_n]$ on an n-fold tensor product of graded vector spaces is via superpermutations. Thus, spelling this out, the action of the simple transposition $s_i$ on the complex $M_\bullet^{[n]}$ is $$s_i \cdot (m_1 \otimes \hdots \otimes m_i \otimes m_{i+1} \otimes \hdots m_n) = (-1)^{\deg(m_i) \deg(m_{i+1})} m_1\otimes \hdots \otimes m_{i+1}\otimes m_{i}\otimes \hdots m_n,$$ where $\deg(m_i) = |m_i| + |m_i|_h$ where $|m_i|$ denotes the inner graded degree of $m_i$ and $|m_i|_h$ denotes the homological degree of $m_i$.
Now, the following lemma shows that the functor $(\cdot)^{[n]}$ behaves well with respect to homotopies. (This is not immediately obvious since $(\cdot)^{[n]}$ is not linear and chain homotopies involve linear combinations of maps.)
\[lem:homotopy\] Let $C_\bullet,D_\bullet$ be complexes of $(A_1,A_2)$-bimodules, and suppose that $f,g: C_\bullet \longrightarrow D_\bullet$ are homotopic maps. Then $f^{[n]},g^{[n]}:C_\bullet^{[n]}\longrightarrow D_\bullet^{[n]}$ are homotopic.
By assumption, there exists a chain homotopy $h$ with $f-g = d_{D}h + hd_{C}$. We set $$h' = \sum_{i+j=n-1} (f^{\otimes i} \otimes h \otimes g^{\otimes j}, 1).$$ Then one can check that $f^{[n]}-g^{[n]} = d_{D^{[n]}}h' + h'd_{C^{[n]}}$.
Graded 2-categories
-------------------
A graded additive $\k$-linear 2-category $\K$ is a category enriched over graded additive $\k$-linear categories. This means that for any two objects $A,B \in \K$ the Hom category ${{\rm Hom}}_{\K}(A,B)$ is a graded additive $\k$-linear category. Moreover, the composition map ${{\rm Hom}}_{\K}(A,B) \times {{\rm Hom}}_{\K}(B,C) \to {{\rm Hom}}_{\K}(A,C)$ is a graded additive $\k$-linear functor.
[**Example.**]{} Suppose $B_n$ is a sequence of graded $\k$-algebras indexed by $n \in \N$. Then one can define a 2-category $\K$ whose objects (0-morphisms) are indexed by $\N$, the 1-morphisms are graded $(B_m,B_n)$-bimodules and the 2-morphisms are maps of graded $(B_m,B_n)$-bimodules.
A graded additive $\k$-linear 2-functor $F: \K \to \K'$ is a (weak) 2-functor that maps the Hom categories ${{\rm Hom}}_{\K}(A,B)$ to ${{\rm Hom}}_{\K'}(FA,FB)$ by additive functors that commute with the auto-equivalence $\la 1 \ra$.
An additive category $\mathcal{C}$ is said to be idempotent complete when every idempotent 1-morphism splits in $\mathcal{C}$. Similarly, we say that the additive 2-category $\K$ is idempotent complete when the Hom categories ${{\rm Hom}}_{\K}(A,B)$ are idempotent complete for any pair of objects $A, B \in \K$, (so that all idempotent 2-morphisms split). All 2-categories in this paper will be idempotent complete.
### The homotopy 2-category
If $\K$ is an additive $\k$-linear 2-category then one can define its homotopy 2-category $\Kom(\K)$ as follows. The objects are the same. The 1-morphisms are unbounded complexes of 1-morphisms in $\K$ while the 2-morphisms are maps of complexes. Two complexes of 1-morphisms are then deemed isomorphic if they are homotopy equivalent.
[**Example.**]{} Denote by $\catC_{a}$ the 2-category of algebras. Then in $\Kom(\catC_{a})$:
- objects are algebras over $\k$,
- 1-morphisms from $A$ to $B$ are complexes of $(A,B)$-bimodules,
- 2-morphisms are chain maps up to homotopy.
Combining Lemmas \[lem:\[n\]functor\] and \[lem:homotopy\] implies the following.
For each $n \in \N$, the 2-functor $(\cdot) \mapsto (\cdot)^{[n]}$ defines an endofunctor of the 2-category $\Kom(\catC_{a})$.
The above endofunctors appeared earlier in [@K], which emphasized their relevance for constructing group actions on categories.
### Triangulated 2-categories
A graded triangulated category is a graded category equipped with a triangulated structure where the autoequivalence $\la 1 \ra$ takes exact triangles to exact triangles. We denote the homological shift by $[\cdot]$ where $[1]$ denotes a downward shift by one.
A graded triangulated $\k$-linear 2-category ${\K'}$ is a category enriched over graded triangulated $\k$-linear categories. This means that for any two objects $A,B \in \K'$ the Hom category ${{\rm Hom}}_{\K'}(A,B)$ is a graded additive $\k$-linear triangulated category.
[**Example.**]{} If $\K$ is a $\k$-linear 2-category then $\Kom(\K)$ is a triangulated 2-category. In the remainder of the paper this extra triangulated structure of $\Kom(\K)$ will not play a role and will usually be ignored.
Quantum Heisenberg algebras {#sec:hei}
===========================
Here we recall the quantum Heisenberg algebra $\h$ and its Fock space representation. We will denote the quantum integer by $$[n] := t^{-n+1} + t^{-n+3} + \dots + t^{n-3} + t^{n-1}.$$
The traditional presentation for the quantum Heisenberg algebra is as a unital algebra generated by $a_i(n)$, where $i \in \I$ and $n \in \Z \setminus \{0\}$. The relations are $$\label{rel:as}
a_i(m) a_j(n) - a_j(n)a_i(m) = \delta_{m,-n} [n \la i,j \ra] \frac{[n]}{n}.$$ When $q=1$, this presentation specializes to the standard presentation of the non-quantum Heisenberg algebra.
For our purposes, a more convenient presentation of $\h$ takes as generators $\{P_i^{(n)},Q_i^{(n)} \}_{i \in \I, n \geq 0}$ subject to the following relations: $$\begin{aligned}
P_i^{(n)} P_j^{(m)} &=& P_j^{(m)} P_i^{(n)} \text{ and } Q_i^{(n)}Q_j^{(m)} = Q_j^{(m)} Q_i^{(n)} \text{ for all } i,j \in \I, \\
Q_i^{(n)} P_j^{(m)} &=&
\begin{cases}
\sum_{k \ge 0} [k+1] P_i^{(m-k)} Q_i^{(n-k)} & \text{ if } i=j, \\
P_j^{(m)} Q_i^{(n)} + P_j^{(m-1)} Q_i^{(n-1)} & \text{ if } \la i,j \ra = -1 \\
P_j^{(m)} Q_i^{(n)} & \text{ if } \la i,j \ra = 0.
\end{cases}\end{aligned}$$ By convention $P_i^{(0)} = Q_j^{(0)} = 1$ and $P_i^{(k)} = Q_i^{(k)} = 0$ when $k < 0$ so the summations in the relations above are all finite. Notice that $\h$ has a natural $\Z$-grading where $\deg P_i^{(n)} = n$ and $\deg Q_i^{(n)} = -n$.
An explicit isomorphism between these two presentations is given in [@CL1] and [@CL2 Section 3.1].
The Fock space {#sec:fock}
--------------
Let $\h^- \subset \h$ denote the subalgebra generated $\{ Q_i^{(n)} \}_{i \in I, n \ge 0}$. Let $\mbox{triv}_0$ denote the trivial (one-dimensional) representation of $\h^-$, where all $Q_i^{(n)}$ ($n>0$) act by zero. Then $V_{Fock} := {{\rm{Ind}}}_{\h^-}^{\h}(\mbox{triv}_0)$ is called the Fock space representation of $\h$.
The Fock space has a basis given by elements of the form $P_{i_k}^{(n_k)} \dots P_{i_1}^{(n_1)}(v)$ where $v$ is a vector spanning $\mbox{triv}_0$. This gives a decomposition $V_{Fock} = \oplus_{n \ge 0} V_{Fock}(n)$. To simplify notation, we will denote $P_{i_k}^{(n_k)} \dots P_{i_1}^{(n_1)}(v)$ by $P_{i_k}^{(n_k)} \dots P_{i_1}^{(n_1)}$.
For any partition $\l \vdash n$ on can define $P_i^{(\l)}$ using Giambelli’s formula as the determinant $$[P_i^{\lambda}] = \mbox{det}_{kl} [P_i^{(\l_k + l-k)}].$$ For example, $P_i^{(1^2)} = P_iP_i - P_i^{(2)}$. See [@CL1 Section 7] for more details.
The braid group action on $V_{Fock}$ {#subsec:braidfock}
------------------------------------
We now describe a braid group action on $V_{Fock}$. Since $V_{Fock}$ is multiplicatively generated by elements $P_i^{(n)}$ it suffices to describe this action on these generators and extend multiplicatively. On generators, the action is given by $$\sigma_i(P_j^{(n)}) =
\begin{cases}
(-t^{-2})^n P_i^{(1^n)} & \text{ if } i=j \\
\sum_{k=0}^n (-t^{-1})^{n-k} P_j^{(k)} P_i^{(n-k)} & \text{ if } \la i,j \ra = -1 \\
P_j^{(n)} & \text{ if } \la i,j \ra = 0
\end{cases}$$ and $$\sigma_i^{-1}(P_j^{(n)}) =
\begin{cases}
(-t^{2})^n P_i^{(1^n)} & \text{ if } i=j \\
\sum_{k=0}^n (-t)^{n-k} P_j^{(k)} P_i^{(n-k)} & \text{ if } \la i,j \ra = -1 \\
P_j^{(n)} & \text{ if } \la i,j \ra = 0.
\end{cases}$$
\[prop:fockspace\] The endomorphisms $ \sigma_i $ and $ \sigma_i^{-1} $ for $ i \in I$ define a representation of the braid group $\Br(D)$ on each weight space $V_{Fock}(n)$ of the Fock space.
This is a consequence of Proposition \[prop:2\] and Remark \[rem:2\] following it.
It is already interesting to see this braid action on various basis vectors, as in the example below.
\[lemma1\] Suppose $i,j,k \in I$ are different with $\la i,j \ra = -1 = \la j,k \ra$ and $\la i,k \ra = 0$. Then $$\begin{aligned}
\sigma_i \sigma_j \sigma_i(P_i^{(n)}) &=& t^{-3n} P_j^{(1^n)} \\
\sigma_j \sigma_i \sigma_j(P_k^{(n)}) &=& \sum_{a=0}^n \sum_{b=0}^{n-a} (-1)^b t^{-2(n-a)+b} P_{k}^{(a)} P_{j}^{(b)} P_{i}^{(n-a-b)}\end{aligned}$$
By definition we have $$\sigma_i \sigma_j \sigma_i (P_i^{(n)}) = (-t^{-2})^n \sum_{a=0}^n \sum_{b=0}^{n-a} (-1)^{a+n} t^{-2n+b} P_j^{(1^b)} P_i^{(a)} P_i^{(1^{n-a-b})}.$$ This simplifies to give $t^{-3n} P_j^{(1^n)}$ if we use the identity $P_i^{(m)} P_i^{(1^n)} = P_i^{(m,1^n)} + P_i^{(m+1,1^{n-1})}$ (see for instance [@CL2 Prop. 1]). The second identity is similar but more involved so we omit the proof.
The action of $ \sigma_i^{\pm 1} $ on the usual generators $ a_j(-n) $ of the Fock space has a somewhat easier description. The proof is a straightforward calculation using the generating functions in [@CL1 Sect. 2.2.1].
$$\sigma_i(a_j(-n)) =
\begin{cases}
-t^{-2n} a_i(-n) & \text{ if } i=j \\
a_j(-n)+(-1)^nt^{-n} a_i(-n) & \text{ if } \la i,j \ra = -1 \\
a_j(-n) & \text{ if } \la i,j \ra = 0
\end{cases}$$
and $$\sigma_i^{-1}(a_j(-n)) =
\begin{cases}
-t^{2n} a_i(-n) & \text{ if } i=j \\
a_j(-n)+(-1)^nt^{n} a_i(-n) & \text{ if } \la i,j \ra = -1 \\
a_j(-n) & \text{ if } \la i,j \ra = 0.
\end{cases}$$
2-representations of $\h$ and the braid complex {#sec:2heis}
===============================================
In this section we review some facts about 2-representations of $\h$ and define the braid complex $\T_i \1_n$.
The 2-category $\H$ {#sec:2cat}
-------------------
In [@CL1] we introduced a 2-category $\H^\Gamma$ associated to any finite subgroup $\Gamma \subset SL_2(\C)$. In our current language this 2-category is associated to the pair $(D,\ep)$ where $D$ is the affine Dynkin diagram corresponding to $\Gamma$ by the McKay correspondence and $\ep$ is an appropriately chosen orientation of $D$.
That definition generalizes with no effort to give a 2-category $\H^D_\ep$ associated to any simply laced Dynkin diagram $D$ and orientation $\ep$. More precisely, $\H^D_\ep$ is the (idempotent closure of) the additive, graded, $\k$-linear 2-category where
- 0-morphisms (objects) are indexed by the integers $\Z$,
- 1-morphisms consist of the identity 1-morphisms $\1_n$ of $n \in \Z$ and compositions, direct sums and grading shifts of $\P_i \1_n: n \rightarrow n+1$ and $\1_n \Q_i: n+1 \rightarrow n$ for $i \in I$,
- The 2-morphisms are generated by adjunction maps, making $\P_i$ and $\Q_i$ bi-adjoint up to shift, together with certain maps $X_i^j\in {{\rm Hom}}(\P_i,\P_j)$ and $T_{ij}\in {{\rm Hom}}(\P_j\P_i,\P_i\P_j)$ satisfying a series of relations.
The generating 2-morphisms and their relations were described diagrammatically in [@CL1] and reviewed in [@CL2 Section 3.2]. In the interest of space, we have elected not to spell them out again. However, the essential algebraic structure of these 2-morphisms is straightforward to summarize, and is enough for the purposes of the current paper:
- The 1-morphisms $\P_i$ and $\Q_i$ are left and right adjoint to one another, up to a grading shift.
- For each $n\geq 0$, there is a natural injective map $$B^D_\ep(n) \longrightarrow \End((\bigoplus_{i \in I} \P_i)^n).$$ In particular, for each $i\in I$ there is an embedding $\k[S_n]\longrightarrow \End(\P_i^n)$. Since $\H^D_\ep$ is idempotent complete, for any partition $\l\vdash n$ we can define the 1-morphism $\P_i^{(\l)}$ as the image of the idempotent $e_\l \in \k[S_{|\l|}]$ acting on $\P_i^{|\l|}$.
From hereon we will fix $D$ and $\ep$ and denote $\H_\ep^D$ simply by $\H$ in order to simplify notation.
2-representation of $\h$ {#sec:2rep}
------------------------
A [*2-representation of $\h$*]{} consists of a graded, idempotent complete $\k$-linear category $\K$ where
- 0-morphisms are graded, $\k$-linear, additive categories $\D(n)$,
- 1-morphisms are (certain types of) functors between these categories,
- 2-morphisms are natural transformations of these functors
together with a 2-functor $\H \rightarrow \K$. We also require that the space of 2-morphisms between any two 1-morphisms in $\K$ be finite dimensional and that ${{\rm Hom}}_\K(\1_n, \1_n \la \ell \ra)$ is zero if $\ell < 0$ and one-dimensinal if $\ell=0$.
The fact that the space of maps between any two 1-morphisms is finite dimensional means that the Krull-Schmidt property holds. Thus any 1-morphism has a unique direct sum decomposition (see section 2.2 of [@Rin]). Note that if $\K$ satisfies the Krull-Schmidt property then so does $\Kom(\K)$.
A 2-representation of $\h$ is said to be *integrable* if $\1_n=0$ are zero for $n \ll 0$. For example, in [@CL1] we constructed an integrable 2-representation where $n$ corresponded to the category of coherent sheaves on the Hilbert scheme of points $\mbox{Hilb}^n(\widehat{\C^2/\Gamma})$ where $\Gamma \subset SL_2(\C)$ is the finite subgroup associated to our Dynkin diagram.
The 2-representation $\K_{Fock}$
--------------------------------
The 2-representation $\K_{Fock}$ categorifies the Fock space representation $V_{Fock}$. It consists of
- 0-morphisms: $n$ indexes the category of projective $B^D_\ep(n)$-modules.
- 1-morphisms: $(B^D_\ep(n), B^D_\ep(n'))$-bimodules which are direct summands of tensor products of the bimodules $\P_i \1_n$ and $\1_n \Q_i$.
- 2-morphisms: all maps of bimodules.
The 2-functor $\H \rightarrow \K_{Fock}$ was defined in [@CL1 Section 9].
Some technical facts {#sec:technical}
--------------------
We gather some useful facts about 2-representations of $\h$.
\[lem:1\] For an arbitrary partition $\l$ we have $$\Q_i^{(\l)} \P_i \cong \P_i \Q_i^{(\l)} \bigoplus_{\l' \subset \l} \Q_i^{(\l')} \la -1,1 \ra \text{ and } \Q_i \P_i^{(\l)} \cong \P_i^{(\l)} \Q_i \bigoplus_{\l' \subset \l} \P_i^{(\l')} \la -1,1 \ra$$ where the sums are over all $\l' \subset \l$ with $|\l'| = |\l|-1$.
Lemma 3.3 [@CL2].
\[lem:2\] Suppose $\l,\l',\mu$ and $\mu'$ are partitions such that $|\l| > |\l'|$ and $|\mu|>|\mu'|$. Then $\dim {{\rm Hom}}( \P_i^{(\l)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 1 \ra) \le 1$ with equality if and only if $\l' \subset \l$ and $\mu' \subset \mu$ with $|\l| = |\l'|+1$ and $|\mu|=|\mu'|+1$. In this case, the map is spanned by the composition $$\P_i^{(\l)} \Q_i^{(\mu)} \longrightarrow \P_i^{(\l')} \P_i \Q_i \Q_i^{(\mu')} \longrightarrow \P_i^{(\l')} \Q_i^{(\mu')} \la 1 \ra$$ where the second map is given by adjunction.
Lemma 3.4 [@CL2].
\[lem:3\] Consider partitions $\l,\l',\mu,\mu'$ such that $|\l|-|\l'|=2=|\mu|-|\mu'|$. Then $$\label{eq:hom}
{{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra)$$ is zero unless $\l' \subset \l$ and $\mu' \subset \mu$ in which case its dimension is equal to $$\begin{cases}
2 \hspace{.5cm} \text{ if } \l \setminus \l' \text{ and } \mu \setminus \mu' \text{ both consist of two boxes in different rows and columns, } \\
0 \hspace{.5cm} \text{ if } \l \setminus \l' \text{ consists of two boxes in same row (resp. column) } \\
\hspace{.6cm} \text{ while } \mu \setminus \mu' \text{ consists of two boxes in same column (resp. row), } \\
1 \hspace{.5cm} \text{ otherwise. }
\end{cases}$$
First one notes that a map in (\[eq:hom\]) consists of a composition of two degree $1$ maps. Since the only degree $1$ maps involving only vertex $i$ are induced from the adjunction map $\P_i\Q_i \rightarrow \id\la 1 \ra$, if follows that any map in (\[eq:hom\]) factors $$\P_i^{(\l)} \Q_i^{(\mu)} \rightarrow \P_i^{|\l|-1}\Q_i^{|\mu|-1} \la 1 \ra \rightarrow \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra.$$ Since $\P_i^{|\l|-1}\Q_i^{|\mu|-1} \la 1 \ra$ is isomorphic to a direct sum of indecomposable terms $\P_i^{(\l'')} \Q_i^{(\mu'')} \la 1 \ra$ for partitions $\l''$, $\mu''$, we see that the space of maps in (\[eq:hom\]) is spanned by maps which factor through some $\P_i^{(\l'')} \Q_i^{(\mu'')} \la 1 \ra$. Subsequently, by Lemma \[lem:2\], it follows that (\[eq:hom\]) is zero unless $\l' \subset \l$ and $\mu' \subset \mu$.
Now, we proceed by induction on $|\l|+|\mu|$. We prove the first case above, the others follow similarly.
First, suppose that there exists some $\l' \not\subset \nu \subset \l$. Then $\P_i \P_i^{(\nu)} \cong \P_i^{(\l)} \oplus_{\gamma} \P_i^{(\gamma)}$. Subsequently $${{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra) \cong {{\rm Hom}}(\P_i \P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra)$$ since, for all $\gamma$ in the sum above, ${{\rm Hom}}(\P_i^{(\gamma)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra) = 0$ because $\l' \not\subset \gamma$. Hence $$\begin{aligned}
(\ref{eq:hom})
&\cong& {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \Q_i \P_i^{(\l')} \Q_i^{(\mu')} \la 1 \ra) \\
&\cong& {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i \Q_i^{(\mu')} \la 1 \ra) \oplus_{\rho \subset \l'} {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\rho)} \Q_i^{(\mu')} \la 0,2 \ra). \end{aligned}$$ The left term above is zero since $\l' \not\subset \nu$. Using induction, every term in the summation on the right is also zero with the exception of the $\rho$ which satisfies $\rho \subset \nu$. The result now follows by induction.
Now, suppose that no such $\nu$ as above exists. Thus means that $\l$ is a partition whose Young diagram is a union of at most two rectangles, as illustrated below: $$\begin{tikzpicture}[>=stealth]
\draw (-1,-.5) node {$\lambda =$};
\draw (0,0) -- (2,0)[very thick];
\draw (2,0) -- (2,-.5)[very thick];
\draw (0,0) -- (0,-1)[very thick];
\draw (0,-1) -- (1,-1)[very thick];
\draw (1,-.5) -- (1,-1)[very thick];
\draw (1,-.5) -- (2,-.5)[very thick];
\end{tikzpicture}$$ Now, we choose $\nu \subset \l$ but this time $\l' \subset \nu$ so that once again we have $\P_i \P_i^{(\nu)} \cong \P_i^{(\l)} \oplus_{\gamma} \P_i^{(\gamma)}$. On the one hand we have $$\begin{aligned}
{{\rm Hom}}(\P_i \P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra)
&\cong& (\ref{eq:hom}) \oplus_\gamma {{\rm Hom}}(\P_i^{(\gamma)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra) \\
&\cong& (\ref{eq:hom}) \oplus \k^{2 \# \{\gamma\}}\end{aligned}$$ where the second line follows by applying the first step above and then induction. On the other hand, by adjunction $$\begin{aligned}
& & {{\rm Hom}}(\P_i \P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i^{(\mu')} \la 2 \ra) \\
&\cong& {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \Q_i \P_i^{(\l')} \Q_i^{(\mu')} \la 1 \ra) \\
&\cong& {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\l')} \Q_i \Q_i^{(\mu')} \la 1 \ra) \oplus_{\rho \subset \l'} {{\rm Hom}}(\P_i^{(\nu)} \Q_i^{(\mu)}, \P_i^{(\rho)} \Q_i^{(\mu')} \la 0,2 \ra) \\
&\cong& (\k \oplus \k) \oplus (\k^{2(\# \{\gamma\} - 1)} \oplus \k \oplus \k) \end{aligned}$$ where the last line requires a quick case by case analysis of the possible $\rho$’s. Comparing these two expressions gives us that $(\ref{eq:hom}) \cong \k^2$ and the induction is complete.
\[cor:3\] Suppose $\nu \subset \mu \subset \l$ are partitions with $|\l| = |\mu|+1 = |\nu|+2$ such that $\l \setminus \nu$ consists of two boxes which are not in the same row or column. Then the composition $$\label{eq:cor3}
\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\mu)} \Q_i^{(\mu^t)} \la 1 \ra \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra$$ consisting of maps from Lemma \[lem:2\] is nonzero.
Fix $\l,\nu$ as above. Any map $\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra$ must factor as $$\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\nu)} \P_i \P_i \Q_i \Q_i \Q_i^{(\nu^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra$$ where the right most map is given by two adjunctions. Now, since $${{\rm Hom}}(\P_i^{(2)} \Q_i^{(1^2)}, \1 \la 2 \ra) = 0 = {{\rm Hom}}(\P_i^{(1^2)} \Q_i^{(2)}, \1 \la 2 \ra)$$ this map factors as $$\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\nu)} \P_i^{(2)} \Q_i^{(2)} \Q_i^{(\nu^t)} \oplus \P_i^{(\nu)} \P_i^{(1^2)} \Q_i^{(1^2)} \Q_i^{(\nu^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra.$$ Now, ${{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)}, \P_i^{(\nu)} \P_i^{(2)} \Q_i^{(2)} \Q_i^{(\nu^t)}) \cong \k \cong {{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)}, \P_i^{(\nu)} \P_i^{(1^2)} \Q_i^{(1^2)} \Q_i^{(\nu^t)})$ and, by Lemma \[lem:3\], $\dim_\k {{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)}, \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra) = 2$. Hence the compositions $$\begin{aligned}
&& \P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\nu)} \P_i^{(2)} \Q_i^{(2)} \Q_i^{(\nu^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra \\
&& \P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\nu)} \P_i^{(1^2)} \Q_i^{(1^2)} \Q_i^{(\nu^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra\end{aligned}$$ are both nonzero. Fortunately, the unique map $\P_i^{(\l)} \rightarrow \P_i^{(\nu)} \P_i^{(2)}$ factors through $\P_i^{(\mu)} \P_i$ and likewise $\Q_i^{(\l^t)} \rightarrow \Q_i^{(2)} \Q_i^{(\nu^t)}$ factors through $\Q_i \Q_i^{(\mu)}$ for any $\nu \subset \mu \subset \l$. Hence $$\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\mu)} \P_i \Q_i \Q_i^{(\mu^t)} \rightarrow \P_i^{(\nu)} \P_i \P_i \Q_i \Q_i \Q_i^{(\nu^t)} \rightarrow \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra$$ is nonzero and the result follows since this composition is the same as the one in (\[eq:cor3\]).
The braid complex $\T_i \1_n$ {#sec:cpx}
-----------------------------
Suppose $\K$ is some integrable 2-representation of $\h$. The main object of study in this paper is the following complex of 1-morphisms $$\label{eq:cpx}
\T_i \1_n := \left[ \dots \rightarrow \bigoplus_{\l \vdash d} \P_i^{(\l)} \Q_i^{(\l^t)} \la -d \ra \1_n \rightarrow \bigoplus_{\l \vdash d-1} \P_i^{(\l)} \Q_i^{(\l^t)} \la -d+1 \ra \1_n \rightarrow \dots \rightarrow \P_i \Q_i \la -1 \ra \1_n\rightarrow \1_n \right].$$ which lives naturally in $\Kom(\K)$. The right hand term $\1_n$ of this complex is in cohomological degree zero. Notice that since $\K$ is integrable this complex is finite.
The differential in (\[eq:cpx\]) is defined as the composition $$\label{eq:diff}
\P_i^{(\l)} \Q_i^{(\l^t)} \rightarrow \P_i^{(\mu)} \P_i \Q_i \Q_i^{(\mu^t)} \rightarrow \P_i^{(\mu)} \Q_i^{(\mu^t)} \la 1 \ra$$ where the first map is inclusion and the second is given by adjunction (note that we must have $\mu \subset \l$ in order for this map to be nonzero). The inclusion map is unique but only up to multiple. Likewise, by Lemma \[lem:2\], the composition is unique but only up to multiple. Fortunately, Proposition \[prop:Tcpx\] and Remark \[rem:1\] below shows that there is a unique way (up to homotopy) to choose these multiples in order to get an indecomposable complex.
Note that we did not check directly that the compositions in (\[eq:diff\]) define a differential ([*i.e.*]{} square to zero). It is possible to check this directly by generalizing the statement in Lemma \[lem:3\] but we avoid doing this extra work because later we will conclude for free that there exists an indecomposable complex with terms as in (\[eq:cpx\]). Then Proposition \[prop:Tcpx\] will tell us that the differentials are indeed given by (\[eq:diff\]).
\[prop:Tcpx\] Any complex whose terms are the same as those of $\T_i$ and which is indecomposable in $\Kom(\K)$ is homotopic to $\T_i$.
Let us fix $n \in \Z$ and look at $\T_i \1_n$. For any $\l,\mu$ such that ${{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)} \1_n, \P_i^{(\mu)} \Q_i^{(\mu^t)} \1_n \la 1 \ra) = 1$ fix a spanning map $f_{\l,\mu}$. A complex as in (\[eq:cpx\]) will have differentials of the form $a_{\l,\mu} f_{\l,\mu}$ for some scalars $a_{\l,\mu} \in \k$. We need to show that
1. if the complex in (\[eq:cpx\]) is indecomposable then $a_{\l,\mu} \ne 0$
2. if $\{ a_{\l,\mu} \}$ and $\{ a'_{\l,\mu} \}$ are two choices of scalars then they are equivalent via some homotopy.
[**Proof of (1).**]{} We proceed by (decreasing) induction on the cohomological degree. The base case is the map $a_{1,1} f_{1,1}: \P_i \Q_i \1_n \la -1 \ra \rightarrow \1_n$. Clearly $a_{1,1} \ne 0$ because otherwise $\T_i \1_n$ would decompose. Now suppose $a_{\beta,\alpha} \ne 0$ for all $\beta$ with $|\beta| > \ell$ (for some $\ell$).
[**Claim.**]{} If $a_{\l,\mu} = 0$ for some $\l$ with $|\l| = \ell$ then all differentials out of $\P_i^{(\l)} \Q_i^{(\l^t)} \1_n$ are zero.
To see this consider another differential $a_{\l,\mu'} f_{\l,\mu'}$. These two maps are part of a unique skew-commutative square $$\label{eq:commsquare}
\xymatrix{
& \P_i^{(\mu)} \Q_i^{(\mu^t)} \1_{n} \la 1 \ra \ar[dr]^{a_{\mu,\nu} f_{\mu,\nu}} & \\
\P_i^{(\l)} \Q_i^{(\l^t)} \1_n \ar[ru]^{a_{\l,\mu} f_{\l,\mu} = 0} \ar[rd]^{a_{\l,\mu'} f_{\l,\mu'}} & & \P_i^{(\nu)} \Q_i^{(\nu^t)} \1_n \la 2 \ra \\
& \P_i^{(\mu')} \Q_i^{({\mu'}^t)} \1_n \la 1 \ra \ar[ru]^{a_{\mu',\nu} f_{\mu',\nu}} &
}$$ By induction the right two maps are nonzero, which means that $a_{\mu',\nu}=0$ since $f_{\l',\nu} f_{\l,\mu'} \ne 0$ by Lemma \[lem:3\]. This proves our claim.
Now, since $\T_i$ is indecomposable and all maps out of $\P_i^{(\l)} \Q_i^{(\l^t)} \1_n$ are zero there must be a sequence of nonzero differentials which connect $\P_i^{(\l)} \Q_i^{(\l^t)} \1_n$ to some other $\P_i^{(\mu)} \Q_i^{({\mu}^t)} \1_n$. Such a path is depicted by the solid arrows in the figure below.
$$\begin{tikzpicture}[>=stealth]
\draw (0,0) -- (1,.5)[->] [very thick];
\draw (1,.5) -- (2,1)[->] [very thick];
\draw (2,1) -- (3,1.5)[->] [very thick];
\draw (3.75,1.65) node {$\P_i^{(\l)}\Q_i^{(\l^t)}$};
\draw (4.5,1.5) -- (5.5,2)[->] [very thick];
\draw (4.5,1.5) -- (5.5,1.5)[->] [very thick];
\draw (4.5,1.5) -- (5.5,1)[->] [very thick];
\draw (5,2) node {$0$};
\draw (5.25,1.7) node {$0$};
\draw (5,1) node {$0$};
\draw (0,0) -- (.9,-.45)[->] [very thick];
\draw (0,-1) -- (.9,-.55)[->] [very thick];
\draw (-1,-1.5) -- (0,-1)[->] [very thick];
\draw (-1,-1.5) -- (0,-2)[->] [very thick];
\draw (0,-2) -- (1,-2.5)[->] [very thick];
\draw (1,-2.5) -- (2,-3)[->] [very thick];
\draw (2,-3) -- (3,-2.5)[->] [very thick];
\draw (3.85,-2.6) node {$\P_i^{(\mu)}\Q_i^{(\mu^t)}$};
\draw (4.5,-3) -- (5.5,-2.5)[->] [very thick];
\draw (4.5,-3) -- (5.5,-3)[->] [very thick];
\draw (4.5,-3) -- (5.5,-3.5)[->] [very thick];
\draw (5,-2.5) node {$\neq0$};
\draw (0,-1) -- (.9,-1.45)[->] [dashed,very thick];
\draw (1,-1.5) -- (1.9,-1.95)[->] [dashed,very thick];
\draw (2.1,-2.05) -- (2.9,-2.5)[->] [dashed,very thick];
\draw (0,-2) -- (.9,-1.55)[->] [dashed,very thick];
\draw (1,-2.5) -- (1.9,-2.05)[->] [dashed,very thick];
\draw (.4,-1.6) node {$g_1$};
\draw (1.4,-2.1) node {$g_2$};
\draw (2.4,-2.5) node {$g_k$};
\draw (-.5,-1) node {$f_0$};
\draw (-.5,-2) node {$g_0$};
\draw (.55,-1) node {$f_1$};
\draw (1.7,-1.5) node {$f_2\hdots$};
\draw (2.55,-2) node {$f_k$};
\end{tikzpicture}$$ By assumption, in the path connecting $\P_i^{(\l)} \Q_i^{(\l^t)} \1_n$ and $\P_i^{(\mu)} \Q_i^{(\mu^t)} \1_n$, the outward differentials (and in particular the map $g_k$ at the end) are assumed to be nonzero. Without loss of generality we may assume that the path is minimal with respect to the area to its right. Now consider the square formed by $f_0,g_0,f_1,g_1$ (note that $f_1,g_1$ are uniquely determined by $f_0,g_0$). The skew-commutativity of this square means that $f_1$ and $g_1$ are either both zero or both nonzero. If they were both nonzero then one would obtain a smaller path which uses $f_1,g_1$ in place of $f_0,g_0$. So we must have $f_1=0=g_1$.
Now, the skew-commutativity of the subsequent squares means that $g_1=0 \Rightarrow g_2=0 \Rightarrow \dots \Rightarrow g_k = 0$ which is a contradiction. Thus $a_{\l,\mu} \ne 0$ and we are done.
[**Proof of (2).**]{} By (1) we know that $a_{1,1} \ne 0 \ne a'_{1,1}$. So we can rescale $\P_i \Q_i \1_n$ (the second term from the right in $\T_i \1_n$) so that $a_{1,1} = a'_{1,1}$. Now suppose by induction that $a_{\alpha,\beta} = a'_{\alpha,\beta}$ for all $|\beta| > \ell$. For any $\l$ with $|\l| = \ell$ and differential $\P_i^{(\l)} \Q_i^{(\l^t)} \1_n \rightarrow \P_i^{(\mu)} \Q_i^{(\mu^t)} \1_n$ we can rescale $\P_i^{(\l)} \Q_i^{(\l^t)}$ so that $a_{\l,\mu} = a'_{\l,\mu}$.
[**Claim.**]{} For any other differential out of $\P_i^{(\l)} \Q_i^{(\l^t)}$ we have $a_{\l,\mu'} = a'_{\l,\mu'}$. To see this consider again the square in (\[eq:commsquare\]). By induction we know that $a_{\mu,\nu} = a'_{\mu,\nu}$ and $a_{\mu',\nu} = a'_{\mu',\nu}$. Then by the rescaling above we also have $a_{\l,\mu} = a'_{\l,\mu}$ so the skew-commutativity of the square also gives $a_{\l,\mu'} = a'_{\l,\mu'}$.
Thus we can continue this way and scale things so that $a_{\l,\mu} = a'_{\l,\mu}$ for all $\mu \subset \l$ (which proves (2)).
\[rem:1\] To define the differentials in (\[eq:cpx\]) one can choose arbitrary nonzero maps $a_{\l,\mu} f_{\l,\mu}: \P_i^{(\l)} \Q_i^{(\l^t)} \1_n \rightarrow \P_i^{(\mu)} \Q_i^{(\mu^t)} \la 1 \ra \1_n$ and then rescale them as in the proof above to obtain a complex ([*i.e.*]{} so the square in (\[eq:commsquare\]) is skew-commutative).
The braid relations {#sec:braidrels}
===================
The braid relations in $\K_{Fock}$ {#sec:braidfock}
----------------------------------
We first work with the 2-category $\Kom(K_{Fock})$. To simplify notation we will denote $B := B^D_\ep(1)$ and $B_i := Be_{i,1}$ and ${}_iB := e_{i,1} B$ the natural $(B,\k)$ and $(\k,B)$ bimodules. Define the complex $\Sigma_i \1_1$ of $(B,B)$-bimodules as $$\Sigma_i \1_1 := B_i \otimes_\k {}_iB \la -2 \ra \rightarrow B$$ where the map is multiplication (or equivalently given by a cap) and the right hand term $B$ is in cohomological degree zero. Similarly, we define $$\Sigma_i^{-1} \1_1 := B \rightarrow B_i \otimes_\k {}_iB \la 2 \ra$$ where the map is given by a cap and the left hand term $B$ is in cohomological degree zero.
\[prop:braids\] The complexes $\Sigma_i \1_1$ satisfy the braid relations of $\Br(D)$ where the inverse of $\Sigma_i \1_1$ is the complex $\Sigma_i^{-1} \1_1$. In other words, in the homotopy category of $(B,B)$-bimodules we have the following homotopy equivalences
- $\Sigma_i \Sigma_i^{-1} \1_1 \xrightarrow{\sim} \1_1$ and $\Sigma_i^{-1} \Sigma_i \1_1 \xrightarrow{\sim} \1_1$,
- $\Sigma_i \Sigma_j \Sigma_i \1_1 \xrightarrow{\sim} \Sigma_j \Sigma_i \Sigma_j \1_1$ if $\la i,j \ra = -1$,
- $\Sigma_i \Sigma_j \1_1 \xrightarrow{\sim} \Sigma_j \Sigma_i \1_1$ if $\la i,j \ra = 0$.
These relations are proven in [@HK], following earlier work [@KS] in type $A$.
\[cor:braids\] In the homotopy category $\Kom(\K_{Fock})$ we have the following homotopy equivalences
- $\Sigma_i^{[n]} ({\Sigma_i^{-1}})^{[n]} \1_n \xrightarrow{\sim} \1_n$ and $({\Sigma_i^{-1}})^{[n]} \Sigma_i^{[n]} \1_n \xrightarrow{\sim} \1_n$,
- $\Sigma_i^{[n]} \Sigma_j^{[n]} \Sigma_i^{[n]} \1_n \xrightarrow{\sim} \Sigma_j^{[n]} \Sigma_i^{[n]} \Sigma_j^{[n]} \1_n$ if $\la i,j \ra = -1$,
- $\Sigma_i^{[n]} \Sigma_j^{[n]} \1_n \xrightarrow{\sim} \Sigma_j^{[n]} \Sigma_i^{[n]} \1_n$ if $\la i,j \ra = 0$.
This is an immediate consequence of the naturality of the wreath functor $(\cdot) \mapsto (\cdot)^{[n]}$ as discussed in section \[sec:wreath\]. For example, the homotopy equivalence $\Sigma_1^{-1} \Sigma_1 \1_1 \xrightarrow{\sim} \1_1$ induces $$\Sigma_i^{[n]} (\Sigma_i^{-1})^{[n]} \1_n \cong (\Sigma_i \Sigma_i^{-1} \1_1)^{[n]} \xrightarrow{\sim} (\1_1)^{[n]} = \1_n.$$
The next theorem relates $\Sigma_i^{[n]} \1_n$ and the complex $\T_i \1_n$ defined in (\[eq:cpx\]).
\[thm:braids\] The complex $\Sigma_i^{[n]}$ is isomorphic to the complex of $(B^D_\ep(n),B^D_\ep(n))$-bimodules $$\label{eq:cpx2}
\bigoplus_{\l \vdash n} \P_i^{(\l)} \Q_i^{(\l^t)} \la -n \ra \1_n \rightarrow \dots \rightarrow \bigoplus_{\l \vdash d} \P_i^{(\l)} \Q_i^{(\l^t)} \la -d \ra \1_n \rightarrow \dots \rightarrow \P_i \Q_i \la -1 \ra \1_n\rightarrow \1_n$$ defined in section \[sec:cpx\].
Proof of Theorem \[thm:braids\]
-------------------------------
The statement of the Theorem \[thm:braids\] only involves one vertex of the Dynkin diagram $D$. To simplify notation we will assume that $D$ contains only one vertex so that $B_\ep^D \cong \k[t]/t^2$. Notice that $\deg t = 2$ so $B_\ep^D$ is commutative (not supercommutative). We will use the notation $B := B_\ep^D$ and $B_n := B_\ep^D(n) = B^{\otimes n} \rtimes \k[S_n]$. Thus $\Sigma \1_1 = B' \rightarrow B$ as a $(B,B)$-bimodule where we denote $B' := B \otimes_\k B$.
By definition the term of the complex $\Sigma_i^{[n]}$ in cohomological degree $-d$ is $\bigoplus_{\ul} B_\ul \rtimes \k[S_n]$ where $\ul = (\ell_1, \dots, \ell_d)$ with $0 \le \ell_1 < \dots < \ell_d \le n$ and $B_\ul$ is a tensor product over $\k$ of $B$s and $B'$s where the $B'$s occur exactly in positions $\ell_1, \dots, \ell_d$. For example, $$B_{(2,3)} = B \otimes_\k B' \otimes_\k B' \otimes_\k B \text{ where } n=4, d=2.$$
The tensor product above is that of supervector spaces where $\deg B = 0$ and $\deg B' = 1$. So, for instance, the involution $(23) \in S_4$ acts on $B_{(2,3)}$ by $$(23) \cdot (b_1 \otimes b_2 \otimes b_3 \otimes b_4) \mapsto - (b_1 \otimes b_3 \otimes b_2 \otimes b_4)$$ while $(12) \in S_4$ acts as a map $B_{(2,3)} \rightarrow B_{(1,3)}$ by $$(12) \cdot (b_1 \otimes b_2 \otimes b_3 \otimes b_4) \mapsto (b_2 \otimes b_1 \otimes b_3 \otimes b_4).$$
\[prop:1\] There is an isomorphism of $(B_n,B_n)$-bimodules $$\bigoplus_{\l \vdash d} B_n e_\l \otimes_{B_{n-d}} e_{\l^t} B_n \xrightarrow{\sim} \bigoplus_{\ul} B_\ul \rtimes \k[S_n]$$ where $B_{n-d}$ acts on $B_n$ via the embedding $$B^{\otimes n-d} \ni b_1 \otimes \dots \otimes b_{n-d} \mapsto 1 \otimes \dots \otimes 1 \otimes b_1 \otimes \dots \otimes b_{n-d} \in B^{\otimes n}$$ and $e_\l \in \k[S_d] \subset B_d \subset B_n$ acts on the first $d$ factors (so it commutes with $B_{n-d}$).
Consider the map $\phi: B_n \otimes_{B_{n-d}} B_n \rightarrow \oplus_{\ul} B_\ul \rtimes \k[S_n]$ given by $$(1^{\otimes n},1) \otimes (1^{\otimes n},1) \mapsto (1^{\otimes d} \otimes 1^{\otimes n-d},1) \in B_{(1,2,\dots,d)} = (B')^{\otimes d} \otimes B^{\otimes n-d} \subset B_{\ul}.$$ This map is well defined since if $(b_1 \otimes \dots \otimes b_{n-d},\sigma) \in B_{n-d}$ then $$\begin{aligned}
\phi((1^{\otimes d} \otimes b_1 \otimes \dots \otimes b_{n-d},\sigma) \otimes (1^{\otimes n},1))
& = (1^{\otimes d} \otimes b_1 \otimes \dots \otimes b_{n-d},\sigma) \\
& = \phi((1^{\otimes n},1) \otimes (1^{\otimes d} \otimes b_1 \otimes \dots \otimes b_{n-d},\sigma)).\end{aligned}$$ Moreover, since $(1^{\otimes d} \otimes 1^{\otimes n-d},1) \in B_{(1,2,\dots,d)}$ generates $B_{\ul}$ as a $(B_n,B_n)$-bimodule $\phi$ is also surjective.
Now, $$B_n \otimes_{B_{n-d}} B_n \cong \bigoplus_{\l,\l' \vdash d} \left( B_n e_\l \otimes_{B_{n-d}} e_{\l'} B_n \right)^{\oplus d_\l d_{\l'}}$$ where $d_\l$ (resp. $d_{\l'}$) is the dimension of the irreducible $\k[S_d]$-module $V_\l$ (resp. $V_{\l'}$) indexed by the partition $\l$ (resp. $\l'$). Now, for $s_i \in S_d$ we have $$\phi((1^{\otimes n}, s_i) \otimes (1^{\otimes n}, 1)) = - (1^{\otimes n}, s_i) = (1^{\otimes n}, \tau(s_i))$$ where the minus sign is because $s_i$ acts on $B' \otimes_\k B'$ by $s_i \cdot (1 \otimes 1) = - (1 \otimes 1)$. Subsequently, $$\phi((1^{\otimes n}, e_{\l}) \otimes (1^{\otimes n}, e_{\l'})) = (1^{\otimes n}, \tau(e_{\l}) e_{\l'}) = \delta_{\l^t,\l'} (1^{\otimes n}, e_{\l^t}).$$ This means that $B_n e_\l \otimes_{B_{n-d}} e_{\l'} B_n$ is in the kernel of $\phi$ unless $\l' = \l^t$. We conclude that $$\label{eq:iso}
\phi: \bigoplus_{\l \vdash d} B_n e_{\l} \otimes_{B_{n-d}} e_{\l^t} B_n \longrightarrow \oplus_{\ul} B_{\ul} \rtimes \k[S_n]$$ is surjective.
To show that the map in (\[eq:iso\]) is an isomorphism we compute the dimensions over $\k$ of both sides. On the one hand $\dim_\k B_\ul = (\dim_\k B')^d (\dim_\k B)^{n-d} = 4^d 2^{n-d}$. Thus $$\label{eq:1}
\dim_\k \left( \oplus_\ul B_\ul \rtimes \k[S_n] \right) = \binom{n}{d} 2^{n+d} n!.$$ On the other hand, $$\dim_\k (B_n e_\l) = \dim_\k B_n \cdot \frac{ \dim_\k V_\l}{d!} = \frac{ 2^n n! \dim_\k V_\l}{d!}$$ and likewise $\dim_\k (e_{\l^t} B_n) = \frac{2^n n! \dim_\k V_\l}{d!}$. Since $B_n e_\l$ and $e_{\l^t} B_n$ are free $B_{n-d}$ modules it follows that $$\dim_{\k} \left( B_n e_{\l} \otimes_{B_{n-d}} e_{\l^t} B_n \right) = \frac{1}{\dim_\k B_{n-d}} \frac{2^{2n} n! n! (\dim_\k V_\l)^2}{d!d!} = \binom{n}{d} 2^{n+d} n! \frac{(\dim_\k V_\l)^2}{d!}.$$ Summing over all $\l \vdash d$ and using that $\sum_{\l \vdash d} (\dim_\k V_\l)^2 = d!$ we get the same dimension as in (\[eq:1\]). Thus (\[eq:iso\]) must be an isomorphism.
Theorem \[thm:braids\] now follows by combining Proposition \[prop:1\] with Proposition \[prop:Tcpx\] which says that any indecomposable complex such as that in (\[eq:cpx2\]) is unique up to homotopy.
\[cor:braids2\] In $\Kom(\K_{Fock})$ the complexes from (\[eq:cpx2\]) satisfy the braid relations of $\Br(D)$.
The braid relations in integrable 2-representations
---------------------------------------------------
We now consider an arbitrary integrable 2-representation $\K$ of $\h$ and prove Theorem \[thm:main1\] using Theorem \[thm:braids\]. We will show that $\T_i \T_i^{-1} \1_n \xrightarrow{\sim} \1_n$ in $\Kom(\oH)$ (the proof of the other braid relations is similar).
The composition $\T_i \T_i^{-1} \1_n$ is a (finite) complex of 1-morphisms in $\oH$. Decompose each term in this complex into indecomposables of the form $\P_i^{(\l)} \Q_i^{(\mu)} \la \ell \ra \1_n$ where $\ell \in \Z$ and $\l,\mu$ are partitions. Since $\End^k(\P_i^{(\l)} \Q_i^{(\mu)} \la \ell \ra \1_n)$ is zero if $k < 0$ and one-dimensional if $k=0$ we can restrict $\T_i \T_i^{-1} \1_n$ to these terms to get a complex of the form $\P_i^{(\l)} \Q_i^{(\mu)} \la \ell \ra \1_n \otimes_\k V_\bullet$ where $V_\bullet$ is a complex of vector spaces.
Using the Cancellation Lemma \[lem:cancel\] it suffices to show that $V_\bullet$ is exact (unless $\l = \mu = \emptyset$ and $\ell = 0$ in which case $V_\bullet$ should have one-dimensional cohomology in degree zero). To see this consider the image of $\T_i \T_i^{-1} \1_n$ in $\Kom(\K_{Fock})$. By Theorem \[thm:braids\] this complex is homotopic to $\1_n$. Thus the image of $V_\bullet$ is exact and so $V_\bullet$ must also be exact.
\[lem:cancel\] Let $ X, Y, Z, W, U, V$ be six objects in an additive category and consider a complex $$\label{eq:6.5}
\dots \rightarrow U \xrightarrow{u} X \oplus Y \xrightarrow{f} Z \oplus W \xrightarrow{v} V \rightarrow \dots$$ where $f = \left( \begin{matrix} A & B \\ C & D \end{matrix} \right)$ and $u,v$ are arbitrary morphisms. If $D: Y \rightarrow W$ is an isomorphism, then (\[eq:6.5\]) is homotopic to a complex $$\begin{aligned}
\label{eq:new}
\dots \rightarrow U \xrightarrow{u} X \xrightarrow{A-BD^{-1}C} Z \xrightarrow{v|_Z} V \rightarrow \dots\end{aligned}$$
The following result is a slight generalization of a lemma which Bar-Natan [@BN] calls “Gaussian elimination”. For a proof see Lemma 6.2 of [@CL2].
Convolution in triangulated 2-representations and Hilbert schemes
=================================================================
Consider an affine Dynkin diagram where $\Gamma \subset SL_2(\C)$ is the finite subgroup associated to it via the McKay correspondence. Let $X_\Gamma = \widehat{\C^2/\Gamma}$ be the minimal resolution and $X^{[n]}_{\Gamma}$ the Hilbert scheme of $n$ points.
In [@CL1] we constructed a Heisenberg 2-represention on the derived categories of coherent sheaves $\oplus_{n \ge 0} D_{\C^\times}(X_\Gamma^{[n]})$. Subsequently, Theorem \[thm:main1\] induces an action of the affine braid group on $\Kom(D_{\C^\times}(X_\Gamma^{[n]}))$. We would like to explain now how this action descends to one on $D_{\C^\times}(X_\Gamma^{[n]})$. The main result we prove is the following.
\[thm:geom\] For each $n \geq 0$, the complexes $\Sigma_i \1_n$ have a unique convolution $$\Conv(\Sigma_i \1_n) \in D_{\C^\times \times \C^\times}(X_\Gamma^{[n]} \times X_\Gamma^{[n]}).$$ These convolutions define an action of the affine braid group on $D_{\C^*}(X_\Gamma^{[n]})$.
The existence of such an affine braid group action is known by combining [@ST; @P]. The convolutions above give another interpretation of this action by showing that it arises from a categorical Heisenberg action.
Convolutions
------------
Let $A_\bullet = A_0 \xrightarrow{f_1} A_1 \rightarrow \cdots \xrightarrow{f_n} A_n $ be a sequence of objects and morphisms in a triangulated category $ \mathcal{D}$ such that $ f_{i+1} \circ f_i = 0 $. Such a sequence is called a complex.
A *(right) convolution* of a complex $A_\bullet $ is any object $B$ such that there exist
1. objects $ A_0 = B_0, B_1, \dots, B_{n-1}, B_n = B $ and
2. morphisms $g_j : B_{j-1} \rightarrow A_j$, $h_j : A_j \rightarrow B_j $ (with $h_0 = id$)
such that $B_{j-1} \xrightarrow{g_j} A_i \xrightarrow{h_j} B_j$ is a distinguished triangle for each $i$ and $ g_j \circ h_{j-1} = f_j $. Such a collection of data is called a *Postnikov system*. We will denote $B_n$ by $\Conv(A_\bullet)$. When $n=1$ then $B_n$ is isomorphic to the usual cone.
[@CK1 Prop. 8.3]\[th:uniquecone\] Consider a complex $A_\bullet$.
1. If ${{\rm Hom}}(A_j[k], A_{j+k+1}) = 0 $ for all $ j \ge 0, k \ge 1 $, then any two convolutions of $ (A_\bullet, f_\bullet) $ are isomorphic.
2. If $ {{\rm Hom}}(A_j[k], A_{j+k+2}) = 0 $ for all $ j \ge 0, k \ge 1 $, then $ (A_\bullet, f_\bullet) $ has a convolution.
\[prop:complexzero\] Suppose $\D$ is a triangulated category which satisfies the Krull-Schmidt property. If $A_\bullet$ is a complex of objects in $\D$ which is homotopic to zero then $\Conv(A_\bullet) \cong 0$.
The proof is by induction on the length of the complex. The base case is trivial.
Now, if $A_\bullet = A_0 \xrightarrow{f_1} A_1 \xrightarrow{} B_\bullet$ is homotopic to zero then there exists a map $g_1: A_1 \rightarrow A_0$ such that $g_1 \circ f_1 = \id_{A_0}$. Since $\D$ is Krull-Schmidt this means that $A_1 \cong A_0 \oplus A_1'$ and we can rewrite $A_\bullet$ as $$A_0 \xrightarrow{(f_1',f_1'')} A_0 \oplus A_1' \xrightarrow{h} B_\bullet$$ where $f_1'$ is an isomorphism. Now, if we take the first cone we obtain a commutative diagram $$\begin{aligned}
\xymatrix{
A_0 \ar[d] \ar[rr]^{(f_1',f_1'')} & & \ar[d]^{(\alpha,\beta)} A_0 \oplus A_1' \ar[rr]^{h} & & B_\bullet \ar[d]^\id \\
0 \ar[rr] & & \Cone(f_1',f_1'') \ar[rr]^u & & B_\bullet
}\end{aligned}$$ where $u$ is some map. Note that $\Conv(A_\bullet) \cong \Conv(\Cone(f_1',f_1'') \xrightarrow{u} B_\bullet)$ so, by induction, it suffices to show that $\Cone(f_1',f_1'') \xrightarrow{u} B_\bullet$ is homotopic to zero.
Since $f_1'$ is an isomorphism this means $\beta$ is an isomorphism. Thus, precomposing with $\beta$ we get $$\label{eq:5}
[\Cone(f_1',f_1'') \xrightarrow{u} B_\bullet] \cong [A_1' \xrightarrow{u \circ \beta} B_\bullet].$$ Note that by commutativity of the square $u \circ \beta = h$.
On the other hand, using the cancellation lemma \[lem:cancel\] $$[A_0 \xrightarrow{(f_1',f_1'')} A_0 \oplus A_1' \xrightarrow{h} B_\bullet] \cong [0 \rightarrow A_1' \xrightarrow{h} B_\bullet].$$ Thus, combining this with (\[eq:5\]) we get that $\Cone(f_1',f_1'') \xrightarrow{u} B_\bullet$ is homotopic to zero.
Proof of Theorem \[thm:geom\]
-----------------------------
First we show that $\Sigma_i \1_n$ has a unique convolution. By Proposition \[th:uniquecone\] it suffices to check that for any partitions $\l,\mu,\nu$ with $|\l|-|\mu| \ge 2$ and $|\l|-|\nu| \ge 3$ we have $$\label{eq:6}
{{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)}, \P_i^{(\mu)} \Q_i^{(\mu^t)} \la 1 \ra) = 0 = {{\rm Hom}}(\P_i^{(\l)} \Q_i^{(\l^t)}, \P_i^{(\nu)} \Q_i^{(\nu^t)} \la 2 \ra)$$ where $\la 1 \ra = [1]$. In the 2-category $\H$ this is clear because any map in the first (resp. second) ${{\rm Hom}}$ space above, must contain two (resp. three) adjunction maps and must therefore have degree at least two (resp. three). On the other hand, this can also be seen by using adjunction and the commutation relations in $\H$ to reduce the statements above to the statement that $\End(\1_0,\1_0 \la s \ra) = 0$ for $s < 0$. Since this is also true in our 2-representation consisting of coherent sheaves on $X_\Gamma^{[n]}$ the vanishing of (\[eq:6\]) holds there too.
It remains to show that these unique convolutions $\Conv(\Sigma_i \1_n)$ satisfy the affine braid relations. This follows from the affine braid relations satisfied in $\Kom(\H)$. We illustrate this by proving that $$\Conv(\Sigma_i \1_n) \circ \Conv(\Sigma_i^{-1} \1_n) \cong \1_n.$$ First, we have that $\Sigma_i \1_n \circ \Sigma_i^{-1} \1_n \cong \1_n$ in the homotopy category. This means that we have a map $\Sigma_i \1_n \circ \Sigma_i^{-1} \1_n \rightarrow \1_n$ whose cone is homotopic to zero. By Proposition \[prop:complexzero\] we know that any convolution of this cone is zero and hence there is an isomorphism $$\Conv(\Sigma_i \1_n \circ \Sigma_i^{-1} \1_n) \xrightarrow{\sim} \1_n$$ for any choice of convolution (we do not know that it has a unique convolution). On the other hand, $\Conv(\Sigma_i \1_n) \circ \Conv(\Sigma_i^{-1} \1_n)$ is [*some*]{} convolution of $\Sigma_i \1_n \circ \Sigma_i^{-1} \1_n$. Thus $$\Conv(\Sigma_i \1_n) \circ \Conv(\Sigma_i^{-1} \1_n) \cong \Conv(\Sigma_i \1_n \circ \Sigma_i^{-1} \1_n) \cong \1_n$$ and we are done.
\[rem:conv\] Although Theorem \[thm:geom\] involves the triangulated category of coherent sheaves on a surface, the same proof works to show that the conclusions in that theorem hold for any (graded) triangulated category where $${{\rm Hom}}(\1_n,\1_n \la \ell \ra) =
\begin{cases}
0 & \text{ if } \ell < 0 \\
\C & \text{ if } \ell = 0 \\
\text{finite dimensional } & \text{ if } \ell > 0.
\end{cases}$$
A larger group
--------------
As mentioned in the introduction, $D_{\C^\times}(X_\Gamma)$ carries an action of the affine braid group defined using Seidel-Thomas twists [@ST]. The affine braid group action constructed above coincides with its lift from $D_{\C^\times}(X_\Gamma)$ to $D_{\C^\times}(X_\Gamma^{[n]})$ using the results in [@P].
On the other hand, there is actually a larger group acting on $D_{\C^\times}(X_\Gamma^{[n]})$. It is generated by certain complexes very similar to our $\Sigma_i \1_n$. These complexes are briefly discussed in section \[sec:vertexops\] (equation (\[eq:eg2\]) is an example of such a complex). However, the autoequivalence such a complex generates is not the lift of any automorphism of $D_{\C^\times}(X_\Gamma)$.
A braid group action on $\Kom(\H)$ {#sec:braidH}
==================================
In this section we define an abstract action of $ \Br(D) $ on $\Kom(\H')$ where $ \H'$ is the full subcategory of $ \H$ generated by the $ \P_i $. The quotient of $\H' $ by the ideal generated by $\1_n$ for $n < 0$ may be thought of as another categorification of Fock space.
Recall that the 2-category $\H$ contains generating 2-morphisms $X_j^k \colon \P_j \rightarrow \P_k \la 1 \ra$ for $ \la j,k \ra=-1$ and $ T_{jk} \colon \P_j \P_k \rightarrow \P_k \P_j $ for any $j,k \in I$. In [@CL1] we encode these maps diagrammatically as a dot and a crossing respectively. To define a braid group action on $\H'$ we need to explain how the generators $\s_i^{\pm 1}$ act on 1-morphisms $\P_j$ and $\Q_j$ as well as on 2-morphisms $X_j^k$ and $T_{jk}$. We will then extend this action monoidally – for example, $\s_i(\P_j\P_j) = \s_i(\P_j) \s_i(\P_j)$.
In the appendix we will prove the following result.
\[thm:main2\] The actions of $\s_i^{\pm 1}$ defined in sections \[sec:action1\]–\[sec:action5\] induce 2-endofunctors of the 2-category $\Kom(\H')$ which satisfy the braid relations of $\Br(D)$.
One can extend the results of this section to describe an action of $\Br(D)$ on the entire category $\Kom(\H)$, rather than on just the upper half $\Kom(\H')$. However, this would require checking even more relations and our interest in this paper is to understand braid group actions arising from integrable 2-representations of $\H$. For this purpose, Theorem \[thm:main2\] is sufficient.
The action of $\s_i^{\pm 1}$ on 1-morphisms {#sec:action1}
-------------------------------------------
As usual, we will use $[k]$ to denote a cohomological shift to the left by $k$ and $\la k \ra$ to denote the internal grading shift of $\H$.
We define $$\s_i (\P_j) :=
\begin{cases}
\P_i \la -2 \ra \oplus \P_i \xrightarrow{(X_i^{i} \ \ 1_i)} \P_i & \text{ if } i=j \in I \\
\P_i \la -1 \ra \xrightarrow{X_i^j} \P_j & \text{ if } \la i,j \ra = -1 \\
\P_j & \text{ if } \la i,j \ra = 0
\end{cases}$$ where the right hand terms are all in homological degree zero. Likewise, we define $$\s_i^{-1}(\P_j) :=
\begin{cases}
\P_i \xrightarrow{(1_i \ \ X_i^{i})} \P_i \oplus \P_i \la 2 \ra & \text{ if } i=j \in I \\
\P_j \xrightarrow{X_j^i} \P_i \la 1 \ra & \text{ if } \la i,j \ra = -1 \\
\P_j & \text{ if } \la i,j \ra = 0
\end{cases}$$
where this time the left hand terms are all in degree zero.
The action of $\s_i$ on $X$’s {#sec:action2}
-----------------------------
Suppose $\la i,j \ra = -1$. We define $$\xymatrix{
\s_i(\P_i \la -1 \ra) \ar[d]_{\s_i(X_i^{j})} & = & \P_i \la -3 \ra \oplus \P_i \la -1 \ra \ar[rr]^{\hspace{.2in} (X_i^i \hspace{.2cm} 1) } \ar[d]_{(0 \hspace{.2cm} 1_i)} & & \P_i \la -1 \ra \ar[d]^{X_i^j} \\
\s_i(\P_{j}) & = & \P_i \la -1 \ra \ar[rr]^{X_i^{j}} & & \P_{j}
}$$ $$\xymatrix{
\s_i(\P_{j} \la -1 \ra) \ar[d]_{\s_i(X_{j}^i)} & = & \P_i \la -2 \ra \ar[rr]^{X_i^{j}} \ar[d]_{(\epsilon_{ij} 1_i \hspace{.2cm} 0)} & & \P_{j} \la -1 \ra \ar[d]^{X_j^i} \\
\s_i(\P_i) & = & \P_i \la -2 \ra \oplus \P_i \ar[rr]^{\hspace{.2in} ( X_i^i 1_i)} & & \P_i }$$ where the rightmost columns are both in cohomological degree zero.
Next, suppose further that $j \ne k$ and $\la j,k \ra = -1$, $\la i,k \ra = 0$. Then define $$\xymatrix{
\s_i(\P_{j} \la -1 \ra) \ar[d]_{\s_i(X_{j}^{k})} & = & \P_i \la -2 \ra \ar[r]^{X_i^{j}} \ar[d]^{}& \P_{j} \la -1 \ra \ar[d]^{X_{j}^{k}} \\
\s_i(\P_{k}) & = & 0 \ar[r]^{} & \P_{k}
}$$ $$\xymatrix{
\s_i(\P_{k}) \ar[d]_{ \s_i(X_{k}^{j})} & = & 0 \ar[r]^{} \ar[d]^{} & \P_{k} \la -1 \ra \ar[d]^{X_{k}^{j}} \\
\s_i(\P_{j}) & = & \P_i \la -1 \ra \ar[r]^{X_i^{j}} & \P_{j}
}$$ where again the rightmost columns are in cohomological degree zero.
An unusual situation is when $i,j,k$ form a triangle, meaning $\la i,j \ra = \la j,k \ra = \la i,k \ra = -1$. In this case we define $$\xymatrix{
\s_i(\P_j) \ar[d]_{\s_i(X_j^k)} & = & \P_i \la -1 \ra \ar[r]^{X_i^j} \ar[d]^{0} & \P_j \ar[d]^{X_j^k} \\
\s_i(\P_k) & = & \P_i \ar[r]^{X_i^k} & \P_k \la 1 \ra
}$$ where the rightmost column is in cohomological degree zero.
Finally, if $\la i,j \ra = \la i,k \ra = 0$ then $\s_i(X_j^k) = X_j^k$.
The action of $\s_i^{-1}$ on $X$’s {#sec:action3}
----------------------------------
Suppose $ \la i, j \ra = -1 $. We define $$\xymatrix{
\s_i^{-1}(\P_i \la -1 \ra) \ar[d]^{\s_i^{-1}(X_i^j)} & = & \P_i \la -1 \ra \ar[d]^{X_i^j} \ar[rr]^{(1_i \hspace{.2cm} X_i^i) \hspace{.6cm}} & & \P_i \la -1 \ra \oplus \P_i \la 1 \ra \ar[d]^{(0 \hspace{.2cm} \epsilon_{ij} 1_i)} \\
\s_i^{-1}(\P_j) & = & \P_{j} \ar[rr]^{X_j^i} & & \P_i \la 1 \ra
}$$
$$\xymatrix{
\s_i^{-1}(\P_j \la -1 \ra) \ar[d]^{\s_i^{-1}(X_j^i)} & = & \P_j \la -1 \ra \ar[d]^{X_j^i} \ar[rr]^{X_j^i} && \P_i \ar[d]^{(1_i \hspace{.2cm} 0)} \\
\s_i^{-1}(\P_i) & = & \P_{i} \ar[rr]^{(1_i \hspace{.2cm} X_i^i)} && \P_i \oplus \P_i \la 2 \ra
}$$
where the leftmost columns are in cohomological degree zero.
In the case when $ i,j,k $ form a triangle so that $ \la i, j \ra = \la j, k \ra = \la i, k \ra = -1 $ we define $$\xymatrix{
\s_i^{-1}(\P_j \la -1 \ra) \ar[d]^{\s_i^{-1}(X_j^k)} & = & \P_j \la -1 \ra \ar[r]^{X_j^i} \ar[d]^{X_j^k} & \P_i \ar[d]^{0} \\
\s_i^{-1}(\P_k) & = & \P_k \ar[r]^{X_k^i} & \P_i \la 1 \ra
}$$ where the leftmost column is in cohomological degree zero.
Next, suppose $ \la j, k \ra = -1, \la i, j \ra = -1, \la i, k \ra = 0 $. Then define
$$\xymatrix{
\s_i^{-1}(\P_j \la -1 \ra) \ar[d]^{\s_i^{-1}(X_j^k)} & = & \P_{j} \la -1 \ra \ar[d]^{X_{j}^{k}} \ar[r]^{X_{j}^i} & \P_i \ar[d]\\
\s_i^{-1}(\P_k) & = & \P_{k} \ar[r]^{} & 0
}.$$
$$\xymatrix{
\s_i^{-1}(\P_k \la -1 \ra) \ar[d]^{\s_i^{-1}(X_k^j)} & = & \P_{k} \la -1 \ra \ar[d]^{X_{k}^{j}} \ar[r]^{} & 0 \ar[d] \\
\s_i^{-1}(\P_j) & = & \P_{j} \ar[r]^{X_{j}^i} & \P_i \la 1 \ra
}.$$
where the leftmost columns are in cohomological degree zero.
Finally if $ \la i, j \ra = \la i, k \ra = 0 $ then define $ \s_i^{-1}(X_j^k) = X_j^k $.
The action of $\s_i$ on $T$’s {#sec:action4}
-----------------------------
We define $ \s_i(T_{ii}) $ as the map of complexes $$\xymatrix{
\P_i \P_i \la -4,-2,-2,0 \ra \ar[r]^{A} \ar[dd]_{- \begin{pmatrix} T_{ii} & 0 & 0 & 0 \\ 0 & 0 & T_{ii} & 0 \\ 0 & T_{ii} & 0 & 0 \\ 0 & 0 & 0 & T_{ii} \end{pmatrix}} & \P_i \P_i \la -2,0,-2,0 \ra \ar[rr]^{B} \ar[dd]^{\begin{pmatrix} 0 & 0 & T_{ii} & 0 \\ 0 & 0 & 0 & T_{ii} \\ T_{ii} & 0 & 0 & 0 \\ 0 & T_{ii} & 0 & 0 \end{pmatrix}}&& \P_i \P_i \ar[dd]^{T_{ii}} \\
\\
\P_i \P_i \la -4,-2,-2,0 \ra \ar[r]^{A} & \P_i \P_i \la -2,0,-2,0 \ra \ar[rr]^{B} && \P_i \P_i
}$$ where, for instance, $\P_i \P_i \la -4,-2,-2,0 \ra$ is shorthand for $\P_i \P_i \la -4 \ra \oplus \P_i \P_i \la -2 \ra \oplus \P_i \P_i \la -2 \ra \oplus \P_i \P_i$ and the rightmost column is in homological degree zero. The maps $A$ and $B$ are given by $$\begin{aligned}
A &=&
\begin{pmatrix}
-1_i X_i^i & -1_i 1_i & 0 & 0 \\
0 & 0 & -1_i X_i^i & -1_i 1_i \\
X_i^i 1_i & 0 & 1_i 1_i & 0 \\
0 & -X_i^i 1_i & 0 & 1_i 1_i
\end{pmatrix} \\
B &=&
\begin{pmatrix}
X_i^i 1_i & 1_i 1_i & 1_i X_i^i & 1_i 1_i
\end{pmatrix}.\end{aligned}$$
Now suppose $\la i,j \ra = -1$. Then we define $\s_i(T_{ij})$ by $$\xymatrix{
\s_i(\P_i \P_j) \ar[dd]^{\s_i(T_{ij})} & = & \P_i \P_i \la -3 \ra \oplus \P_i \P_i \la -1 \ra \ar[r]^{A \hspace{.3in}} \ar[dd]^{\begin{pmatrix} -T_{ii} & 0 \\ 0 & -T_{ii} \end{pmatrix}} & \P_i \P_j \la -2 \ra \oplus \P_i \P_j \oplus \P_i \P_i \la -1 \ra \ar[rr]^{\hspace{.5in} B}
\ar[dd]^{\begin{pmatrix} T_{ij} & 0 & 0 \\ 0 & T_{ij} & 0 \\ 0 & 0 & T_{ij} \end{pmatrix}}&& \P_i \P_j \ar[dd]^{T_{ij}} \\
\\
\s_i(\P_j \P_i) & = & \P_i \P_i \la -3 \ra \oplus \P_i \P_i \la -1 \ra \ar[r]^{C \hspace{.3in}} & \P_j \P_i \la -2 \ra \oplus \P_j \P_i \oplus \P_i \P_i \la -1 \ra \ar[rr]^{\hspace{.5in} D} && \P_j \P_i
}$$ where the rightmost column is in cohomological degree zero and $$\begin{aligned}
A &=
\begin{pmatrix}
-1_i X_i^k & 0 \\
0 & -1_i X_i^k \\
X_i^i 1_i & 1_i 1_i
\end{pmatrix}
\ \ \
C = \begin{pmatrix}
X_i^k 1_i & 0 \\
0 & X_i^k 1_i \\
-1_i X_i^i & -1_i 1_i
\end{pmatrix} \\
B &=
\begin{pmatrix}
X_i^i 1_k & 1_i 1_k & 1_i X_i^k
\end{pmatrix}
\ \ \
D =
\begin{pmatrix}
1_k X_i^i & 1_k 1_i & X_i^k 1_i
\end{pmatrix}.\end{aligned}$$
Likewise, if $\la i,j \ra = -1$ then we define $\s_i(T_{ji})$ by $$\xymatrix{
\s_i(\P_j \P_i) \ar[dd]^{\s_i(T_{ji})} & = & \P_i \P_i \la -3 \ra \oplus \P_i \P_i \la -1 \ra \ar[r]^{A \hspace{.3in}} \ar[dd]^{\begin{pmatrix} -T_{ii} & 0 \\ 0 & -T_{ii} \end{pmatrix}} & \P_j \P_i \la -2 \ra \oplus \P_j \P_i \oplus \P_i \P_i \la -1 \ra \ar[rr]^{\hspace{.5in} B}
\ar[dd]^{\begin{pmatrix} T_{ji} & 0 & 0 \\ 0 & T_{ji} & 0 \\ 0 & 0 & T_{ii} \end{pmatrix}}&& \P_j \P_i \ar[dd]^{T_{ji}} \\
\\
\s_i(\P_i \P_j) & = & \P_i \P_i \la -3 \ra \oplus \P_i \P_i \la -1 \ra \ar[r]^{C \hspace{.3in}} & \P_i \P_j \la -2 \ra \oplus \P_i \P_j \oplus \P_i \P_i \la -1 \ra \ar[rr]^{\hspace{.5in} D} && \P_i \P_j
}$$ where the rightmost column is in cohomological degree zero and $$\begin{aligned}
A &=
\begin{pmatrix}
X_i^j 1_i & 0 \\
0 & X_i^j 1_i \\
-1_i X_i^i & -1_i 1_i
\end{pmatrix}
\ \ \ \ \
C=
\begin{pmatrix}
-1_i X_i^j & 0 \\
0 & -1_i X_i^j \\
X_i^i 1_i & 1_i 1_i
\end{pmatrix} \\
B &= \begin{pmatrix} 1_j X_i^i & 1_j 1_i & X_i^j 1_i \end{pmatrix} \ \ \ \
D = \begin{pmatrix} X_i^i 1_j & 1_i 1_j & 1_i X_i^j \end{pmatrix}.\end{aligned}$$
On the other hand, if $\la i,j \ra = 0$ then $\s_i(T_{ij})$ is given by $$\xymatrix{
\s_i(\P_i \P_j) \ar[dd]_{\s_i(T_{ij})} & = & \P_i \P_j \la -2 \ra \oplus \P_i \P_j \ar[rr]^{\hspace{.3in} \begin{pmatrix} X_i^i 1_j & 1_i 1_j \end{pmatrix}} \ar[dd]^{\begin{pmatrix} T_{ij} & 0 \\ 0 &T_{ij} \end{pmatrix}} && \P_i \P_j \ar[dd]^{T_{ij}} \\
\\
\s_i(\P_j \P_i) & = & \P_j \P_i \la -2 \ra \oplus \P_j \P_i \ar[rr]^{\hspace{.3in} \begin{pmatrix} 1_j X_i^i & 1_j 1_i \end{pmatrix}} && \P_j \P_i
}$$ where the rightmost column is in cohomological degree zero.
Likewise, if $\la i,j \ra = 0$ then $\s_i(T_{ji})$ is given by $$\xymatrix{
\s_i(\P_j \P_i) \ar[dd]^{\s_i(T_{ji})} & = & \P_j \P_i \la -2 \ra \oplus \P_j \P_i \ar[rr]^{\hspace{.3in} \begin{pmatrix} 1_j X_i^i & 1_j 1_i \end{pmatrix}} \ar[dd]^{\begin{pmatrix} T_{ji} & 0 \\ 0 &T_{ji} \end{pmatrix}} && \P_j \P_i \ar[dd]^{T_{ji}}\\
\\
\s_i(\P_i \P_j) & = & \P_i \P_j \la -2 \ra \oplus \P_i \P_j \ar[rr]^{\hspace{.3in} \begin{pmatrix} X_i^i 1_j & 1_i 1_j \end{pmatrix}} && \P_i \P_j
}$$ where the rightmost column is in cohomological degree zero.
Next, suppose $\la i,j \ra = -1 = \la i,k \ra$. Define $\s_i(T_{jk})$ by $$\xymatrix{
\s_i(\P_j \P_k) \ar[dd]^{\s_i(T_{jk})} & = & \P_i \P_i \la -2 \ra \ar[rr]^{\begin{pmatrix} 1_i X_i^k \\ X_i^j 1_i \end{pmatrix} \hspace{.5in}} \ar[dd]^{T_{ii}} && \P_i \P_k \la -1 \ra \oplus \P_j \P_i \la -1 \ra \ar[rrr]^{\hspace{.5in} \begin{pmatrix} X_i^j 1_k & 1_j X_i^k \end{pmatrix}} \ar[dd]^{\begin{pmatrix} 0 & T_{ji} \\ T_{ik} & 0 \end{pmatrix}} &&& \P_j \P_k \ar[dd]^{T_{jk}} \\
\\
\s_i^{}(\P_j \P_k) & = &\P_i \P_i \la -2 \ra \ar[rr]^{\begin{pmatrix} 1_i X_i^j \\ X_i^k 1_i \end{pmatrix} \hspace{.5in}} && \P_i \P_j \la -1 \ra \oplus \P_k \P_i \la -1 \ra \ar[rrr]^{\hspace{.5in} \begin{pmatrix} X_i^k 1_j & 1_k X_i^j \end{pmatrix}} &&& \P_k \P_j
}$$ where the rightmost column is in cohomological degree zero.
On the other hand, if $\la i,j \ra = -1$ and $\la i,k \ra = 0$ then $\s_i(T_{jk})$ is defined by $$\xymatrix{
\s_i(\P_j \P_k) \ar[d]_{\s_i(T_{jk})} & = & \P_i \P_k \la -1 \ra \ar[r]^{X_i^j 1_k} \ar[d]^{T_{ik}} & \P_j \P_k \ar[d]^{T_{jk}} \\
\s_i(\P_k \P_j) & = & \P_k \P_i \la -1 \ra \ar[r]^{1_k X_i^j} & \P_k \P_j
}$$ while $\s_i(T_{kj})$ is defined by $$\xymatrix{
\s_i(\P_k \P_j) \ar[d]_{\s_i(T_{kj})} & = & \P_k \P_i \la -1 \ra \ar[r]^{1_k X_i^j} \ar[d]^{T_{ki}} & \P_k \P_j \ar[d]^{T_{kj}} \\
\s_i(\P_j \P_k) & = & \P_i \P_k \la -1 \ra \ar[r]^{X_i^j 1_k} & \P_j \P_k
}$$ where both the rightmost columns are in cohomological degree zero.
Finally, if $\la i, j \ra = 0 = \la i, k \ra$ then $\s_i(T_{jk}) = T_{jk}$.
The action of $ \s_i^{-1} $ on $T$’s {#sec:action5}
------------------------------------
We define $ \s_i^{-1}(T_{ii}) $ as a map of complexes: $$\xymatrix{
\P_i \P_i \ar[r]^{\hspace{-.8in} A} \ar[dd]^{T_{ii}} & \P_i \P_i \oplus \P_i \P_i \la 2 \ra \oplus \P_i \P_i \oplus \P_i \P_i \la 2 \ra \ar[rr]^{\hspace{.5in} B} \ar[dd]^{\begin{pmatrix} 0 & 0 & T_{ii} & 0 \\ 0 & 0 & 0 & T_{ii} \\ T_{ii} & 0 & 0 & 0 \\ 0 & T_{ii} & 0 & 0 \end{pmatrix}} && \P_i \P_i \la 0, 2, 2, 4 \ra \ar[dd]^{-\begin{pmatrix} T_{ii} & 0 & 0 & 0 \\ 0 & 0 & T_{ii} & 0 \\ 0 & T_{ii} & 0 & 0 \\ 0 & 0 & 0 & T_{ii} \end{pmatrix}}\\
\\
\P_i \P_i \ar[r]^{\hspace{-.8in} A} & \P_i \P_i \oplus \P_i \P_i \la 2 \ra \oplus \P_i \P_i \oplus \P_i \P_i \la 2 \ra \ar[rr]^{\hspace{.5in} B} && \P_i \P_i \la 0, 2, 2, 4 \ra
}$$ where the leftmost column is in cohomological degree zero. The maps $ A $ and $ B $ are given by
$$A = \begin{pmatrix}
1_i 1_i \\
1_i X_i^i \\
1_i 1_i \\
X_i^i 1_i
\end{pmatrix}
\ \ \ \text{ and } \ \ \
B =
\begin{pmatrix}
1_i 1_i & 0 & -1_i 1_i & 0 \\
0 & 1_i 1_i & -1_i X_i^i & 0 \\
X_i^i 1_i & 0 & 0 & -1_i 1_i \\
0 & X_i^i 1_i & 0 & -1_i X_i^i
\end{pmatrix}.$$
Now suppose $ \la i, j \ra = -1 $. Then we define $ \s_i^{-1}(T_{ij}) $ as a map of complexes by $$\xymatrix{
\s_i^{-1}(\P_i \P_j) \ar[dd]^{\s_i^{-1}(T_{ij})} &=& \P_i \P_j \ar[r]^{A \hspace{.6in}} \ar[dd]^{T_{ij}} & \P_i \P_j \oplus \P_i \P_j \la 2 \ra \oplus \P_i \P_i \la 1 \ra \ar[rr]^{B}
\ar[dd]^{\begin{pmatrix} T_{ij} & 0 & 0 \\ 0 & T_{ij} & 0 \\ 0 & 0 & T_{ii} \end{pmatrix}}&& \P_i \P_i \la 1 \ra \oplus \P_i \P_i \la 3 \ra \ar[dd]^{\begin{pmatrix} -T_{ii} & 0 \\ 0 & -T_{ii} \end{pmatrix}} \\
\\
\s_i^{-1}(\P_j \P_i) &=& \P_j \P_i \ar[r]^{C \hspace{.6in}} & \P_j \P_i \oplus \P_j \P_i \la 2 \ra \oplus \P_i \P_i \la 1 \ra \ar[rr]^{D} && \P_i \P_i \la 1 \ra \oplus \P_i \P_i \la 3 \ra
}$$ where the leftmost column is in cohomological degree zero and
$$\begin{aligned}
A &=
\begin{pmatrix}
1_i 1_k \\
X_i^i 1_k \\
1_i X_k^i
\end{pmatrix}
\ \ \
B =
\begin{pmatrix}
-1_i X_k^i & 0 & 1_i 1_i \\
0 & -1_i X_k^i & X_i^i 1_i
\end{pmatrix}
\\
C&=
\begin{pmatrix}
1_k 1_i \\
1_k X_i^i \\
X_k^i 1_i
\end{pmatrix}
\ \ \
D =
\begin{pmatrix}
X_k^i 1_i & 0 & -1_i 1_i \\
0 & X_k^i 1_i & -1_i X_i^i
\end{pmatrix}.\end{aligned}$$
Likewise, if $ \la i, j \ra = -1 $ then we define $ \s_i^{-1}(T_{ji}) $ by $$\xymatrix{
\s_i^{-1}(\P_j \P_i) \ar[dd]^{\s_i^{-1}(T_{ji})} &=& \P_j \P_i \ar[r]^{A \hspace{.6in}} \ar[dd]^{T_{ji}} & \P_j \P_i \oplus \P_j \P_i \la 2 \ra \oplus \P_i \P_i \la 1 \ra \ar[rr]^{B}
\ar[dd]^{\begin{pmatrix} T_{ji} & 0 & 0 \\ 0 & T_{ji} & 0 \\ 0 & 0 & T_{ii} \end{pmatrix}}&& \P_i \P_i \la 1 \ra \oplus \P_i \P_i \la 3 \ra \ar[dd]^{\begin{pmatrix} -T_{ii} & 0 \\ 0 & -T_{ii} \end{pmatrix}} \\
\\
\s_i^{-1}(\P_i \P_j) &=& \P_i \P_j \ar[r]^{C \hspace{.6in}} & \P_i \P_j \oplus \P_i \P_j \la 2 \ra \oplus \P_i \P_i \la 1 \ra \ar[rr]^{D} && \P_i \P_i \la 1 \ra \oplus \P_i \P_i \la 3 \ra
}$$ where the leftmost column is in degree zero and $$\begin{aligned}
A &=
\begin{pmatrix}
1_j 1_i \\
1_j X_i^i \\
X_j^i 1_i
\end{pmatrix}
\ \ \
B =
\begin{pmatrix}
X_j^i 1_i & 0 & -1_i 1_i \\
0 & X_j^i 1_i & -1_i X_i^i
\end{pmatrix}
\\
C&=
\begin{pmatrix}
1_i 1_j \\
X_i^i 1_j \\
1_i X_j^i
\end{pmatrix}
\ \ \
D =
\begin{pmatrix}
-1_i X_j^i & 0 & 1_i 1_i \\
0 & -1_i X_j^i & X_i^i 1_i
\end{pmatrix}.\end{aligned}$$
On the other hand, if $ \la i, j \ra = 0 $ then $ \s_i^{-1}(T_{ij}) $ and $ \s_i^{-1}(T_{ji}) $ are given by $$\xymatrix{
\s_i^{-1}(\P_i \P_j) \ar[dd]^{\s_i^{-1}(T_{ij})} &=& \P_i \P_j \ar[rr]^{( 1_i 1_j \ \ X_i^i 1_j ) \hspace{.4in}} \ar[dd]^{T_{ij}} && \P_i \P_j \oplus \P_i \P_j \la 2 \ra \ar[dd]^{\begin{pmatrix} T_{ij} & 0 \\ 0 & T_{ij} \end{pmatrix}}\\
\\
\s_i^{-1}(\P_j \P_i) &=& \P_j \P_i \ar[rr]^{( 1_j 1_i \ \ 1_j X_i^i ) \hspace{.4in}} && \P_j \P_i \oplus \P_k \P_i \la 2 \ra
}$$ $$\xymatrix{
\s_i^{-1}(\P_j \P_i) \ar[dd]^{\s_i^{-1}(T_{ji})} &=& \P_j \P_i \ar[rr]^{( 1_j 1_i \ \ 1_j X_i^i ) \hspace{.4in}} \ar[dd]^{T_{ji}} && \P_j \P_i \oplus \P_j \P_i \la 2 \ra \ar[dd]^{\begin{pmatrix} T_{ji} & 0 \\ 0 & T_{ji} \end{pmatrix}}\\
\\
\s_i^{-1}(\P_i \P_j) &=& \P_i \P_j \ar[rr]^{( 1_i 1_j \ \ X_i^i 1_j) \hspace{.4in}} && \P_i \P_j \oplus \P_i \P_j \la 2 \ra
}$$ where the leftmost columns are in cohomological degree zero.
Next, suppose $ \la i, j \ra = -1 = \la i, k \ra $. Define $ \s_i^{-1}(T_{jk}) $ by $$\xymatrix{
\s_i^{-1}(\P_j \P_k) \ar[dd]^{\s_i^{-1}(T_{jk})} &=& \P_j \P_k \ar[rrr]^{( X_j^i 1_k \ \ 1_j X_k^i) \hspace{.5in}} \ar[dd]^{T_{jk}} &&& \P_i \P_k \la 1 \ra \oplus \P_j \P_i \la 1 \ra \ar[rrrr]^{\hspace{.5in}( 1_i X_k^i \ \ X_j^i 1_i )} \ar[dd]^{\begin{pmatrix} 0 & T_{ji} \\ T_{ik} & 0 \end{pmatrix}} &&&& \P_i \P_i \la 2 \ra \ar[dd]^{T_{ii}} \\
\\
\s_i^{-1}(\P_k \P_j) &=& \P_k \P_j \ar[rrr]^{( X_k^i 1_j \ \ 1_k X_j^i )\hspace{.5in}} &&& \P_i \P_j \la 1 \ra \oplus \P_k \P_i \la 1 \ra \ar[rrrr]^{\hspace{.5in} ( 1_i X_j^i \ \ X_k^i 1_i)} &&&& \P_i \P_i \la 2 \ra
}$$ where the leftmost column is in cohomological degree zero.
On the other hand, if $ \la i, j \ra = - 1 $ and $ \la i,k \ra = 0 $ then $ \s_i^{-1}(T_{jk}) $ and $ \s_i^{-1}(T_{kj}) $ are defined by $$\xymatrix{
\s_i^{-1}(\P_j \P_k) \ar[d]^{\s_i^{-1}(T_{jk})} &=& \P_j \P_k \ar[r]^{X_j^i 1_k} \ar[d]^{T_{jk}} & \P_i \P_k \la 1 \ra \ar[d]^{T_{ik}}\\
\s_i^{-1}(\P_k \P_j) &=& \P_k \P_j \ar[r]^{1_k X_j^i} & \P_k \P_i \la 1 \ra
}$$ $$\xymatrix{
\s_i^{-1}(\P_k \P_j) \ar[d]^{\s_i^{-1}(T_{kj})} &=& \P_k \P_j \ar[r]^{1_k X_j^i} \ar[d]^{T_{kj}} & \P_k \P_i \la 1 \ra \ar[d]^{T_{ki}}\\
\s_i^{-1}(\P_j \P_k) &=& \P_j \P_k \ar[r]^{X_j^i 1_k} & \P_i \P_k \la 1 \ra
}$$ where both leftmost columns are in cohomological degree zero.
Finally if $ \la i, j \ra = 0 = \la i, k \ra $ then $ \s_i^{-1}(T_{jk}) = T_{jk} $.
\[prop:welldefined\] The definitions above give well defined endomorphisms of the 2-category $\Kom(\H)$.
Some homotopy equivalences
--------------------------
Some of the definitions above can be simpified, as we now explain. The reason we do not use these simpler definitions is that in practice they are more difficult to work with when checking the braid relations in the next section.
The complex $\s_i(\P_i)$ is homotopy equivalent to $\P_i \la -2 \ra [1]$ via the maps $\nu_{\P_i}$ and $\bar{\nu}_{\P_i}$ defined as follows $$\xymatrix{
\s_i(\P_i) \ar[d]_{\nu_{\P_i}} & = & \P_i \la -2 \ra \oplus \P_i \ar[rr]^{\hspace{.5cm} (X_i^i \hspace{.1in} 1_i)} \ar[d]_{(1_i \hspace{.1cm} 0)} & & \P_i \ar[d] \\
\P_i \la -2 \ra [1] \ar[d]_{\bar{\nu}_{\P_i}} & = & \P_i \la -2 \ra \ar[rr] \ar[d]_{(1_i \hspace{.1cm} -X_i^i)} & & 0 \ar[d] \\
\s_i(\P_i) & = & \P_i \la -2 \ra \oplus \P_i \ar[rr]^{\hspace{.5cm} (X_i^i \hspace{.1in} 1_i)} & & \P_i
}$$ Clearly $\nu_{\P_i} \bar{\nu}_{\P_i}$ is the identity map. On the other hand, $\bar{\nu}_{\P_i} \nu_{\P_i} $ is homotopic to the identity using $(0 \hspace{.1cm} -1_i): \P_i \rightarrow \P_i \la -2 \ra \oplus \P_i$.
Likewise, $\s_i^{-1}(\P_i)$ is homotopy equivalent to $\P_i \la 2 \ra [-1]$ via the maps $\zeta_{\P_i}$ and $\bar{\zeta}_{\P_i}$ defined by $$\xymatrix{
\s_i^{-1}(\P_i) \ar[d]_{{\zeta}_{\P_i}} & = & \P_i \ar[d] \ar[rr]^{\hspace{-.3cm} (1_i \hspace{.1cm} X_i^i)} & & \P_i \oplus \P_i \la 2 \ra \ar[d]^{(-X_i^i \hspace{.1cm} 1_i)} \\
\P_i \la 2 \ra [-1] \ar[d]_{\bar{\zeta}_{\P_i}} & = & 0 \ar[d] \ar[rr] & & \P_i \la 2 \ra \ar[d]^{(0 \hspace{.1cm} 1_i)} \\
\s_i^{-1}(\P_i) & = & \P_i \ar[rr]^{\hspace{-.3cm}(1_i \hspace{.1cm} X_i^i)} & & \P_i \oplus \P_i \la 2 \ra
}$$
Using these homotopy equivalences one simplify some of the definitions above. In particular, if $\la i,j \ra = -1$ then we get $$\xymatrix{
\nu_{\P_i} \s_i(\P_i \la -1 \ra) \ar[d]_{\s_i(X_i^j) \circ \bar{\nu}_{\P_i}} & = & \P_i \la -3 \ra \ar[r] \ar[d]^{-X_i^i}& 0 \ar[d] \\
\s_i(\P_j) & = & \P_i \la -1 \ra \ar[r]^{X_i^{j}} & \P_{j}
}$$ $$\xymatrix{
\s_i(\P_j \la -1 \ra) \ar[d]_{\nu_{\P_i} \circ \s_i(X_j^i)} & = & \P_i \la -2 \ra \ar[r]^{X_i^{j}} \ar[d]^{\epsilon_{ij} 1_i}& \P_{j} \la -1 \ra \ar[d] \\
\nu_{\P_i} \s_i(\P_i) & = & \P_i \la -2 \ra \ar[r]^{} & 0
}$$ Moreover, it is easy to check that $$(\nu_{\P_i} \nu_{\P_i}) \circ \s_i(T_{ii}) \circ (\bar{\nu}_{\P_i} \bar{\nu}_{\P_i}) = -T_{ii} \colon \P_i \P_i \la -4 \ra [2] \rightarrow \P_i \P_i \la -4 \ra [2].$$ while if $\la i,j \ra = -1$ then $$\xymatrix{
(\nu_{\P_i} \s_i(\P_i)) \s_i (\P_j) \ar[d]_{(1_{\s_i(\P_j)} \nu_{\P_i}) \circ (\s_i(T_{ij})) \circ (\bar{\nu}_{\P_i} 1_{\s_i(\P_j)})} & = & \P_i \P_i \la -3 \ra \ar[r]^{-1_i X_i^j} \ar[d]^{-T_{ii}} & \P_i \P_j \la -2 \ra \ar[d]^{T_{ij}} \\
\s_i(\P_j) (\nu_{\P_i} \s_i(\P_i)) & = & \P_i \P_i \la -3 \ra \ar[r]^{X_i^j 1_i} & \P_j \P_i \la -2 \ra
}$$ $$\xymatrix{
\s_i(\P_j) (\nu_{\P_i} \s_i(\P_i)) \ar[d]_{(\nu_{\P_i} 1_{\s_i(\P_j)}) \circ (\s_i(T_{ji})) \circ (1_{\s_i(\P_j)} \bar{\nu}_{\P_i})} & = & \P_i \P_i \la -3 \ra \ar[r]^{X_i^j 1_i} \ar[d]^{-T_{ii}} & \P_j \P_i \la -2 \ra \ar[d]^{T_{ji}} \\
(\nu_{\P_i} \s_i(\P_i)) \s_i(\P_j) & = & \P_i \P_i \la -3 \ra \ar[r]^{-1_i X_i^j} & \P_i \P_j \la -2 \ra
}$$ where the rightmost columns are in cohomological degree $-1$.
Finally, if $\la i,j \ra = 0$ then $$\begin{aligned}
1_j \nu_{\P_i} \circ \s_i(T_{ij}) \circ \bar{\nu}_{\P_i}1_j = T_{ij}: & \P_i \P_j \la -2 \ra [1] \rightarrow \P_j \P_i \la -2 \ra [1] \\
\nu_{\P_i} 1_j \circ \s_i(T_{ji}) \circ 1_j \bar{\nu}_{\P_i} = T_{ji}: & \P_j \P_i \la -2 \ra [1] \rightarrow \P_i \P_j \la -2 \ra [1].\end{aligned}$$
There are similar simplifications involving $\s_i^{-1}$ which we omit.
Some remarks and conjectures {#sec:remarks}
============================
The action on 1-morphisms in $\Kom(\H')$
----------------------------------------
The action of $\Br(D)$ on $\Kom(\H')$ from section \[sec:braidH\] was defined explicitly only on 1-morphisms $\P_i$. On some more general 1-morphisms it acts as follows.
\[prop:2\] The braid group action from section \[sec:braidH\] on the 2-category $\Kom(\H')$ acts on $\P_j^{(n)}$ by $$\s_i(\P_j^{(n)}) \cong
\begin{cases}
\P_i^{(1^n)} \la -2n \ra [n] & \text{ if } i=j \\
\left[\P_i^{(n)} \la -n \ra \rightarrow \P_i^{(n-1)} \P_j \la -n+1 \ra \rightarrow \dots \rightarrow \P_i \P_j^{(n-1)} \la -1 \ra \rightarrow \P_j^{(n)} \right] & \text{ if } \la i,j \ra = -1 \\
\P_j^{(n)} & \text{ if } \la i,j \ra = 0
\end{cases}$$ where the rightmost term is in cohomological degree zero and by $$\s_i^{-1}(\P_j^{(n)}) \cong
\begin{cases}
\P_i^{(1^n)} \la 2n \ra [-n] & \text{ if } i=j \\
\left[\P_j^{(n)} \rightarrow \P_j^{(n-1)} \P_i \la 1 \ra \rightarrow \dots \rightarrow \P_j \P_i^{(n-1)} \la n-1 \ra \rightarrow \P_i^{(n)} \la n \ra \right] & \text{ if } \la i,j \ra = -1 \\
\P_j^{(n)} & \text{ if } \la i,j \ra = 0
\end{cases}$$ where the leftmost term is in degree zero. The differentials in the complexes are the unique morphisms (up to a scalar) given by the compositions $$\begin{aligned}
& \P_i^{(r)} \P_j^{(n-r)} \rightarrow \P_i^{(r-1)} \P_i \P_j^{(n-r)} \xrightarrow{IX_i^jI} \P_i^{(r-1)} \P_j \P_j^{(n-r)} \la 1 \ra \rightarrow \P_i^{(r-1)} \P_j^{(n-r+1)} \la 1 \ra \\
& \P_i^{(r)} \P_j^{(n-r)} \rightarrow \P_i^{(r)} \P_j \P_j^{(n-r-1)} \xrightarrow{IX_j^iI} \P_i^{(r)} \P_i \P_j^{(n-r-1)} \la 1 \ra \rightarrow \P_i^{(r+1)} \P_j^{(n-r-1)} \la 1 \ra.\end{aligned}$$
In order to compute $ \s_i(\P_j^{(n)}) $ one must compute $ \s_i(\P_j^n) $ and determine the image of $ \s_i(e_{(n)}) $ where $ e_{(n)} $ is the trivial idempotent in $ S_n $ acting on $n$ strands colored by $ j $.
The case that $ \la i, j \ra = 0 $ is trivial since $ \s_i(\P_j^n) = \P_j^n $ and $ \s_i(T_{jj}) = T_{jj} $. Thus the image of $ \s_i(e_{(n)}) $ on $ \P_j^n $ is $ \P_j^{(n)} $ by definition.
If $i=j$ then, up to homotopy, $ \s_i(\P_i^n) \cong \P_i^n \la -2n \ra [n] $ and $ \s_i(e_{(n)}) = e_{(1^n)} $ since $ \s_i(T_{ii}) = -T_{ii} $. Thus the image of $ \s_i(e_{(n)}) $ on $ \s_i(\P_i^n) $ is $ \P_i^{(1^n)} \la -2n \ra [n] $.
If $\la i, j \ra = -1$ then $\s_i(\P_j) = [\P_i \la -1 \ra \rightarrow \P_j]$ which means that $\s_i(\P_j^n) $ is now a complex. To simplify notation, for a sequence $ {\bf d} = (d_1, \ldots, d_n) $ where each entry is $ i $ or $ j $ we denote $ \P_{\bf d} = \P_{d_1} \cdots \P_{d_n} $. Then the term in $\s_i(\P_j^n)$ lying in cohomological degree $-r$ (where $r \ge 0$) is $\bigoplus_{\bf d} \P_{\bf d} \la -r \ra$ where $r$ of the entries of $ {\bf d} $ are $i$ and $n-r$ are $j$.
Since $S_n$ permutes the entries of $ {\bf d} $ we may consider the subgroup $ S_{\bf d} \subset S_n $ which stabilizes ${\bf d}$. Let ${\bf d'}$ be another sequence where $ r $ of the entries are $ i $ and $ n-r $ are $ j $. Then, using the definition of $\sigma_i(T_{jj})$ from section \[sec:action4\], the 2-morphism $\s_i(e_{(n)}) $ induces a map $ \P_{\bf d} \la -r \ra \rightarrow \P_{\bf d'} \la -r \ra $ which (up to a nonzero scalar) is the sum over all elements in the coset of $ S_n / S_{\bf d} $ which transforms $ {\bf d} $ to $ {\bf d'} $.
Since for any ${\bf d,d'}$ as above there is an isomorphism $ \phi \colon \P_{\bf d} \rightarrow \P_{\bf d'} $ such that $ \s_i(e_{(n)})_{|\P_{\bf d}} = \s_i(e_{(n)})_{|\P_{\bf d'}} \circ \phi $ it suffices to consider the compute the image of $ \s_i(e_{(n)}) $ on $ \P_{({\bf i},{\bf j})} $ where $({\bf i}, {\bf j}) := (\underbrace{i, \ldots, i,}_r \underbrace{j, \ldots, j}_{n-r})$.
So consider now $\s_i(e_{(n)})_{|\P_{\bb}}: \P_{\bb} \rightarrow \bigoplus_{\bf d} \P_{\bf d}$. We must show that the image is isomorphic to $\P_i^{(r)} \P_j^{(n-r)}$. Let $ w_{{\bb}, {\bf d}} $ be a minimal length representative in this coset. The component of this map $\P_{\bb} \rightarrow \P_{\bf d}$ is a sum over all elements in the coset of $ S_n / S_r \times S_{n-r} $ which transforms $ {\bb} $ to $ {\bf d} $.
Consider the composition $ \P_{\bb} \rightarrow \bigoplus_{\bf d} \P_{\bf d} \rightarrow \bigoplus_{\bf d} \P_{\bb} $ where the first map is the restriction of $ \s_i(e_{(n)}) $ and the second map is the isomorphism $ \bigoplus_{\bf d} w_{{\bb}, {\bf d}}^{-1} $. The composition is the diagonal map where each entry is the sum over all elements in $S_r \times S_{n-r}$. Thus the image of $ \s_i(e_{(n)}) $ in $ \P_{\bb} $ is $ \P_i^{(r)} \P_j^{(n-r)}$.
For example, take $ r=2, n=3 $ so that $ {\bb} = (i,i,j) $. Then the restriction of $ \s_i(e_{(3)}) $ to $ \P_{(i,i,j)} $ is the map $$\begin{pmatrix}
1+s_1 \\
s_2 + s_2 s_1 \\
s_1 s_2 s_1 + s_1 s_2
\end{pmatrix}
\colon \P_{(i,i,j)} \rightarrow \P_{(i,i,j)} \oplus \P_{(i,j,i)} \oplus \P_{(j,i,i)}$$ where the $ s_1,s_2 $ are the standard transpositions. Composing this map with $$\bigoplus_{\bf d} w_{{\bb}, {\bf d}}^{-1} =
\begin{pmatrix}
1 & & \\
& s_2 & \\
& & s_2 s_1
\end{pmatrix}
\colon \P_{(i,i,j)} \oplus \P_{(i,j,i)} \oplus \P_{(j,i,i)} \rightarrow \P_{(i,i,j)} \oplus \P_{(i,i,j)} \oplus \P_{(i,i,j)}$$ we obtain $$\begin{pmatrix}
1+s_1 \\
1+s_1 \\
1+s_1
\end{pmatrix}
\colon \P_{(i,i,j)} \rightarrow \P_{(i,i,j)} \oplus \P_{(i,i,j)} \oplus \P_{(i,i,j)}.$$ The computation of $ \s_i^{-1}(\P_j^{(n)}) $ is similar so we omit it.
\[rem:2\] It was shown in [@CL1] that $\K_{Fock}$ categorifies $V_{Fock}$ (the Fock space). This means that there exists an isomorphism $$\Phi: K_0(\Kom(\K_{Fock})) = K_0(\K_{Fock}) \rightarrow V_{Fock}$$ which takes $\P_i^{(n)} \1_0 \mapsto P_i^{(n)}$ in the notation of section \[subsec:braidfock\]. Subsequently, Proposition \[prop:2\] induces the action of $\Br(D)$ on $V_{Fock}$ described in Proposition \[prop:fockspace\].
Consider the zig-zag algebra $B$ of Dynkin type $A_k$. In Section 4 of [@KS], a complex of finitely generated, projective $B$-modules is associated to any “admissible” curve in the $k$-punctured disk. Equivalently, this is a complex of $\P$’s in $\Kom(\H')$ (for example $\P_1 \la -1 \ra \rightarrow \P_2$). It would be interesting to generalize that result as follows: to any “admissible” $n$-tuple of curves in the $k$-punctured disk, associate a complex of elements in $\H'$ where each term is a sum of a product of exactly $n$ $\P$’s (for example, the complexes appearing in the statement of Proposition \[prop:2\]).
A conjectural intertwiner {#sec:intertwiner}
-------------------------
At this point we have two braid group actions – one is via the complexes $\T_i$ and the other is directly on the 2-category $\Kom(\H)$. We conjecture that these two actions are related as follows.
\[conj:intertwiner\] Consider a 2-representation $\K$ of $\H$ and denote by $\R$ an arbitrary 1-morphism in $\Kom(\H)$. Then $ \s_i(\R) \circ \T_i \cong \T_i \circ \R $.
In particular, this means that the braid group action on $\Kom(\H)$ from section \[sec:braidH\] is just conjugation using the complexes $\T_i$.
Vertex operators and braid groups {#sec:vertexops}
---------------------------------
In this section we suppose that our Dynkin diagram $D$ is of affine type (and still simply laced). Denote by $\g$ the associated affine Lie algebra. In [@CLa] we defined what it means to have a 2-representation of $\g$. Roughly, this consists of a 2-category where the objects are indexed by weights of $\g$, 1-morphisms are generated by $\E_i^{(r)}$ and $\F_i^{(r)}$ where $i \in I, r \in \N$ and there are various 2-morphisms with relations. This definition is analogous to the one from this paper for 2-representations of $\h$.
Suppose $\K$ is a integrable 2-representation of $\h$. In [@CL2] we showed that $\Kom(\K)$ can be given the structure of a 2-representation of $\g$. This categorifies the Frenkel-Kac-Segal vertex operator construction. Roughly, we did the following.
- We defined $\1_\l \mapsto \1_n$ if $\l = w \cdot \Lambda_0 - n \delta$ for some Weyl element $w$ and $\1_\l \mapsto 0$ otherwise. Here $\l$ is a weight, $\Lambda_0$ is the fundamental weight corresponding to the affine node and $\delta$ is the imaginary root.
- We mapped $\E_i,\F_i$ as follows $$\begin{aligned}
\label{eq:vertex1}
\E_i \1_\l &\mapsto \left[ \dots \rightarrow \P_i^{(l)} \Q_i^{(1^{k+l})} \la -l \ra \rightarrow \dots \rightarrow \P_i \Q_i^{(1^{k+1})} \la -1 \ra \rightarrow \Q_i^{(1^{k})} \right] \\
\label{eq:vertex2}
\1_\l \F_i &\mapsto \left[ \P_i^{(1^{k})} \rightarrow \P_i^{(1^{k+1})} \Q_i \la 1 \ra \rightarrow \dots \rightarrow \P_i^{(1^{k+l})} \Q_i^{(l)} \la l \ra \rightarrow \dots \right]\end{aligned}$$ if $k := \la \l, \alpha_i \ra + 1 \ge 0$ (and to similar complexes if $k < 0$).
We argued that this action extends to give a 2-representation of $\g$. This means that there also exist complexes for divided power $\E_i^{(r)} \1_\l$ and $\1_\l \F_i^{(r)}$. We gave the following conjectural explicit description of these complexes (proven when $r=1,2$). Although we did not identify these complexes explicitly (apart from the cases $r=1,2$) we did conjecture that in general, if $k := \la \l, \alpha_i \ra + r \ge 0$, then $$\label{eq:E}
\E_i^{(r)} \1_\l :=
\left[ \dots \rightarrow \bigoplus_{w(\mu) \le r, |\mu|=l} \P_i^{(\mu^t)} \Q_i^{(r^k, \mu)} \la -l \ra \rightarrow \dots \rightarrow \P_i \Q_i^{(r^k,1)} \la -1 \ra \rightarrow \Q_i^{(r^k)} \right] \la - \binom{r}{2} \ra [\binom{r}{2}].$$ for certain explicit differentials. Here the direct sum is over all partitions $\mu$ of size $|\mu|$ which fit in a box of width $r$ ($w(\mu)$ denotes the width of $\mu$). We also conjectured similar formulas if $k \le 0$ and likewise for $\1_\l \F_i^{(r)}$.
### Associated braid group actions
Given any 2-representation of $\g$, we considered in [@CK2] the Rickard complex defined by $$\label{eq:T}
\sT_i \1_\l := \left[ \dots \rightarrow \E_i^{(-\la \l, \alpha_i \ra + s)} \F_i^{(s)} \la -s \ra \1_\l \rightarrow \dots \rightarrow \E_i^{(- \la \l,\alpha_i \ra + 1)} \F_i \la -1 \ra \1_\l \rightarrow \E_i^{(- \la \l,\alpha_i \ra)} \1_\l \right]$$ if $\la \l,\alpha_i \ra \le 0$ (and similarly if $\la \l,\alpha_i \ra \ge 0$). We then showed [@CK2 Theorem 2.10] that these complexes satisfy the braid relations in $\Br(D)$. Notice that the domain and range of $\sT_i \1_\l$ are given by $$\sT_i \1_\l: \l \rightarrow s_i \cdot \l \ \ \text{ where } \ \ s_i \cdot \l = \l - \la \l, \alpha_i \ra \alpha_i.$$ If $\1_\l \mapsto \1_n$ under the map from [@CL2] then it is easy to check that $\1_{s_i \cdot \l} \mapsto \1_n$. Thus, if we compose the complexes for $\sT_i$ from (\[eq:T\]) with those for $\E$’s and $\F$’s (from (\[eq:vertex1\]) and (\[eq:vertex2\])) then we obtain complexes $\sT_i \in \Kom(\K)$ with domain and range $n$.
[**Example 1.**]{} Suppose $\1_\l \mapsto \1_n$ under the map from [@CL2] with $\la \l, \alpha_i \ra = -n$. Then, the complex from (\[eq:E\]) with $k=0$ and $r=n$ gives the following expression for $\E_i^{(n)} \1_\l$ $$\label{eq:E'}
\left[ \dots \rightarrow \bigoplus_{w(\mu) \le n, |\mu|=l} \P_i^{(\mu^t)} \Q_i^{(\mu)} \la -l \ra \1_n \rightarrow \dots \rightarrow \P_i \Q_i \la -1 \ra \1_n \rightarrow \1_n \right] \la - \binom{n}{2} \ra [\binom{n}{2}].$$ Notice that the terms in (\[eq:E’\]) are zero if $|\mu| > n$ so the extra condition that $w(\mu) \le n$ is not necessary. Subsequently, $\E_i^{(n)} \1_\l$ is the same as our complex $\T_i \1_n$. On the other hand, it is not difficult to check that in this case $\F_i^{(s)} \1_\l = 0$ for any $s > 0$. Thus, the expression in (\[eq:T\]) simplifies to give $\sT_i \1_\l = \E_i^{(n)} \1_\l$. Thus, $\sT_i \1_\l = \T_i \1_n$ and, using [@CK2], we recover the braiding of the $\T_i \1_n$ (Theorem \[thm:main1\]).
[**Example 2.**]{} Suppose $\la \l, \alpha_i \ra = 1$ and that, under the map in [@CL2], $\1_\l \mapsto \1_2$. Then $$\sT_i \1_\l = \left[ \F_i^{(2)} \E_i \la -1 \ra \1_\l \rightarrow \F_i \1_\l \right]: \l \rightarrow s_i \cdot \l.$$ Using the definitions in [@CL2], one can check that $$\begin{aligned}
\F_i \1_\l &\mapsto \left[ \1_2 \rightarrow \P_i \Q_i \la 1 \ra \1_2 \rightarrow \P_i^{(1^2)} \Q_i^{(2)} \la 2 \ra \1_2 \right] \\
\F_i^{(2)} \1_{\l + \alpha_i} &\mapsto \P_i^{(2)} \la 1 \ra [-1] \1_0 \ \ \text{ and } \ \ \E_i \1_\l \mapsto \Q_i^{(1^2)} \1_2.\end{aligned}$$ Combining this together gives that $$\P_i^{(1^2)} \Q_i^{(2)} [-1] \1_2 \rightarrow \left[ \1_2 \rightarrow \P_i \Q_i \la 1 \ra \1_2 \rightarrow \P_i^{(2)} \Q_i^{(1^2)} \la 2 \ra \1_2 \right].$$ This collapses to give a complex of the form $$\label{eq:eg2}
\left[ \P_i^{(1^2)} \Q_i^{(2)} \1_2 \oplus \1_2 \rightarrow \P_i \Q_i \la 1 \ra \1_2 \rightarrow \P_i^{(2)} \Q_i^{(1^2)} \la 2 \ra \1_2 \right].$$ Notice that this complex, is [*not*]{} of the form of $\T_i \1_2$.
### In conclusion
Using [@CK2], and assuming the conjectural expressions for $\E_i^{(r)}$ and $\F_i^{(r)}$ from [@CL2] (like the one in (\[eq:E\])) we recover the main result in this paper. On the other hand, as Example 2 above illustrates, the full categorical action from [@CL2] and the braid group action it induces via [@CK2] gives us a larger collection of complexes in $\Kom(\H)$ which satisfy the braid relations ((\[eq:eg2\]) is an example of one such complex). It would be interesting to explicitly identify all these complexes in $\Kom(\H)$ directly.
Proof of Proposition \[prop:welldefined\]
=========================================
What one needs to check is that the image of any two equivalent 2-morphisms ([*i.e.*]{} related by some 2-relation) under any $\s_i^{\pm 1}$ are identical.
There are many 2-relations so we will not check all of them in this paper. We illustrate by checking one of the most difficult relations, namely $$\s_i(T_{jj} 1_j) \circ \s_i(1_j T_{jj}) \circ \s_i (T_{jj} 1_j) = \s_i(1_j T_{jj}) \circ \s_i (T_{jj} 1_j) \circ \s_i(1_j T_{jj})$$ whenever $\la i,j \ra = -1$.
By direct computation, $\s_i(T_{jj} 1_j) \circ \s_i(1_j T_{jj}) \circ \s_i(T_{jj} 1_j)$ is the following map of complexes: $$\xymatrix{
\P_i \P_i \P_i \la -3 \ra \ar[r]^{a} \ar[dd]^{A} & {\begin{matrix} \P_i \P_i \P_j \la -2 \ra \\ \oplus \\ \P_i \P_j \P_i \la -2 \ra \\ \oplus \\ \P_j \P_i \P_i \la -2 \ra \end{matrix}} \ar[r]^{b} \ar[dd]^{B} & {\begin{matrix} \P_i \P_j \P_j \la -1 \ra \\ \oplus \\ \P_j \P_i \P_j \la -1 \ra \\ \oplus \\ \P_j \P_j \P_i \la -1 \ra \end{matrix}} \ar[r]^{c} \ar[dd]^{C} & \P_j \P_j \P_j \ar[dd]^{D} \\
\\
\P_i \P_i \P_i \la -3 \ra \ar[r]^{d} & {\begin{matrix} \P_i \P_i \P_j \la -2 \ra \\ \oplus \\ \P_i \P_j \P_i \la -2 \ra \\ \oplus \\ \P_j \P_i \P_i \la -2 \ra \end{matrix}} \ar[r]^{e} & {\begin{matrix} \P_i \P_j \P_j \la -1 \ra \\ \oplus \\ \P_j \P_i \P_j \la -1 \ra
\\ \oplus \\ \P_j \P_j \P_i \la -1 \ra \end{matrix}} \ar[r]^{f} & \P_j \P_j \P_j
}$$ where $$a =
\begin{pmatrix}
1_i 1_i X_i^j \\
1_i X_i^j 1_i \\
X_i^j 1_i 1_i \\
\end{pmatrix}
\hspace{.2in}
b =
\begin{pmatrix}
1_i X_i^j 1_j & 1_i 1_j X_i^j & 0 \\
X_i^j & 0 & 1_j 1_i X_i^j \\
0 & X_i^j 1_j 1_i & 1_j X_i^j 1_i
\end{pmatrix}
\hspace{.2in}
c =
\begin{pmatrix}
X_i^j 1_j 1_j & 1_j X_i^j 1_j & 1_j 1_j X_i^j
\end{pmatrix}$$ $$d =
\begin{pmatrix}
1_i X_i^j 1_i \\
1_i 1_i X_i^j \\
X_i^j 1_i 1_i \\
\end{pmatrix}
\hspace{.2in}
e =
\begin{pmatrix}
1_i 1_j X_i^j & 1_i X_i^j 1_j & 0 \\
X_i^j & 0 & 1_j X_i^j 1_i \\
0 & X_i^j 1_i 1_j & 1_j 1_i X_i^j
\end{pmatrix}
\hspace{.2in}
f =
\begin{pmatrix}
X_i^j 1_j 1_j & 1_j 1_j X_i^j & 1_j X_i^j 1_j
\end{pmatrix}$$ $$A=
\begin{pmatrix}
(T_{ii} 1_i) \circ (1_i T_{ii}) \circ (T_{ii} 1_i)
\end{pmatrix}
\hspace{3.4in}$$ $$B =
\begin{pmatrix}
0 & (T_{ji} 1_i) \circ (1_j T_{ii}) \circ (T_{ij} 1_i) & 0 \\
0 & 0 & (T_{ii} 1_j) \circ (1_i T_{ji}) \circ (T_{ji} 1_i) \\
(T_{ij} 1_i) \circ (1_i T_{ij}) \circ (T_{ii} 1_j) & 0 & 0
\end{pmatrix}$$ $$C =
\begin{pmatrix}
0 & 0 & (T_{ji} 1_j) \circ (1_j T_{ji}) \circ (T_{jj} 1_i) \\
(T_{jj} 1_i) \circ (1_j T_{ij}) \circ (T_{ij} 1_j) & 0 & 0 \\
0 & (T_{ij} 1_j) \circ (1_i T_{jj}) \circ (T_{ji} 1_j) & 0
\end{pmatrix}$$ $$D=
\begin{pmatrix}
(T_{jj} 1_j) \circ (1_j T_{jj}) \circ (T_{jj} 1_j)
\end{pmatrix}.
\hspace{3.3in}$$ Similarly, $ \s_i(1_j T_{jj}) \circ \s_i(T_{jj} 1_j) \circ \s_i(1_j T_{jj}) $ is a map of complexes: $$\xymatrix{
\P_i \P_i \P_i \la -3 \ra \ar[r]^{a} \ar[dd]^{A'} & {\begin{matrix} \P_i \P_i \P_j \la -2 \ra \\ \oplus \\ \P_i \P_j \P_i \la -2 \ra \\ \oplus \\ \P_j \P_i \P_i \la -2 \ra \end{matrix}} \ar[r]^{b} \ar[dd]^{B'} & {\begin{matrix} \P_i \P_j \P_j \la -1 \ra \\ \oplus \\ \P_j \P_i \P_j \la -1 \ra \\ \oplus \\ \P_j \P_j \P_i \la -1 \ra \end{matrix}} \ar[r]^{c} \ar[dd]^{C'} & \P_j \P_j \P_j \ar[dd]^{D'} \\
\\
\P_i \P_i \P_i \la -3 \ra \ar[r]^{d} & {\begin{matrix} \P_i \P_i \P_j \la -2 \ra \\ \oplus \\ \P_i \P_j \P_i \la -2 \ra \\ \oplus \\ \P_j \P_i \P_i \la -2 \ra \end{matrix}} \ar[r]^{e} & {\begin{matrix} \P_i \P_j \P_j \la -1 \ra \\ \oplus \\ \P_j \P_i \P_j \la -1 \ra
\\ \oplus \\ \P_j \P_j \P_i \la -1 \ra \end{matrix}} \ar[r]^{f} & \P_j \P_j \P_j
}$$ where $$A'=
\begin{pmatrix}
(1_i T_{ii}) \circ (T_{ii} 1_i) \circ (1_i T_{ii})
\end{pmatrix}
\hspace{3.4in}$$ $$B' =
\begin{pmatrix}
0 & (1_i T_{ij}) \circ (T_{ii} 1_j) \circ (1_i T_{ji}) & 0 \\
0 & 0 & (1_i T_{ji}) \circ (T_{ji} 1_i) \circ (1_j T_{ii}) \\
(1_j T_{ii}) \circ (T_{ij} 1_i) \circ (1_i T_{ij}) & 0 & 0
\end{pmatrix}$$ $$C' =
\begin{pmatrix}
0 & 0 & (1_i T_{jj}) \circ (T_{ji} 1_j) \circ (1_j T_{ji}) \\
(1_j T_{ij}) \circ (T_{ij} 1_j) \circ (1_i T_{jj}) & 0 & 0 \\
0 & (1_j T_{ji}) \circ (T_{jj} 1_i) \circ (1_j T_{ij}) & 0
\end{pmatrix}$$ $$D'=
\begin{pmatrix}
(1_j T_{jj}) \circ (T_{jj} 1_j) \circ (1_j T_{jj})
\end{pmatrix}.
\hspace{3.3in}$$ Equalities of matrices $ A=A', B=B', C=C', D=D' $ follow from the three strand relation in the category.
Proof of Theorem \[thm:main2\]
==============================
To prove Theorem \[thm:main2\] one needs to fix isomorphisms $\s_i \s_i^{-1} M \rightarrow M$, $\s_i^{-1} \s_i^{} M \rightarrow M$, $ \s_i \s_{i+1} \s_i M \rightarrow \s_{i+1} \s_i \s_{i+1} M $ on all generating 1-morphisms of $\Kom(\H')$ and then show that these isomorphisms are natural with respect to all generating 2-morphisms.
### Reidemeister 2 relations on 1-morphisms
Recall the homotopy equivalences $$\nu_{\P_i} : \s_i(\P_i) \xrightarrow{\sim} \P_i \la -2 \ra [1] \ \ \text{ and } \ \ \zeta_{\P_i}: \s_i^{-1}(\P_i) \xrightarrow{\sim} \P_i \la 2 \ra [-1].$$ We can use these to define isomorphisms $$\s_i^{-1} \s_i(\P_i) \xrightarrow{\s_i^{-1}(\nu_{\P_i})} \s_i^{-1}(\P_i \la -2 \ra [1]) \xrightarrow{\zeta_{\P_i}} \P_i \ \ \text{ and } \ \ \s_i \s_i^{-1}(\P_i) \xrightarrow{\s_i(\zeta_{\P_i})} \s_i(\P_i \la 2 \ra [-1]) \xrightarrow{\nu_{\P_i}} \P_i.$$
On the other hand, if $\la i,j \ra = -1$ then we use the following maps $$\xymatrix{
\s_i^{-1} \s_i (\P_j) \ar[d] & = & \P_i \la -1 \ra \ar[d] \ar[rr]^{\hspace{-1cm} (X_i^j \ \ 1 \ \ X_i^i)} & & \P_j \oplus \P_i \la -1 \ra \oplus \P_i \la 1 \ra \ar[d]^{(1 \ \ -X_i^j \ \ 0)} \ar[rr]^{\hspace{1cm} (X_j^i \ \ 0 \ \ -\epsilon_{ij})} & & \P_i \la 1 \ra \ar[d] \\
\P_j \ar[d] & = & 0 \ar[d] \ar[rr] && \P_j \ar[rr] \ar[d]^{(1 \ \ 0 \ \ \ep_{ij} X_j^i)} && 0 \ar[d] \\
\s_i^{-1} \s_i (\P_j) & = & \P_i \la -1 \ra \ar[rr]^{\hspace{-1cm} (X_i^j \ \ 1 \ \ X_i^i)} & & \P_j \oplus \P_i \la -1 \ra \oplus \P_i \la 1 \ra \ar[rr]^{\hspace{1cm} (X_j^i \ \ 0 \ \ -\epsilon_{ij})} & & \P_i \la 1 \ra
}$$ where the rightmost column is in cohomological degree one. The vertical composition is $$\xymatrix{
\P_i \la -1 \ra \ar[rr]^{} \ar[d]^{0} & & \P_j \oplus \P_i \la -1 \ra \oplus \P_i \la 1 \ra \ar[rr]^{} \ar[d]^{\gamma_0} \ar[lld]_{D_0} & & \P_i \la 1 \ra \ar[d]^{0} \ar[lld]_{D_1}\\
\P_i \la -1 \ra \ar[rr]^{} & & \P_j \oplus \P_i \la -1 \ra \oplus \P_i \la 1 \ra \ar[rr]^{} & & \P_i \la 1 \ra
}$$ where $\gamma_0 = \begin{pmatrix} 1 & -X_i^j & 0 \\ 0 & 0 & 0 \\ \epsilon_{ij} X_i^j & -X_i^i & 0 \end{pmatrix}$. Here $D_0 = (0 \ -1 \ 0)$ and $D_1 = (0 \ 0 \ \epsilon_{ij})$ give a homotopy between this composition and the identity map.
### Reidemeister 3 relations on 1-morphisms
\[braidp\_i\] For $ \la i, j \ra = -1$ there is an isomorphism $$\gamma_{\P_i} \colon \s_i \s_{j} \s_i(\P_i) \rightarrow \s_{j} \s_i \s_{j}(\P_i).$$
One checks that the map of complexes $ \beta \colon \s_i \s_{j}(\P_i) \rightarrow \P_j \la -1 \ra $ given by $$\xymatrix{
\P_i \la -2 \ra \ar[rr]^{\begin{pmatrix}
-\epsilon_{ij} \\
0 \\
X_i^j
\end{pmatrix}\hspace{.4in}} \ar[d] & & \P_i \la -2 \ra \oplus \P_i \la 1 \ra \oplus \P_j \la -1 \ra \ar[rr]^{\hspace{.7in} \begin{pmatrix}
X_i^i & 1 & X_j^i
\end{pmatrix}} \ar[d]^{\begin{pmatrix}
\epsilon_{ij} X_i^j & 0 & 1
\end{pmatrix}} & & \P_i \ar[d] \\
0 \ar[rr] & & \P_j \ar[rr] & & 0
}$$ is a homotopy equivalence with inverse map $ \bar{\beta} $. Then $ \beta \circ \s_i \s_j(\nu_{\P_i}) \colon \s_i \s_j \s_i(\P_i) \rightarrow \P_j \la -3 \ra [2] $ is a homotopy equivalence.
Similarly $ \nu_{\P_{j}} \circ \s_{j}(\beta_{\P_{j}}) \colon \s_{j} \s_i \s_{j}(\P_i) \rightarrow \P_{j} \la -3 \ra [-2] $ is a homotopy equivalence. Finally we define $$\gamma_{\P_i} = -\s_{i+1}(\bar{\beta}_{\P_{i+1}}) \circ \bar{\nu}_{\P_{i+1}} \circ \beta_{\P_{i+1}} \circ \s_i \s_{i+1} (\nu_{\P_i}).$$
For $ \la i, j \ra = -1$ there is an isomorphism $$\gamma_{\P_{j}} \colon \s_i \s_{j} \s_i(\P_{j}) \rightarrow \s_{j} \s_i \s_{j}(\P_{j}).$$
The proof of this is similar to that of Proposition \[braidp\_i\].
Next, if $\la i,j \ra = -1 = \la j,k \ra$ and $\la i,k \ra =0$ one needs to write down a homotopy equivalence $\s_i \s_j \s_i(\P_k) \xrightarrow{\sim} \s_j \s_i \s_j(\P_k)$. A direct calculation shows that $$\begin{aligned}
\s_i \s_j \s_i (\P_k) &= \left[ \P_i \la -2 \ra \xrightarrow{X_i^j} \P_j \la -1 \ra \xrightarrow{X_j^k} \P_k \right] \\
\s_j \s_i \s_j (\P_k) &= \left[ \P_j \la -3 \ra \xrightarrow{A} \P_j \la -3 \ra \oplus \P_j \la -1 \ra \oplus \P_i \la -2 \ra \xrightarrow{B} \P_j \la -1 \ra \oplus \P_j \la -1 \ra \xrightarrow{C} \P_k \right]\end{aligned}$$ where $$A = \begin{pmatrix} 1 & 0 & X_j^i \end{pmatrix}
\hspace{.2in}
B = \begin{pmatrix} 0 & -1 & 0 \\ X_j^j & 1 & X_i^j \end{pmatrix}
\hspace{.2in}
C = \begin{pmatrix} X_j^k & X_j^k \end{pmatrix}.$$ It is not difficult to check that the following maps are homotopy equivalences: $$\xymatrix{
\s_i \s_j \s_i (\P_k) \ar[d]^{\phi} & = & \P_j \la -3 \ra \ar[r]^-{A} \ar[d] & \P_j \la -3 \ra \oplus \P_j \la -1 \ra \oplus \P_i \la -2 \ra \ar[r]^-{B} \ar[d]^{(-X_j^i \ \ 0 \ \ 1)} & \P_j \la -1 \ra \oplus \P_j \la -1 \ra \ar[r]^-{C} \ar[d]^{(1 \ \ 1)} & \P_k \ar[d]^{1} \\
\s_j \s_i \s_j(\P_k) \ar[d]^{\bar{\phi}} & = & 0 \ar[r] \ar[d] & \P_i \la -2 \ra \ar[r]^-{X_i^j} \ar[d]^{(0 \ \ -X_i^j \ \ 1)} & \P_j \la -1 \ra \ar[r]^-{X_j^k} \ar[d]^{(1 \ \ 0)} & \P_k \ar[d]^{1} \\
\s_i \s_j \s_i (\P_k) & = & \P_j \la -3 \ra \ar[r]^-{A} & \P_j \la -3 \ra \oplus \P_j \la -1 \ra \oplus \P_i \la -2 \ra \ar[r]^-{B} & \P_j \la -1 \ra \oplus \P_j \la -1 \ra \ar[r]^-{C} & \P_k
}$$
Finally, if $i,j,k$ form a triangle then there is a more complicated homotopy equivalence $\s_i \s_j \s_i (\P_k) \xrightarrow{\sim} \s_j \s_i \s_j (\P_k)$ which we omit (the interested reader can contact the authors for more details).
### Reidemeister 2 relations on 2-morphisms
First, assuming $\la j,k \ra = -1$, one needs to check that the following diagrams commute $$\xymatrix{
\s_i^{-1} \s_i (\P_k \la -1 \ra) \ar[rr]^{ \s_{i}^{-1} \s_i (X_k^j)} \ar[d] & & \s_i^{-1} \s_i (\P_j) \ar[d] \\
\P_k \la -1 \ra \ar[rr]^{X_k^j} & & \P_j
}
\hspace{1in}
\xymatrix{
\s_i^{} \s_i^{-1} (\P_k \la -1 \ra) \ar[rr]^{ \s_{i}^{} \s_i^{-1} (X_k^j)} \ar[d] & & \s_i^{} \s_i^{-1} (\P_j) \ar[d] \\
\P_k \la -1 \ra \ar[rr]^{X_k^j} & & \P_j.
}$$ where the vertical maps are those from the previous section.
Next, for any $i,j,k$, one needs to check the commutativity of the following squares $$\xymatrix{
\s_i^{-1} \s_i (\P_j \P_k) \ar[rr]^{\s_i^{-1} \s_i (T_{jk})} \ar[d] && \s_i^{-1} \s_i (\P_j \P_k) \ar[d] \\
\P_j \P_k \ar[rr]^{T_{jk}} && \P_j \P_k
}
\hspace{1in}
\xymatrix{
\s_i^{} \s_i^{-1} (\P_j \P_k) \ar[rr]^{\s_i^{} \s_i^{-1} (T_{jk})} \ar[d] && \s_i^{} \s_i^{-1} (\P_j \P_k) \ar[d] \\
\P_j \P_k \ar[rr]^{T_{jk}} && \P_j \P_k. }$$
### Reidemeister 3 relations on 2-morphisms
First, assuming $ \la i,j \ra =-1$ and $\la l,k \ra = -1$, one needs to check that the following diagram commutes $$\xymatrix{
\s_i \s_{j} \s_i (\P_{k} \la -1 \ra) \ar[rrr]^{\s_i \s_{i+1} \s_i (X_{k}^{l})} \ar[d]& & & \s_i \s_{j} \s_i (\P_{l}) \ar[d]\\
\s_{j} \s_{i} \s_{j} (\P_{k} \la -1 \ra) \ar[rrr]^{\s_{j} \s_{i} \s_{j} (X_{k}^{l})} & & & \s_{j} \s_{i} \s_{j} (\P_{l})
}.$$
Next, for any $i,j,k,l$ with $ \la i,j \ra =-1$, one needs to show that the following diagram commutes $$\xymatrix{
\s_i \s_{j} \s_i (\P_{l} \P_k) \ar[rrr]^{\s_i \s_{j} \s_i (T_{lk})} \ar[d]^{\gamma_{\P_l} \gamma_{\P_k}} & & & \s_i \s_{j} \s_i (\P_{k} \P_l) \ar[d]^{\gamma_{\P_k} \gamma_{\P_l}}\\
\s_{j} \s_{i} \s_{j} (\P_{l} \P_k) \ar[rrr]^{\s_{j} \s_{i} \s_{j} (T_{lk})} & & & \s_{j} \s_{i} \s_{j} (\P_{k} \P_l)
}.$$
Checking that all these diagrams commute breaks up into many cases. Each case, though not difficult, is a bit tedious (the interested reader can contact the authors for more details about these calculations).
[E-G-S]{}
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- '6Dtabflow.bib'
date: January 2016
title: 6D RG Flows and Nilpotent Hierarchies
---
Introduction \[sec:INTRO\]
==========================
Renormalization group flows constitute a foundational element in the study of quantum field theory. As fixed points of these flows, conformal field theories are especially important in a range of physical phenomena. One of the surprises from string theory is that suitable decoupling limits lead to the construction of conformal fixed points in more than four spacetime dimensions [@Witten:1995zh].
Though the full list of conformal field theories is still unknown, there has recently been significant progress in classifying six-dimensional superconformal field theories (SCFTs). A top down classification of 6D SCFTs via compactifications of F-theory has been completed in [@Heckman:2013pva; @DelZotto:2014hpa; @Heckman:2014qba; @DelZotto:2014fia; @Heckman:2015bfa] (see also [@Bhardwaj:2015xxa] and [@Bhardwaj:2015oru] as well as the holographic classification results of reference [@Apruzzi:2013yva]).[^1] An important element in this work is that in contrast to lower-dimensional systems, all of these SCFTs have a simple universal structure given (on its partial tensor branch) by a generalized quiver gauge theory consisting of a single spine of quiver nodes joined by links which in [@DelZotto:2014hpa] were dubbed “conformal matter.” There can also be a small amount of decoration by such links on the ends of this generalized quiver.
With such a list in place, the time is ripe to extract more detailed properties of these theories. Though the absence of a Lagrangian construction is an obstruction, it is nevertheless possible to extract some precision data such as the anomaly polynomial [@Harvey:1998bx; @Ohmori:2014pca; @Ohmori:2014kda], the scaling dimensions of certain protected operators [@Heckman:2014qba] and the structure of the partition vector and its relation to the spectrum of extended defects [@Tachikawa:2013hya; @DelZotto:2015isa].
It is also natural to expect that there is an overarching structure governing possible RG flows between conformal fixed points. In recent work [@Heckman:2015ola], the geometry of possible deformations of the associated Calabi–Yau geometry of an F-theory compactification has been used to characterize possible flows between theories, and has even been used to give a proof by brute force(i.e. sweeping over a large list of possible flows) of $a$- and $c$-theorems in six dimensions [@Heckman:2015axa] (see also [@Cordova:2015fha] and [@Beccaria:2015ypa; @Cremonesi:2015bld]). In this geometric picture, there are two general classes of flows parameterized by vevs for operators of the theory. On the tensor branch, we consider vevs for the real scalars of 6D tensor multiplets, which in the geometry translate to volumes of $\mathbb{P}^{1}$’s in the base of an F-theory model. On the Higgs branch, we consider vevs for operators which break the $SU(2)$ R-symmetry of the SCFT. Geometrically, these correspond to complex structure deformations. There are also mixed branches. Even so, a global picture of how to understand the network of flows between theories remains an outstanding open question.
Motivated by the fact that all 6D SCFTs are essentially just generalized quivers, our aim in this note will be to study possible RG flows for one such class of examples in which the decoration on the left and right of a generalized quiver is minimal. These are theories which in M-theory are realized by a stack of $k$ M5-branes probing the transverse geometry $\mathbb{R}_{\bot
}\times\mathbb{C}^{2}/\Gamma_{ADE}$, i.e. the product of the real line with an ADE singularity. In F-theory they are realized by a single linear chain of $-2$ curves in the base which are wrapped by seven-branes with gauge group of corresponding ADE type, in which there is a non-compact ADE seven-brane on the very left and one on the very right as well. In M-theory, we reach the SCFT point by making all the M5-branes coincident on the $\mathbb{R}_{\bot}$ factor (while still probing the orbifold singularity), while in F-theory this is obtained by collapsing all of the $-2$ curves to zero volume.
One of the interesting features of these models is that on the partial tensor branch, i.e. where we separate all M5-branes along the transverse real line, and in F-theory where we resolve all $-2$ curves, we can recognize that there are additional degrees of freedom localized along defects of a higher-dimensional bulk theory. Indeed, from the F-theory perspective, the degenerations of the elliptic fibration at these points needs to be accompanied by additional blowups in the base, leading to conformal matter. The reason for the suggestive terminology is twofold. First, the actual structure of the geometries constructed from M- and F-theory has the appearance of a generalized quiver. Second, and perhaps more importantly, there is a precise notion in the F-theory description of activating complex structure deformations at the places where conformal matter is localized. For example, the breaking pattern for a conformal matter system with $E_{8}\times E_{8}$ global symmetry to a system with only $E_{7}\times
E_{7}$ global symmetry is given by:$$y^{2}=x^{3}+\alpha u^{3}v^{3}x+u^{5}v^{5}.$$ Such deformations trigger a decrease in the total number of tensor multiplets, and also break the UV R-symmetry, with another emerging in the IR.
Since the structure of tensor branch flows is immediately captured by the geometry of the F-theory model, i.e. Kähler resolutions of the base, we shall primarily focus on Higgs branch flows. Part of our aim will be to develop a general picture of how vevs for conformal matter generate RG flows.
Along these lines, we provide supporting evidence for this picture of conformal matter vevs, and use it as a way of characterizing the induced flows for 6D SCFTs. In more detail, we consider the class of theories called $$\label{eq:T}
\mathcal{T}(G, \mu_{L},\mu_{R}, k)$$ in reference [@DelZotto:2014hpa]. They are parameterized by a choice of ADE group $G$; by a pair of nilpotent elements $\mu_{L}$ and $\mu_{R}$ in the complexification $\mathfrak{g}_{\mathbb{C}}$ of the Lie algebra of $G$; and by a positive integer $k$. In the M-theory realization, $k$ is the number of M5-branes, and $\mu_L$, $\mu_R$ specify Nahm pole data of a 7D super Yang-Mills theory. In the F-theory description, the theories (\[eq:T\]) represent a chain of $-2$ curves with gauge group $G$ on each of them, with a T-brane[@Donagi:2003hh; @Cecotti:2010bp] (see also [@Donagi:2011jy; @Anderson:2013rka; @Heckman:2010qv; @Collinucci:2014qfa; @Collinucci:2014taa]) on each flavor curve at an end of the chain. The $\mu_L$ and $\mu_R$ appear as residues of a Higgs field for a Hitchin system on these flavor curves. These residues are in turn captured by operator vevs of the low energy effective field theory [@Anderson:2013rka; @Beasley:2008dc]. This provides the basic link between boundary data and the vevs of operators associated with conformal matter.
The first result of this paper is an explicit identification of the tensor branch for the theories $\mathcal{T}(G, \mu_{L},\mu_{R}, k)$ of line (\[eq:T\]). To reach this, we shall find it convenient to take $k$ in (\[eq:T\]) sufficiently large so that the effects of $\mu_{L}$ and $\mu_{R}$ decouple, so our aim will be to capture the effects of flows associated with just a single nilpotent element of the flavor symmetry algebra. For $G=SU(N)$ or $SO(2N)$, nilpotent elements can be parameterized by partitions (i.e. Young diagrams). For $G=E_n$, one cannot use partitions any more: their analogues are called *Bala–Carter labels* (for a review of B–C labels, see for example [@NILPbook Chap. 8] or [@Chacaltana:2012zy]).[^2]
Secondly, we will find that the well-known partial ordering on nilpotent elements also leads to a class of theories which can be connected by an RG flow:$$\label{eq:RG}
\mu < \nu\Rightarrow\text{RG\ Flow: }\mathcal{T}(\mu)\rightarrow
\mathcal{T}(\nu).$$
To provide further evidence in favor of our proposal, we also consider related theories where the left flavor symmetry is replaced by a non-simply laced algebra. We get to such theories by first doing a blowdown of some curves on the tensor branch for the $\mathcal{T}(G, \mu_{L},\mu_{R}, k)$ theories which are then followed by a further vev for conformal matter. In this case, the flavor symmetry does not need to be a simply laced ADE type algebra, but can also be a non-simply laced BCFG algebra. All of this is quite transparent on the F-theory side, and we again expect a parametrization of flows in terms of nilpotent hierarchies. We find that this is indeed the case, again providing highly non-trivial evidence for our proposal.
Another outcome from our analysis is that by phrasing everything in terms of *algebraic* data of the 6D SCFT flavor symmetry, we can also read off the unbroken flavor symmetry, i.e. those symmetry generators which commute with our choice of nilpotent element. This provides a rather direct way to determine the resulting IR flavor symmetry which is different from working with the associated F-theory geometry. Indeed, there are a few cases where we find that the geometric expectation from F-theory predicts a flavor symmetry which is a proper subalgebra of the flavor symmetry found through our field theoretic analysis. This is especially true in the case of *abelian* flavor symmetries. For more details on extracting the geometric contribution to the flavor symmetry, see e.g. [@Bertolini:2015bwa].
The rest of this paper is organized as follows. In section \[sec:CONFMATT\] we give a brief overview of some elements of conformal matter and how it arises in both M- and F-theory constructions. After this, in section \[sec:CLASSICAL\] we give a first class of examples based on flows involving 6D SCFTs where the flavor symmetry is a classical algebra. For the $SU$-type flavor symmetries, there is a beautiful realization of nilpotent elements in terms of partitions of a brane system. This is also largely true for the $SO$- and $Sp$-type algebras as well, though there are a few cases where this correspondence breaks down. When this occurs, we find that there is still a flow, but that some remnants of exceptional algebras creep into the description of the 6D SCFT because of the presence of conformal matter in the system. After this, we turn in section \[sec:EXCEPTIONAL\] to flows for theories with an exceptional flavor symmetry. In some cases there is a realization of these flows in terms of deformations of $(p,q)$ seven-branes, though in general, we will find it more fruitful to work in terms of the algebraic characterization of nilpotent orbits. Section \[sec:SHORT\] extends these examples to “short” generalized quivers where the breaking patterns of different flavor symmetries are correlated, and in section \[sec:FLAVOR\] we explain how this algebraic characterization of flavor breaking sometimes leads to different predictions for the flavor symmetries of a 6D SCFT compared with the geometric realization. In section \[sec:CONC\] we present our conclusions and potential directions for future research. Some additional material on the correspondence between nilpotent orbits for exceptional algebras and the corresponding F-theory SCFTs is provided in an Appendix.
Conformal Matter \[sec:CONFMATT\]
=================================
In this section we discuss some of the salient features of 6D conformal matter introduced in references [@DelZotto:2014hpa; @Heckman:2014qba], and the corresponding realization of these systems in both M- and F-theory. We also extend these considerations, explaining the sense in which conformal matter vevs provide a succinct way to describe brane recombination in non-perturbatively realized configurations of intersecting seven-branes.
Recall that to get a supersymmetric vacuum in 6D Minkowski space, we consider F-theory compactified on an elliptically fibered Calabi–Yau threefold. Since we are interested in a field theory limit, we always take the base of the elliptic model to be non-compact so that gravity is decoupled. Singularities of the elliptic fibration lead to divisors in the base, i.e. these are the loci where seven-branes are wrapped. When the curve is compact, this leads to a gauge symmetry in the low energy theory, and when the curve is non-compact, we instead have a flavor symmetry.
In F-theory, we parameterize the profile of the axio-dilaton using the Minimal Weierstrass model:$$y^{2}=x^{3}+fx+g,$$ where here, $f$ and $g$ are sections of bundles defined over the base. As explained in reference [@Heckman:2013pva], the non-Higgsable clusters of reference [@Morrison:2012np] can be used to construct the base for the tensor branch of all 6D SCFTs. The basic idea is that a collapsed $-1$ curve in isolation defines the E-string theory,that is, a theory with an $E_{8}$ flavor symmetry. By gauging an appropriate subalgebra of this flavor symmetry, we can start to produce larger bases, provided these additional compact curves are part of a small list of irreducible building blocks known as non-Higgsable clusters.
For the present work, we will not need to know much about the structure of these non-Higgsable clusters, so we refer the interested reader for example to [@Heckman:2013pva; @Morrison:2012np] for further details. The essential feature we require is that the self-intersection of a curve — or a configuration of curves — dictates the minimal gauge symmetry algebra supported over the curve. In some limited situations, additional seven-branes can be wrapped over some of these curves. Let us briefly recall the minimal gauge symmetry for the various building blocks of an F-theory base:$$\begin{aligned}
\text{single curve} & \text{: }\overset{\mathfrak{su}_{3}}{3}\text{,}\overset{\mathfrak{so}_{8}}{4}\text{,}\overset{\mathfrak{f}_{4}}{5}\text{,}\overset{\mathfrak{e}_{6}}{6}\text{,}\overset{\mathfrak{e}_{7}}{7}\text{,}\overset{\mathfrak{e}_{7}}{8}\text{,}\overset{\mathfrak{e}_{8}}{9}\text{,}\overset{\mathfrak{e}_{8}}{10}\text{,}\overset{\mathfrak{e}_{8}}{11}\text{,}\overset{\mathfrak{e}_{8}}{12}\\
\text{two curves} & \text{: }\overset{\mathfrak{su}_{2}}{2}\text{
}\overset{\mathfrak{g}_{2}}{3}\\
\text{three curves} & \text{: }2\text{ }\overset{\mathfrak{sp}_{1}}{2}\text{
}\overset{\mathfrak{g}_{2}}{3}\text{,}\overset{\mathfrak{su}_{2}}{2}\text{
}\overset{\mathfrak{so}_{7}}{3}\text{ }\overset{\mathfrak{su}_{2}}{2}.\end{aligned}$$ In some cases, there are also matter fields localized at various points of these curves. This occurs, for example, for a half hypermultiplet in the $\mathbf{56}$ of an $\mathfrak{e}_7$ gauge algebra supported on a $-7$ curve, and also occurs for a half hypermultiplet in the $\mathbf{2}$ of an $\mathfrak{su}_2$ gauge algebra supported on the $-2$ curve of the non-Higgsable cluster $2,3$. When the fiber type is minimal, we shall leave these matter fields implicit.
For non-minimal fiber enhancements, we indicate the corresponding matter fields which arise from a collision of the compact curve supporting a gauge algebra, and a non-compact component of the discriminant locus. We use the notation $[N_f = n]$ and $[N_s = n]$ to indicate $n$ hypermultiplets respectively in the fundamental representation or spinor representation (as can happen for the $\mathfrak{so}$-type gauge algebras). Note that when the representation is pseudo-real, $n$ can be a half-integer. We shall also use the notation $[G]$ to indicate a corresponding non-abelian flavor symmetry which is localized in the geometry.[^3]
One of the hallmarks of 6D SCFTs is the generalization of the conventional notion of hypermultiplets to “conformal matter.” An example of conformal matter comes from the geometry: $$y^{2}=x^{3}+u^{5}v^{5}.$$ At the intersection point, the order of vanishing for $f$ and $g$ becomes too singular, and blowups in the base are required. Let us list the minimal conformal matter for the collision of two ADE singularities which are the same [@Heckman:2013pva; @DelZotto:2014hpa; @Heckman:2014qba; @Morrison:2012np; @Bershadsky:1996nu]: $$\begin{aligned}
& \lbrack E_{8}]1,2,2,3,1,5,1,3,2,2,1[E_{8}]\\
& \lbrack E_{7}]1,2,3,2,1[E_{7}]\\
& \lbrack E_{6}]1,3,1[E_{6}]\\
& \lbrack SO_{2n}]\overset{\mathfrak{sp}_{n}}{1}[SO_{2n}]\\
& \lbrack SU_{n}]\cdot\lbrack SU_{n}].\end{aligned}$$ In the case of the collision of D-type symmetry algebras, there are also half hypermultiplets localized at the $\mathfrak{so}/\mathfrak{sp}$ intersections, and in the case of the A-type symmetry algebras, we have a conventional hypermultiplet in the bifundamental representation.
Given this conformal matter, we can then proceed to gauge these flavor symmetries to produce longer generalized quivers. Assuming that the flavor symmetries are identical, we can then label these theories according to the number of gauge groups $(k-1)$:$$[G_{0}]-G_{1}-....-G_{k-1}-[G_{k}],$$ in the obvious notation. Implicit in the above description is the charge of the tensor multiplets paired with each such gauge group factor. In F-theory, we write the partial tensor branch for this theory as: $$\lbrack G]\overset{\mathfrak{g}}{2}...\overset{\mathfrak{g}}{2}[ G ],$$ i.e. there are $(k-1)$ compact $-2$ curves, each with a singular fiber type giving a corresponding gauge group of ADE type, and on the left and the right we have a flavor symmetry supported on a non-compact curve. This is a partial tensor branch because at the collision of two components of the discriminant locus, the elliptic fiber ceases to be in Kodaira-Tate form. Indeed, such a collision point is where the conformal matter of the system is localized. Performing the minimal required number of blowups in the base to reach a model where all fibers remain in Kodaira-Tate form, we get the tensor branch of the associated conformal matter. Let us note that this class of theories also has a straightforward realization in M-theory via $k$ spacetime filling M5-branes probing the transverse geometry $\mathbb{R}_{\bot}\times\mathbb{C}^{2}/\Gamma_{G}$, where $\Gamma_{G}$ is a discrete ADE subgroup of $SU(2)$. In that context, the conformal matter is associated with localized edge modes which are trapped on the M5-brane.
Starting from such a configuration, we can also consider various boundary conditions for our configuration. In the context of intersecting seven-branes, these vacua are dictated by the Hitchin system associated with the $G_{0}$ and $G_{k}$ flavor branes. In particular, the collision point between $G_{0}$ and $G_{1}$ and that between $G_{k-1}$ and $G_{k}$ allows us to add an additional source term for the Higgs field at these punctures:$$\label{HitchinSource}
\overline{\partial}\Phi_{0}=\mu_{0,1}^{(L)}\text{ }\delta_{G_{0}\cap G_{1}}\text{ \ \ and \ \ }\overline{\partial}\Phi_{k}=\mu_{k-1,k}^{(R)}\text{
}\delta_{G_{k-1}\cap G_{k}},$$ where the $\delta$’s denote $(1,1)$-form delta functions with support at the collision of the two seven-branes. Here, $\mu_{0,1}^{(L)}$ and $\mu
_{k-1,k}^{(R)}$ are elements in the complexifications of the Lie algebras $\mathfrak{g}_{0}$ and $\mathfrak{g}_{k}$. The additional subscripts indicate that these elements are localized at the intersection point of two seven-branes. Moreover, the superscript serves to remind us that the source is really an element of the left or right symmetry algebra. In what follows, we shall often refer to these nilpotent elements as $\mu_{L}$ and $\mu_{R}$ in the obvious notation.
When the collision corresponds to ordinary localized matter, there is an interpretation in terms of the vevs of these matter fields [@Anderson:2013rka; @Beasley:2008dc]. More generalized source terms localized at a point correspond to vevs for conformal matter [@DelZotto:2014hpa; @Heckman:2014qba]. Let us also note that similar considerations apply for the boundary conditions of 7D super Yang Mills-theory, and so can also be phrased in M-theory as well.
In principle, there can also be more singular source terms on the righthand sides of line (\[HitchinSource\]). Such higher order singularities translate in turn into higher (i.e. degree 2 or more) order poles for the Higgs field at a given puncture. Far from the marked point, these singularities are subleading contributions to the boundary data of the intersecting seven-brane configuration so we expect that the effects of possible breaking patterns (as captured by the residue of the Higgs field simple pole) will suffice to parameterize possible RG flows.
Proceeding in this way, we see that for each collision point, we get two such source terms, which we can denote by $\left( \mu_{i,i+1}^{(L)},\mu
_{i,i+1}^{(R)}\right) $. On general grounds, we expect that the possible flows generated by conformal matter vevs are specified by a sequence of such pairs. Even so, these sequences are rather rigid, and in many cases simply stating $\mu_{0,1}^{(L)}$ and $\mu_{k-1,k}^{(R)}$ is typically enough to specify the flow.[^4] In this case, the invariant data is really given by the conjugacy class of the element in the flavor symmetry algebra, i.e. the orbit of the element inside the complexified Lie algebra [@DelZotto:2014hpa; @Gaiotto:2014lca].
In the context of theories with weakly coupled hypermultiplets, the fact that neighboring hypermultiplet vevs are coupled together through D-term constraints means that specifying one set of vevs will typically propagate out to additional vev constraints for matter on neighboring quiver nodes. Part of our aim in this note will be to determine what sorts of constraints are imposed by just the leftmost element of such a sequence. Given a sufficiently long generalized quiver gauge theory, the particular elements $\mu_{L}$ and $\mu_{R}$ can be chosen independently from one another [@DelZotto:2014hpa]. For this reason, we shall often reference the flow for a theory by only listing the leftmost quiver nodes:$$\lbrack G_{0}]-G_{1}-G_{2}-....$$
Now, although the M-theory realization is simplest in the case where the left flavor symmetry is of ADE-type, there is no issue in the F-theory realization with performing a partial tensor branch flow to reach more general flavor symmetries of BCFG-type. Indeed, to reach such configurations we can simply consider the corresponding non-compact seven-brane with this symmetry. From the perspective of the conformal field theory, we can reach these cases by starting with a theory with ADE flavor symmetry and flowing through a combination of Higgs and tensor branch flows. We shall therefore view these flavor symmetries on an equal footing with their simply laced cousins.
What then are the available choices for our boundary data $\mu\in
\mathfrak{g}_{\mathbb{C}}$? It is helpful at this point to recall that any element of a simple Lie algebra can be decomposed into a semi-simple and nilpotent part:$$\mu=\mu_{s}+\mu_{n},$$ so that for any representation of $\mathfrak{g}_{\mathbb{C}}$, the image of $\mu_{s}$ is a diagonalizable matrix, and $\mu_{n}$ is nilpotent. Geometrically, the contribution from the semi-simple elements is described by an unfolding which is directly visible in the complex geometry.
Less straightforward is the contribution from the nilpotent elements. Indeed, such T-brane contributions (so-named because they often look like upper triangular matrices) have a degenerate spectral equation, and as such do not appear directly in the deformations of the complex geometry. Rather, they appear in the limiting behavior of deformations associated with the Weil intermediate Jacobian of the Calabi–Yau threefold and its fibration over the complex structure moduli of the threefold [@Anderson:2013rka]. For flows between SCFTs, however, the key point is that all we really need to keep track of is the relevant hierarchies of scales induced by such flows. This is where the hyperkahler nature of the Higgs branch moduli space, and in particular its geometric avatar becomes quite helpful. We recall from [@Anderson:2013rka] that there is a direct match between the geometric realization of the Higgs branch moduli space of the seven-brane gauge theory in terms of the fibration of the Weil intermediate Jacobian of the Calabi-Yau threefold over the complex structure moduli. In this picture, the base of the Hitchin moduli space is captured by complex structure deformations. Provided we start at a smooth point of the geometric moduli space, we can interpret this in the associated Hitchin system as a diagonalizable Higgs field vev. As we approach singular points in the geometric moduli space, we can thus reach T-brane configurations. From the geometric perspective, however, this leads to the *same* endpoint for an RG flow, so we can either label the resulting endpoint of the flow by a nilpotent orbit of the flavor symmetry group or by an explicit F-theory geometry. Said differently, T-brane vacua do not lead to non-geometric phases for 6D SCFTs [@Heckman:2015bfa]. One of our aims will be to determine the *explicit* Calabi–Yau geometry for the F-theory SCFT associated with a given nilpotent orbit.[^5]
In general, given a nilpotent element $\mu \in \mathfrak{g}_{\mathbb{C}}$ a semisimple Lie algebra, the Jacobson–Morozov theorem tells us that there is a corresponding homomorphism $$\label{JacMor}
\rho:\mathfrak{sl}(2,\mathbb{C})\rightarrow\mathfrak{g}_{\mathbb{C}}$$ where the nilpotent element $\mu$ defines a raising operator in the image. The commutant subalgebra of ${\mathrm{Im}}(\rho)$ in $\mathfrak{g}_{\mathbb{C}}$ then tells us the unbroken flavor symmetry for this conformal matter vev. Though a microscopic characterization of conformal matter is still an outstanding open question, we can therefore expect that an analysis of symmetry breaking patterns can be deduced using this purely algebraic characterization. Indeed, more ambitiously, one might expect that once the analogue of F- and D-term constraints have been determined for conformal matter, we can use such conformal matter vevs as a pragmatic way to extend the characterization of bound states of perturbative branes in terms of such breaking patterns. From this perspective one can view the analysis of the present paper as determining these constraints for a particular class of operator vevs.
One of the things we would like to determine are properties of the IR fixed point associated with a given nilpotent orbit. For example, we would like to know both the characterization on the tensor branch, as well as possible flavor symmetries of the system. As explained in reference [@Heckman:2015ola], a flow from a UV SCFT to an IR SCFT in F-theory is given by some combination of Kähler and complex structure deformations. In all the flows, we will indeed be able to track the rank of the gauge groups, as well as the total number of tensor multiplets for each proposed IR theory. The decrease in the rank of gauge groups (on the tensor branch) translates to a less singular elliptic fiber, and is a strong indication of a complex structure deformation. So, to verify that we have indeed realized a flow, it will suffice to provide an explicit match between a given nilpotent orbit and a corresponding F-theory geometry where the tensor branch of the SCFT is given by a smaller number of tensor multiplets and a smaller gauge group.
With this in mind, our plan in much of this note will be to focus on the flows induced by nilpotent elements, i.e. T-branes, and to determine the endpoints of these flows. An added benefit of this analysis will be that by tracking the commutant subalgebra of the parent flavor symmetry, we will arrive at a proposal for the unbroken flavor symmetry for these theories.
E-String Flows {#ssec:estring}
--------------
As we have already mentioned, one of the important structural features of 6D SCFTs is that on their tensor branch, they are built up via a gluing construction using the E-string theory. As one might expect, the RG flows associated with this building block will therefore be important in our more general discussion of flows induced by conformal matter vevs.
With this in mind, let us recall a few additional features of this theory. Recall that in M-theory, the rank $k$ E-string theory is given by $k$ M5-branes probing an $E_{8}$ Hořava–Witten nine-brane [@Horava:1995qa; @Horava:1996ma]. In F-theory, it is realized on the tensor branch by a collection of curves in the base:$$\text{E-string theory base: }[E_{8}]\underset{k}{\text{ }\underbrace{1,2,...,2}}.$$ We reach the 6D SCFT by collapsing all of these curves to zero size. Now, provided $k<12$, we also get an SCFT by gauging this $E_{8}$ flavor symmetry. This gauge group is supported on a $-12$ curve:$$\overset{\mathfrak{e}_{8}}{(12)}\underset{k}{\text{ }\underbrace{1,2,...,2}}.$$ Starting from the UV SCFT, we reach various IR fixed points by moving onto a partial Higgs branch. These have the interpretation of moving onto the Higgs branch of the 6D SCFT. In the heterotic picture, we can picture this as moving onto various branches of the multi-instanton moduli space. For example, we can consider moving some of the small instantons to a different point of the $-12$ curve. This complex structure deformation amounts to partitioning the small instantons into separate chains (after moving onto the tensor branch for the corresponding fixed point): $$\overset{\mathfrak{e}_{8}}{(12)}\underset{k}{\text{ }\underbrace{1,2,...,2}}\rightarrow\text{ }\underset{l}{\underbrace{2,...,2,1}}\overset{\mathfrak{e}_{8}}{(12)}\underset{k-l}{\text{ }\underbrace{1,2,...,2}},\label{manure}$$ and as can be verified by an analysis of the corresponding anomaly polynomials, this does indeed define an RG flow [@Heckman:2015ola]. In equations, the deformation of the singular Weierstrass model for the UV theory to the less singular IR theory is given by:$$y^{2}=x^{3}+u^{5}v^{k}\rightarrow x^{3}+u^{5}(v-v_{1})^{l}(v-v_{2})^{k-l},$$ where $u=0$ denotes the $\mathfrak{e}_{8}$ locus, and $v=v_{1}$ and $v=v_{2}$ indicate the two marked points on $u=0$ where the small instantons touch this seven-brane.
We can also consider dissolving the instantons back into flux in the $\mathfrak{e}_{8}$ gauge theory. Geometrically, this is described by a sequence of blowdowns involving the $-1$ curve, which in turn increases the self-intersection of its neighboring curves by $+1$. Moving to a generic point of complex structure moduli then Higgses the $\mathfrak{e}_{8}$ down to a lower gauge symmetry (on the tensor branch). For example, after combining four small instantons we reach a $-8$ curve with an $\mathfrak{e}_{7}$ gauge symmetry:$$\overset{\mathfrak{e}_{8}}{(12)}\underset{k}{\text{ }\underbrace{1,2,...,2}}\rightarrow\text{ }\overset{\mathfrak{e}_{7}}{(8)}\underset{k-4}{\text{
}\underbrace{1,2,...,2}}.$$
An important feature of this class of deformations is that they are localized. What this means is that when we encounter larger SCFT structures, the same set of local deformations will naturally embed into more elaborate RG flows, and can be naturally extended to small instanton tails attached to other curves of self-intersection $-x$.
For example, in all cases other than the A-type symmetry algebras, we will encounter examples of a blowdown of a $-1$ curve, and a corresponding complex structure deformation. Additionally, in the case of the exceptional flavor symmetries, we will sometimes have to consider small instanton maneuvers of the type given in line (\[manure\]): $$...(x)\text{ }1,2,...\rightarrow...\underset{1}{(x)}\text{ }1,...,$$ that is, we move one of the small instantons to a new location on the $-x$ curve. Doing this may in turn require further deformations, since now the curve touching the $-1$ curve on the right is now closer to the $-x$ curve.
Flows for Classical Flavor Symmetries \[sec:CLASSICAL\]
=======================================================
As a warmup for our general analysis, in this section we consider the case of RG flows parameterized by nilpotent orbits of the classical algebras of $SU$-, $SO$- and $Sp$-type. Several aspects of nilpotent elements of the classical algebras can be found in [@NILPbook], and we shall also follow the discussion found in [@Chacaltana:2012zy].
There is a simple algebraic characterization of all nilpotent orbits of $\mathfrak{sl}(N,\mathbb{C})$. First, note that given an $N\times N$ nilpotent matrix we can then decompose it (in a suitable basis) as a collection of nilpotent Jordan blocks of size $\mu_{i}\times\mu_{i}$. Without loss of generality, we can organize these from largest to smallest, i.e. $\mu_{1}\geq...\geq\mu_{N}\geq0$, so we also define a partition, i.e. a choice of Young diagram. Note that we allow for the possibility that some $\mu_{i}$ are zero. When this occurs, it simply means that the partition has terminated earlier for some $l\leq N$. Similar considerations also hold for the other classical algebras with a few restrictions [@NILPbook]:$$\begin{aligned}
\mathfrak{so} & :\text{even multiplicity of each even }\mu_{i}\\
\mathfrak{sp} & :\text{even multiplicity of each odd }\mu_{i},\end{aligned}$$ where we note that if all $\mu_{i}$ are even for $\mathfrak{so}(2N,\mathbb{C})$, we get two nilpotent elements which are related to each other by a $\mathbb{Z}_2$ outer automorphism of the algebra.
There is also a natural ordering of these partitions. Given partitions $\mu=(\mu_{1},...,\mu_{N})$ and $\nu=(\nu_{1},...,\nu_{N})$, we say that:$$\label{eq:order}
\mu\geq\nu\text{ \ \ if and only if \ \ }\underset{i=1}{\overset{k}{\sum}}\mu_{i}\geq\underset{i=1}{\overset{k}{\sum}}\nu_{i}\text{ for all \ \ }1\leq
k\leq N.$$ There is a related ordering specified by taking the transpose of a given partition, i.e. by reflecting a Young diagram along a $45$ degree angle (see figure \[fig:transpose\] for an example). The ordering for the transposed partitions reverses the ordering of the original partitions, i.e. we have $\mu > \nu $ if and only if $\mu^T < \nu^T$. Finally, as a point of notation we shall often write a partition in the shorthand $(\mu_{1}^{d_{1}},...,\mu_{l}^{d_{l}})$ to indicate that $\mu_i$ has multiplicity $d_i$.
As an example, see the first column of figure \[fig:su4\] for an example of the ordering of partitions of $N=4$ according to (\[eq:order\]). The diagrams are reverse ordered so that for $\mu < \nu$ (or equivalently for $\mu^T > \nu^T$), the partition $\mu$ appears higher up than $\nu$. Intuitively, if one takes a Young diagram and moves a box at the end of a row to a lower row, one obtains a “smaller” Young diagram. In the example of figure \[fig:su4\], the ordering is total (i.e. any two diagrams can be compared). This ceases to be the case for larger $N$.
[transpose.pdf]{}
Given such a partition, we can also readily read off the unbroken symmetry, i.e. the generators which will commute with this choice of partition. As reviewed for example in [@Chacaltana:2012zy], for a partition $\mu$ where the entry $\mu_{i}$ has multiplicity $d_{i}$, these are:$$\begin{aligned}
\mathfrak{su} & :\mathfrak{g}_{\text{unbroken}}=\mathfrak{s}\left(
\underset{i}{\oplus}\mathfrak{u}(d_{i})\right) \label{suflavor}\\
\mathfrak{so} & :\mathfrak{g}_{\text{unbroken}}=\underset{i\text{
odd}}{\oplus}\mathfrak{so}\left( d_{i}\right) \oplus\underset{i\text{
even}}{\oplus}\mathfrak{sp}\left( d_{i}/2\right) \label{eq:soflavor} \\
\mathfrak{sp} & :\mathfrak{g}_{\text{unbroken}}=\underset{i\text{
even}}{\oplus}\mathfrak{so}\left( d_{i}\right) \oplus\underset{i\text{
odd}}{\oplus}\mathfrak{sp}\left( d_{i}/2\right) , \label{eq:spflavor}\end{aligned}$$ where in the above “$i$ odd” or “$i$ even” is shorthand for indicating that $\mu_i$ is odd or even, respectively.
Observe that in the case of the $\mathfrak{su}$-type flavor symmetries, there is an overall trace condition on a collection of unitary algebras. This leads to a general expectation that such theories will have many $\mathfrak{u}(1)$ symmetry algebra factors. Similarly, for the $\mathfrak{so}$ and $\mathfrak{sp}$ algebras, we get $\mathfrak{so}(2) \simeq \mathfrak{u}(1)$ factors when $d_i = 2$. Such symmetry factors can sometimes be subtle to determine directly from the associated F-theory geometry, a point we return to later on in section \[sec:FLAVOR\].
For $\mathfrak{su}$ gauge groups, there is also a physical realization in terms of IIA suspended brane configurations [@Hanany:1997gh; @Gaiotto:2014lca]; we will return to this picture in subsection \[sub:su\]. For the $\mathfrak{so/sp}$-type gauge algebras, which we will discuss in subsection \[sub:so\], a similar story involves the use of O6 orientifold planes. In these cases, the best we should in general hope for is that the nilpotent elements which embed in a maximal $\mathfrak{su}(N)$ subalgebra can also be characterized in terms of partitions of branes (and their images under the orientifold projection). Indeed, we will see some striking examples where the “naïve” semi-classical intuition fails in a rather spectacular way: Starting from a perturbative IIA configuration, we will generate SCFT flows which land us on non-perturbatively realized SCFTs i.e. those in which the string coupling is order one!
The rest of this section is organized as follows. Mainly focusing on a broad class of examples, we first explain for the $\mathfrak{su}$-type flavor symmetries how hierarchies for nilpotent elements translate to corresponding hierarchies for RG flows. We then turn to a similar analysis for the $\mathfrak{so}_{\text{even}}$ flavor symmetries where we encounter our first examples of flows involving conformal matter vevs. These cases are a strongly coupled analogue of weakly coupled Higgsing, and we shall indeed see that including these flows is necessary to maintain the expected correspondence between nilpotent elements and RG flows. Finally, we turn to the cases of $\mathfrak{so}_{\text{odd}}$ and $\mathfrak{sp}$-type flavor symmetry algebras.
Flows from $\mathfrak{su}_{N}$ {#sub:su}
------------------------------
As a first class of examples, we consider flows starting from the 6D SCFT with tensor branch: $$\label{eq:m0}
\lbrack SU(N)]\overset{\mathfrak{su}_{N}}{2}...\overset{\mathfrak{su}_{N}}{2}[SU(N)],$$ that is, we have colliding seven-branes with a hypermultiplet localized at each point of intersection. One can Higgs each of the two $SU(N)$ flavor symmetries in a way parameterized by two partitions $\mu_L$, $\mu_R$ of $N$; this results in the SCFT ${\cal T}(SU(N),\mu_L, \mu_R,k)$, where $(k-1)$ is the number of gauge groups in (\[eq:m0\]).
These theories can be realized in terms of D6-branes suspended in between NS5-branes [@Hanany:1997gh; @Gaiotto:2014lca]. At the very left and right, these D6-branes attach to D8-branes, and the choice of boundary condition on each D8-brane is controlled by Nahm pole data,which in turn dictates the flavor symmetry for the resulting 6D SCFT. These Nahm poles are boundary conditions for the Nahm equations living on the D6-brane worldvolume; they describe a “fuzzy funnel," namely a fuzzy sphere configuration on the D6s which expands into a D8. This description is T-dual to the Hitchin pole description of section \[sec:CONFMATT\].
As described in the introduction, we will at first consider theories where the number of gauge groups $(k-1)$ is sufficiently large enough so that the effect of Higgsing the left and right flavor groups are decoupled. (We will comment on the situation where that does not happen in section \[sec:SHORT\].) Given partitions $\mu_L$ and $\mu_R$ for the theory ${\cal T}(SU(N), \mu_L, \mu_R, k)$, there is a straightforward algorithm for determining the associated suspended brane configuration [@DelZotto:2014hpa; @Hanany:1997gh; @Gaiotto:2014lca] (for a longer review, see also section 2 of [@Cremonesi:2015bld]). To illustrate, let us focus on the left partition $\mu_L=\mu$. Consider now the transposed Young diagram $\mu^{T}=(\mu_{1}^{T},...,\mu_{N}^{T})$. The gauge groups are now given by $SU(N_i(\mu))$, with $$\label{eq:Ni}
\mu_{i}^{T} = N_i - N_{i-1},$$ where $N_0 = 0$. The gauge group $SU(N_i)$ also has $f_i$ hypermultiplets in the fundamental representation. Anomaly cancellation requires $2 N_i=N_{i-1}+N_{i+1}+f_i$. So in fact the function $i\mapsto N_i$ is convex; moreover, the $f_i$ are equal to the jump in the slope of this function. This accounts for the presence of the product flavor symmetry factors in (\[suflavor\]). See figure \[fig:su4\] for a depiction of the suspended brane configurations, associated partitions and quivers for the $N=4$ case.
Let us now verify that if we have a two partitions $\mu$ and $\nu$ such that $\mu < \nu$, that there is then a corresponding RG flow between the theories, i.e. $\mathcal{T(\mu)}\rightarrow\mathcal{T(\nu)}$. For each choice of partition, we get a sequence of gauge groups: $$\left\{ N_{i}(\mu)\right\}_{i}\text{ \ \ and \ \ }\left\{ N_{i}(\nu)\right\}_{i}.$$ From (\[eq:Ni\]) we see that $N_i(\mu)= \sum_{j=1}^i \mu_j^T$. So the condition that $\mu < \nu$, or $\mu^T > \nu^T$, translates to a related condition on the values of each of these ranks:$$N_{i}(\mu) \geq N_{i}(\nu).$$ In some cases this condition is vacuously true since $N_{i}(\nu)$ may be zero after initiating some breaking pattern. The resulting nilpotent hierarchy therefore directly translates back to allowed RG flows for our system. We also note that this correspondence between hierarchies and RG flows applies even for partitions of different sizes. More precisely, for theories with a different number of boxes in the respective Young diagrams, we first consider the transposed partition, and then use the partial ordering for these partitions. In other words, $\mu^{T} > \nu^{T}$ implies the existence of an RG flow between the corresponding theories even if $|\mu| \neq |\nu|$.
Flows from $\mathfrak{so}_{\text{even}}$ {#sub:so}
----------------------------------------
One of the significant simplifications in studying RG flows for the theories with $\mathfrak{su}$-type flavor symmetries is that there is a direct match between nilpotent orbits of the flavor symmetry and geometric maneuvers for the configuration of suspended branes. This is mainly due to the fact that the resulting theories on the tensor branch have conventional matter fields. In all other cases, we will inevitably need to include the effects of vevs for conformal matter.
As a first example of this type, we now turn to examples where the flavor symmetry on one side of our generalized quiver theory is an $\mathfrak{so}_{\text{even}}$-type flavor symmetry. One way to engineer these examples is to consider the case of a stack of M5-branes probing a D-type orbifold singularity. In the F-theory realization, we then get our UV theory on the tensor branch:$$\lbrack SO(2N)]\overset{\mathfrak{sp}_{N}}{1}\text{ }\overset{\mathfrak{so}_{2N}}{4}\text{ }\overset{\mathfrak{sp}_{N}}{1}...\overset{\mathfrak{so}_{2N}}{4}\text{ }\overset{\mathfrak{sp}_{N}}{1}[SO(2N)].$$ We shall primarily focus on the effects of nilpotent flows associated with just one flavor symmetry factor, so we will typically assume a sufficiently large number of tensor multiplets are present to make such genericity assumptions.
Because this is still a classical algebra, all of the nilpotent orbits are labeled by a suitable partition of $2N$, but where each even entry occurs with even multiplicity. Additionally, there is clearly a partial ordering of these partitions. However, in this case we can expect the breaking patterns to be more involved in part because now, we can also give vevs to conformal matter. Indeed, we shall present examples where matter in a spinor representation inevitably makes an appearance. In the IIA setup, these odditiesformally require the presence of a negative number of branes in a suspended brane configuration, as shown in Figure \[SO10brane\]. In such cases, we must instead pass to the F-theory realization of these models.
[1.0]{} \[fig:so10a\]
[1.0]{} \[fig:so10b\]
[1.0]{} \[fig:so10c\]
We shall primarily focus on some illustrative examples. Figure \[fig:so8flows\] summarizes RG flows among theories ${\cal T}(SO(8),\mu_L,\mu_R,k)$ where we vary $\mu_L$ and for simplicity hold fixed $\mu_R=(1^{8})$. In figures \[fig:SO10\] and \[fig:SO12\] we show similar diagrams for ${\cal T}(SO(10),\mu_L,1^{10},k)$ and ${\cal T}(SO(12),\mu_L,1^{12},k)$, respectively. As already noted in section \[sec:CONFMATT\], we omit the flavors which are implicit for theories with minimal fiber types, i.e. those which arise on non-Higgsable clusters (e.g. $2,3$ and $7$).
All of the flows we consider are associated with motion on the Higgs branch. So, even though these flows are parameterized by nilpotent orbits (i.e. T-branes), the hyperkahler structure of the Higgs branch ensures that we can also understand these flows in terms of a complex structure deformation [@Heckman:2015bfa]. It is easiest to exhibit them after shrinking some $-1$ curves present on the tensor branch. For example, in the first flow of figure \[fig:SO10\], one can shrink the leftmost $-1$ curve on both the $(1^{10})$ and $(2^2,1^6)$ configuration. The complex deformation from $(1^{10})$ to $(2^2, 1^6)$ is actually realized by a two-parameter family of deformations which in Tate form (see e.g. [@Bershadsky:1996nh]) is given by the Weierstrass model: $$y^2 + (u + \epsilon_1) v xy + (uv)^2 y = x^3 + (uv) x^2 + (u + \epsilon_2) (u^2 v^3) x + (uv)^4,$$ so that when $\epsilon_1 , \epsilon_2 = 0$, we realize the original $(1^{10})$ configuration, while for $\epsilon_{1}, \epsilon_2 \neq 0$, we have an $\mathfrak{su}_4 \simeq \mathfrak{so}_6$ flavor symmetry localized along $u = 0$, with an $\mathfrak{so}_{10}$ localized along $v = 0$. The appearance of the two unfolding parameters is instructive and illustrates that specifying a T-brane configuration imposes further restrictions on the allowed deformations. Indeed, although there is no nilpotent generator which breaks $\mathfrak{so}_{10}$ to either $\mathfrak{so}_8$ or $\mathfrak{su}_5$, there are of course semisimple generators which do.
In principle, one could proceed in this way for all the flows in figures \[fig:so8flows\], \[fig:SO10\], and \[fig:SO12\]. In practice, however, one can speed up the computation by using information coming from field theory, from anomaly cancellation, and from the known properties of the E-string theory. The possible gauge algebras on a curve, depending on its self-intersection number, are listed for example in [@Heckman:2015bfa Pages 45–46]. The expected representations of the matter fields, and the corresponding flavor group symmetries acting on them, are listed in [@Bertolini:2015bwa Table 5.1]. For example, we sometimes encounter an $\mathfrak{so}$-type gauge theory on a $-4$, $-3$, $-2$ and $-1$ curve. Anomaly cancellation uniquely fixes the spectrum of hypermultiplets transforming in a non-trivial representation of the gauge symmetry algebra. For a $-4$ curve with $\mathfrak{so}$-type gauge algebra, all matter transforms in the fundamental representation. For the last three cases, there are always spinor representations, and the number of spinors is $16/d_{s}$, $32/d_{s}$ and $48/d_{s}$ respectively, where $d_{s}$ is the dimension of the irreducible spinor representation of this algebra. Note that this also places an upper bound on the rank of the gauge groups, i.e. the maximal rank $\mathfrak{so}$-type algebra for a system with spinors is in these cases respectively $\mathfrak{so}(12)$, $\mathfrak{so}(13)$ and $\mathfrak{so}(12)$.
Finally, one should keep in mind that the E-string living on an empty $-1$ curve has an $E_8$ flavor symmetry; thus, when we gauge a product subalgebra, we necessarily have $\mathfrak{g}_1 \times \mathfrak{g}_2 \subset \mathfrak{e}_8$ (see [@Heckman:2013pva]). If this subalgebra is not maximal, we also expect there to be a residual flavor symmetry given by the commutant subalgebra.
Let us now turn to some examples. The first flow of figure \[fig:SO10\] simply corresponds to giving a vev to a fundamental hypermultiplet for the leftmost $\mathfrak{sp}_1$ gauge algebra, breaking it completely. One ends up with an E-string, and $\mathfrak{so}_6 \oplus \mathfrak{so}_{10}\subset \mathfrak{so}_{16}$ is indeed a subalgebra of $\mathfrak{e}_8$. The gauge algebra $\mathfrak{so}_{10}$ on the leftmost $-4$ curve should still have 2 fundamental hypermultiplets; given the presence of the $\mathfrak{sp}_1$ on the right, we deduce the presence of a “side link” of conformal matter with flavor symmetry $SU(2)$. In the next step, we can give a vev to this side link, which breaks $\mathfrak{so}_{10}\to \mathfrak{so}_9$ and leads to $(3,1^7)$; or alternatively we can shrink the empty $-1$ curve. In this second case, the $-4$ curve becomes a $-3$ curve, and now the $\mathfrak{so}_{10}$ should support three fundamental hypers; again, given the presence of the $\mathfrak{sp}_1$ on the right, we deduce an $\mathfrak{sp}_2$ flavor symmetry. We can now iterate the process until no further Higgsing is possible; this leads to figure \[fig:SO10\].
The diagram precisely corresponds with the ordering of partitions, in agreement with line (\[eq:RG\]). That we achieve a perfect match between the hierarchies of nilpotent elements and a corresponding hierarchy of RG flows again provides strong evidence for our proposed picture of RG flows induced by conformal matter vevs.
As an examples of a Higgsing operation, consider the SCFT with tensor branch: $$[SO(12)] \,\, \overset{\mathfrak{sp}_{2}}{1} \,\, \overset{\mathfrak{so}_{12}}{4}\,\,\overset{\mathfrak{sp}_{2}}{1} \,\, ....$$ We can flow to another SCFT in the IR by activating a vev for a fundamental hypermultiplet of the leftmost $\mathfrak{sp}_2$ gauge algebra. The resulting tensor branch for this IR SCFT is then: $$[SO(8)] \,\, \overset{\mathfrak{sp}_{1}}{1} \,\, \underset{[Sp(1)]}{\overset{\mathfrak{so}_{12}}{4}}\,\,\overset{\mathfrak{sp}_{2}}{1} \,\, ....$$ The hypermultiplet in the bifundamental representation, i.e. the $\frac{1}{2}\bf(4,12)$ decomposes as $\frac{1}{2}\textbf{(2,12)}\oplus\textbf{(1,12)}$, yielding the single fundamental on the leftmost $\mathfrak{so}_{12}$ of the IR theory, which transforms under a global $Sp(1)$ symmetry.
We can also see that vevs of conformal matter can sometimes drive us away from a perturbative IIA realization of the tensor branch. For example, by starting on the tensor branch, we can collapse the leftmost $-1$ curve of the configuration: $$[SO(7)] \,\, {1}\,\,\overset{\mathfrak{so}_{9}}{4}\,\,\overset{\mathfrak{sp}_{1}}{1} \,\, ...$$ so a vev for conformal matter can trigger a flow to the configuration with tensor branch: $$[SU(2) \times SU(2)] \,\,\overset{\mathfrak{so}_{9}}{3}\,\,\overset{\mathfrak{sp}_{1}}{1} \,\, ....$$ That is, collapsing the $-1$ curve converts the $-4$ curve to a $-3$ curve and the remnants of conformal matter not eaten by the Higgs mechanism show up as matter in possibly “exotic” representations. In this case, a spinor and a fundamental of $\mathfrak{so}_9$ appear on the -3 curve after blowdown. Note that at the SCFT point, we are always dealing with collapsed curves anyway, so we should properly view this as a complex structure deformation. Such deformations may also involve collapsing $-1$ curves located in the interior of the tensor branch quiver. For instance, the bottom flow in figure \[fig:so8flows\] corresponds to a blowdown of the leftmost $-1$ curve of the theory $${\overset{\mathfrak{su}_2}2} \, {\overset{\mathfrak{g_{2}}}3} \,\, 1\,\, \overset{\mathfrak{so_{8}}}4 \,\, 1\,\, ...[SO(8)]$$ and produces a theory with quiver $$2 \,\, \overset{\mathfrak{su}_2}2 \,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1\,\, \overset{\mathfrak{so_{8}}}4 \,\, 1 \,\, ...[SO(8)]$$ Note that the $-3$ curve of the UV theory has become the second $-2$ curve of the IR theory, and the leftmost $-4$ curve of the UV theory has become the $-3$ curve of the IR theory.
Additionally, recall that partitions of $2N$ with only even entries give rise to two distinct nilpotent orbits of $\mathfrak{so}(2N)$, which are related to each other by outer automorphism. However, matching with the hierarchy of RG flows reveals that these distinct nilpotent elements do *not* give rise to distinct 6D SCFTs. Thus, we conclude that RG flows parametrized by nilpotent orbits related by an outer automorphism lead to physically equivalent IR fixed points. This is illustrated most poignantly in the $\mathfrak{so}(8)$ case shown in figure \[fig:so8flows\]: here, not only the two $(2^4)$ orbits but also the $(3,1^5)$ partition are related by the triality outer automorphism (likewise for $(4^2)_I$, $(4^2)_{II}$ and $(5,1^3)$).[^6] We see that in both cases, all three of these nilpotent orbits correspond to the same 6D SCFT. In the $\mathfrak{so}(10)$ and $\mathfrak{so}(12)$ figures, we therefore display only a single theory for each partition.
We also observe that just as in the case of theories with an $\mathfrak{su}$-type flavor symmetry, we can extend the nilpotent hierarchy to partitions with a different number of boxes, i.e. by working in terms of the transposed Young diagrams: $$\mu^T > \nu^T \Rightarrow T(\mu) \rightarrow T(\nu).$$ For instance, comparing the list of $SO(10)$ theories with the list of $SO(12)$ theories, we see that there is clearly a flow from the $(2^2,1^8) $ theory of $SO(12)$ to the $(1^{10})$ theory of $SO(10)$, as expected since $(2^2, 1^8)^T > (1^{10})^T$. However, there is no flow that will take us from the $(2^4, 1^4)$ theory of $SO(12)$ to the $(1^{10})$ theory of $SO(10)$, and indeed $(2^4, 1^4)^T \ngtr (1^{10})^T$.
\(1) \[startstop\] [ $
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\(2) \[startstop, below of=1\] [ $
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\(3) \[startstop, below of=2\] [ $
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\(5) \[startstop, below of=3\] [ $
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$]{};
\(6) \[startstop, below of=5\] [ $
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$]{};
\(7) \[startstop, below of=6\] [ $
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$]{};
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$]{};
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$]{};
\(9) \[startstop, below of=7\] [ $
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$]{};
\(10) \[startstop, below of=9\] [ $
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$]{};
\(1) – (2); (2) – (3); (2) – (4); (3) – (5); (2) – (4b); (4) – (5); (5) – (6); (4b) – (5); (6) – (7); (6) – (7b); (7) – (9); (7b) – (9); (6) – (8); (8) – (9); (9) – (10);
\(1) \[startstop\] [ $
1^{10}: [SO(10)]\,\,\overset{\mathfrak{sp_1}}1 \,\, {\overset{\mathfrak{so_{10}}}4} \,\, \overset{\mathfrak{sp_1}}1 \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(2) \[startstop, below of=1\] [ $
2^2,1^6: [SO(6)]\,\, 1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{10}}}4} \,\, \overset{\mathfrak{sp_1}}1 \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(3) \[startstop, below of=2\] [ $
3,1^7: [SO(7)]\,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(4) \[startstop, right of=3,xshift=7cm\] [ $
2^4,1^2:[Sp(2)]\,\, \underset{[N_s=1]}{\overset{\mathfrak{so_{10}}}3} \,\, \overset{\mathfrak{sp_1}}1 \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(5) \[startstop, below of=3\] [ $
3,2^2,1^3: [SU(2) \times SU(2)]\,\, \overset{\mathfrak{so_{9}}}3 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(6) \[startstop, below of=5\] [ $
3^2,1^4:[SU(2) \times SU(2)]\,\,\overset{\mathfrak{so_{8}}}3 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(7) \[startstop, below of=6\] [ $
3^2,2^2:[SU(2) ]\,\, \overset{\mathfrak{so_{7}}}3 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(8) \[startstop, below of=7\] [ $
3^3,1: \overset{\mathfrak{g_2}}3 \,\, \underset{[SU(2)]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, ...[SO(10)]
$]{};
\(9) \[startstop, right of=7, xshift=6cm\] [ $
5,1^5: [Sp(2) ]\,\, \overset{\mathfrak{so_7}}3 \,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_1}}1} \,\, ...[SO(10)]
$]{};
\(10) \[startstop, below of=8\] [ $
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$]{};
\(11) \[startstop, right of=10, xshift=6cm\] [ $
5,2^2,1: [SU(2) ] \,\, \overset{\mathfrak{g_2}}3 \,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_1}}1} \,\, ...[SO(10)]
$]{};
\(12) \[startstop, below of=10\] [ $
5,3,1^2: \overset{\mathfrak{su_3}}3 \,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_1}}1} \,\, ...[SO(10)]
$]{};
\(13) \[startstop, below of=12\] [ $
5^2: \overset{\mathfrak{su_2}}2 \,\, \overset{\mathfrak{so_{7}}}3 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_1}}1} \,\, \overset{\mathfrak{so_{10}}}4 \,\, ...[SO(10)]
$]{};
\(14) \[startstop, right of=13, xshift=5cm\] [ $
7,1^3: \overset{\mathfrak{su_2}}2 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{7}}}3} \,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, ...[SO(10)]
$]{};
\(15) \[startstop, below of=13\] [ $
7,3: {\overset{\mathfrak{su_2}}2} \,\, \overset{\mathfrak{g_2}}3 \,\, 1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, ...[SO(10)]
$]{};
\(16) \[startstop, below of=15\] [ $
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\(1) – (2); (2) – (3); (2) – (4); (3) – (5); (4) – (5); (5) – (6); (6) – (7); (7) – (8); (6) – (9); (8) – (10); (8) – (11); (9) – (11); (10) – (12); (11) – (12); (12) – (13); (12) – (14); (13) – (15); (14) – (15); (15) – (16);
\(1) \[startstop, yshift=-2cm\] [ $
1^{12}: [SO(12)]\,\,\overset{\mathfrak{sp_2}}1 \,\, {\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(2) \[startstop, below of=1\] [ $
2^2, 1^{8}: [SO(8)]\,\,\overset{\mathfrak{sp_1}}1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(3) \[startstop, below of=2\] [ $
2^4,1^4: [SO(4)]\,\, 1 \,\, \underset{[Sp(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(4) \[startstop, right of=3,xshift=8cm\] [ $
3,1^9: [SO(9)]\,\, \overset{\mathfrak{sp}_1}1 \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(5) \[startstop, below of=3\] [ $
2^6: [Sp(3)]\,\, {\overset{\mathfrak{so_{12}}}3} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(6) \[startstop, right of=5, xshift=8cm\] [ $
3,2^2,1^5: [SO(5)] \,\, 1\,\, \underset{[Sp(1)]}{\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(7) \[startstop, below of=6\] [ $
3^2,1^6: [SO(6)] \,\, 1\,\, {\overset{\mathfrak{so_{10}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(8) \[startstop, below of=5\] [ $
3,2^4,1: [Sp(2)] \,\, {\overset{\mathfrak{so_{11}}}3} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(9) \[startstop, below of=8\] [ $
3^2,2^2,1^2: [Sp(1)] \,\, {\overset{\mathfrak{so_{10}}}3} \,\, \overset{\mathfrak{sp_2}}1 \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(10) \[startstop, below of=9\] [ $
3^3,1^3: [Sp(1)] \,\, {\overset{\mathfrak{so_{9}}}3} \,\, \underset{[SO(3)]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(12) \[startstop, below of=10\] [ $
4^2,1^4:
[SU(2) \times SU(2)] \,\, \overset{\mathfrak{so}_{8}}3 \,\, \overset{\mathfrak{sp_1}}1 \,\, \underset{[Sp(1)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(11) \[startstop, right of=12, xshift=7cm\] [ $
3^4:
\overset{\mathfrak{so}_{7}}3 \,\, \underset{[SO(4)]}{\overset{\mathfrak{sp_2}}1} \,\, {\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(13) \[startstop, below of=12\] [ $
4^2,2^2:
[SU(2)] \,\, \overset{\mathfrak{so}_{7}}3 \,\, {\overset{\mathfrak{sp_1}}1} \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(14) \[startstop, below of=13\] [ $
4^2,3,1:
[SU(2)] \,\, \overset{\mathfrak{g}_{2}}3 \,\, \overset{\mathfrak{sp_1}}1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(15) \[startstop, right of=14, xshift=7cm\] [ $
5,1^7:
[SO(7)] \,\, 1 \,\, \overset{\mathfrak{so}_{9}}4 \,\, \overset{\mathfrak{sp_1}}1 \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(16) \[startstop, below of=15, xshift=-1cm\] [ $
5,2^2,1^3:
[SU(2) \times SU(2)] \,\, \overset{\mathfrak{so}_{9}}3 \,\, \overset{\mathfrak{sp_1}}1 \,\, \overset{\mathfrak{so_{11}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(17) \[startstop, below of=16, xshift=-1cm\] [ $
5,3,1^4:
[SU(2) \times SU(2)] \,\, \overset{\mathfrak{so}_{8}}3 \,\, \overset{\mathfrak{sp_1}}1 \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(18) \[startstop, below of=17, xshift=-1cm\] [ $
5,3,2^2:
[SU(2)] \,\, \overset{\mathfrak{so}_{7}}3 \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(19) \[startstop, below of=18,xshift=-2cm\] [ $
5,3^2,1:
\overset{\mathfrak{g}_{2}}3 \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(20) \[startstop, below of=19\] [ $
5^2,1^2:
\overset{\mathfrak{su}_{3}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{10}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(21) \[startstop, below of=20, xshift=-3cm\] [ $
6^2:
\overset{\mathfrak{su}_{2}}2 \,\, \overset{\mathfrak{so}_{7}}3 \,\, \overset{\mathfrak{sp_1}}1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(22) \[startstop, right of=21, xshift=6cm\] [ $
7,1^5:
[Sp(2)] \,\, \overset{\mathfrak{so}_{7}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(23) \[startstop, below of=22, xshift=-1cm\] [ $
7,2^2,1:
[Sp(1)] \,\, \overset{\mathfrak{g}_{2}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(24) \[startstop, below of=23, xshift=-1cm\] [ $
7,3,1^2:
\overset{\mathfrak{su_3}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(25) \[startstop, below of=24, xshift=-4cm\] [ $
7,5:
\overset{\mathfrak{su_2}}2 \,\, \overset{\mathfrak{so_7}}3 \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(26) \[startstop, right of=25, xshift=6cm\] [ $
9,1^3:
\overset{\mathfrak{su_2}}2 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_7}}3} \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(27) \[startstop, below of=25,xshift=3cm\] [ $
9,3:
\overset{\mathfrak{su_2}}2 \,\, \overset{\mathfrak{g_2}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(28) \[startstop, below of=27\] [ $
11,1:
2\,\, \overset{\mathfrak{su_2}}2 \,\, \overset{\mathfrak{g_2}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(12)]
$]{};
\(1) – (2); (2) – (3); (2) – (4); (3) – (5); (3) – (6); (4) – (6); (5) – (8); (6) – (7); (6) – (8); (7) – (9); (8) – (9); (9) – (10); (10) – (11); (10) – (12); (11) – (13); (12) – (13); (13) – (14); (12) – (17); (7) – (15); (15) – (16); (10) – (16); (16) – (17); (13) – (18); (14) – (19); (17) – (18); (18) – (19); (19) – (20); (20) – (21); (17) – (22); (22) – (23); (19) – (23); (23) – (24); (20) – (24); (24) – (25); (21) – (25); (24) – (26); (25) – (27); (26) – (27); (27) – (28);
> >
Flows from $\mathfrak{so}_{\text{odd}}$ and $\mathfrak{sp}_{N}$ {#ssec:SOodd}
---------------------------------------------------------------
Finally, we come to the analysis of flows involving the non-simply laced classical algebras $\mathfrak{so}(2N+1)$ and $\mathfrak{sp}(N)$. In these cases, we do not directly reach the desired flavor symmetry from M5-branes probing an ADE singularity. Rather, we must first consider the case of a partial tensor branch flow and / or some contribution from conformal matter vevs. For example, to reach the $\mathfrak{sp}$-type flavor symmetries, we can start from:$$\lbrack SO(2N)]\overset{\mathfrak{sp}_{N}}{1}\text{ }\overset{\mathfrak{so}_{2N}}{4}\text{ }\overset{\mathfrak{sp}_{N}}{1}...\overset{\mathfrak{so}_{2N}}{4}\text{ }\overset{\mathfrak{sp}_{N}}{1}[SO(2N)],$$ and by decompactifying the leftmost and rightmost $-1$ curves, we reach the system:$$\lbrack Sp(N)]\overset{\mathfrak{so}_{2N}}{4}\text{ }\overset{\mathfrak{sp}_{N}}{1}...\overset{\mathfrak{so}_{2N}}{4}[Sp(N)].$$ In the case of an $SO(2N+1)$ flavor symmetry we can also start from a theory with $SO(2N+2p)$ flavor symmetry. For sufficiently large $p$, we can then reach the desired $SO(2N+1)$ flavor symmetry by activating a conformal matter vev associated with the partition $(2p-1,1^{2N+1})$. See figures \[fig:sp3flows\] and \[fig:so9flows\] for examples of the flow diagrams and associated F-theory models for these systems.
\(1) \[startstop, xshift=-1cm\] [ $
1^6: [Sp(3)] \,\, \overset{\mathfrak{so_{14}}}4 \,\, \overset{\mathfrak{sp_{3}}}1 \,\, \overset{\mathfrak{so_{14}}}4 \,\, ... \,\, [Sp(3)]
$]{};
\(2) \[startstop, below of=1\] [ $
2,1^4: [Sp(2)] \,\, \overset{\mathfrak{so_{13}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_{3}}}1} \,\, \overset{\mathfrak{so_{14}}}4 \,\, ... \,\, [Sp(3)]
$]{};
\(3) \[startstop, below of=2\] [ $
2^2,1^2: [Sp(1)] \,\, \overset{\mathfrak{so_{12}}}4 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_{3}}}1} \,\, \overset{\mathfrak{so_{14}}}4 \,\, ... \,\, [Sp(3)]
$]{};
\(4) \[startstop, below of=3\] [ $
2^3: \overset{\mathfrak{so_{11}}}4 \,\, \underset{[SO(3)]}{\overset{\mathfrak{sp_{3}}}1} \,\, \overset{\mathfrak{so_{14}}}4 \,\, ... \,\, [Sp(3)]
$]{};
\(5) \[startstop, below of=4, xshift=5cm\] [ $
3^2: \overset{\mathfrak{so_{10}}}4 \,\, {\overset{\mathfrak{sp_{2}}}1} \,\, \underset{[Sp(1)]}{\overset{\mathfrak{so_{14}}}4} \,\, \overset{\mathfrak{sp_{3}}}1 \,\, {\overset{\mathfrak{so_{14}}}4}\,\, ... \,\, [Sp(3)]
$]{};
\(6) \[startstop, below of=4, xshift=-5cm\] [ $
4,1^2: [Sp(1)] \,\, \overset{\mathfrak{so_{11}}}4 \,\, {\overset{\mathfrak{sp_{2}}}1} \,\, {\overset{\mathfrak{so_{13}}}4} \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_{3}}}1} \,\, {\overset{\mathfrak{so_{14}}}4}\,\, ... \,\, [Sp(3)]
$]{};
\(7) \[startstop, below of=6, xshift=5cm\] [ $
4,2: \overset{\mathfrak{so_{10}}}4 \,\, \underset{[N_f = \frac{1}{2}]}{\overset{\mathfrak{sp_{2}}}1} \,\, {\overset{\mathfrak{so_{13}}}4} \,\, \underset{[N_f = \frac{1}{2}]}{\overset{\mathfrak{sp_{3}}}1} \,\, {\overset{\mathfrak{so_{14}}}4}\,\, ... \,\, [Sp(3)]
$]{};
\(8) \[startstop, below of=7\] [ $
6: \overset{\mathfrak{so_{9}}}4 \,\, {\overset{\mathfrak{sp_{1}}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_{2}}}1 \,\, {\overset{\mathfrak{so_{13}}}4}\,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_{3}}}1} \,\, {\overset{\mathfrak{so_{14}}}4}\,\, ... \,\, [Sp(3)]
$]{};
\(1) – (2); (2) – (3); (3) – (4); (4) – (5); (4) – (6); (5) – (7); (6) – (7); (7) – (8);
\(1) \[startstop, xshift=-2cm\][ $
1^9: [SO(9)]\,\, \overset{\mathfrak{sp}_1}1 \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(2) \[startstop, below of=1, xshift=0cm\] [ $
2^2,1^5: [SO(5)] \,\, 1\,\, \underset{[Sp(1)]}{\overset{\mathfrak{so_{11}}}4} \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(3) \[startstop, below of=2, xshift=4cm\] [ $
3,1^6: [SO(6)] \,\, 1\,\, {\overset{\mathfrak{so_{10}}}4} \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(4) \[startstop, right of=3, xshift=-10cm\] [ $
2^4,1: [Sp(2)] \,\, \underset{[N_s=\frac12]}{\overset{\mathfrak{so_{11}}}3} \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(5) \[startstop, below of=4, xshift=5cm\] [ $
3,2^2,1^2: [Sp(1)] \,\, \underset{[N_s=1]}{\overset{\mathfrak{so_{10}}}3} \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(6) \[startstop, below of=5\] [ $
3^2,1^3: [Sp(1)] \,\, {\overset{\mathfrak{so_{9}}}3} \,\, \underset{[SO(3)]}{\overset{\mathfrak{sp_2}}1} \,\, \overset{\mathfrak{so_{12}}}4 \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(7) \[startstop, below of=6, xshift=4cm\] [ $
3^3:
\overset{\mathfrak{so}_{7}}3 \,\, \underset{[SO(4)]}{\overset{\mathfrak{sp_2}}1} \,\, {\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(8) \[startstop, below of=7, xshift=0cm\] [ $
4^2,1:
[SU(2)] \,\, \overset{\mathfrak{g}_{2}}3 \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_1}}1} \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{12}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(9) \[startstop,right of=7, xshift=-10cm\] [ $
5,1^4:
[SU(2) \times SU(2)] \,\, \overset{\mathfrak{so}_{8}}3 \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(10) \[startstop, below of=9, xshift=0cm\] [ $
5,2^2:
[SU(2)] \,\, \overset{\mathfrak{so}_{7}}3 \,\, \underset{[N_f=\frac12]}{\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(11) \[startstop, below of=10, xshift=4cm\] [ $
5,3,1:
\overset{\mathfrak{g}_{2}}3 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(12) \[startstop, below of=11, xshift=0cm\] [ $
7,1^2:
\overset{\mathfrak{su_3}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(13) \[startstop, below of=12\] [ $
9:
\overset{\mathfrak{su_2}}2 \,\, \overset{\mathfrak{g_2}}3 \,\, 1 \,\, {\overset{\mathfrak{so_{9}}}4} \,\, {\overset{\mathfrak{sp_1}}1} \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \overset{\mathfrak{sp_2}}1 \,\, ... [SO(9)]
$]{};
\(1) – (2); (2) – (3); (2) – (4); (4) – (5); (3) – (5); (5) – (6); (6) – (7); (7) – (8); (7) – (10); (6) – (9); (9) – (10); (8) – (11); (10) – (11); (11) – (12); (12) – (13);
Exceptional Flavor Symmetries \[sec:EXCEPTIONAL\]
=================================================
In the previous section we focused on examples with classical flavor symmetry algebras where there is a combinatorial construction of all nilpotent orbits in terms of partitions of positive integers (with suitable restrictions).
But we have also seen that for all cases other than the A-type flavor symmetry, conformal matter vevs can sometimes drive us to a conformal fixed point where spinor representations are present, indicating that the construction really requires non-perturbative elements (i.e., an embedding in F-theory).
Now, in the case of flows from a theory with exceptional flavor symmetries, we must resort to the F-theory realization from the start. Nevertheless, we still expect that some (but not all!) of the RG flows induced by nilpotent orbits can be understood in terms of partitions of perturbative D7-branes. For example, in the terminology of [@Gaberdiel:1997ud], a seven-brane with $E_{8}$ gauge symmetry is given by a non-perturbative bound state of seven-branes of different $(p,q)$ type, i.e. $A^{7}BC^{2}$. In a suitable duality frame, the $A$-type seven-branes are just the perturbative D7-branes, and so we can expect some of the nilpotent orbits to be described by partitions of these seven seven-branes. By a similar token, there are six such seven-branes for $E_{7}$ and five for $E_{6}$. Nevertheless, there are also more general nilpotent orbits which do not appear to admit such a simple characterization in terms of partitions.
To deal with this more general class of nilpotent orbits, and to verify that we indeed get a corresponding match with hierarchies expected from RG flows, we will instead need to rely on some results from the Bala–Carter (B–C) theory of nilpotent orbits for exceptional algebras. The main point is that for each nilpotent element $\mu\in\mathfrak{g}_{\mathbb{C}}$, we get a corresponding homomorphism via the Jacobson–Morozov theorem (see line (\[JacMor\])). So, to characterize possible homomorphisms, we simply need to specify the embedding in a subalgebra of $\mathfrak{g}_{\mathbb{C}}$. Indeed, there is also a notion of partial ordering for these nilpotent orbits, which is reviewed in great detail in reference [@Chacaltana:2012zy]. For this reason, we should expect there to be a similar correspondence between nilpotent orbits and RG flows.
Since there is a finite list of nilpotent orbits for each exceptional flavor symmetry, we can explicitly determine the induced flow for each case. For the simply laced algebras $E_{6}$, $E_{7}$ and $E_{8}$, our starting point will be a long generalized quiver of the form:$$\begin{aligned}
& \lbrack E_{6}]-E_{6}-E_{6}-...,\\
& \lbrack E_{7}]-E_{7}-E_{7}-...,\\
& \lbrack E_{8}]-E_{8}-E_{8}-...,\end{aligned}$$ i.e. we take a stack of M5-branes probing an E-type singularity. The links here $-$ denote the corresponding conformal matter for these systems. In F-theory terms, the resolved theory on the tensor branch for each of these cases is:$$\begin{aligned}
& [E_{6}]\,1\overset{\mathfrak{su}_{3}}{3}1\overset{\mathfrak{e}_{6}}{6}...,\label{esixconfmatt}\\
& \lbrack E_{7}]\,1\text{ }\overset{\mathfrak{su}_{2}}{2}\text{ }\overset{\mathfrak{s0}_{7}}{3}\text{ }\overset{\mathfrak{su}_{2}}{2}1\overset{\mathfrak{e}_{7}}{8}...,\\
& [E_{8}]\,1\text{ }2\text{ }\overset{\mathfrak{sp}_{1}}{2}\text{
}\overset{\mathfrak{g}_{2}}{3}\text{ }1\text{ }\overset{\mathfrak{f}_{4}}{5}\text{ }1\text{ }\overset{\mathfrak{g}_{2}}{3}\text{ }\overset{\mathfrak{sp}_{1}}{2}\text{ }2\text{ }1\text{ }(\overset{\mathfrak{e}_{8}}{12})..., \label{eeightconfmatt}$$ We can also reach SCFTs with non-simply laced flavor symmetry algebras $\mathfrak{g}_{2}$ and $\mathfrak{f}_{4}$ by decompactifying the $-3$ and $-5$ curves of the $(E_{8},E_{8})$ conformal matter system:$$\begin{aligned}
& \lbrack G_{2}]\overset{\mathfrak{sp}_{1}}{2}\text{ }2\text{ }1\text{
}(\overset{\mathfrak{e}_{8}}{12})...,\\
& \lbrack F_{4}]\text{ }1\text{ }\overset{\mathfrak{g}_{2}}{3}\text{
}\overset{\mathfrak{sp}_{1}}{2}\text{ }2\text{ }1\text{ }(\overset{\mathfrak{e}_{8}}{12})....\end{aligned}$$ In these cases, the $...$ indicates that we continue beyond this point with a sequence of $E_{8}$ gauge groups with conformal matter between each such factor.
The rest of this section is organized as follows. We begin by giving an analysis of the nilpotent orbits of the simply laced exceptional algebras and the corresponding F-theory models associated with each such element. Using Bala–Carter theory, we also determine the flavor symmetries expected from the commutant of the nilpotent orbit in the parent flavor symmetry algebra and compare it with those flavor symmetries visible on the tensor branch of an F-theory model. We then turn to a similar analysis for the non-simply laced exceptional algebras.
Flows from $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$, $\mathfrak{e}_{8}$
---------------------------------------------------------------------
Let us begin with an analysis of the flows for the exceptional algebras $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$. Proceeding as in the previous examples, we start from the theories (\[esixconfmatt\])–(\[eeightconfmatt\]) and break the flavor symmetry on the left in various ways while holding fixed the flavor symmetry on the right. That is, we consider the theories ${\cal T}(E_n,\mu_L, \mu_R,k)$ obtained by varying $\mu_L$ whilst holding $\mu_R$ fixed and trivial. We now show how the hierarchy on nilpotent orbits determines hierarchies of RG fixed points.
For $\mathfrak{e}_6$ we show the results in a diagram similar to the ones given so far, in figure \[fig:E6\]. In the cases with $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$ flavor symmetry, the full list of nilpotent hierarchies does not easily fit on a few pages, but is presented for example in [@Chacaltana:2012zy App. C]. Thus in Appendix \[app:nilp\] we give the full list of Bala–Carter labels, the corresponding global flavor symmetries (expected from the commutants of ${\mathrm{Im}}(\rho)$ in $\mathfrak{g}_{\mathbb{C}}$; see (\[JacMor\])) and the corresponding realization in an F-theory model, with the understanding that there is an RG flow whenever there is an ordering relation between the corresponding label as in [@Chacaltana:2012zy App. C], in agreement with line (\[eq:RG\]).
The methods we used to produce these results are the same as the ones for the previous tables, as described in section \[sub:so\]. Once again, each flow corresponds to a complex deformation, which can be exhibited most easily by shrinking some $-1$ curve; for example, the very first flow corresponds to the deformation $y^2 = x^3 + (u^2+ \epsilon x)^2 v^2$. At $\epsilon=0$ this describes a collision between an $\mathfrak{e}_6$ at $u = 0$ and an $\mathfrak{su}_3$ curve at $v = 0$; for $\epsilon\neq 0$ the $u = 0$ curve instead supports an $\mathfrak{su}_6$ gauge algebra. Once again, however, it is quicker to use a combination of field theory techniques and F-theory intuition. There is a new type of Higgs flow that did not appear earlier: see for example the flows $3A_1\to A_2$ or $D_4(a_1)\to D_4$ in figure \[fig:E6\]. This type of flow was discussed around [@Heckman:2015ola Eq.(4.24)]. In $3A_1$, we can shrink the leftmost $-1$ curve, we reveal another $-1$ curve; if we also shrink that one as well, we have a special point on the leftmost $\mathfrak{e}_6$ curve of multiplicity 2. The flow consists of going to a more generic situation where there are two special points of multiplicity 1; blowing them up produces two separate $-1$ curves touching the $\mathfrak{e}_6$ curve, which we see in the $A_2$ theory. These are examples of “small instanton maneuvers” of the type encountered in subsection \[ssec:estring\].
Another new point is that in some examples the flavor symmetry expected from the B-C labels refines the “naïve” expectation one would have from just treating subsectors of a field theory on its tensor branch in isolation. In some cases, this also conforms with restrictions on non-abelian flavor symmetries expected from F-theory considerations. In other cases, however, we find that —especially for abelian symmetry factors— the B-C label analysis provides a systematic way to extract such flavor symmetries which are difficult to deduce using other techniques. We develop this point further in section \[sec:FLAVOR\].
An important aspect of the tight match found here is that in general, we find several gauge groups of $\mathfrak{e}_{n}$ type will generically be Higgsed in a given flow by conformal matter vevs. This is not altogether surprising since related phenomena are already present for models with weakly coupled hypermultiplets. Indeed in the quivers in the third column of figure \[fig:su4\] we see that the ranks of the gauge groups decrease in an RG flow not only in the rightmost position. There, it is a consequence of the fact that there will typically be a propagating sequence of D-term constraints.
\(1) \[startstop, xshift=-1cm\] [ $
0: [E_6] \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(2) \[startstop, below of=1\] [ $
A_1: [SU(6)] \,\, \overset{\mathfrak{su_{3}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(3) \[startstop, below of=2\] [ $
2 A_1: [SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(4) \[startstop, below of=3\] [ $
3 A_1:[SU(2)] \,\, 2 \,\, \underset{[SU(3)]}1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(5) \[startstop, xshift=0cm, below of=4\] [ $
A_2: [SU(3)] \,\, 1 \,\, \underset{[SU(3)]}{\underset{1}{\overset{\mathfrak{e_{6}}}6}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(6) \[startstop, xshift=0cm, below of=5\] [ $
A_2 + A_1: [SU(3)] \,\, 1 \,\, {\underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]
$]{};
\(7) \[startstop, xshift=-4cm, below of=6\] [ $
2 A_2: [G_2] \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(8) \[startstop, xshift=6cm, below of=6\] [ $
A_2 +2 A_1: [SU(2)]\,\,\overset{\mathfrak{e_{6}}}4 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(9) \[startstop, xshift=-10cm, below of=8\] [ $
2 A_2 + A_1: [SU(2)]\,\,\overset{\mathfrak{f_{4}}}4 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \, \, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(10) \[startstop, xshift=0cm, below of=8\] [ $
A_3: [Sp(2)] \,\, {\overset{\mathfrak{so_{10}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(11) \[startstop, xshift=-6cm, below of=10\] [ $
A_3+A_1: [SU(2)] \,\, {\overset{\mathfrak{so_{9}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(12) \[startstop, xshift=0cm, below of=11\] [ $
D_4(a_1): {\overset{\mathfrak{so_{8}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(13) \[startstop, xshift=-4cm, below of=12\] [ $
A_4: [SU(2)] \,\, {\overset{\mathfrak{so_{7}}}3} \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(14) \[startstop, xshift=0cm, below of=13\] [ $
A_4+A_1: {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]
$]{};
\(15) \[startstop, xshift=8cm, right of=14\] [ $
D_4: {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \underset{[SU(3)]}{\underset{1}{\overset{\mathfrak{e_{6}}}6}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(16) \[startstop, xshift=0cm, below of=14\] [ $
A_5: [SU(2)] \,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\,\overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(17) \[startstop, xshift=0cm, below of=15\] [ $
D_5 (a_1): {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(18) \[startstop, xshift=-5cm, below of=17\] [ $
E_6 (a_3): {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(19) \[startstop, xshift=0cm, below of=18\] [ $
D_5: {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(20) \[startstop, xshift=0cm, below of=19\] [ $
E_6 (a_1): {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(21) \[startstop, xshift=0cm, below of=20\] [ $
E_6: 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]
$]{};
\(1) – (2); (2) – (3); (3) – (4); (4) – (5); (5) – (6); (6) – (7); (6) – (8); (7) – (9); (8) – (9); (8) – (10); (10) – (11); (9) – (11); (11) – (12); (12) – (13); (13) – (14); (12) – (15); (14) – (16); (14) – (17); (15) – (17); (16) – (18); (17) – (18); (18) – (19); (19) – (20); (20) – (21);
> >
\(1) \[startstop, xshift=-1cm\] [ $
1: [G_2] \,\, \overset{\mathfrak{su}_2}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[G_2]
$]{};
\(2) \[startstop, below of=1\] [ $
A_1: [SU(2)] \,\, 2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ... [G_2]
$]{};
\(3) \[startstop, below of=2\] [ $
\widetilde A_1: [SU(2)] \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[G_2]
$]{};
\(4) \[startstop, below of=3\] [ $
G_2(a_1): \overset{\mathfrak{e_{8}}}{9} \,\, 1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[G_2]
$]{};
\(5) \[startstop, below of=4\] [ $
G_2: \overset{\mathfrak{e_{7}}}{8} \,\, 1 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[G_2]
$]{};
\(1) – (2); (2) – (3); (3) – (4); (4) – (5);
Flows from $\mathfrak{f}_{4}$, $\mathfrak{g}_{2}$
-------------------------------------------------
Finally, as a last class of examples, we also consider flows induced by nilpotent orbits for the non-simply laced algebras $\mathfrak{f}_{4}$ and $\mathfrak{g}_{2}$. Actually, we can reach all of these flows by first considering a nilpotent orbit which has commutant subalgebra $\mathfrak{f}_{4}$ and $\mathfrak{g}_{2}$, and then adding an additional nilpotent element which embeds in this subalgebra. This is quite similar to our analysis of flavor symmetries of $\mathfrak{so}_{\text{odd}}$ type. Alternatively, we can work out the F-theory geometries obtained from such nilpotent orbits. The results of this final set of analyses, along with the partially ordered set of RG flows / nilpotent elements is displayed in figures \[fig:g2flows\] and \[fig:f4flows\].
As a curiosity, we also notice that the diagrams for $\mathfrak{f}_{4}$ and $\mathfrak{g}_{2}$ can be embedded into the one for $\mathfrak{e}_8$. The reason is that both $\mathfrak{f}_{4}$ and $\mathfrak{g}_{2}$ appear in the $E_8-E_8$ conformal matter theory. In the $\mathfrak{e}_8$ nilpotent hierarchy, the theory labeled $D_4$ (see the table of appendix \[app:E8\]), for example, is almost identical to the theory labeled 1 in figure \[fig:f4flows\]; the only difference is that the leftmost $\mathfrak{e}_8$ is on a $-11$ curve rather than on a $-12$ curve. Starting from this $D_4$ theory, then, we can reproduce all the flows that appear in the $\mathfrak{f}_4$ diagram of figure \[fig:f4flows\]; the theories of that figure have almost identical avatars in the $\mathfrak{e}_8$ nilpotent hierarchy. We show the correspondence in figure \[fig:subF4G2\]. One can check in [@Chacaltana:2012zy Table 19] that the theories shown in that diagram are indeed in the correct inclusion relation for $\mathfrak{e}_8$. Thus, the $\mathfrak{f}_4$ nilpotent hierarchy is isomorphic to a sub-hierarchy of the $\mathfrak{e}_8$ nilpotent hierarchy. Similarly, one can check that the $\mathfrak{g}_2$ is also a sub-hierarchy of the $\mathfrak{f}_4$ hierarchy, as also summarized in figure \[fig:subF4G2\].
\(1) \[startstop, xshift=-1cm\] [ $
1: [F_4] \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(2) \[startstop, below of=1\] [ $
A_1: [Sp(3)] \,\, \overset{\mathfrak{g_{2}}}2 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(3) \[startstop, below of=2\] [ $
\widetilde A_1: [SU(4)] \,\, \overset{\mathfrak{su_{3}}}2 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(4) \[startstop, below of=3\] [ $
A_1 + \widetilde A_1: [SO(4)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, \underset{[N_f=1]}{\overset{\mathfrak{su_{2}}}2} \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(5) \[startstop, below of=4, xshift=-4cm\] [ $
A_2: \overset{\mathfrak{su_{1}}}2 \,\, \underset{[SU(3)]}{\overset{\mathfrak{su_{2}}}2} \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(6) \[startstop, below of=4, xshift=4cm\] [ $
\widetilde A_2: [G_2] \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ...[F_4]
$]{};
\(7) \[startstop, below of=6, yshift=-2cm\] [ $
\widetilde A_2 +A_1: [SU(2)] \,\, 2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ...[F_4]
$]{};
\(8) \[startstop, below of=5\] [ $
A_2 + \widetilde A_1: [SU(2)] \,\,2\,\, 2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ...[F_4]
$]{};
\(9) \[startstop, below of=8\] [ $
B_2: [SU(2)] \,\, 2 \,\, 1\,\, \underset{[SU(2)]}{\underset{2}{\underset{1}{\overset{\mathfrak{e_{8}}}{12}}}} \,\, 1 \,\, ...[F_4]
$]{};
\(10) \[startstop, below of=9, xshift=5cm\] [ $
C_3(a_1): [SU(2)] \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{10} \,\, 1 \,\, ...[F_4]
$]{};
\(11) \[startstop, below of=10, xshift=0cm\] [ $
F_4(a_3): \overset{\mathfrak{e_{8}}}{8} \,\, 1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[F_4]
$]{};
\(12) \[startstop, below of=11, xshift=-4cm\] [ $
B_3: \overset{\mathfrak{e_{7}}}{8} \,\, \underset{[SU(2)]}1 \,\, 2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[F_4]
$]{};
\(13) \[startstop, below of=11, xshift=4cm\] [ $
C_3: [SU(2)] \,\, 1 \,\, \overset{\mathfrak{e_{7}}}{8} \,\, 1 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[F_4]
$]{};
\(14) \[startstop, below of=13, xshift=-4cm, yshift=.1cm\] [ $
F_4(a_2): \overset{\mathfrak{e_{7}}}{7} \,\, 1 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[F_4]
$]{};
\(15) \[startstop, below of=14, xshift=0cm, yshift=.2cm\] [ $
F_4(a_1): {\overset{\mathfrak{e_{6}}}{6}} \,\, 1 \,\, {\overset{\mathfrak{su_3}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1\,\, ...[F_4]
$]{};
\(16) \[startstop, below of=15, xshift=0cm, yshift=.3cm\] [ $
F_4: \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, {\overset{\mathfrak{g_2}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 2 \,\, 1 \,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1\,\, ...[F_4]
$]{};
\(1) – (2); (2) – (3); (3) – (4); (4) – (5); (4) – (6); (5) – (8); (6) – (7); (8) – (7); (8) – (9); (7) – (10); (9) – (10); (10) – (11); (11) – (12); (11) – (13); (12) – (14); (13) – (14); (14) – (15); (15) – (16);
\(1) \[startstop, xshift=-1cm\] [ $
1 \to D_4 $]{};
\(2) \[startstop, below of=1\] [ $
A_1 \to D_4 + A_1
$]{};
\(3) \[startstop, below of=2\] [ $
\widetilde A_1 \to D_5(a_1)
$]{};
\(4) \[startstop, below of=3\] [ $
A_1 + \widetilde A_1 \to D_5(a_1)+A_1
$]{};
\(5) \[startstop, below of=4, xshift=-4.2cm, yshift=.5cm\] [ $
A_2\to D_4+ A_2
$]{};
\(6) \[startstop, below of=4, xshift=4.2cm, yshift=.5cm\] [ $
\widetilde A_2\to \underline{E_6(a_3)}
$]{};
\(7) \[startstop, below of=6, yshift=-.5cm\] [ $
\widetilde A_2 +A_1\to \underline{E_6(a_3)+A_1}
$]{};
\(8) \[startstop, below of=5\] [ $
A_2 + \widetilde A_1\to D_5(a_1)+A_2
$]{};
\(9) \[startstop, below of=8,xshift=.5cm\] [ $
B_2 \to D_6(a_2)
$]{};
\(10) \[startstop, below of=4, yshift=-2.5cm\] [ $
C_3(a_1)\to \underline{E_7(a_5)}
$]{};
\(11) \[startstop, below of=10, xshift=0cm\] [ $
F_4(a_3)\to \underline{E_8(a_7)}
$]{};
\(12) \[startstop, below of=11, xshift=-2cm\] [ $
B_3 \to A_6+A_1
$]{};
\(13) \[startstop, below of=11, xshift=2cm\] [ $
C_3\to E_7(a_4)
$]{};
\(14) \[startstop, below of=13, xshift=-2cm\] [ $
F_4(a_2)\to \underline{D_5+A_2}
$]{};
\(15) \[startstop, below of=14, xshift=0cm\] [ $
F_4(a_1)\to E_6(a_1)+A_1
$]{};
\(16) \[startstop, below of=15, xshift=0cm\] [ $
F_4 \to E_6+A_1
$]{};
\(1) – (2); (2) – (3); (3) – (4); (4) – (5); (4) – (6); (5) – (8); (6) – (7); (8) – (7); (8) – (9); (7) – (10); (9) – (10); (10) – (11); (11) – (12); (11) – (13); (12) – (14); (13) – (14); (14) – (15); (15) – (16);
Short Quivers {#sec:SHORT}
=============
Up to this point, we have assumed that the generalized quivers of our 6D SCFTs were sufficiently long to Higgs the left and right of the quiver independently. Strictly speaking, even when this is not the case we can continue to parameterize all flows according to two independent nilpotent orbits. However, the resulting flow will then contain various redundancies since the data associated with this pair will inevitably become correlated. Our plan in this section will be to extend our analysis of flows to theories where this happens, which we will call “short quivers.”
The picture is clearest in the case of flows from $\mathfrak{su}_N$. Here, the allowed Higgsings are characterized by a partition on the left of the quiver and a partition on the right. The non-redundant data of such flows is captured by a pair of partitions of equal size. Moreover, each column of a partition corresponds to the change in gauge group rank between neighboring nodes. If there are $(k-1)$ tensor multiplets in the theory, then there can be up to $k$ changes in the rank of the associated symmetry algebra (including the leftmost and rightmost flavor symmetries). So, there are at most total $k$ columns in the two partitions. For a large quiver $k \gg N$, and the restriction on the number of columns of the partition simply comes from the size of each partition, $N$. For small quivers, on the other hand, the requirement that the total number or columns should be at most $k$ places important constraints.
As an example, we list the theories with three tensor multiplets and partitions of size three: $$\begin{gathered}
(1^3): [SU(3)] \,\, \overset{\mathfrak{su3}}2 \,\,\overset{\mathfrak{su3}}2 \,\, \overset{\mathfrak{su3}}2 \,\, [SU(3)] : (1^3) \\
(1^3): [SU(3)] \,\, \overset{\mathfrak{su3}}2 \,\, \underset{[N_f=1]}{\overset{\mathfrak{su3}}2} \,\, \overset{\mathfrak{su2}}2 \,\, [SU(1)] : (2,1)\\
(1^3): [SU(4)] \,\, \overset{\mathfrak{su3}}2 \,\,\overset{\mathfrak{su2}}2 \,\, \overset{\mathfrak{su1}}2 : (3) \\
(2,1): [SU(1)] \,\, \overset{\mathfrak{su2}}2 \,\, \underset{[SU(2)]}{\overset{\mathfrak{su3}}2} \,\, \overset{\mathfrak{su2}}2 \,\, [SU(1)] : (2,1)\end{gathered}$$ where for the purposes of uniformity with higher rank examples we have listed the (trivial) flavor symmetry factor $SU(1)$ which in F-theory is associated with a component of the discriminant locus with $I_1$ fiber type.
For longer quivers, we could also consider the flows corresponding to partitions $\mu_L = (3), \mu_R =(2,1)$ and $\mu_L=(3) , \mu_R=(3)$. However, since we only have three hypermultiplets in the case at hand, we are constrained to consider pairs of partitions with no more than four columns, so we need not concern ourselves with such flows.
Similar comments apply for the $BCDEFG$ theories. We illustrate it with a discussion of $E_6$ nilpotent orbits. Here, the analog to the “number of columns of the partition" in the $\mathfrak{su}_N$ case is the distance that the breaking pattern propagates into the interior of the quiver, that is, the number of $E_6$ gauge group factors which are (partially) broken. For instance, the nilpotent orbits in figure \[fig:E6\] with B–C labels $0, A_1, 2 A_1, 3 A_1, A_2, A_1+A_1$, and $A_2 + 2 A_1$ do not introduce any breaking into the interior of the quiver. Even for a theory with a single $\mathfrak{e}_6$ node, it is possible to trigger an RG flow from any of these nilpotent orbits on the left or the right. Two such examples are $$\begin{gathered}
A_1 : [SU(6)] \,\, \overset{\mathfrak{su}_3}2 \,\, 1 \,\, \underset{[N_f=1]}{\overset{\mathfrak{e}_6}5} \,\, 1 \,\, [SU(3)] : A_2 + A_1 \\
3 A_1: [SU(2)] \,\, 2 \,\, \underset{[SU(3)]}1 \,\, {\overset{\mathfrak{e}_6}6} \,\, 1 \,\, \overset{\mathfrak{su}_3}3 \,\, 1 \,\,[E_6] : 0\end{gathered}$$ On the other hand, nilpotent orbits such as the one of B–C label $D_5$ propagate several nodes into the interior of the quiver. For quivers with a single $\mathfrak{e}_6$ node, we can ignore these nilpotent elements.
Global Symmetries in 6D SCFTs {#sec:FLAVOR}
=============================
One of the important aspects of the characterization of RG flows in terms of nilpotent orbits is that this is *algebraic* data directly associated with a conformal fixed point. Assuming the absence of an emergent flavor symmetry in the IR, we can then use the labelling by nilpotent orbits to read off the flavor symmetry for IR fixed points.
Indeed, we have performed a match between a particular class of 6D SCFTs and nilpotent orbits for classical and exceptional algebras. In many cases, the global symmetry which is manifest on the tensor branch matches to what is expected from the nilpotent orbit. An example is the theory $$[SU(6)] \,\, \overset{\mathfrak{su_{3}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6],$$ which corresponds to the nilpotent orbit of $E_6$ with B–C label $A_1$. However, there are other instances in which the global symmetry of a 6D SCFT cannot be easily determined from the theory on the tensor branch. In particular, as discussed in [@Bertolini:2015bwa], there are instances in which the expected field theoretic global symmetry does not match the global symmetry predicted by F-theory. An example is the theory with tensor branch, $$[SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6].
\label{so7example}$$ This is the theory associated with nilpotent orbit of $E_6$ with B–C label $2 A_1$. The “naïve” field theoretic expectation is that there should be an $SO(8)$ acting on the eight half-hypermultiplets of $SU(2)$, whereas F-theory only permits an $\mathfrak{so}(7)$ flavor curve to meet the $\mathfrak{su}(2)$ gauge algebra. However, in [@Ohmori:2015pia], it was argued that the naïve field theoretic expectation is wrong in this instance, and the correct global symmetry of the field theory matches the prediction from F-theory, with the eight half-hypermultiplets transforming in the spinor of $\mathfrak{so}(7)$. We note that this also matches the global symmetry predicted from the data of the corresponding nilpotent orbit.
This example dealt with the simple case of an $I_2$ Kodaira fiber type over the leftmost $-2$ curve. But the business of determining global symmetries for 6D SCFTs becomes even more involved once we consider theories with $I_1$, $II$, $III$, and $IV$ fiber types. The fibers $I_0$, $I_1$, and $II$ all lead to trivial gauge algebras; $I_2$ and $III$ both lead to $\mathfrak{su}(2)$ gauge algebras; and the split $I_3$ and $IV$ fibers both lead to $\mathfrak{su}(3)$ gauge algebras. Nevertheless, the expectation from geometry is that they lead to different global symmetries [@Bertolini:2015bwa; @Morrison:2016djb]. This leads to the natural question: do theories with distinct fiber types but identical gauge algebras lead to distinct 6D SCFTs? If not, what is the correct global symmetry for these theories? If so, does the F-theory prediction always match the global symmetry seen in field theory?
The analysis of the present paper sheds light on these questions. We expect that the continuous component of the global symmetry of a 6D SCFT can be read off directly from the commutant of the nilpotent orbit. Indeed, in all cases in which the global symmetry of the 6D SCFT is well understood, including the subtle case of line (\[so7example\]), we find this is indeed the case.[^7] Under the assumption that this holds generally, we compare the global symmetries of the 6D SCFTs to the F-theory prediction. We find that the global symmetry group of a 6D SCFT always contains the global symmetry group predicted by F-theory, and in many cases this containment is proper. We also find no evidence that theories with identical gauge algebras but distinct fiber types should correspond to distinct 6D SCFTs up to different numbers of free hypermultiplets.
For a first example, consider the theory corresponding to the $E_7$ nilpotent orbit of B–C label $A_3+A_2+A_1$, $$[SU(2)] \,\, {\overset{\mathfrak{e_{7}}}5} \,\, 1 \,\, ...$$ The global symmetry here is evidently $SU(2)$, rotating the three half-hypermultiplets of $\mathfrak{e}_7$ as a triplet, but as was shown in [@Bertolini:2015bwa], F-theory does not permit any flavor curves to meet a curve carrying gauge algebra $\mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7$, or $\mathfrak{e_8}$. Instead, it appears that a flavor symmetry emerges at the origin of the tensor branch (i.e. the SCFT point of the moduli space), matching the field-theoretic expectation (c.f. Table 5.1 of [@Bertolini:2015bwa]) rather than the F-theory prediction.
A similar story arises in the case of the $E_7$ theory corresponding to B–C label $2A_2$. This theory has $G_2 \times SU(2)$ global symmetry. The gauge algebras of the theory may be realized in several different ways within F-theory, two of which are as follows: $$[I_0^{*,ns}] \,\, {\overset{IV^{ns}}2} \,\, {\overset{II}2}\,\, \overset{I_0}1 \,\, \overset{III^*}8 \,\, ...$$ $$[I_3] \,\, {\overset{I_2}2} \,\, {\overset{I_1}2}\,\, \underset{[I_2]}{\overset{I_0}1} \,\, \overset{III^*}8 \,\, ...
\label{2A2}$$ Here, the Kodaira fiber types in brackets are supported on non-compact flavor curves. The first theory has a $G_2$ flavor symmetry living on the non-compact curve with fiber type $I_0^{*,ns}$, but no non-Abelian flavor curve may touch the curve of self-intersection $-1$ with $I_0$ fiber type [@miranda1986extremal; @miranda1990persson; @persson1990configurations]. In the second case, on the other hand, an $SU(2)$ flavor curve of Kodaira type $I_2$ does touch the $I_0$ curve, but the global symmetry on the left is reduced from $G_2$ to $SU(3)$. Thus, there is one F-theory configuration in which the $G_2$ flavor symmetry on the left is apparent and one F-theory configuration in which the $SU(2)$ flavor symmetry below is apparent, but there is no F-theory configuration in which the full $G_2 \times SU(2)$ symmetry is realized. It appears that upon flowing to the IR, the flavor symmetry acting on the hypermultiplets of this theory is the maximal symmetry group acting on those hypermultiplets in any F-theory realization of the model.
Consider next the theory corresponding to B–C label $ A_2 + 2 A_1$: $$[SO(4)] \,\, {\overset{\mathfrak{su_{2}}}2} \,\, \underset{[SU(2)]}{\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ...$$ Here, the flavor symmetry expected from F-theory is simply $SU(2) \times SU(2)$, coming from a non-compact $I_2$ flavor curve hitting each of the two $-2$ curves with $\mathfrak{su}_2$ gauge algebras. However, the symmetry is enhanced from $\mathfrak{su}_2 \times \mathfrak{su}_2$ to $\mathfrak{su}_2 \times \mathfrak{su}_2 \times \mathfrak{su}_2$.
The theory corresponding to the $E_8$ orbit with B–C label $D_4 + A_2$ has an $SU(3)$ global symmetry: $$2\,\, \underset{[SU(3)]}{\overset{\mathfrak{su}_2}2 } \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$$ We should think of this $SU(3)$ as rotating three hypermultiplets charged under the $\mathfrak{su}_2$ gauge symmetry. An additional half-hypermultiplet of the ${\mathfrak}{su_2}$ lives at the intersection with each unpaired $-2$ tensor.
Another important point is that theories with identical gauge algebras never show up as distinct nilpotent orbits. The two F-theory models of (\[2A2\]) provide one such example. Another particularly interesting case is the $E_7$ nilpotent orbit with B–C label $A_3 + 2 A_1$: $$[SU(2)] \,\, 2 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1 \,\, ...$$ The gauge algebras shown can be realized in F-theory with either a $I_0$ fiber, an $I_1$ fiber, or a $II$ fiber on the empty $-2$ curve. The fact that these do not correspond to different nilpotent orbits of $E_7$ is a possible indication that all three of these F-theory realizations give the same 6D SCFT up to different numbers of free hypermultiplets.[^8] If we decompactify all base curves besides this $-2$ curve (corresponding to a flow along the tensor branch), we are left with a theory of just a $-2$ curve of fiber type $I_0$, $I_1$, and $II$, respectively. Assuming that all three of these fiber types do indeed give the same 6D SCFT before this tensor branch flow, we find that the resulting 6D SCFTs after the flow must be identical as well (modulo free hypermultiplets). Thus, we conjecture that the interacting sector of these three theories are the same and given by the $A_1$ $(2,0)$ 6D SCFT.
Of course, the other possibility is that these distinct F-theory models do give rise to distinct 6D SCFTs, but that only one of them can be realized by an RG flow parameterized by a nilpotent orbit. This would itself be a rather surprising result. Determining which solution is the correct one is left as a question for future study.
As a final set of comments, we note that we have also presented evidence for IR fixed points with *abelian* flavor symmetries, a fact which is quite straightforward using the algebraic data of nilpotent orbits. By contrast, identifying such symmetry factors from a geometric perspective can sometimes be subtle. Roughly speaking, we would like to associate such abelian symmetry factors with non-compact components of the discriminant locus supporting a singular $I_1$ fiber. Observe, however, that at least for *gauge* theories (i.e. fibers supported on compact curves), an $I_n$ fiber is expected to realize an $\mathfrak{su}_n$ rather than $\mathfrak{u}_n$ gauge algebra. The distinction boils down to the fact that for a 6D SCFT on its tensor branch, this additional $\mathfrak{u}(1)$ factor is anomalous, and so inevitably decouples anyway via the Stückelberg mechanism. For flavor symmetries, however, there is a priori no such issue. Indeed, in many of the examples encountered earlier, we can clearly see that the presence of an additional $\mathfrak{u}(1)$ correlates tightly with such $I_n$ fibers. We have also seen that in some breaking patterns, there is an overall tracelessness condition, for example with flavor symmetry algebras such as $\mathfrak{s}(\mathfrak{u}(n_1) \oplus ... \oplus \mathfrak{u}(n_l))$. We take this to mean that these $\mathfrak{u}(1)$ flavor symmetries can in general be delocalized in the geometry, that is, they are spread over multiple components of the discriminant locus.
For this reason, we have not assigned the presence of $\mathfrak{u}(1)$’s to specific locations in the diagrams of figures \[fig:so8flows\]–\[fig:f4flows\], as we did for non-abelian symmetries. Their presence can be read off from (\[eq:soflavor\]) and (\[eq:spflavor\]) for figures \[fig:so8flows\]–\[fig:so9flows\], and is shown explicitly in the tables of appendix \[app:nilp\]. In many cases, there is a clear guess as to the origin of the abelian symmetries, coming from the presence of a hypermultiplet localized at the collision of a compact curve with a non-compact curve. In other cases, they are associated to an E-string which has a gauged subgroup of $E_8$ whose commutant has one or more $\mathfrak{u}(1)$. For example, for the $E_6$ nilpotent orbit $2A_1$ in appendix \[app:E6\] (or figure \[fig:E6\]) we see an E-string with gauged subalgebra $\mathfrak{su}_2 \oplus \mathfrak{e}_6$; or in theory $(3^2,1^2)$ we see an E-string with gauged subalgebra $\mathfrak{su}_3 \oplus \mathfrak{so}_8$.
It would be interesting to further explore the extent to which such abelian flavor symmetry factors (both continuous and discrete) can be deduced more directly from the geometric perspective.
Conclusions \[sec:CONC\]
========================
In this note we have studied renormalization group flows between 6D SCFTs induced by vevs for conformal matter. Focusing on the case of T-brane vacua i.e. those vacua labeled by the orbits of nilpotent elements of a flavor symmetry algebra, we have first of all established a direct correspondence between certain nilpotent orbits, and a class of F-theory geometries. An important aspect of this analysis is that the natural notion of partial ordering of elements in the nilpotent cone of a simple Lie algebra has a direct physical interpretation in terms of hierarchies of renormalization group flows. Moreover, we have also used this algebraic data to calculate the unbroken flavor symmetry of the IR fixed point. To reinforce this point, we have considered explicit examples of generalized quiver theories with flavor symmetries of type ABCDEFG. We have used these examples to study global symmetries in 6D SCFTs, finding that the global symmetry read off from the nilpotent orbit can be larger than the global symmetry predicted from F-theory. In the remainder of this section we discuss some avenues for future investigation.
In the case of $\mathfrak{su}_N$ and $\mathfrak{so}_{\text{even}}$ theories, we remarked that by taking transposed partitions, our nilpotent hierarchy of RG flows extends to flows between theories of different maximal gauge group rank such as $$[SU_{10}]-SU_{10}-...-SU_{10}-[SU_{10}]\rightarrow\lbrack SU_{9}]-SU_{9}-...-SU_{9}-[SU_{9}].$$ It would be interesting to extend this analysis to exceptional algebras. Establishing this sort of correspondence in more detail would provide an opportunity to potentially map out the full class of possible RG flows from a UV parent theory. This would bring us significantly closer to the ambitious goal of classifying *all* RG flows between 6D SCFTs.
In our analysis, we primarily focused on theories which have a sufficiently large number of tensor multiplets. Indeed, the parent theories we have started with all have known holographic duals which take the form $AdS_{7}\times
S^{4}/\Gamma_{ADE}$. The effects of the nilpotent element vevs are primarily confined to a small region of the quiver theory, which in the holographic dual will correspond (in units where the radius of the sphere is one) to an order $1/N$ size effect. It would be quite interesting to confirm this picture directly in the holographic dual, perhaps by evaluating a protected quantity such as the conformal anomalies of the 6D SCFT.
Finally, it would be interesting to also study how the data of conformal matter vevs as parameterized by nilpotent orbits shows up in little string theories (see e.g. [@Bhardwaj:2015oru]). We arrive at examples of little string theories by compactifying M5-branes on the background $S^{1}\times\mathbb{C}^{2}/\Gamma_{ADE}$. When we do so, the independent data about partitions used to label possible flows are now identified, and always appear with gauge group factors rather than flavor group factors (there are none for the circular quivers). This in turn means that the purely local perturbations induced by a choice of partition now propagate out to the entire generalized quiver, providing a rather novel window into flows for more general 6D theories.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank J. Distler, T. Dumitrescu, S. Gukov, N. Mekareeya, D.R. Morrison and C. Vafa for helpful discussions. We also thank the Simons Center for Geometry and Physics 2015 summer workshop for hospitality during the initial stages of this project. JJH also thanks the theory groups at Columbia University, the ITS at the CUNY graduate center, and the CCPP at NYU for hospitality during the completion of this work. The work of JJH is supported by NSF CAREER grant PHY-1452037. JJH also acknowledges support from the Bahnson Fund at UNC Chapel Hill as well as the R. J. Reynolds Industries, Inc. Junior Faculty Development Award from the Office of the Executive Vice Chancellor and Provost at UNC Chapel Hill. The work of TR is supported by NSF grant PHY-1067976. TR is also supported by the NSF GRF under DGE-1144152. AT is supported in part by INFN, by the MIUR-FIRB grant RBFR10QS5J String Theory and Fundamental Interactions, and by the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) - ERC Grant Agreement n. 307286 (XD-STRING).
Nilpotent Flows for E-type Flavor Symmetries {#app:nilp}
============================================
In this Appendix we collect the full list of nilpotent orbits for exceptional E-type flavor symmetries, and the corresponding F-theory model associated with each such flow. We also present the unbroken flavor symmetry for each such model which is predicted by the choice of a nilpotent element.
$E_6$ Nilpotent Orbits {#app:E6}
----------------------
The $E_6$ Nilpotent orbits are as follows. The nilpotent hierarchy is given in figure \[fig:E6\].
B–C Label Global Symmetry Theory
---------------- ----------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 $E_6$ $[E_6] \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$A_1$ $SU(6)$ $[SU(6)] \,\, \overset{\mathfrak{su_{3}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$2 A_1$ $ Spin(7)\times U(1)$ $[SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$3 A_1 $ $ SU(3) \times SU(2)$ $[SU(2)] \,\, 2 \,\, \underset{[SU(3)]}1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$A_2$ $SU(3) \times SU(3)$ $[SU(3)] \,\, 1 \,\, \underset{[SU(3)]}{\underset{1}{\overset{\mathfrak{e_{6}}}6}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$A_2 + A_1$ $SU(3)\times U(1)$ $[SU(3)] \,\, 1 \,\, {\underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, ...[E_6]$
$2 A_2$ $G_2$ $[G_2] \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$A_2 +2 A_1$ $ SU(2)\times U(1)$ $[SU(2)]\,\,{\overset{\mathfrak{e_{6}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$2 A_2 + A_1 $ $ SU(2)$ $ [SU(2)]\,\,{\overset{\mathfrak{f_{4}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$A_3$ $Sp(2)\times U(1)$ $ [Sp(2)] \,\, {\overset{\mathfrak{so_{10}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$A_3+A_1$ $ SU(2)\times U(1)$ $ [SU(2)] \,\, {\overset{\mathfrak{so_{9}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$D_4(a_1)$ $U(1)^2$ $ {\overset{\mathfrak{so_{8}}}4} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$A_4$ $SU(2)\times U(1)$ $ [SU(2)] \,\, {\overset{\mathfrak{so_{7}}}3} \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$A_4+A_1 $ $U(1) $ $ {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\, ...[E_6]$
$D_4$ $ SU(3)$ $ {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \underset{[SU(3)]}{\underset{1}{\overset{\mathfrak{e_{6}}}6}} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$A_5$ $ SU(2)$ $ [SU(2)] \,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\,\overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$D_5 (a_1)$ $ U(1)$ ${\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$E_6 (a_3)$ $ 1 $ $ {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$D_5$ $U(1)$ $ {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$E_6 (a_1)$ 1 $ {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
$E_6$ 1 $2 \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{g_2}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\,\overset{\mathfrak{e_{6}}}6 \,\, 1 \,\,...[E_6]$
\[E6list\]
$E_7$ Nilpotent Orbits
----------------------
The $E_7$ Nilpotent orbits are as follows. The nilpotent hierarchy can be found for example in [@Chacaltana:2012zy Table 16].
B–C Label Global Symmetry Theory
------------------ ----------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 $E_7$ $[E_7] \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{so_{7}}}3 \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_1$ $SO(12)$ $[SO(12)] \,\, \overset{\mathfrak{sp_{1}}}1 \,\, \overset{\mathfrak{so_{7}}}3 \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$2A_1$ $SO(9)\times SU(2)$ $[SO(9)] \,\, 1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{7}}}3} \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$(3A_1)'$ $Sp(3)\times SU(2)$ $[Sp(3)] \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{7}}}2} \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$(3A_1)''$ $F_4$ $[F_4] \,\, 1 \,\, {\overset{\mathfrak{g_{2}}}3} \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$4A_1$ $Sp(3)$ $[Sp(3)] \,\, {\overset{\mathfrak{g_{2}}}2} \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_2$ $SU(6)$ $[SU(6)] \,\, {\overset{\mathfrak{su_{4}}}2} \,\, \overset{\mathfrak{su_{2}}}2\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_2+A_1$ $SU(4)\times U(1)$ $[SU(4)] \,\, {\overset{\mathfrak{su_{3}}}2} \,\, \underset{[N_f=1]}{\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_2+2 A_1$ $SU(2) \times SU(2) \times SU(2)$ $[SO(4)] \,\, {\overset{\mathfrak{su_{2}}}2} \,\, \underset{[SU(2)]}{\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$2A_2$ $G_2 \times SU(2)$ $[G_2] \,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{su_{1}}}2}\,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_2+ 3A_1$ $G_2$ $ {\overset{\mathfrak{su_{1}}}2} \,\, \underset{[G_2]}{\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$2A_2+ A_1$ $SU(2) \times SU(2)$ $[SU(2)] \,\, {2} \,\, 2 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_3$ $SO(7) \times SU(2)$ $[SO(7)] \,\, \overset{\mathfrak{so}_7}2 \,\, 1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1 \,\, ... [E_7]$
$(A_3 +A_1)'$ $SU(2) \times SU(2) \times SU(2)$ $[SU(2)] \,\, 2 \,\, \underset{[SU(2)]}1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1 \,\, ... [E_7]$
$(A_3 +A_1)''$ $SO(7) $ $[SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1 \,\, ... [E_7]$
$A_3 +2A_1$ $SU(2) \times SU(2) $ $[SU(2)] \,\, 2 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1 \,\, ... [E_7]$
$D_4(a_1)$ $SU(2) \times SU(2) \times SU(2)$ $[SU(2)] \,\,1 \,\, \overset{[SU(2)]}{\overset{1}{\underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}}}} \,\, 1 \,\, ... [E_7]$
$D_4(a_1)+A_1$ $SU(2) \times SU(2)$ $[SU(2)] \,\,1 \,\, \overset{[SU(2)]}{\overset{1}{{\overset{\mathfrak{e_{7}}}7}}} \,\, 1 \,\, ... [E_7]$
$A_3+A_2$ $SU(2) \times U(1)$ $[SU(2)] \,\, 1 \,\, \underset{[N_f=1]}{\overset{\mathfrak{e_{7}}}6} \,\, 1 \,\, ... [E_7]$
$D_4$ $Sp(3)$ $[Sp(3)] \,\, {\overset{\mathfrak{so_{12}}}4} \,\,{\overset{\mathfrak{sp_{1}}}1} \,\,{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{1}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_3+A_2+A_1$ $SU(2) $ $[SU(2)] \,\, {\overset{\mathfrak{e_{7}}}5} \,\, 1 \,\, ... [E_7]$
$A_4$ $SU(3) \times U(1)$ $[SU(3)] \,\, 1 \,\, {\overset{\mathfrak{e_{6}}}6} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{so_{7}}}3 \,\, ... [E_7]$
$A_4+A_1$ $U(1)^2 $ $ \underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{so_{7}}}3 \,\, ... [E_7]$
$D_4+A_1$ $Sp(2)$ $[Sp(2)] \,\, {\overset{\mathfrak{so_{11}}}4} \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp_{1}}}1} \,\,{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$D_5(a_1)$ $SU(2)\times U(1)$ $[Sp(1)] \,\, {\overset{\mathfrak{so_{10}}}4} \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp_{1}}}1} \,\,{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_4+A_2$ $1 $ $ {\overset{\mathfrak{f_{4}}}5} \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{so_{7}}}3 \,\, ... [E_7]$
$A_5''$ $G_2$ $ [G_2]\,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, ... [E_7]$
$A_5+A_1$ $SU(2)$ $ [Sp(1)]\,\, {\overset{\mathfrak{f_{4}}}4} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, ... [E_7]$
$D_5(a_1) + A_1$ $SU(2)$ $ {\overset{\mathfrak{so_{9}}}4} \,\, \underset{[SO(3)]}{\overset{\mathfrak{sp_{1}}}1} \,\,{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$A_5'$ $SU(2) \times SU(2)$ $[SU(2)]\,\, {\overset{\mathfrak{so_{9}}}4} \,\, 1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2}\,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$D_6(a_2) $ $SU(2)$ $ [SU(2)] \,\, {\overset{\mathfrak{so_{9}}}4} \,\, 1 \,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$E_6(a_3) $ $SU(2)$ $ {\overset{\mathfrak{so_{8}}}4} \,\, 1 \,\, \underset{[SU(2)]}{\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$E_7(a_5) $ $1$ $ {\overset{\mathfrak{so_{8}}}4} \,\, 1 \,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1 \,\, ... [E_7]$
$D_5 $ $SU(2) \times SU(2)$ $ [SU(2)]\,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1 \,\, ... [E_7]$
$A_6 $ $SU(2) $ $ [SU(2)]\,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1 \,\, ... [E_7]$
$D_6(a_1) $ $SU(2) $ $ {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\,{2}\,\, \underset{[SU(2)]}1 \,\, {\overset{\mathfrak{e_{7}}}8} \,\, 1 \,\, ... [E_7]$
$D_5 +A_1$ $SU(2) $ $ {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1 \,\, ... [E_7]$
$E_7(a_4)$ $1$ $ {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1 \,\, ... [E_7]$
$D_6$ $SU(2) $ $[SU(2)] \,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1\,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, ... [E_7]$
$E_6(a_1)$ $U(1)$ $ {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, {\overset{\mathfrak{e_{6}}}6} \,\, 1\,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, ... [E_7]$
$E_6$ $SU(2) $ $ {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1\,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, ... [E_7]$
$E_7(a_3)$ $1 $ $ {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1\,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, ... [E_7]$
$E_7(a_2)$ $1 $ $ {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1\,\, {\overset{\mathfrak{su_{2}}}2} \,\, {\overset{\mathfrak{so_{7}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, ... [E_7]$
$E_7(a_1)$ $1 $ $ {\overset{\mathfrak{su_{2}}}2}\,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1\,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, ... [E_7]$
$E_7$ $1 $ $ 2 \,\, {\overset{\mathfrak{su_{2}}}2}\,\, {\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\, {\overset{\mathfrak{f_{4}}}5} \,\, 1\,\, {\overset{\mathfrak{g_{2}}}3} \,\, {\overset{\mathfrak{su_{2}}}2} \,\, 1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, ... [E_7]$
: 6D SCFTs associated with $E_7$ nilpotent orbits.[]{data-label="E7list"}
$E_8$ Nilpotent Orbits {#app:E8}
----------------------
The $E_8$ Nilpotent orbits are as follows. The nilpotent hierarchy can be found for example in [@Chacaltana:2012zy Table 19].
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B–C Label Global Symmetry Theory
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0 $E_8$ $[E_8] \,\, 1\,\, 2 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_1$ $E_7$ $[E_7] \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$2 A_1$ $SO(13)$ $[SO(13)] \,\, \overset{\mathfrak{sp_{1}}}1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$3 A_1$ $F_4 \times SU(2)$ $[F_4] \,\, 1 \,\, \underset{[Sp(1)]}{\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_2$ $E_6$ $[E_6] \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$4 A_1$ $Sp(4)$ $[Sp(4)] \,\, \overset{\mathfrak{g_{2}}}2 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_2+A_1$ $SU(6)$ $[SU(6)] \,\, \overset{\mathfrak{su_{3}}}2 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_2+2A_1$ $SO(7) \times SU(2)$ $[SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_3$ $SO(11)$ $[SO(11)] \,\, \overset{\mathfrak{sp_{1}}}1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_2+3 A_1$ $G_2 \times SU(2)$ $[SU(2)] \,\, 2 \,\, \underset{[G_2]}1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$2 A_2$ $G_2 \times G_2$ $[G_2] \,\, 1 \,\, \underset{[G_2]}{\underset{1}{\overset{\mathfrak{f_{4}}}5}} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$2 A_2+A_1$ $G_2 \times SU(2)$ $[G_2] \,\, 1 \,\, \underset{[Sp(1)]}{\overset{\mathfrak{f_{4}}}4} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_3+A_1$ $SO(7) \times SU(2)$ $[SO(7)] \,\, 1 \,\,
\underset{[SU(2)] }{\overset{\mathfrak{so}_9}4} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ 2A_2+2 A_1$ $Sp(2)$ $[Sp(2)] \,\, {\overset{\mathfrak{f}_4}3} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_4(a_1)$ $SO(8)$ $[SO(8)] \,\, 1 \,\, {\overset{\mathfrak{so}_8}4} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_3 + 2 A_1$ $Sp(2) \times SU(2)$ $[Sp(2)] \,\, \underset{[SU(2)]}{\overset{\mathfrak{so}_9}3} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_4(a_1)+A_1$ $SU(2) \times SU(2) \times SU(2)$ $[SU(2) \times SU(2) \times SU(2)] \,\, {\overset{\mathfrak{so}_8}3} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_4$ $F_4$ $[F_4] \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ A_3+ A_2$ $Sp(2)\times U(1)$ $[Sp(2)] \,\, \overset{\mathfrak{so}_7}3 \,\, 1 \,\, \overset{\mathfrak{g}_2}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_4$ $SU(5)$ $[SU(5)] \,\, \overset{\mathfrak{su}_4}2 \,\, \overset{\mathfrak{su}_3}2 \,\, \overset{\mathfrak{su}_2}2 \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_3+A_2+A_1$ $SU(2) \times SU(2)$ $[Sp(1)] \,\, \overset{\mathfrak{g}_2}3 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{g}_2}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_4+A_1$ $Sp(3)$ $[Sp(3)] \,\, \overset{\mathfrak{g}_2}2 \,\, \overset{\mathfrak{su}_2}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ D_4(a_1)+A_2$ $SU(3) $ $ \overset{\mathfrak{su}_3}3 \,\, \underset{[SU(3)]}1 \,\, \overset{\mathfrak{g}_2}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_4+A_1$ $SU(3)\times U(1)$ $[SU(3)] \,\, \overset{\mathfrak{su}_3}2 \,\, \underset{[N_f=1]}{\overset{\mathfrak{su}_3}2} \,\, \overset{\mathfrak{su}_2}2 \,\, \overset{\mathfrak{su_{1}}}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ 2 A_3$ $Sp(2)$ $\overset{\mathfrak{su}_2}2 \,\, \underset{[Sp(2)]}{\overset{\mathfrak{g}_2}2} \,\, \overset{\mathfrak{su}_2}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_5(a_1)$ $SU(4)$ $[SU(4)] \,\, \overset{\mathfrak{su}_3}2 \,\, \overset{\mathfrak{su}_2}2 \,\, \overset{\mathfrak{su}_1}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ A_4 + 2 A_1$ $SU(2)\times U(1)$ $\underset{[N_f=1]}{\overset{\mathfrak{su}_2}2}\,\, \underset{[SU(2)]}{\overset{\mathfrak{su}_3}2 } \,\, \overset{\mathfrak{su}_2}2 \,\, \overset{\mathfrak{su}_1}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ A_4 + A_2$ $SU(2) \times SU(2)$ $[SO(4)] \,\, \overset{\mathfrak{su}_2}2\,\, {\overset{\mathfrak{su}_2}2 } \,\, \underset{[N_f=1]}{\overset{\mathfrak{su}_2}2} \,\, \overset{\mathfrak{su}_1}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_5(a_1) + A_1$ $SU(2) \times SU(2)$ $[SO(4)] \,\, {\overset{\mathfrak{su}_2}2 } \,\, \underset{[N_f=1]}{\overset{\mathfrak{su}_2}2} \,\, \overset{\mathfrak{su}_1}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$A_4+A_2+ A_1$ $ SU(2)$ $ {\overset{\mathfrak{su}_1}2 } \,\, \underset{[N_f=1]}{\overset{\mathfrak{su}_2}2} \,\, \underset{[N_f=1]}{\overset{\mathfrak{su}_2}2} \,\, \overset{\mathfrak{su}_1}2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$A_5$ $ G_2 \times SU(2)$ $ [G_2] \,\, {\overset{\mathfrak{su}_2}2} \,\, 2 \,\, 1\,\, \underset{[SU(2)]}{\underset{2}{\underset{1}{\overset{\mathfrak{e_{8}}}{12}}}} \,\, 1 \,\, ... [E_8]$
$A_4+A_3$ $ SU(2)$ $[SU(2)] \,\, 2\,\, {2 } \,\, 2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$D_4+A_2$ $ SU(3)$ $ 2\,\, \underset{[SU(3)]}{\overset{\mathfrak{su}_2}2 } \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_6(a_3)$ $ G_2$ $ [G_2] \,\, {\overset{\mathfrak{su}_2}2 } \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{10} \,\, 1 \,\, ... [E_8]$
$A_5+A_1$ $ SU(2) \times SU(2)$ $ [SU(2)] \,\, {2} \,\, 2 \,\, 1\,\, \underset{[SU(2)]}{\underset{2}{\underset{1}{\overset{\mathfrak{e_{8}}}{12}}}} \,\, 1 \,\, ... [E_8]$
$D_5(a_1) + A_2$ $ SU(2)$ $ [SU(2)] \,\, 2\,\, {2} \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_6(a_3) + A_1$ $ SU(2)$ $ [SU(2)] \,\, {2} \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{10} \,\, 1 \,\, ... [E_8]$
$D_6(a_2) $ $ SU(2) \times SU(2)$ $ [SU(2)] \,\, 2 \,\, 1\,\, \underset{[SU(2)]}{\underset{2}{\underset{1}{\overset{\mathfrak{e_{8}}}{11}}}} \,\, 1 \,\, ... [E_8]$
$D_5 $ $ SO(7)$ $ [SO(7)] \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1\,\, \overset{\mathfrak{e_{7}}}8 \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_7(a_5) $ $ SU(2)$ $ [SU(2)] \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{9} \,\, 1 \,\, ... [E_8]$
$D_5 +A_1$ $ SU(2) \times SU(2)$ $ [SU(2)] \,\, 2 \,\, \underset{[SU(2)]}1 \,\, \overset{\mathfrak{e_{7}}}8 \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_8(a_7)$ $ 1$ $ \overset{\mathfrak{e_{8}}}{7} \,\, 1 \,\, 2 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$D_6(a_1)$ $ SU(2) \times SU(2)$ $ [SU(2)] \,\, 1 \,\, \underset{[SU(2)]}{\underset{1}{\overset{\mathfrak{e_{7}}}8}} \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$A_6$ $ SU(2) \times SU(2)$ $ [SU(2)] \,\, 1 \,\, {\overset{\mathfrak{e_{7}}}8} \,\, \underset{[SU(2)]}1\,\, 2\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_7(a_4)$ $ SU(2)$ $ [SU(2)] \,\, 1 \,\, \overset{\mathfrak{e_{7}}}7 \,\, 1\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$A_6+A_1$ $ SU(2) $ $ \overset{\mathfrak{e_{7}}}7 \,\, \underset{[SU(2)]}1\,\, 2\,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_6(a_1)$ $ SU(3) $ $ [SU(3)] \,\, 1 \,\, {\overset{\mathfrak{e_{6}}}6} \,\, 1\,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$D_5+A_2$ $ U(1) $ $\underset{[N_f=1]}{\overset{\mathfrak{e_{7}}}6} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\,\overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$D_7(a_2)$ $ U(1) $ ${\overset{\mathfrak{e_{6}}}6} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\,\overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$ E_6$ $G_2$ $[G_2] \,\, 1 \,\, {\overset{\mathfrak{f}_4}5} \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ A_7$ $SU(2)$ ${\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, \underset{[Sp(1)]}{\overset{\mathfrak{g_{2}}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ E_6(a_1)+A_1$ $U(1)$ $\underset{[N_f=1]}{\overset{\mathfrak{e_{6}}}5} \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ E_8(b_6)$ $1$ ${\overset{\mathfrak{f_{4}}}5} \,\, 1 \,\, {\overset{\mathfrak{su_{3}}}3} \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_6$ $Sp(2)$ $ [Sp(2)] \,\, \overset{\mathfrak{so_{11}}}4 \,\, \overset{\mathfrak{sp}_1}1 \,\, \overset{\mathfrak{so_{9}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ E_7(a_3)$ $SU(2)$ $ [Sp(1)] \,\, \overset{\mathfrak{so_{10}}}4 \,\, \underset{[N_f=\frac{1}{2}]}{\overset{\mathfrak{sp}_1}1} \,\, \overset{\mathfrak{so_{9}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ D_7(a_1)$ $U(1)$ $ \overset{\mathfrak{so_{9}}}4 \,\, \underset{[N_f=1]}{\overset{\mathfrak{sp}_1}1} \,\, \overset{\mathfrak{so_{9}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ E_6 +A_1$ $SU(2)$ $ [Sp(1)] \,\, \overset{\mathfrak{f_{4}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ E_7(a_2)$ $SU(2)$ $ [Sp(1)] \,\, \overset{\mathfrak{so_{9}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$ E_8(a_6)$ $1$ $ \overset{\mathfrak{so_{8}}}4 \,\, {1} \,\, \overset{\mathfrak{so_{8}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{12} \,\, 1 \,\, ... [E_8]$
$ E_8(b_5)$ $1$ $ \overset{\mathfrak{so_{8}}}4 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_7(a_1)$ $ SU(2)$ $ [SU(2)] \,\, \overset{\mathfrak{so}_7}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, {\overset{\mathfrak{e_{7}}}8} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$D_7$ $ SU(2) $ $ \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \underset{[SU(2)]}{\underset{2}{\underset{1}{\overset{\mathfrak{e_{8}}}{12}}}} \,\, 1 \,\, ... [E_8]$
$E_8(a_5)$ $ 1 $ $ \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{10} \,\, 1 \,\, ... [E_8]$
$E_8(b_4)$ $ 1$ $ \overset{\mathfrak{g}_2}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, {\overset{\mathfrak{e_{7}}}8} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_7$ $SU(2)$ $[SU(2)] \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_8(a_4)$ $ 1$ $ \overset{\mathfrak{su}_3}3 \,\, 1 \,\, {\overset{\mathfrak{e_{6}}}6} \,\, 1 \,\, \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_8(a_3)$ $1$ $ \overset{\mathfrak{su_{3}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_8(a_2)$ $ 1$ $\overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{so}_7}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 1 \,\, {\overset{\mathfrak{e_{7}}}8} \,\, 1 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, ... [E_8]$
$E_8(a_1)$ $1$ $ {\overset{\mathfrak{su_{2}}}2} \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
$E_8$ $1$ $2 \,\, \overset{\mathfrak{su_{2}}}2 \,\, \overset{\mathfrak{g_{2}}}3 \,\, 1 \,\, \overset{\mathfrak{f_{4}}}5 \,\, 1 \,\, \overset{\mathfrak{g_{2}}}3 \,\, \overset{\mathfrak{su_{2}}}2 \,\, 2 \,\, 1\,\, \overset{\mathfrak{e_{8}}}{11} \,\, 1 \,\, ... [E_8]$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: 6D SCFTs associated with $E_8$ nilpotent orbits.[]{data-label="E8list"}
[^1]: There are still a few outlier theories which appear consistent with field theoretic constructions, and also admit an embedding in perturbative IIA string theory (see e.g. [@Hanany:1997gh]). As noted in [@Bhardwaj:2015oru], these will likely yield to an embedding in a non-geometric phase of F-theory since the elements of these constructions are so close to those obtained in geometric phases of F-theory.
[^2]: As a brief aside, let us note that the case of $k$ sufficiently large leads to a class of (singular) M-theory duals for these theories in which the T-brane data is localized near the orbifold fixed points of the classical gravity dual [@DelZotto:2014hpa]. For $G=SU(N)$, these theories also have a IIA realization [@Hanany:1997gh; @Gaiotto:2014lca] and a non-singular holographic dual [@Apruzzi:2013yva; @Apruzzi:2015wna].
[^3]: Here we do not distinguish between the algebra and the global structure of the flavor symmetry group.
[^4]: Note that when we initiate a larger breaking pattern from say $[E_8]-E_8-....-E_8-[E_8]$ to $[E_7]-E_7-....-E_7-[E_7]$, the effects of the breaking pattern are not localized and propagate from one end of the generalized quiver to the other. We leave a detailed analysis of such flows for future work.
[^5]: A related class of explicit F-theory models classified by group theoretic data was studied in references [@DelZotto:2014hpa; @Heckman:2015bfa]. Though this data is purely geometric on the F-theory side, in the dual heterotic description, we have small instantons of heterotic string theory on an ADE singularity in which the boundary data of the small instantons leads to different classes of 6D SCFTs. This boundary data is classified by homomorphisms from discrete ADE subgroups of $SU(2)$ to $E_8$.
[^6]: One way to see this triality is to note that the weighted Dynkin diagrams associated with these nilpotent orbits are related by permutation of the three external nodes [@NILPbook Page 84].
[^7]: Note that this match holds for $SO(2N+1)$ nilpotent orbits only after we take into account the subtlety of $SO(2N+1) \subset SO(2N+2p)$ for small $p$ discussed in section \[ssec:SOodd\].
[^8]: The requirement that Higgs branch flows preserve gravitational anomalies fixes the number of free hypermultiplets, which means that our RG flow analysis will be unable to distinguish between two F-theory models that give 6D SCFTs differing only by a number of free hypermultiplets.
|
---
abstract: |
We point out that, in the context of the SM, $|V^2_{13}| + | V^2_{23}|$ is expected to be large, of order one. The fact that $|V^2_{13}| + |V^2_{23}| \approx 1.6 \times 10^{-3}$ motivates the introduction of a symmetry S which leads to $V_{CKM} ={1\>\!\!\!\mathrm{I}} $, with only the third generation of quarks acquiring mass. We consider two scenarios for generating the mass of the first two quark generations and full quark mixing. One consists of the introduction of a second Higgs doublet which is neutral under S. The second scenario consists of assuming New Physics at a high energy scale , contributing to the masses of light quark generations, in an effective field theory approach. This last scenario leads to couplings of the Higgs particle to $s\overline s$ and $c \overline c$ which are significantly enhanced with respect to those of the SM. In both schemes, one has scalar-mediated flavour- changing neutral currents which are naturally suppressed. Flavour violating top decays are predicted in the second scenario at the level $ \mbox{Br} (t \rightarrow h c )
\geq 5\times 10^{-5}$.
---
CERN-TH-2016-039\
[**[What if the Masses of the First Two Quark Families are not Generated by the Standard Higgs?]{}**]{}
F. J. Botella $^{a,c}$ [^1], G. C. Branco $^{b, c}$ [^2], M. N. Rebelo $^{b, c}$ [^3], and J. I. Silva-Marcos $^b$[^4],
[*$^a$ Departament de F' isica Teòrica and IFIC, Universitat de València-CSIC, E-46100, Burjassot, Spain.*]{}\
[*$^b$ Departamento de Física and Centro de F' isica Te' orica de Part' iculas (CFTP), Instituto Superior T' ecnico (IST), U. de Lisboa (UL), Av. Rovisco Pais, P-1049-001 Lisboa, Portugal.\
*$^c$ Theory Department, CERN, CH 1211 Geneva 23, Switzerland**]{}
The recent discovery of the Higgs particle at LHC, rendered even more urgent to understand the mechanism responsible for the generation of fermion masses and mixing. In the framework of the Standard Model (SM), fermion masses arise exclusively through Yukawa interactions and the Brout-Englert-Higgs mechanism is responsible for both gauge symmetry breaking and the generation of fermion masses. Some of the outstanding questions one may ask, include :
i\) Two of the salient flavour features in the quark sector are the strong hierarchy of quark masses and the fact that the $V_{CKM}$ matrix is close to the identity. In the framework of the SM, can one conclude that these two features are related in some way? How can one understand small quark mixing in the SM ?
ii\) In the SM, all fermion masses are generated through the vacuum expectation value (vev) of the Standard Higgs. Alternatively, one may consider a scenario where the Standard Higgs only gives mass to the third generation, while the masses of the two first generations originate from another source. A crucial question is : how can this alternative scenario be tested at LHC and future accelerators?
In this paper, we address the above two questions. With respect to (i), we show that actually in the SM the “natural" value of $(|V^2_{13}| + |
V^2_{23}|)$ is large, of order one. In order to address this question, we study in detail quark mixing in the extreme chiral (EC) limit, where only the third generation of quarks acquires mass, while $m_d$, $m_s$, $m_u$, $%
m_c $ remain massless. We do the analysis in the context of the Standard Model (SM) and some of its extensions. We will show that in the SM in the EC limit the generic situation is having non-trivial mixing parametrised by an angle with a free value, not fixed in the SM context. Without loss of generality, one can identify this angle with the $V_{23}$ entry. Therefore, the fact that experimentally $|V_{23}| = 4.09 \times 10^{-2}$, is entirely unnatural within the framework of the SM. In fact, the smallness of $|V_{23}|$ may be interpreted as a hint from experiment, indicating that one should find a symmetry or a principle which may account for the smallness of $|V_{23}|$.
In our analysis, we start with the most general rank one matrices $M_u$, $
M_d $, taking into account that in the SM the flavour structure of the Yukawa couplings generating the up and down quark mass matrices are entirely independent. The appearance of a non-trivial mixing even in the EC limit case, corresponds to a misalignment of the two mass matrices $M_u$, $M_d$, in flavour space. We define a dimensionless weak basis (WB) invariant denoted A which provides a measure of this misalignment. In the EC limit, this invariant A varies from 0 to 1 , with 0 corresponding to exact alignment and 1 to total misalignment.
With respect to question (ii) we consider the possibility that in leading order the SM Higgs only gives mass to the third generation. This is achieved in a natural way through the introduction of a discrete symmetry S which leads to quark mass matrices of rank one, aligned in flavour space. We then conjecture that the generation of the mass of the first two generations arises from a different source. If this new source is just another Higgs doublet and if one assumes that the new doublet is neutral with respect to the symmetry S, then one is led to a flavour structure analogous to what one encounters in a class of the BGL-type models [@Branco:1996bq], [Botella:2009pq]{}, which have been extensively analysed in the literature [@Botella:2011ne], [@Botella:2012ab], [@Bhattacharyya:2013rya] [@Botella:2014ska], [@Bhattacharyya:2014nja], [@Botella:2015hoa], [@Sher:2016rhh]. If, on the other hand, the new contribution arises in the framework of an effective field theory where the New Physics (NP) particles have been integrated out, then assuming that this NP contribution is of order $m_s$ and $m_c$ in the down and up sectors, one can estimate the couplings of the Standard Higgs to $t\overline{t}$, $b\overline{b}$, $c%
\overline{c}$, $s\overline{s}$. It turns out that the couplings to $t%
\overline{t}$, $b\overline{b}$ do not differ much from those in the SM, but the couplings to $c\overline{c}$, $s\overline{s}$ are significantly enhanced with respect to those in the SM.
*Mixing in the EC limit:* We analyse quark mixing in the EC limit, where the quark mass matrices $M_{d}$ and $M_{u}$ are rank one matrices generated by two independent Yukawa coupling matrices $Y_{d}$, $Y_{u}$. Therefore, $M_{d}$, $M_{u}$ can be written: $$M_{d}={U_{L}^{d}}^{\dagger }\ \mbox{diag}(0,0,m_{b})\ {U_{R}^{d}},\qquad
M_{u}={U_{L}^{u}}^{\dagger }\ \mbox{diag}(0,0,m_{t})\ {U_{R}^{u}}
\label{loo}$$One does not loose generality by considering the specific ordering of $m_{b}$, $m_{t}$ in Eq. (\[loo\]), since a permutation changing these positions can always be included in the unitary matrices $U_{L,R}^{d,u}$. The quark mixing matrix appearing in the charged weak interactions is given by $V^{0}={%
U_{L}^{u}}^{\dagger }{U_{L}^{d}}$ and it is at this stage an arbitrary mixing matrix. Taking into account that in the EC limit the first two generations are massless, one can make an arbitrary redefinition of the light quark masses through a unitary transformation of the type: $$W_{u,d}=\left[
\begin{array}{cc}
X_{u,d} & 0 \\
0 & 1
\end{array}
\right] \label{lii}$$ where $X_{u,d}$ are $2\times 2$ unitary matrices. Under this transformation $%
V^{0}$ transforms as $V^{0}\rightarrow V^{\prime }=W_{u}^{\dagger }\ VW_{d}$. One has the freedom to choose $X_{u,d}$ at will to diagonalize the $%
2\times 2$ upper left sector of $V^{\prime }$ leading to $|V_{12}^{\prime
}|=|V_{21}^{\prime }|=0$. Unitarity of $V^{\prime }$ leads then to the constraint ${V^{\prime }}_{13}^{\ast }V_{23}^{\prime }=0$. One can then choose, without loss of generality, $V_{13}^{\prime }=0$ and $V_{CKM}$ becomes then an orthogonal matrix, with mixing only between the second and third generation, characterised by an angle $\alpha $, with $|V_{23}^{\prime
}|=|V_{32}^{\prime }|=|\sin \alpha |$. The important point that we wish to emphasise is that this mixing in the EC limit of the SM, is arbitrary. The smallness of $|V_{13}|^{2}+|V_{23}|^{2}$ in the SM, in general, cannot be related to the smallness of the mass ratios $m_{i}^{2}/m_{3}^{2}$ where $%
i=1,2$. Therefore, in the framework of the SM the observed smallness of $%
|V_{23}|\approx 10^{-2}$, provides a hint for the presence of a flavour symmetry.
*An Invariant Measure of Alignment:* Experimentally one encounters in the quark sector $V_{CKM}\approx {1\>\!\!\!\mathrm{I}}$ which corresponds to an alignment of the quark mass matrices in flavour space. It is useful to have an invariant measure of the mixing defined in terms of the mass matrices when written in an arbitrary weak basis. This can be done by defining the following weak basis invariant [@Branco:2011aa]: $$A\equiv \frac{1}{2}trB^{2},\quad \mbox{with}\quad B=h_{d}-h_{u} \label{a}$$ where the build blocks are the two matrices: $$h_{d}=\frac{H_{d}}{tr[H_{d}]},\qquad h_{u}=\frac{H_{u}}{tr[H_{u}]} \label{h}$$ with the notation $H_{u,d}\equiv M_{u,d}M_{u,d}^{\dagger }$. By construction, one has $trh_{d}=trh_{u}=1$. Given the two rank one matrices $%
M_{d,u}$, described before, corresponding to the EC limit one obtains: $$A\equiv \frac{1}{2}trB^{2}=|V_{23}|^{2}+|V_{13}|^{2} \label{que}$$ The result of Eq. (\[que\]) is exact in the EC limit. The invariant $A$ still gives a measure of the size of mixing when the first two generations acquire mass, and in this case we have $A\approx
|V_{23}|^{2}+|V_{13}|^{2}+O(m_{s}/m_{b})^{4}$
*Obtaining Small Mixing Through a Symmetry:* As stated before, mixing in the EC limit is parametrised by an arbitrary mixing angle involving two generations. In general, in the SM there is no reason to assume that this mixing angle is either close to zero or maximal, in fact it can take any value. It is possible to introduce a symmetry which leads to the vanishing of this mixing. Without loss of generality, this angle can parametrise mixing between the second and the third generations. Let us consider the following symmetry S, in the context of the particle content of the SM, with only one Higgs doublet. $$Q_{L3}^{0}\rightarrow \exp {(i\tau )}\ Q_{L3}^{0}\ ,\quad
u_{R3}^{0}\rightarrow \exp {(i2\tau )}u_{R3}^{0}\ ,\quad \phi \rightarrow
\exp {(i\tau )} \phi \ , \quad \tau \neq 0, \pi \label{S symetry up quarks}$$ where $Q_{Lj}^{0}$ is a left-handed quark doublet and $\Phi$ is the Higgs doublet. All other fermions transform trivially under S. This symmetry leads to the following pattern of texture zeros for the Yukawa couplings: $$Y_d = \left[%
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
\times & \times & \times%
\end{array}%
\right], \qquad Y_u = \left[%
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \times%
\end{array}%
\right] \label{ydyu}$$ which clearly lead to $V_{CKM}$ equal to the identity. The matrices of Eq. (\[ydyu\]) are written in the WB chosen by the symmetry. The matrix $Y_d$ can be written in the same form as $Y_{u}$ by means of a rotation of the right-handed down quarks, which simply corresponds to a different choice of WB.
Next we present possible ways of extending this scenario in order to generate the masses of the first two generations of quarks, as required experimentally, without generating large mixing and keeping the Higgs mediated flavour changing neutral currents (HFCNC) under control.
*Generating the Masses of the First Two Quark Generations:* At this stage, one has to address the question of the origin of the masses of the first two generations. The discovery of the Higgs particle at LHC and the study of its production and decay has shown that the vev of the Higgs field gives the dominant contribution to the masses of the fermions of the third generation, namely to the top and bottom quarks, as well as the $\tau $-lepton. It is conceivable that the masses of the quarks of the first two generations arise from a different source, [@Babu:1999me], [Giudice:2008uua]{}, [@Goudelis:2011un], [@Perez:2015aoa], [Altmannshofer:2015esa]{}, [@Ghosh:2015gpa], [@Bauer:2015kzy], so that the quak mass matrices have the form: $$M=M^{(0)}+M^{(1)} \label{m}$$ where $M^{(0)}$ is generated by the vev of the standard Higgs $\phi $ and $%
M^{(1)}$ may arise from the vev of a second Higgs $\phi ^{\prime }$ or from other unspecified source. In either case, the fact that there are two different sources giving contributions to the mass of quarks of a given charge, leads to scalar mediated flavour-changing neutral currents (FCNC). These currents are naturally suppressed in both of the scenarios we consider below, once the experimental values of the $V_{CKM}$ entries are taken into account.
*Adding a Second Higgs Doublet:* The simplest possibility to generate masses for the first two generations, is through the addition of a second doublet $\phi ^{\prime }$ which is neutral under S. In this case, the contribution of $\phi ^{\prime }$ to the quark mass matrix is of the form: $$M_{d}^{(1)}=\frac{v^{\prime }}{\sqrt{2}}\left[
\begin{array}{ccc}
\times & \times & \times \\
\times & \times & \times \\
0 & 0 & 0%
\end{array}%
\right] ;\qquad M_{u}^{(1)}=\frac{v^{\prime }}{\sqrt{2}}\left[
\begin{array}{ccc}
\times & \times & 0 \\
\times & \times & 0 \\
0 & 0 & 0
\end{array}
\right] \label{m1}$$ This structure coincides with what one encounters in a class of BGL models [@Branco:1996bq]. It has been shown that in this model, the full flavour structure only depends on $V_{CKM}$ and thus the model obeys the Minimal Flavour Violation [@Buras:2000dm], [@D'Ambrosio:2002ex], [Botella:2009pq]{} principle. In this model there are FCNC but they are naturally suppressed by small $V_{CKM}$ elements.
In this context, there are two types of BGL models: (1) top models described by Eqs. (\[S symetry up quarks\]), (\[ydyu\]) and (\[m1\]) with FCNC only in the down sector and (2) bottom models with the rôle of up and down quarks interchanged. This second class of models give rise to FCNC’s only in the up sector. From low energy flavour data the scale of new physics in top models can be quite light at a few hundred GeV [@Botella:2014ska]. Bottom like models introduce new scales close to the TeV region [@Botella:2014ska]. Flavour conserving or flavour blind Higgs observables can be accommodated in both categories because the new couplings, compared to the SM couplings, i.e., the coupling modifiers $\kappa_{Z}$ , $\kappa_{W}$, $\kappa_{t}$, $\kappa_{\tau }$, $\kappa_{b}$, $\kappa_{g}$ and $\kappa_{\gamma }$ may deviate from 1 at the percent level. These models have been extensively studied in the literature [@Bhattacharyya:2013rya] [@Botella:2014ska], [@Bhattacharyya:2014nja], [@Botella:2015hoa], [@Sher:2016rhh]. In the Higgs sector the most relevant prediction specific to top models is the decay $h\rightarrow b\bar{s}+s\bar{b}$ with branching ratios at most between $10^{-3}$ and $10^{-2}$ [@Botella:2015hoa] The bottom models predict the rare top decay $t\rightarrow hc$ with a branching ratio of at most $10^{-3}$ [@Botella:2015hoa] In both classes of models these predictions can be correlated with $h\rightarrow \mu \bar{\tau}+\tau \bar{\mu}$ occuring at a branching ratio which can reach at most $10^{-2}$
*Generating light quark masses from New Physics at a high energy scale:* Here, we consider that only one Higgs doublet is introduced in the framework of the SM and introduce the symmetry $S$ of Eq. (\[S symetry up quarks\]) which implies that only the third generation of quarks acquire mass, with $V_{CKM}={1\>\!\!\!\mathrm{I}}$. We shall consider that the quark masses of the first two generations arise from New Physics contributing to the Yukawa couplings in leading order through an effective six-order operator of the form: $${\mathcal{L}}_{eff}=- \left( Y_{d}^{(1)}\right) _{jk}\frac{\phi ^{\dagger
}\phi }{\Lambda ^{2}}\bar{Q}_{L_{j}}^{0}d_{R_{k}}^{0}\phi - \left(
Y_{u}^{(1)}\right) _{jk}\frac{\phi ^{\dagger }\phi }{\Lambda ^{2}}\bar{Q}%
_{L_{j}}^{0}u_{R_{k}}^{0}\tilde{\phi} \label{eff1}$$ The Yukawa and quark mass matrices have then the form [@Goudelis:2011un]: $$\sqrt{2}Y_{d,u}=Y_{d,u}^{(0)}+3Y_{d,u}^{(1)}\ \left( \frac{v^{2}}{\Lambda ^{2}}
\right) \qquad M_{d,u}=v\left[ Y_{d,u}^{(0)}+Y_{d,u}^{(1)}\ \left( \frac{%
v^{2}}{\Lambda ^{2}}\right) \right] \label{esta}$$The fact that $Y_{d,u}$ are not proportional to $M_{d,u}$ leads to Higgs mediated Flavour-Changing-Neutral -Currents (FCNC). At this stage, it is useful to estimate the size of the new mass scale $\Lambda $. From Eq. ([esta]{}) and taking into account that $v=174$ GeV, $m_{t}=173$ GeV one obtains $(Y_{u}^{(0)})_{tt}\approx 1$. Assuming $Y_{u}^{(1)}\approx
(Y_{u}^{(0)})_{tt}$, one obtains $\Lambda =\left[ \frac{Y_{u}^{(1)}v^{3}}{%
m_{c}}\right] ^{1/2}\approx 2$ TeV, so the new mass scale is of the order of a few TeV. For the down quark sector, taking into account that $m_{b}\approx
4.2$ GeV, $m_{s}\approx 0.095$ GeV, one obtains $(Y_{d}^{(0)})_{bb}\approx
\frac{m_{b}}{v}\approx 0.02$, $Y_{d}^{(1)}\approx 0.07$. Note that in the present framework one obtains $|V_{23}|\approx O(m_{s}/m_{b})$ but one does not provide an explanation for the smallness of $m_{b}$/$m_{t}$. We will show that the potentially dangerous FCNC are naturally suppressed in the present framework. The down quark mass matrix is diagonalised by : $$U_{dL}^{\dagger }\left[ Y_{d}^{(0)}+Y_{d}^{(1)}\frac{v^{2}}{\Lambda ^{2}}%
\right] U_{dR}=\frac{D_{d}}{v} \label{u}$$ where $D_{d}\equiv \mbox{diag}(m_{d},m_{s},m_{b})$, with an analogous expression for the up sector. In the quark mass eigenstate basis, the Yukawa coupling matrix becomes: $$\sqrt{2} Y_{d}^{m}= \sqrt{2}\left( U_{dL}^{\dagger }Y_{d}U_{dR} \right) =
\frac{3D_{d}}{v}-2U_{dL}^{\dagger}Y_{d}^{(0)}U_{dR} \label{eq12}$$ At this stage, it is useful to write $U_{dL}^{\dagger }Y_{d}^{(0)}U_{dR}$ explicitly. Taking into account that $Y_{d}^{(0)}=\mbox{diag}(0,0,\frac{m_{b}%
}{v})$, one obtains: $$\left( U_{dL}^{\dagger }Y_{d}^{(0)}U_{dR}\right) _{jk}=(U_{dL}^{\ast
})_{3j}(U_{dR})_{3k}\frac{m_{b}}{v} \label{eq13}$$ with an analogous expression for the up sector. The strength of the Higgs couplings $Y_{d}^{m}$ is controlled by Eqs. (\[eq12\]), (\[eq13\]) and one has to take into account the very strict bounds on flavour violating scalar couplings, which can be derived from $K^0 -\overline{K^0}$ , $B_d -\overline{B_d}$ , $B_s -\overline{B_s}$, $D^0 -\overline{D^0}$ mixings. These bounds have been recently analysed in [@Blankenburg:2012ex]. From $B_s -\overline{B_s}$ mixing, one derives bounds on $| (U^\ast_{dL})_{32} (U_{dR})_{33}| $ and $| (U^\ast_{dL})_{33} (U_{dR})_{32}| $ which taking into account that $|(U_{dL})_{33}| \approx 1$ and $|(U_{dR})_{33}| \approx 1$, lead to $|(U_{dL})_{32}| \leq 1.4 \times 10^{-2}$. Similarly, one derives from $B_d -\overline{B_d}$ mixing the bound $|(U_{dL})_{31}| \leq 3 \times 10^{-3}$. It is remarkable that these bounds lead to $| (U^\ast_{dL})_{31} (U_{dR})_{32}| \simeq | (U^\ast_{dL})_{32} (U_{dR})_{31}|
\leq 4.4 \times 10^{-4}$ which guarantees that the Higgs contribution to $K^0 -\overline{K^0}$ mixing is sufficiently suppressed, to conform to the strict experimental bound. Flavour-changing scalar couplings in the up-sector are controlled by $| (U^\ast_{uL})_{3i} (U_{uR})_{3j}| $. On the other hand, $U_{uL}$ is constrained to be in a region which can generate the observed $V_{CKM} = (U^\dagger_{uL}U_{dL})$. Once these constraints are taken into account, one predicts, in the present framework, the strength of flavour-changing decays of the top quark, namely $$\mbox{Br} (t \rightarrow h c )
\geq 5\times 10^{-5}
\label{brbr}$$ It is interesting to notice that in this framework the previously analysed new flavour changing Higgs contributions all arise from the third column of the matrices $U_{dL}$, $U_{dR}$, $U_{uL}$ and $U_{uR}$.
So far we have only discussed the off-diagonal Higgs couplings. In the diagonal couplings, one has to distinguish between the couplings of the third generation (i.e. $t\bar{t}h$ and $b\bar{b}h$) and those of the two light generations. Taking into account that $|(U_{dL}^{\ast })_{33}(U_{dR})_{33}|\approx 1$ and also $|(U_{uL}^{\ast })_{33}(U_{uR})_{33}|\approx 1$ it is clear that the couplings of the third quark generation coincide with those in the SM. On the contrary, for the first two generations, one has a significant enhancement by a factor of $3$, leading, for example to: $$\Gamma (h\rightarrow q\bar{q})\approx 9\Gamma ^{SM}(h \rightarrow q \bar{q})
\label{gama} \qquad q = d, s, c .
\label{eq16}$$ At this stage, the following comment is in order. For the down sector, the experimental constraints from meson mixing are very strict and the prediction of Eq. (\[eq16\]) for $q= d, s$, is solid. For $c\bar{c}$ although the enhancement of Eq. (\[eq16\]) holds for most of the allowed parameter space, there are regions of allowed parameter space, where the enhancement is not as strong. So far, we have only discussed the quark sector. Note that the observed lepton flavour mixing is large and therefore there is no motivation to opt for the symmetry S to act in the lepton sector in a way analogous to the quark sector, since this would lead to $V_{PMNS}={1\>\!\!\!\mathrm{I}}$ in leading order, in contrast to experiment. We shall assume that leptons are neutral with respect to S, which leads to couplings of the Higgs particle to leptons which coincide with those of the SM.
Taking into account that $\Gamma ^{SM}(h\rightarrow \bar{c}c)/\Gamma
^{SM}(h\rightarrow \mbox{all})\sim 3\%$, assuming Eq. (\[eq16\]) and that the other relevant decay channels do not change, we get $\Gamma (h\rightarrow \mbox{all})\approx
1.23\Gamma ^{SM}(h\rightarrow \mbox{all})$. This result gives rise to a definitive prediction for the signal strength parameters $\mu ^{f}$ [ATLASCMS]{} in the decay channels $f=\gamma \gamma ,ZZ,WW,\tau \bar{\tau},b
\bar{b}$: $$\mu ^{f}=\frac{\Gamma (h\rightarrow f)\ \Gamma ^{SM}(h\rightarrow \mbox{all})
}{\Gamma (h\rightarrow \mbox{all})\ \Gamma ^{SM}(h\rightarrow f)}\approx 0.81
\label{mu}$$ compatible with the combined ATLAS CMS analysis [@ATLASCMS]. Looking at the coupling modifiers analysis $\kappa_{f}$ we have as in the SM no modification of the couplings to the relevant channels $$\kappa_Z = \kappa_W = \kappa_t = \kappa_\tau = \kappa_b = \kappa_g =
\kappa_\gamma = 1
\label{k}$$ but the large enhancement in the undetected $c\bar{c}$ channel contributes to the so-called beyond the SM branching ratio $BR_{BSM}\sim 18.8\%$ in perfect agreement with the $34\%$ joint upper bound from ATLAS and CMS at $%
95\%$ C.L. [@ATLASCMS].\
*TeV completion:* A possible TeV completion of the present model, can be implemented in the framework of an extension of the SM where one adds three $Q = - 1/3$ vector-like quarks, $D_{\alpha}$, and three $Q = 2/3$ vector-like quarks, $U_{\beta}$, isosinglets of $SU(2)$, to the spectrum of the SM.
We introduce the symmetry S considered in Eq. (\[S symetry up quarks\]), with the standard like quarks transforming as before. The symmetry S is spontaneously broken by the vev of the Higgs field and we also allow for a soft-breaking term $\overline{D}_{L\beta }d_{R3}$ with a similar soft-breaking term for the up sector. The leading higher order operators are the ones in Eq. (\[eff1\]), which, in the down sector, are generated through the diagram of Fig. 1.
{width="100.00000%"}
It can be shown that a realistic quark mass spectrum and a correct pattern of quark mixing can be generated, although a full description goes beyond the scope of this paper [@future].\
[*In summary, the crucial points of our paper are:*]{}
- Contrary to what may be a common belief, in the SM, the natural value of $|V_{23}|^{2}+|V_{13}|^{2}$ is of order one. In the SM, without an additional symmetry, the smallness of $V_{CKM}$ mixings cannot be derived from the observed strong hierarchy of quark masses.
- We point out that the fact that $|V_{23}|^{2}+|V_{13}|^{2}$ is small may be considered as a hint of Nature suggesting the introduction of a symmetry S. We have given an example of such a symmetry, which leads to $V_{CKM}={1\>\!\!\!\mathrm{I}}$ with only the third quark generation acquiring mass.
-We have suggested two different scenarios to generate the masses of the two lighter quark generations. One of them, consists of the introduction of a second Higgs doublet, which is neutral under S. This framework leads to a BGL type-model [@Branco:1996bq], [@Botella:2009pq] which have been analysed in the literature. Another scenario consists of assuming that New Physics at a high energy scale, contributes to the light quark masses in an effective field theory approach. This scenario leads to the following striking predictions which can be tested at LHC-run 2, as well as in other future accelerators: The diagonal Higgs quark couplings of the 3rd generation, i.e. tth and bbh, essentially coincide with those of the SM. The diagonal Higgs couplings of the lighter quarks are enhanced with respect to those of the SM, by about a factor of three, with the most significant effect of this enhancement given by Eq. (\[gama\]). In this framework one predicts Higgs mediated flavour violating top decays, as indicated in Eq. (\[brbr\])
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank the CERN Theory Department for hospitality and partial financial support. We thank Luca Fiorini for interesting discussions. This work is partially supported by Spanish MINECO under grant FPA2015-68318-R, and SEU-2014-0398, by Generalitat Valenciana under grant GVPROMETEOII 2014-049 and by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through the projects CERN/FIS-NUC/0010/2015, and CFTP-FCT Unit 777 (UID/FIS/00777/2013) which are partially funded through POCTI (FEDER), COMPETE, QREN and EU. The authors also acknowledge the hospitality of Universidad de Valencia, IFIC, and CFTP at IST Lisboa during visits for scientific collaboration.
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[^1]: fbotella@uv.es
[^2]: gbranco@tecnico.ulisboa.pt
[^3]: rebelo@tecnico.ulisboa.pt
[^4]: juca@cftp.ist.utl.pt
|
---
abstract: |
Near-ambient pressure XPS and STM experiments are performed to study the intercalation of oxygen and nitrogen at different partial gas pressures and different temperatures in the graphene/Ni/Ir(111) system of different morphologies. We performed detailed experiments on the investigation of the chemical state and topography of graphene, before and after gas intercalation, depending on the amount of pre-intercalated Ni in graphene/Ir(111). It is found that only oxygen can be intercalated under graphene in all considered cases, indicating the role of the intra-molecular bonding strength and possibility of gas molecules dissociation on different metallic surfaces on the principal possibility and on the mechanism of intercalation of different species under graphene.\
This document is the unedited author’s version of a Submitted Work that was subsequently accepted for publication in J. Phys. Chem. C., copyright $\copyright$ American Chemical Society after peer review. To access the final edited and published work see doi: 10.1021/acs.jpcc.9b01407.
author:
- 'Changbao Zhao,$^{1,2}$ Jiayi Li,$^{1,2}$ Jiuxiang Dai,$^{1,3}$ Elena Voloshina,$^{4,}$[^1] Yuriy Dedkov,$^{4,}$[^2] Yi Cui$^{1,}$[^3]'
title: 'Intercalation of O$_2$ and N$_2$ in the Graphene/Ni Interfaces of Different Morphology'
---
Introduction {#intr}
============
The epitaxial growth of different 2D materials, like graphene (gr), h-BN, MoS$_2$, etc. on different substrates [@Dedkov:2015kp; @Auwarter:2018kf; @Li:2016hn] gives an opportunity to study the influence of low-dimensionality on the reactivity of the studied systems, i.e. the effects of the so-called 2D confined catalysis [@Deng:2016ch; @Fu:2016gt]. These effects include (but not limited to) intercalation of different species in the space between 2D material and support, catalysis under 2D cover, growth under 2D cover of adlayers with the artificial crystallographic and electronic structures, solution intercalation, etc. Here as examples one can consider intercalation of different elements (metals and non-metals) [@Dedkov:2015kp; @Batzill:2012; @Dedkov:2001; @Shikin:2000; @Emtsev:2011fo; @Verbitskiy:2015kq], intercalation of molecules (like CO, H$_2$O, C$_{60}$) [@Feng:2012il; @Politano:2016jb; @Gupta:2014he; @Shikin:2000a; @Monazami:2015gg; @Zhang:2015hm; @Emmez:2014cl; @Gurarslan:2014bx], stabilization of the pseudomorphic growth of metals on different substrates [@Dedkov:2008e; @Pacile:2013jc; @Decker:2013ch], CO oxidation under graphene or h-BN on Pt(111) [@Zhang:2015hm; @Yao:2014hy] (see for review Refs. and references therein).
Previous experimental and theoretical works on the confined catalysis dealt with the epitaxial 2D layers grown on the bulk support and the catalytic activity of such systems was taken as is without attempts to modify their properties via change of system morphology or its electronic properties. Here, one can assume that, e.g. the intercalation of guest-metal atoms in the epitaxial gr/metal or gr/semiconductor interface might tune the properties of these interfaces. One can think, for example, on the creation of additional reaction centres from atoms or clusters of guest-metal and/or on the formation of the new artificial gr/guest-metal interface, where the system morphology and the low-dimensionality of the intercalated metal will influence the catalytic properties of the studied systems. For example, considering the artificial gr/$n$ML-Ni/Ir(111) or gr/$n$ML-Co/Ir(111) systems, which can be formed via intercalation of Ni or Co, respectively, in gr/Ir(111) [@Pacile:2013jc; @Vu:2016ei], several factors, like gr-metal lattice mismatch, graphene morphology, quantum size effects in the electronic structure of the Ni or Co layer, might influence the catalytic activity of such gr/Ni or gr/Co interfaces.
Here we present the experiments on the oxygen and nitrogen intercalation in gr/$n$ML-Ni/Ir(111), where thickness of the Ni layer is varied from sub-monolayer to bulk-like thick film. Our near-ambient pressure x-ray photoelectron spectroscopy (XPS) and scanning tunnelling microscopy (STM) studies allowed to discriminate between different factors, like system morphology and the electronic structure of thin Ni films, which govern the oxygen intercalation and formation of thin NiO layer under graphene. Surprisingly, our attempts on the N$_2$ intercalation in the gr/Ni/Ir(111) interface in a wide range of the gas pressure and substrate temperature did not succeed, indicating the stronger N-N bond compared to the O-O one and influence of the catalytically active metal substrates under graphene on the molecules decomposition during intercalation. Our experiments demonstrate the possibility to tune the catalytic properties of the gr/metal system in the confined catalysis processes and will stimulate further studies in this research field.
Results and discussions {#ResDisc}
=======================
Oxygen intercalation in gr/Ir(111): NAP-XPS and STM
---------------------------------------------------
We started our studies on the oxygen and nitrogen intercalation in the gr/Ni/Ir(111) system from the well-known example: O$_2$ – intercalation in the parent gr/Ir(111) interface [@Larciprete:2012aaa; @Granas:2012cf]. In our near-ambient-pressure-XPS (NAP-XPS) and NAP-STM studies the Ir(111) substrate was fully covered with a complete graphene layer and the obtained NAP-XPS and STM results are compiled in Figs. \[grOIr\_STM\] and \[grOIr111\_NAP\_XPS\].
Graphene was grown on Ir(111) via exposing its surface to C$_2$H$_4$ at $10^{-6}$mbar and $1400$K. This procedure leads to a controllable growth of a high-quality graphene layer which fully covers Ir(111) as deduced from the large-scale STM experiments. In the STM images (Fig. \[grOIr\_STM\]), gr/Ir(111) displays large fully graphene covered terraces, which are several hundreds nanometer wide and have straight steps following the directions of the graphene moiré structure [@Voloshina:2013dq; @Dedkov:2014di]. This graphene moiré structure is clearly measured in the STM experiments as an additional modulation of the graphene lattice with a periodicity of $25.2$Å (Fig. \[grOIr\_STM\](a)). It is formed on top of Ir(111) due to the relatively large lattice mismatch of $\approx10$% between two in-plane lattices. Previously published experimental data [@Hattab:2011ix] indicate the formation of the rotational domains at the temperature which was used during graphene growth on Ir(111) in our experiments. Despite the fact that we cannot fully exclude the formation of such domains during the preparation routine, our present STM results show that number of the corresponding defects (domain boundaries) is very small (if any).
Oxygen intercalation in high-quality gr/Ir(111) was performed at the O$_2$ partial pressure of $0.1$mbar and the sample temperatures of $200^\circ$C. Fig. \[grOIr\_STM\](b) shows the STM image acquired after $2$min of oxygen exposure at these conditions and one can see that the moiré structure of the gr/Ir(111) system started to become weak. The areas in the STM image shown in panel (b) mark the high-symmetry positions of the gr/Ir(111) structure (namely the ATOP positions) which disappear after described treatment. According to other experiments on the CO intercalation in h-BN/Ru(0001) [@Wu:2018gb] the process of the gas molecules intercalation starts around the defect sites in the 2D layer or/and at the steps and boundaries of the 2D-material/metal structure, where 2D layer becomes flatter after gas molecules penetration under 2D layer. We believe that the similar mechanism can be applied here as it is also supported by the previous studies on the oxygen intercalation in gr/Ir(111) [@Larciprete:2012aaa; @Granas:2012cf]. The prolonged exposure of the gr/Ir(111) system to oxygen or/and the slight increase of the temperature used during this procedure leads to the intercalation of oxygen in all areas leading to the flattening of a graphene layer on Ir(111) (Fig. \[grOIr\_STM\](c)). After oxygen intercalation in gr/Ir(111) the moiré structure of graphene is hardly recognisable in the STM images (Fig. \[grOIr\_STM\](c,d)).
The process of the oxygen intercalation in gr/Ir(111) was further confirmed and studied in the NAP-XPS experiment. Fig. \[grOIr111\_NAP\_XPS\] shows the time evolution of the (a) C$1s$ and (b) O$1s$ emission lines during the exposure of gr/Ir(111) to O$_2$ at the partial pressure of $0.1$mbar and the sample temperature of $200^\circ$C. These data show that oxygen gradually intercalates in gr/Ir(111) at these conditions that is manifested as the shift of the C$1s$ line to smaller binding energies. After complete oxygen intercalation in gr/Ir(111) the binding energy for C$1s$ is $283.7\pm0.1$eV compared to the initial value of $284.2\pm0.1$eV measured before intercalation for gr/Ir(111). Both values for the binding energy are in good agreement with the previously published data [@Larciprete:2012aaa]. At the same time the intensity of the O$1s$ line is monotonically increased with the unchanged binding energy for this line. These data provide a direct spectroscopic evidence that oxygen can diffuse to the gr/Ir(111) interface and adsorb on Ir(111) underneath the graphene overlayer. The shift of the C$1s$ line to the smaller binding energies indicates the strong $p$ doping of a graphene layer in the resulting gr/O/Ir(111) systems as confirmed by the previous angle-resolved photoemission (ARPES) data [@Larciprete:2012aaa]. From the XPS data we found that the full-width-at-the-half-maximum (FWHM) of the C$1s$ line before oxygen intercalation is $0.82$eV and it does not change significantly after oxygen intercalation in gr/Ir(111). From all these results, we conclude that these parameters for the oxygen intercalation (the O$_2$ partial pressure of $0.1$mbar and the sample temperature of $200^\circ$) can be used in our further experiments on the studies of the gas intercalations in the gr/Ni/Ir(111) system. At the same time, according to the XPS and STM data, we conclude that oxygen is uniformly intercalated in the gr/Ir(111) system.
Oxygen intercalation in gr/$1.6$ML-Ni/Ir(111) and gr/Ni(111): NAP-XPS temperature dependence
--------------------------------------------------------------------------------------------
In the further experiments, prior to the oxygen or nitrogen intercalation, the gr/Ir(111) interface was modified via intercalation of Ni layer of different thickness. The intercalation of different amounts of Ni in gr/Ir(111) leads to the formation of the gr/Ni/Ir(111) interfaces of different morphologies: from the strongly buckled graphene layer on $1$ML-Ni/Ir(111) to flat graphene on thick-Ni/Ir(111) [@Pacile:2013jc]. As tests systems for the gas intercalation we chose gr/$1.6$ML-Ni/Ir(111) (strongly buckled graphene) and gr/Ni(111) (flat graphene) systems. The results for the oxygen intercalation in these interfaces are summarized in Fig. \[grONiIr\_TempDep\] where intensities of the C$1s$, O$1s$, and Ni$2p$ lines are shown as a function of the substrate temperature used in these NAP-XPS measurements.
Before oxygen intercalation, the C$1s$ lines for the considered systems have a different shape: for gr/Ni(111) it consists of a single peak located at $284.9\pm0.1$eV, whereas for the gr/$1.6$ML-Ni/Ir(111) systems this line consists of a main peak at $284.9\pm0.1$eV and small shoulder at $284.65\pm0.1$eV (see the respective fit of the C1s XPS spectrum for gr/$1.6$ML-Ni/Ir(111) before oxygen exposure in Fig. \[grONiIr\_TempDep\](a)), which is consistent with the previously published results [@Pacile:2013jc]. (The small bump at $283.3\pm0.1$eV in the C$1s$ spectrum for gr/Ni(111) is due to the small fraction of the Ni$_2$C phase which is rarely avoided during formation of graphene on a Ni(111) single crystal [@Dedkov:2017jn; @Spath:2016db].) For both systems, the emission line for Ni$2p$ represents the spin-orbit split doublet at $852.6\pm0.1$eV and $869.9\pm0.1$eV as well as the correlation satellite peaks which are located $6$eV below every main emission line.
The exposure of both systems to O$_2$ at the partial pressure of $0.1$mbar and different samples temperatures indicates the existence of different processes which define the oxygen intercalation under graphene as well as the oxidation of the underlying Ni. For the first case of the gr/$1.6$ML-Ni/Ir(111) system, the exposure to O$_2$ at room temperature does not lead to any changes of the C$1s$ and Ni$2p$ emission lines (Fig. \[grONiIr\_TempDep\](a,c)) as was also previously observed for other graphene-based intercalation systems [@Dedkov:2008e; @deLima:2014dm; @deCamposFerreira:2018hh]. The similar treatment of the gr/Ni(111) interface leads to the partial intercalation of oxygen already at room temperature as can be visible from the appearance of the shoulder on the low binding energy side of the C$1s$ peak (third spectrum from the bottom in Fig. \[grONiIr\_TempDep\](d)). These conclusions are also supported by the absence of the O$1s$ emission for these conditions in case of gr/$1.6$ML-Ni/Ir(111) (Fig. \[grONiIr\_TempDep\](b)) and its appearance in case of gr/Ni(111) (Fig. \[grONiIr\_TempDep\](e)). The same is valid for Ni$2p$ where the start of the nickel oxidation is detected for gr/Ni(111) at $0.1$mbar and room temperature. From these data we can conclude that the quality and the morphology of a graphene layer for gr/$1.6$ML-Ni/Ir(111) is more uniform compared to gr/Ni(111) and that number of defects in graphene is much smaller for the former system. This leads to the effective protection of the underlying intercalated layer for gr/Ni/Ir. One of the reasons which might promote the intercalation of oxygen under graphene on Ni(111) single crystal is the unavoidable formation of the Ni$_2$C phase which always formed during sample preparation. These phase was also observed in our previous and present experiments as confirmed by STM and XPS.
Further increase of the sample temperature above the room temperature or/and the partial oxygen pressure for both considered systems – gr/$1.6$ML-Ni/Ir(111) and gr/Ni(111) – leads to the effective intercalation of oxygen under graphene and to the oxidation of Ni. In the end of this experiment the binding energy of C$1s$ is $284.1$eV indicating the complete decoupling of graphene from the substrate and the formation of the gr/NiO interface [@Dedkov:2017jn]. It is interesting to note the difference for two systems with regard to the shape of the O$1s$ peak. In the end of the intercalation process it consists of a single component at $529.5\pm0.1$eV with a small shoulder $531.3\pm0.1$eV for gr/$1.6$ML-Ni/Ir(111) and consists of two clear peaks at $529.4\pm0.1$eV and $531.2\pm0.1$eV for gr/Ni(111). The low binding energy component in both cases can be clearly assigned to the emission from O$^{2-}$ in NiO [@Lorenz:2000cx; @Rettew:2011kd]. The second one might be due to the presence of OH$^{-}$ groups on the surface or can be connected with defects in the oxide layer [@Rettew:2011kd; @Tyuliev:1999je; @Roberts:1984du]. Taking into account the fact that before oxygen intercalation the gr/Ni(111) sample contains some fraction of Ni$_2$C we can conclude that the second peak is dominated by the emission from the oxygen atoms associated with the defects in the nickel oxide layer.
Oxygen intercalation in gr/$n$ML-Ni/Ir(111): STM morphology
-----------------------------------------------------------
The morphology of the gr/$n$ML-Ni/Ir(111) system with different amount of intercalated Ni was investigated by means of STM before and after oxygen intercalation. These results are compiled in Fig. \[grONiIr\_STM\] where data for (a-c) gr/$0.5$ML-Ni/Ir(111), (d-f) gr/$1.2$ML-Ni/Ir(111), and (g-i) gr/$1.6$ML-Ni/Ir(111) are shown (left column: morphology before oxygen intercalation; middle and right columns: morphology after oxygen intercalation).
STM results obtained before oxygen intercalation demonstrate high quality of the respective gr/$n$ML-Ni/Ir(111) systems where Ni layers of different thicknesses were successfully completely intercalated in gr/Ir(111). For gr/$0.5$ML-Ni/Ir(111) (Fig. \[grONiIr\_STM\](a)) the areas which are partly modified by Ni can be easily recognized due to the increased corrugation of a graphene layer on Ni which is now pseudomorphically arranged on Ir(111) [@Pacile:2013jc]. Here Ni intercalates at different places of this system – on terraces as well as around step edges. We found that it is possible to intercalate more than $1$ML of Ni in gr/Ir(111) (Fig. \[grONiIr\_STM\](d,g)), where first layer of Ni forms the pseudomorphic layer at the gr/Ir(111) interface and the excess amount of Ni forms mono-, double-, and triple-ML thick islands under graphene. The moiré structure of a graphene layer is preserved only for the double-layer thick Ni islands indicating the lattice relaxation in the Ni layer for thicker layers. In case of a thick Ni layer (more than $20$ML, STM data are not shown) between graphene and Ir(111), the different system preparation was used – $20$ML-thick Ni(111) film was grown on Ir(111) and then graphene was grown via CVD as described in Sec.“Experimental”, leading to the lattice-matched system in this case, similar to graphene on the bulk Ni(111), but with the extremely low fraction of the Ni$_2$C phase.
The intercalation of oxygen in gr/$n$ML-Ni/Ir(111) leads to the drastic changes in morphology due to the formation of NiO at the gr/Ir(111) interface. After oxygen intercalation in gr/$0.5$ML-Ni/Ir(111) two areas can be clearly identified – gr/O/Ir(111) and gr/NiO/Ir(111) (they are marked in Fig. \[grONiIr\_STM\](b)). The first one is very similar to the one presented in Fig. \[grOIr\_STM\](c,d) with a faint moiré structure, whereas for the gr/NiO/Ir(111) patches the corrugation of graphene is much larger due to the formation of the lattice-mismatched NiO/Ir(111) interface which demonstrates the long-range periodicities similar to the one observed for other NiO,CoO/$4d$,$5d$-metal(111) interfaces [@Hagendorf:2006eo; @DeSantis:2011fv; @Franz:2012kn; @Gragnaniello:2013ip]. After intercalation of oxygen in gr/$n$ML-Ni/Ir(111) with thicker Ni layer of more than $1$ML, the STM images demonstrate stripe-like and maze-like long-range ordered structures for the NiO layer under graphene, which is also supported by fast Fourier transformation (FFT) images (Fig. \[grONiIr\_STM\](e-f,h-i)). It can be explained by the in-plane relaxation in the NiO layer as a function of thickness. In all cases the atomic resolution in graphene layer can be easily obtained where a graphene lattice with all 6 carbon atoms is imaged (Fig. \[grONiIr\_STM\]: (c) and inset of (i)) confirming the effective decoupling of graphene from the underlying NiO layer. The decoupling of graphene from NiO is also supported by the present XPS data as well as by the recent ARPES and density-functional theory (DFT) studies [@Dedkov:2017jn] where $p$-doping of graphene was observed.
Oxygen intercalation in gr/$n$ML-Ni/Ir(111): NAP-XPS time dependence
--------------------------------------------------------------------
In order to study the dynamics of the oxygen intercalation process we performed time-dependent NAP-XPS experiments for different gr/$n$ML-Ni/Ir(111) systems and these results are presented for $0.5$ML-Ni (Fig. \[grO05MLNiIr111\_NAP\_XPS\]), $1.2$ML-Ni (Fig. \[grO12MLNiIr111\_NAP\_XPS\]), and $20$ML-Ni (Fig. \[grOthickNiIr111\_NAP\_XPS\]). First two experiments were performed in the same experimental conditions for the oxygen intercalation ($0.1$mbar and $150^\circ$C) and for the thick Ni layer we chose higher temperature ($250^\circ$C) in order to vary one of the parameters in comparison with the experimental conditions for thin Ni layers.
For the gr/$0.5$ML-Ni/Ir(111) system, two C$1s$ components can be identified in the spectra after its fit (Fig. \[grO05MLNiIr111\_NAP\_XPS\](a); fit components are shown for the bottom spectra as thin solid lines) – at $284.9\pm0.1$eV and $284.2\pm0.1$eV, which can be assigned to the graphene covered areas with and without Ni intercalated, respectively. The intercalation of oxygen in this system on the initial stage (time between $0$min and $20$min) leads to the diffusion of oxygen atoms towards Ni islands and to the predominant oxidation of the Ni layer as indicated by the reduction of the intensity of the corresponding C$1s$ component. After this process is nearly complete, the areas where Ni was not intercalated start to fill up with oxygen atoms. The second step is indicated as a gradual shift of the centre of gravity of the whole C$1s$ peak towards lower binding energy and as a simultaneous reduction of its FWHM. Finally this peak has a single component indicating the complete oxygen layer intercalation and that in all areas (above Ir and above formed NiO) graphene layer is in contact with oxygen atoms. The measured at the same time O$1s$ and Ni$2p$ spectra (Fig. \[grO05MLNiIr111\_NAP\_XPS\](b,c)) show the increase of the oxygen signal and complete oxidation of the thin Ni layer already at the initial steps of oxygen intercalation.
In case of gr/$1.2$ML-Ni/Ir(111), as was discussed earlier, the C$1s$ peak consists of the two components which can be assigned to two distinct areas for the strongly buckled graphene layer in this system: the higher and lower binding energy components are due to the larger and smaller interaction strengths of carbon atoms with Ni atoms in a lattice mismatched graphene layer [@Pacile:2013jc]. The oxygen intercalation in this system leads to the gradual decrease of intensities of both gr-Ni-related C$1s$ components with the simultaneous growth of the intensity of the gr-NiO-related component (Fig. \[grO12MLNiIr111\_NAP\_XPS\](a)). These changes indicate that during oxygen-intercalation the oxygen atoms open the boundary between graphene and metal step by step, and finally complete the intercalation with the full oxidation of the Ni layer and its conversion to NiO. This picture is also supported by the changes of intensities of the O$1s$ and Ni$2p$ lines (Fig. \[grO12MLNiIr111\_NAP\_XPS\](b,c)) – the intensity of the correlation satellite ($6$eV below the main line) slowly decreases with the simultaneous growth of the components which can be assigned to the formation of NiO. The behaviour of the shape of the Ni$2p$ emission line for the thin NiO layers which are formed during oxygen intercalation in gr/$0.5$ML-Ni/Ir(111) and gr/$1.2$ML-Ni/Ir(111) is similar to the one observed in the previous experiments on the growth of thin NiO on metallic surfaces, like Ag(001) [@Caffio:2004hg] and Cu(111) [@SnchezAgudo:2000cm], where suppression of the shake-up satellite was observed without big energy shift of the main emission line. In the present experiments the energy shift of the Ni$2p_{3/2}$ peak (and the corresponding spin-orbit split counterpart) of about $300$meV to higher binding energies is observed. From the other side the absence of the big energy shift of the Ni$2p$ line can be assigned to the adsorbing of oxygen above Ni under graphene. However, further experiments and theoretical considerations are required to shed light on this effect.
The time dependent NAP-XPS experiments on oxygen intercalation in the gr/$20$ML-Ni/Ir(111) system show the similar intercalation behaviour like for gr/bulk-Ni(111) (Fig. \[grOthickNiIr111\_NAP\_XPS\]). In this case the single-component C$1s$ peak at $284.9\pm0.1$eV is slowly shifted to the smaller binding energies and reaches $284.1\pm0.1$eV after the intercalation process is complete (Fig. \[grOthickNiIr111\_NAP\_XPS\](a)). At this stage the process of the oxygen intercalation and formation of the interface NiO layer saturates that is indicated by the stop of the change of the position of the C$1s$ line as well as by the stop of the growth of the intensity of the O$1s$ line and the respective components in the Ni$2p$ spectra (Fig. \[grOthickNiIr111\_NAP\_XPS\](b,c)). It is interesting to note that in the O$1s$ spectra for this system the component which is assigned to the presence of OH$^{-}$ groups on the surface or can be connected with defects in the oxide layer appears already on the initial steps of the oxygen intercalation in gr/$20$ML-Ni/Ir(111). The slow decrease of the fraction of this component in the total intensity of the O$1s$ line as a function of time can be assigned with a small reduction of the number of defects in the NiO layer. However, comparison of the respective O$1s$ spectra for all three systems shows that number of defects in the formed NiO layer is much smaller in case of pre-intercalated thin Ni layers in gr/Ir(111).
Attempts of the nitrogen intercalation in gr/Ir(111) and gr/Ni(111): NAP-XPS temperature dependence
---------------------------------------------------------------------------------------------------
It is well known that adsorption of graphene on metallic or semiconducting surfaces always lead to the doping of graphene with its charge neutrality point placed above or below the Fermi level followed by the modification its electronic structure [@Dedkov:2015kp; @Batzill:2012; @Tesch:2018hm]. Decoupling of graphene from the substrate aims to have a neutral graphene with high charge mobility. Different approaches on the intercalation of the atoms of, e.g., Ge, O, or F, partially succeed in this strategy [@Verbitskiy:2015kq] or lead to the strongly $p$-doped graphene for the highly electronegative atoms [@Dedkov:2017jn; @Larciprete:2012aaa; @Granas:2012cf; @Walter:2011gf]. One of the possibilities to reach the charge neutrality of graphene is its decoupling from the substrate by nitrogen atoms [@Caffrey:2015et]. However, up to now our knowledge, there are no studies on the intercalation of nitrogen in the gr/substrate interfaces. Some rare studies on the N-plasma or atomic-N treatment of such interfaces can be found in the literature [@Masuda:2015eg; @VelezFort:2014kh; @Tsai:2014gf; @Chai:2008fc] that leads to the interfaces nitridation or to the N-atom incorporation in a graphene layer.
We have chosen the different approach and performed several attempts to intercalate nitrogen in gr/Ir(111) and gr/Ni(111) at high partial pressure of the N$_2$ gas and at the elevated substrate temperature. Fig. \[NAPXPS\_N2\_grIr111\_grNi111\] shows C$1s$ (a and c) and N$1s$ (b and d) spectra for gr/Ir(111) and gr/Ni(111), respectively, during exposure of these systems to N$_2$ at different gas pressures and temperatures. One can clearly see that in both cases the C$1s$ line does not change during the sample treatment procedure – no binding energy nor intensity changes, indicating the absence of the nitrogen intercalation. The same is true if N$1s$ spectra are considered – the binding energy of this line is $405.1\pm0.1$eV and $405.3\pm0.1$eV for gr/Ir(111) and gr/Ni(111), respectively, that is characterised for N$_2$ molecules [@Kramberger:2013dt]. (The slight difference in the binding energy for the N$1s$ line can be assigned to the different interaction strength at the respective gr-metal interface that can change the graphene-adsorbate interaction [@Huttmann:2015hb].) Moreover, this N$1s$ line disappear after gas is fully pumped from the UHV chamber, again supporting our conclusion on the absence of the nitrogen intercalation in the studied gr-metal interfaces. Our STM data for gr/Ir(111) and gr/Ni(111) (not shown here) collected before and after exposure these surfaces to N$_2$ also do not demonstrate any changes in the sample morphology.
Conclusions
===========
We performed systematic NAP-XPS and STM studies of oxygen and nitrogen intercalation in the gr/Ni/Ir(111) interfaces of different morphologies which can be realized via pre-intercalation of different amounts of Ni. It is found that oxygen can be easily intercalated in all interfaces at the elevated temperature of the substrate and that intercalation is associated with the O$_2$-molecules dissociation. The intercalation of oxygen leads to the effective oxidation of the underlying Ni layer with the formation of the NiO layer of different thickness and morphology. This process leads to the effective decoupling of graphene in all cases with the resulting its strong $p$-doping. Our attempts to intercalate nitrogen at different conditions in gr/Ir(111) and gr/Ni(111) do not lead to the gas intercalation pointing the importance of the gas molecules dissociation at the gr-metal interface. Considering all available data on the gas intercalation in the gr-metal interfaces we can conclude that the bonding energy of atoms in the molecules and the probability to split in atoms at the metallic surface play crucial roles in the process. We suggest that the respective theoretical model has to be built which will allow to rationalise the studies of the systems on the intercalation of different gaseous species in different gr-metal and gr-semiconductor interfaces.
Experimental {#ExpDet .unnumbered}
============
All experiments were performed in a customized station from SPECS Surface Nano Analysis GmbH for near-ambient pressure (NAP) spectroscopy and microscopy experiments consisting of three ultrahigh vacuum (UHV) chambers: NAP-XPS chamber, NAP-STM chamber, and molecular beam epitaxy (MBE) chamber, allowing for a sample transfer without breaking of the vacuum conditions. The background pressure of all three chambers is below than $< 3 \times 10^{-10}$mbar.
The SPECS Aarhus NAP-STM is equipped with a high-pressure reaction cell inside, which is sealed via a Viton$^{\textregistered}$ O-ring when performing high pressure experiments in the range of $10^{-5}$mbar – $100$mbar. The maximum sample temperature during scanning can reach up to $770$K in UHV and about $500$K under near ambient pressure conditions and heating is performed by a halogen lamp heater which is mounted directly on the lid.
The NAP-XPS chamber in the backfilling configuration is equipped with a SPECS PHOIBOS150 NAP hemispherical analyser (with $5$ stages of the differential pumping system) and monochromatic SPECS $\mu$-FOCUS600 NAP x-ray source: AlK$\alpha$ line ($h\nu=1486.6$eV), x-ray spot size on the sample $<250\mu\mathrm{m}$. This system allows NAP-XPS samples investigations in the pressure range for gasses up to $25$mbar with sample temperatures up to $1400$K by laser heating.
As a substrate for our experiments we used Ir(111) and Ni(111) crystals (MaTecK GmbH). Prior to every experimental run, these crystals were cleaned via cycles of Ar$^+$ sputtering ($2$kV, $5 \times 10^{-6}$mbar, $15$min) at room temperature with subsequent oxygen treatment at $700$K ($0.1$mbar, $10$min), followed by ultrahigh vacuum temperature flash for $3$min at $1800$K for Ir(111) and at $1000$K for Ni(111), respectively. Temperature for the graphene growth on Ir(111) or Ni(111) was estimated from the power vs. temperature curve (measured independently) or was measured by a chromel-alumel thermocouple, respectively. No subsurface Ar gas bubbles were observed for metallic surfaces used in our experiments as deduced from the STM and XPS measurements. Graphene overlayers were grown on Ir(111) and Ni(111) surface via chemical vapour deposition (CVD) procedure using C$_2$H$_4$ as described in the literature [@Voloshina:2013dq; @Dedkov:2017jn], which lead to the graphene layers of very high quality as was confirmed by STM and XPS experiments. Thin intercalation Ni layers under graphene on Ir(111) were formed via annealing of the respective Ni/graphene/Ir(111) system at $800$K and the thickness of the intercalated Ni layer was estimated from the subsequent STM and XPS experiments. Intercalation of gases, oxygen and nitrogen, was studied via direct backfilling of the NAP-XPS chamber through the leak valves with the raised sample temperature, which was measured by a chromel-alumel thermocouple or by an infrared pyrometer. XPS spectra were measured directly during studies of the intercalation process in the “live” mode. The positions of the individual components in some spectra were determined after fitting procedure - single components were considered as a convolution of the Lorentzian and Doniach-Sunjic line shapes with the linear background.
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![STM images of gr/Ir(111) before and after oxygen intercalation: (a) gr/Ir(111), $75\times75\,\mathrm{nm}^2$, $U_T=+0.17\,\mathrm{V}$, $I_T=0.68\,\mathrm{nA}$, (b) after exposure of gr/Ir(111) to O$_2$ at $0.1$mbar and $200^\circ$C for $2$min (partial intercalation), $30\times30\,\mathrm{nm}^2$, $U_T=+0.58\,\mathrm{V}$, $I_T=0.42\,\mathrm{nA}$, and (c) after exposure of gr/Ir(111) to O$_2$ at $0.1$mbar and $200^\circ$C for $50$min (complete intercalation), $200\times200\,\mathrm{nm}^2$, $U_T=+1.25\,\mathrm{V}$, $I_T=0.11\,\mathrm{nA}$. Atomically resolved image in (d) shows a zoomed area of (c) with imaging parameters: $7\times7\,\mathrm{nm}^2$, $U_T=+0.01\,\mathrm{V}$, $I_T=1\,\mathrm{nA}$. Circles in (b) mark the high-symmetry position of gr/Ir(111) which disappear after $2$min exposure this system to oxygen.[]{data-label="grOIr_STM"}](fig1.jpg){width="0.7\columnwidth"}
![NAP-XPS intensities of (a) C$1s$ and (b) O$1s$ measured for gr/Ir(111) during its exposure to O$_2$ at $0.1$mar and $200^\circ$C. The respective exposure time for every spectra is marked in the figure legend. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="grOIr111_NAP_XPS"}](fig2.jpg){width="0.7\columnwidth"}
![NAP-XPS intensities of (a,d) C$1s$, (b,e) O$1s$, and (c,f) Ni$2p$ measured for gr/$1.6$ML-Ni/Ir(111) (top row) and gr/Ni(111) (bottom row) during exposure to O$_2$ at fixed pressure - $0.1$mar or $1.0$mbar and different sample temperatures. The respective exposure parameters for every spectra are marked in the figure legend. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="grONiIr_TempDep"}](fig3.jpg){width="0.75\columnwidth"}
![STM images before (left column) and after (middle and right columns) oxygen intercalation in gr/$n$ML-Ni/Ir(111): (a-c) gr/$0.5$ML-Ni/Ir(111), (d-f) gr/$1.2$ML-Ni/Ir(111), and (g-i) gr/$1.6$ML-Ni/Ir(111). Insets of (c) and (f) show the respective FFT images and inset of (i) shows the small scale atomically resolved zoom this image. Imaging parameters: (a) $170\times170\,\mathrm{nm}^2$, $U_T=+0.31\,\mathrm{V}$, $I_T=0.35\,\mathrm{nA}$; (b) $100\times100\,\mathrm{nm}^2$, $U_T=+0.45\,\mathrm{V}$, $I_T=0.41\,\mathrm{nA}$; (c) $6\times6\,\mathrm{nm}^2$, $U_T=+0.01\,\mathrm{V}$, $I_T=0.4\,\mathrm{nA}$; (d) $105\times105\,\mathrm{nm}^2$, $U_T=+0.42\,\mathrm{V}$, $I_T=0.29\,\mathrm{nA}$; (e) $100\times100\,\mathrm{nm}^2$, $U_T=+0.02\,\mathrm{V}$, $I_T=0.27\,\mathrm{nA}$; (f) $10\times10\,\mathrm{nm}^2$, $U_T=+0.03\,\mathrm{V}$, $I_T=0.49\,\mathrm{nA}$; (g) $75\times75\,\mathrm{nm}^2$, $U_T=+0.3\,\mathrm{V}$, $I_T=0.56\,\mathrm{nA}$; (h) $100\times100\,\mathrm{nm}^2$, $U_T=+0.12\,\mathrm{V}$, $I_T=0.62\,\mathrm{nA}$; (i) $30\times30\,\mathrm{nm}^2$, $U_T=+0.01\,\mathrm{V}$, $I_T=1.05\,\mathrm{nA}$; iset of (i) $10\times10\,\mathrm{nm}^2$, $U_T=+0.004\,\mathrm{V}$, $I_T=0.44\,\mathrm{nA}$. Parameters for the oxygen intercalation: (b,c) $0.1$mbar, $150^\circ$C, $60$min, (e,f) $0.1$mbar, $150^\circ$C, $110$min, (h,i) $0.1$mbar, $150^\circ$C, $150$min.[]{data-label="grONiIr_STM"}](fig4.jpg){width="0.7\columnwidth"}
![Variation of the NAP-XPS intensities for (a) C$1s$, (b) O$1s$, and (c) Ni$2p$ measured during exposure of the gr/$0.5$ML-Ni/Ir(111) system to O$_2$ at fixed pressure ($0.1$mar) and fixed sample temperature ($150^\circ$C). The respective exposure time for every spectra is marked in the figure legend. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="grO05MLNiIr111_NAP_XPS"}](fig5.jpg){width="80.00000%"}
![Variation of the NAP-XPS intensities for (a) C$1s$, (b) O$1s$, and (c) Ni$2p$ measured during exposure of the gr/$1.2$ML-Ni/Ir(111) system to O$_2$ at fixed pressure ($0.1$mar) and fixed sample temperature ($150^\circ$C). The respective exposure time for every spectra is marked in the figure legend. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="grO12MLNiIr111_NAP_XPS"}](fig6.jpg){width="0.8\columnwidth"}
![Variation of the NAP-XPS intensities for (a) C$1s$, (b) O$1s$, and (c) Ni$2p$ measured during exposure of the gr/$20$ML-Ni/Ir(111) system to O$_2$ at fixed pressure ($0.1$mar) and fixed sample temperature ($150^\circ$C). The respective exposure time for every spectra is marked in the figure legend. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="grOthickNiIr111_NAP_XPS"}](fig7.jpg){width="0.8\columnwidth"}
![NAP-XPS intensities of (a,c) C$1s$ and (b,d) N$1s$ measured during exposure of gr/Ir(111) and gr/Ni(111), respectively, to N$_2$ at different gas pressures and temperatures of the substrate. All XPS spectra are shifted for clarity. The smoothed line through the experimental points is shown for every spectra.[]{data-label="NAPXPS_N2_grIr111_grNi111"}](fig8.jpg){width="0.5\columnwidth"}
{width="0.5\columnwidth"}
[^1]: Corresponding author. E-mail: voloshina@shu.edu.cn
[^2]: Corresponding author. E-mail: dedkov@shu.edu.cn
[^3]: Corresponding author. E-mail: ycui2015@sinano.ac.cn
|
---
abstract: 'In analysing a well-known hash-coding method, Knuth gave an exact expression for the average number of rejections encountered by players of a variant of musical chairs. We study a variant more closely related to musical chairs itself and deduce the same expression by a purely combinatorial approach.'
author:
- 'Vaughan R. Pratt'
bibliography:
- 'pratt.bib'
date: 'December 1973, revised 1974'
title: 'A combinatorial analysis of the average time for open-address hash coding insertion'
---
In an analysis of the average time to insert an item when using open-address hash-coding, Knuth [@knuth1973 p. 528-530] reduced the problem to the following question about musical chairs. Given $m$ chairs arranged in a circle and numbered clockwise from $0$ to $m-1$, if in turn each of $n$ people arrives at a randomly selected chair (his *initial* chair) and walks clockwise until he finds an empty chair (his *final* chair), what is the average number of *rejections* (chairs found occupied during the search) per player?
For $m \geq n$, Knuth showed that the average number of rejections is $$\frac{1}{2n}\left( \frac{n^{\underline{2}}}{m} + \frac{n^{\underline{3}}}{m^{2}} + \frac{n^{\underline{4}}}{m^{3}} + \ldots \right)$$ where $n^{\underline{i}} = {n\choose{i}}i!$, the number of ways of choosing from $n$ items an ordered subset of $i$ items. Knuth’s analysis is very complex and gives no clue as to why the answer is so simple. Knuth appreciated this issue and asked for an explanation [@knuth1973a].
In the following analysis we reduce the problem yet further, to a game even closer to musical chairs, in which the average number of rejections is the same as in Knuth’s game. We establish a one-to-one correspondence between the set of all rejections encountered, over the whole sample space, and a set whose cardinality is easily established. In this way we eliminate *all* of the involved algebra of Knuth’s proof. Regrettably, despite the absence of algebra, the following combinatorial proof is not as simple as the author first envisaged when he undertook the search for it. A simpler proof would be a valuable contribution to this facet of the theory of hash coding.
In musical chairs, the players move around until the music stops, at which time they all rush for chairs. To change Knuth’s problem to resemble more closely this situation we shall require all $n$ people to have arrived before letting them search for their chairs. In effect, the people are partitioned into $m$ labelled blocks (the chairs providing the labels), some of which may be empty. All blocks then travel around the chairs clockwise at the same speed, each losing one member to each vacant chair it passes until it becomes empty. This process is the *seating process*. To resolve seating conflicts, we rank the players in advance. The reader should have little difficulty in verifying that this version of musical chairs involves the same average number of rejections per player as does Knuth’s version.
For our analysis, we shall count the total number $R$ of rejections suffered by all players, summed over all possible *samples* (assignments of players to initial chairs). The average is then $\frac{R}{nm^{n}}$ (we must assume the players are distinguishable), which implies that $R$ should be $$\frac{1}{2}(n^{\underline{2}}m^{n-1} + n^{\underline{3}}m^{n-2} + n^{\underline{4}}m^{n-3} + ...)$$.
We say that a sample $S$ of $n$ players *matches* a *sub-sample* $T$ of $k$ out of those $n$ players, for $k \leq n$, when every member of the $j$-th block of $T$ is a member of the $j$-th block of $S$, for $j = 0, 1,..., m - 1$. When a sub-sample $P$ of $k$ out of our $n$ players has a block of two people assigned to chairs $c$, $k-2$ blocks, of one person each, assigned to chairs ${c+1,c+2,...,c+k-2}$ (addition mod $m$), and no other blocks, we call $P$ a *$k$-pattern*. Exactly $m^{n-k}$ samples match any given $k$-pattern, and there are $\frac{n^{\underline{k}}m}{2}$ possible $k$-patterns, taking rotations into account. Hence the total number of possible matches of samples with $k$-patterns, for $k \geq 2$, is $$\frac{1}{2}(n^{\underline{2}}m^{n-1} + n^{\underline{3}}m^{n-2} + n^{\underline{4}}m^{n-3} + ...),$$ which equals the total number of rejections given by Knuth’s analysis. We shall give a one-to-one correspondence between rejections and matches, which immediately gives us an alternative derivation of this expression for the number of rejections.
Given a sample $S$ and a particular rejection, say of player $a$ at the seat containing player $z$, we shall construct a particular match between some sample $T$ and some pattern $P$. We shall form $T$ from $S$ merely by relabelling the blocks of $S$, that is, by permuting them with respect to the chairs. The permutation amounts to gathering together a certain subset of the blocks, called the *distinguished* blocks. The block containing player $a$ (call it $b_1$) remains where it is, at chair $c$, and is the first distinguished block. The other distinguished blocks of $S$ (call them $b_2, b_3,...,b_k$ for some $k \geq 1$) are slid around the circle counter-clockwise by interchanging them with the non-distinguished blocks, until blocks $b_1, b_2,...,b_k$ occupy chairs $c, c+1, ..., c+k-1$ respectively. The undistinguished blocks occupy the remaining chairs. Note that, among themselves, the $k$ distinguished blocks retain their relative initial seating order, as do the $m-k$ undistinguished blocks.
We now specify the distinguished blocks. Block $b_1$ has already been specified. Block $b_{i+1}$ is specified by the following procedure.
1. If block $b_i$ contains player $z$, then we are done and $k = i$. We refer to player $z$ as $p_k$.
2. Otherwise, let $d_i$ be the last chair to which $b_i$ lost a member (call him $p_i$) before $b_i$ (possibly exhausted) encountered $z$’s final chair. Then $b_{i+1}$ is the block whose initial chair is $d_i + 1$.
This procedure completes the description of the construction of $T$. Note that there is a question as to whether $d_i$ is always defined, which is dealt with now.
In the following we let $[a, b]$ denote the set of chairs $\{a, a+1, a+2, ..., b\}$, the addition being modulo $m$. We let $(a,b] = [a,b] - \{a\}$, $[a,b) = [a,b] - \{b\}$, and $(a,b) = [a,b] - \{a, b\}$. We let $\underline{b}_i$ denote the initial chair of $b_i$, and the final chair of $z$. A player *sits* in a set $S$ of chairs when his final chair is in $S$. A block of players sits in $S$ when *some* player in that block sits in $S$. A set $C$ of chairs sits in $S$ when *some* block whose initial chair is in $C$ sits in $S$. The dual of “sits” is *sits only*; a block sits only in $S$ when *all* players in the block sit in $S$, and a set $C$ of chairs sits only in $S$ when *all* blocks whose initial chairs are in $C$ sit only in $S$.
The $b_i$’s determined by the procedure satisfy the following lemmas and theorem.
$[\underline{b}_1,\underline{b}_k)$ and $[\underline{b}_k, \underline{z}]$ are disjoint.
If not, then ${\underline{b}_1} \in [\underline{b}_k, \underline{z}]$ But then $a$, which is in $b_1$, would not be rejected by $z$ when $a$ reached $\underline{z}$, a contradiction.
$[\underline{b}_1,\underline{b}_k)$ does not sit in $[\underline{b}_k, \underline{z}]$.
Each chair in $[\underline{b}_k, \underline{z}]$ is visited by an un-exhausted $b_k$ (because it contains $z$) before being visited by any block initially in $[\underline{b}_1,\underline{b}_k)$, by Lemma 1.
- $b_i$ is non-empty for $1 \leq i \leq k$.
- $d_i \in [\underline{b}_i,\underline{b}_k)$ for $1 \leq i < k$.
- $[\underline{b}_1, d_i]$ does not sit in $(d_i, \underline{b}_k]$ for $1 \leq i < k$ during the seating process.
- $b_{i+1} \in (\underline{b}_i,\underline{b}_k]$ for $1 \leq i < k$.
We use induction on $i$.
1. When $i = 1$, $b_1$ contains a. When $i>1$, $\underline{b}_i \in (\underline{b}_{i-1}, \underline{b}_k)$ by (iv). Hence a is rejected at $b_i$ as it progresses towards $\underline{z}$. The player seated at $\underline{b}_i$ must have originated in $[\underline{b}_1,\underline{b}_i]$ for $a$ to be rejected. By (iii) he cannot have originated in $[\underline{b}_1,d_{i-1}]$, which leaves only $\underline{b}_i$. Hence $b_i$ is non-empty.
2. By (i), $d_i$ exists. By the procedure , $d_i \in [\underline{b}_i, \underline{z}]$. By Lemma 2 and (iv), $d_i \notin [\underline{b}_k, \underline{z}]$. Hence $d_i \in [\underline{b}_i, \underline{b}_k]$.
3. By the procedure, $b_i$ does not sit in $(d_i, \underline{b}_k]$. Also $(\underline{b}_i, d_i]$ sits only in $(\underline{b}_i, d_i)$, otherwise $b_i$ would not sit in $d_i$. So $[\underline{b}_i, d_i]$ does not sit in $(d_i, \underline{b}_k]$. By (iii), $[\underline{b}_1, d_{i-1}]$ does not sit in $(d_{i-1}, \underline{b}_k]$, and a portion does not sit in $(d_i, \underline{b}_k]$ by (ii). (For convenience take $[\underline{b}_1, d_0]$ to be empty.) Hence $[\underline{b}_1, d_i]$ does not sit in $(d_i, \underline{b}_k]$.
4. This follows directly from (ii) and the procedure, which makes $\underline{b}_{i+1} = d_i + 1$.
The procedure halts.
By (iv) of Theorem 3, as $i$ increases $b_i$ gets closer to $b_k$.
We have now established that the distinguished blocks $b_1, b_2, ..., b_k$ are well-defined by the procedure. Hence the sample $T$ can now be constructed, by grouping together the distinguished blocks of $S$ as described earlier.
We construct the $(k+1)$-pattern $P$ to be matched by $T$ thus. Player $a$ is assigned to chair $c$ ($b_1$’s initial chair in both $S$ and $T$). For each distinguished block $b_i$ we assign player $p_i$ (as defined in the procedure) to chair $c+i-1$. This completes the construction of $P$. It is trivial to verify that $T$ matches $P$.
We have thus far exhibited a map from rejections to matches. To see that the map is a bijection, we show that its inverse is totally and uniquely defined.
Given a sample $T$ that matches a $(k+1)$-pattern $P$, we show how to reconstruct the corresponding sample $S$. The pattern serves to identify the blocks $b_1, b_2, ..., b_k$. To form $S$, we merge these blocks with the remaining $m-k$ undistinguished blocks of $T$. We leave block $b_1$ where it is, in chair $c$. Let player $a$ (the one who suffers the rejection we are constructing) be the second-ranked of the two players in $P$ at chair $c$, the other of which we call $p_1$. We assign undistinguished (possibly empty) blocks, in the order they appear directly following $b_k$ in $T$, to chairs $c+1, c+2, ...$ until $b_1$ and $b_2$ are sufficiently far apart that in the seating process for S player $p_1$ will be seated before $b_1$ arrives at $b_2$’s initial chair. (Recall that for seating purposes we ranked the players in advance.) This fixes the position for $b_2$. We now insert further undistinguished blocks from T between $b_2$ and $b_3$, to seat player $p_2$ (the player in chair $c+1$ in pattern $P$), and so on until the position for $b_k$ is determined. Player $p_k$ then becomes player $z$. It should be clear that when $m \geq n$ this procedure will never require more than the available number of undistinguished blocks. This completes the demonstration of a one-to-one correspondence between rejections and matches.
In conclusion, we have characterized the rejection aspect of musical chairs in terms of a reasonably natural correspondence with “matching chairs”, in which we are interested in matches between samples and patterns, rather than in rejections of players by occupied chairs. The correspondence is arrived at by the way of an “un-merging” of blocks which is specified by using the notion of “last player seated before rejection.”
|
---
abstract: |
We show that for a generic nullhomotopic simple closed curve $\Gamma$ in the boundary of a compact, orientable, mean convex $3$-manifold $M$ with $H_2(M,{\mathbf{Z}})=0$, there is a unique area minimizing disk $D$ embedded in $M$ with $\partial
D = \Gamma$. We also show that the same is true for nullhomologous curves in absolutely area minimizing surface case.
address: |
Koc University\
Department of Mathematics\
Sariyer, Istanbul 34450 Turkey
author:
- Baris Coskunuzer
title: Generic Uniqueness of Area Minimizing Disks for Extreme Curves
---
\[section\] \[thm\][Lemma]{} \[thm\][Corollary]{}
\[section\]
\[section\]
[^1]
Introduction
============
The Plateau problem asks the existence of an area minimizing disk for a given curve in the ambient manifold $M$. This problem was solved for ${\mathbf{R}}^3$ by Douglas [@Do], and Rado [@Ra] in early 1930s. Later, it was generalized by Morrey [@Mo] for Riemannian manifolds. Then, regularity (nonexistence of branch points) of these solutions was shown by Osserman [@Os], Gulliver [@Gu] and Alt [@Al]. In the early 1960s, the same question was studied for absolutely area minimizing surfaces, i.e. for surfaces that minimize area among all oriented surfaces with the given boundary (without restriction on genus). The geometric measure theory techniques proved to be quite powerful, and De Georgi, Federer-Fleming solved the problem for area minimizing surfaces [@Fe].
Later, the question of embeddedness of the solution was studied by many experts. First, Gulliver-Spruck showed embeddedness for the extreme curves with total curvature less than $4\pi$ in [@GS]. Tomi-Tromba [@TT] and Almgren-Simon [@AS] showed the existence of embedded minimal (not necessarily area minimizing) disks for extreme curves. Then, Meeks-Yau [@MY1] showed that, for extreme boundary curves, area minimizing disks must be embedded. Recently, Ekholm, White, and Wienholtz generalized Gulliver-Spruck embeddedness result by removing extremeness condition from the curves [@EWW].
On the other hand, the number of the solutions was also an active area of research. First, Rado showed that if a curve can be projected bijectively to a convex plane curve, then it bounds a unique minimal disk. Then, Nitsche proved uniqueness of minimal disks for the boundary curves with total curvature less than $4\pi$ in [@Ni]. Then, Tromba [@Tr] showed that a generic curve in ${\mathbf{R}}^3$ bounds a unique area minimizing disk. Then, Morgan [@M] proved a similar result for area minimizing surfaces. Later, White proved a very strong generic uniqueness result for fixed topological type in any dimension [@Wh1]. In particular, he showed that a generic $k$-dimensional, $C^{j,\alpha}$ submanifold of a Riemannian manifold cannot bound two smooth, minimal $(k+1)$-manifolds having the same area.
In this paper, we will give a new generic uniqueness results for both versions of the Plateau problem. Our techniques are simple and topological. The first main result is the following:
[**Theorem 3.2:**]{} Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Then for a generic nullhomotopic (in $M$) simple closed curve $\Gamma$ in $\partial M$, there exists a unique area minimizing disk $D$ in $M$ with $\partial D = \Gamma$.
This theorem is also true for compact, irreducible, orientable, mean convex $3$-manifolds (See Remark 3.2). The second main result is a similar theorem for absolutely area minimizing surfaces.
[**Theorem 4.3:**]{} Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Then for a generic nullhomologous (in $M$) simple closed curve $\Gamma$ in $\partial M$, there exists a unique absolutely area minimizing surface $\Sigma$ in $M$ with $\partial \Sigma = \Gamma$.
These results naturally generalize to noncompact homogeneously regular $3$-manifolds (see the last section).
The short outline of the technique for generic uniqueness is the following: For simplicity, we will focus on the case of the area minimizing disks in a mean convex manifold $M$. Let $\Gamma_0$ be a nullhomotopic (in $M$) simple closed curve in ${\partial M}$. First, we will show that either there exists a unique area minimizing disk $ D_0$ in $M$ with $\partial D_0=\Gamma_0$, or there exist two [*disjoint*]{} area minimizing disks $ D_0^+ , D_0^-$ in $M$ with $\partial D_0^\pm=\Gamma_0$.
Now, take a small neighborhood $N(\Gamma_0)\subset {\partial M}$ which is an annulus. Then foliate $N(\Gamma_0)$ by simple closed curves $\{\Gamma_t\}$ where $t\in(-\epsilon, \epsilon)$, i.e. $N(\Gamma_0) \simeq \Gamma\times (-\epsilon, \epsilon)$. By the above fact, for any $\Gamma_t$ either there exists a unique area minimizing disk $ D_t$, or there are two area minimizing disks $
D_t^\pm$ disjoint from each other. Also, since these are area minimizing disks, if they have disjoint boundary, then they are disjoint by [@MY2]. This means, if $t_1<t_2$, then $ D_{t_1}$ is disjoint and [*below*]{} $ D_{t_2}$ in $M$. Consider this collection of area minimizing disks. Note that for curves $\Gamma_t$ bounding more than one area minimizing disk, we have a canonical region $N_t$ in $M$ between the disjoint area minimizing disks $ D_t^\pm$.
Now, take a finite curve $\beta\subset M$ which is transverse to the collection of these area minimizing disks $\{ D_t\}$ whose boundaries are $\{\Gamma_t\}$. Let the length of this line segment be $C$.
Now, the idea is to consider the [*thickness*]{} of the neighborhoods $N_t$ assigned to the boundary curves $\{\Gamma_t\}$. When $\Gamma_t$ bounds a unique area minimizing disk $D_t$, let $N_t=D_t$ be a degenerate canonical region for $\Gamma_t$. Let $s_t$ be the length of the segment $I_t$ of $\beta$ between $ D_t^+$ and $ D_t^-$, which is the [*width*]{} of $N_t$ assigned to $\Gamma_t$. Then, the curves $\Gamma_t$ bounding more than one area minimizing disk have positive width, and contributes to total thickness of the collection, and the curves bounding a unique area minimizing disk has $0$ width and do not contribute to the total thickness. Since $\sum_{t\in(-\epsilon, \epsilon)} s_t < C$, the total thickness is finite. This implies for only countably many $t\in(-\epsilon, \epsilon)$, $s_t>0$, i.e. $\Gamma_t$ bounds more than one area minimizing disk. For the remaining uncountably many $t\in(-\epsilon, \epsilon)$, $s_t=0$, and there exists a unique area minimizing disk for those $t$. This proves the space of simple closed curves of uniqueness is dense in the space of Jordan curves in ${\partial M}$. Then, we will show this space is not only dense, but also generic.
The organization of the paper is as follows: In the next section we will cover some basic results which will be used in the following sections. In section 3, we will prove the first main result of the paper. Then in section 4, we will show the area minimizing surfaces case. Finally in section 5, we will have some final remarks.
Acknowledgements:
-----------------
I am very grateful to the referee for very valuable comments and suggestions. I would like to thank Brian White and Frank Morgan for very useful conversations.
Preliminaries
=============
In this section, we will overview the basic results which we use in the following sections. First, we should note that Hass-Scott’s very nicely written paper [@HS] would be a great reference for a good introduction for the notions in this paper. We will start with some basic definitions.
An [*area minimizing disk*]{} is a disk which has the smallest area among the disks with the same boundary. An [*absolutely area minimizing surface*]{} is a surface which has the smallest area among all orientable surfaces (with no topological restriction) with the same boundary.
Let $M$ be a compact Riemannian $3$-manifold with boundary. Then $M$ is a [*mean convex*]{} (or sufficiently convex) if the following conditions hold.
- $\partial M$ is piecewise smooth.
- Each smooth subsurface of $\partial M$ has nonnegative curvature with respect to inward normal.
- There exists a Riemannian manifold $N$ such that $M$ is isometric to a submanifold of $N$ and each smooth subsurface $S$ of $\partial M$ extends to a smooth embedded surface $S'$ in $N$ such that $S' \cap M = S$.
A simple closed curve is an [*extreme curve*]{} if it is on the boundary of its convex hull. A simple closed curve is called as [*$H$-extreme curve*]{} if it is a curve in the boundary of a mean convex manifold $M$.
Note that our results in this paper are for $H$-extreme curves which are in the boundary of a fixed $3$-manifold $M$. Since any extreme curve is also $H$-extreme, our results applies to this case as well. Note also that for any smooth embedded curve $\Gamma$, one can find a mean convex (sufficiently thin) solid torus $T_\Gamma$ such that $\Gamma\subset
\partial T_\Gamma$, hence $\Gamma$ is $H$-extreme. So, an $H$-extreme curve should be understood with the mean convex manifold which comes with the definition. However, being extreme for a curve is the property of the curve alone (depends only on the ambient manifold).
Now, we state the main facts which we use in the following sections.
[@MY2], [@MY3] Let $M$ be a compact, mean convex $3$-manifold, and $\Gamma\subset\partial M$ be a nullhomotopic simple closed curve. Then, there exists an area minimizing disk $ D\subset M$ with $\partial D = \Gamma$. Moreover, all such disks are properly embedded in $M$ and they are pairwise disjoint. Also, if $\Gamma_1, \Gamma_2 \subset \partial M$ are disjoint simple closed curves, then the area minimizing disks $ D_1, D_2$ spanning $\Gamma_1, \Gamma_2$ are also disjoint.
There is an analogous fact for area minimizing surfaces, too.
[@Fe], [@HSi], [@Wh2] Let $M$ be a compact, mean convex $3$-manifold, and $\Gamma\subset\partial M$ be a nullhomologous simple closed curve. Then, there exists a smoothly embedded absolutely area minimizing surface $\Sigma\subset M$ with $\partial \Sigma =
\Gamma$.
Now, we state a lemma about the limit of area minimizing disks in a mean convex manifold. Note that we mean that the boundary of the disk is in the boundary of the manifold by being [*properly embedded*]{}.
[@HS] Let $M$ be a compact, mean convex $3$-manifold and let $\{ D_i\}$ be a sequence of properly embedded area minimizing disks in $M$. Then there is a subsequence $\{ D_{i_j}\}$ of $\{ D_i\}$ such that $ D_{i_j} \rightarrow \widehat{D}$, a countable collection of properly embedded area minimizing disks in $\Omega$.
[**Convention:**]{} Throughout the paper, all the manifolds will be assumed to be compact, orientable, mean convex and having trivial second homology, i.e. $H_2(M,{\mathbf{Z}})=0$. We will also assume that all the surfaces are orientable as well.
Generic Uniqueness for Area Minimizing Disks
============================================
In this section, we will prove the generic uniqueness of area minimizing disks for $H$-extreme curves. For this, we first show that for any nullhomotopic simple closed curve in the boundary of a mean convex $3$-manifold, either there exists a unique area minimizing disk spanning the curve, or there are two canonical extremal area minimizing disks which bounds a region containing all other area minimizing disks with same boundary. Similar results also appears in [@MY3], [@Li], [@Wh3] and [@Co].
Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Let $\Gamma$ be a nullhomotopic (in $M$) simple closed curve in $\partial M$. Then either there is a unique area minimizing disk $D$ in $M$ with $\partial
D = \Gamma$, or there are two canonical area minimizing disks $ D^+$ and $ D^-$ in $M$ with $\partial D^\pm = \Gamma$, and any other area minimizing disk in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $D^+$ and $D^-$ in $M$.
Let $M$ be a mean convex $3$-manifold and let $\Gamma\subset\partial M$ be a nullhomotopic simple closed curve. Take a small neighborhood $A$ of $\Gamma$ in $\partial M$, which will be a thin annulus where $\Gamma$ is the core. $\Gamma$ separates the annulus $A$ into two parts, say $A^+$ and $A^-$ by giving a local orientation. Define a sequence of pairwise disjoint simple closed curves $\{\Gamma_i^+\} \subset A^+ \subset \partial M$ such that $\lim\Gamma_i^+ =
\Gamma$. Now, by Lemma 2.1, for any curve $\Gamma_i^+$, there exist an embedded area minimizing disk $ D_i^+$ with $\partial D_i^+ = \Gamma_i^+$. This defines a sequence of area minimizing disks $\{ D_i^+\}$ in $M$. By Lemma 2.3, there exists a subsequence $\{ D_{i_j}^+\}$ converging to a countable collection of area minimizing disks $\widehat{
D}^+$ with $\partial \widehat{ D}^+ = \Gamma$.
We claim that this collection $\widehat{ D}^+$ consists of only one area minimizing disk. Assume that there are two disks in the collection say $ D_a^+$ and $ D_b^+$, and say $ D_a^+$ is [*above*]{} $D_b^+$ (in the positive side of $D_b^+$ in the local orientation). By Lemma 2.1, $D_a^+$ and $D_b^+$ are embedded and disjoint. They have the same boundary $\Gamma\subset \partial M$. $
D_b^+$ is also limit of the sequence $\{ D_i^+\}$. But, since for any area minimizing disk $
D_i^+\subset M$, $\partial D_i^+ =\Gamma_i^+$ is disjoint from $\partial D_a^+ = \Gamma$, $ D_i^+$ disjoint from $ D_a^+$, again by Lemma 2.1. This means $ D_a^+$ is a barrier between the sequence $\{ D_i^+\}$ and $ D_b^+$, and so, $ D_b^+$ cannot be limit of this sequence. This is a contradiction. So $\widehat{ D}^+$ is just one area minimizing disk, say $ D^+$. Similarly, $\widehat{ D}^- = D^-$.
Now, we claim these area minimizing disks $ D^+$ and $ D^-$ are canonical, depending only on $\Gamma$ and $M$, and independent of the choice of the sequence $\{\Gamma_i\}$ and $\{ D_i\}$. Let $\{\gamma_i^+\}$ be another sequence of simple closed curves in $A^+$. Assume that there exists another area minimizing disk $E^+$ with $\partial E^+ = \Gamma$ and $E^+$ is a limit of the sequence of area minimizing disks $E_i^+$ with $\partial E_i^+ = \gamma_i^+ \subset A^+$. By Lemma 2.1, $ D^+$ and $E^+$ are disjoint. Then one of them is above the other one. If $ D^+$ is above $E^+$, then $ D^+$ between the sequence $E_i^+$ and $E^+$. This is because, all $E_i^+$ are disjoint and above $E^+$ as $\partial E_i^+ = \gamma_i$ are disjoint and above $\Gamma$. Similarly, $D^+$ is [*below*]{} $E_i$ for any $i$ (in the negative side of $E_i$ in the local orientation), as $\partial D^+ = \Gamma$ is below the curves $\gamma_i^+\subset A^+$. Now, since $ D^+$ is between the sequence $\{E_i^+\}$ and its limit $E^+$, and $E^+$ and $ D^+$ are disjoint, $ D^+$ will be a barrier for the sequence $\{E_i^+\}$, and so they cannot limit on $E^+$. This is a contradiction. Similarly, $ D^+$ cannot be below $E^+$, so they must be same. Hence, $ D^+$ and $ D^-$ are canonical area minimizing disks for $\Gamma$.
Now, we will show that any area minimizing disk in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $D^+$ and $D^-$ in $M$, i.e. $\partial N \supseteq D^+ \cup D^-$ ($H_2(M,{\mathbf{Z}})=0$). Let $E$ be any area minimizing disk with boundary $\Gamma$. By Lemma 2.1, $E$ is disjoint from $D^+$ and $D^-$. Hence, if $E$ is not in $N$, then it must be completely outside of $N$. So, $E$ is either above $D^+$ or below $D^-$. However, $D^+ = \lim
D_i^+$ and $\Gamma_i^+ \rightarrow \Gamma$ from above. Moreover, again by Lemma 2.1, $E$ must be disjoint from $D_i^+$. Hence, $E$ would be a barrier between the sequence $\{ D_{i_j}^+\}$ and $D^+$ like in previous paragraph. This is a contradiction. Similarly, same is true for $D^-$. Hence, any area minimizing disk in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $D^+$ and $D^-$ in $M$. This also shows that if $D^+ = D^-$, then there exists a unique area minimizing disk in $M$ with boundary $\Gamma$.
The results in [@MY3], [@Li], [@Wh3] are similar to this one in some sense. In those papers, the authors show the “strong uniqueness” property, which says that either an $H$-extreme curve bounds more than one [*minimal disk*]{} in the mean convex manifold $M$ or there is a unique [*minimal surface*]{} bounding the curve which is indeed an area minimizing disk in $M$. Our result is relatively different than the others. In above lemma, we proved that either there exists a unique area minimizing disk in $M$ bounding the $H$-extreme curve, or there are two canonical extremal area minimizing disks in $M$ which bounds a region containing all other area minimizing disks with same boundary.
Now, we prove the main result of the paper.
Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Then for a generic nullhomotopic (in $M$) simple closed curve $\Gamma$ in $\partial M$, there exists a unique area minimizing disk $D$ in $M$ with $\partial
D = \Gamma$. In other words, let $\mathcal{A}$ be the space of nullhomotopic (in $M$) simple closed curves in $\partial
M$ and let $\mathcal{A}'\subset \mathcal{A}$ be the subspace containing the curves bounding a unique area minimizing disk in $M$. Then, $\mathcal{A}'$ is generic in $\mathcal{A}$, i.e. $\mathcal{A}'$ is countable intersection of open dense subsets.
We will prove this theorem in 2 steps.
**Claim 1:** $\mathcal{A}'$ is dense in $\mathcal{A}$ as a subspace of $C^0(S^1,\partial M)$ with the supremum metric.
Let $\mathcal{A}$ be the space of nullhomotopic simple closed curves in $\partial M$. We parametrize this space with $C^0$ parametrizations, and use supremum metric, i.e. $\mathcal{A}= \{\alpha\in
C^0(S^1,\partial M)\ | \ \alpha(S^1) \mbox{ is an embedding, and nullhomotopic in } M\}$.
Now, let $\Gamma_0\in \mathcal{A}$ be a nullhomotopic simple closed curve in $\partial M$. Since $\Gamma_0$ is simple, there exists a small closed neighborhood $N(\Gamma_0)$ of $\Gamma_0$ which is an annulus in $\partial M$. Let $\Gamma:[-\epsilon,\epsilon]\rightarrow \mathcal{A}$ be a small path in $\mathcal{A}$ through $\Gamma_0$ such that $\Gamma(t)=\Gamma_t$ and $\{\Gamma_t\}$ foliates $N(\Gamma)$ with simple closed curves $\Gamma_t$. In other words, $\{\Gamma_t\}$ are pairwise disjoint simple closed curves, and $N(\Gamma_0)=\bigcup_{t\in [-\epsilon,\epsilon]} \Gamma_t$.
By Lemma 3.1, for any $\Gamma_t$ either there exists a unique area minimizing disk $D_t$ in $M$, or there is a canonical region $N_t$ in $M$ between the canonical area minimizing disks $D_t^+$ and $D_t^-$. With abuse of notation, if $\Gamma_t$ bounds a unique area minimizing disk $D_t$ in $M$, define $N_t=D_t$ as a degenerate canonical neighborhood for $\Gamma_t$. Clearly, degenerate neighborhood $N_t$ means $\Gamma_t$ bounds a unique area minimizing disk, and nondegenerate neighborhood $N_s$ means that $\Gamma_s$ bounds more than one area minimizing disk. Note that by Lemma 3.1 and Lemma 2.1, all canonical neighborhoods in the collection are pairwise disjoint.
Now, let $\widehat{N}$ be the union of these canonical neighborhoods $\{N_t\}$, i.e. $\widehat{N} =
\bigcup_{t\in [-\epsilon,\epsilon]}N_t$. Then, $\partial \widehat{N} \supseteq D_\epsilon^+ \cup
N(\Gamma_0) \cup D_{-\epsilon}^-$. Let $p^+$ be a point in $D_\epsilon^+$ and $p^-$ be a point in $D_{-\epsilon}^-$. Let $\beta$ be a finite curve from $p^+$ to $p^-$ intersecting transversely all the canonical neighborhoods in the collection $\widehat{N}$.
Now, for each $t\in[-\epsilon,\epsilon]$, we will assign a real number $s_t\geq 0$. Let $I_t = \beta\cap N_t$, and $s_t$ be the length of $I_t$. Then, if $N_t$ is degenerate (There exists a unique area minimizing disk $D_t$ in $M$ for $\Gamma_t$), then $s_t$ would be $0$. If $N_t$ is nondegenerate ($\Gamma_t$ bounds more than one area minimizing disk), then $s_t > 0$. Also, it is clear that for any $t$, $I_t\subset \beta$ and $I_t\cap I_s=\emptyset$ for any $t\neq s$. Then, $\sum_{t\in[-\epsilon,\epsilon]} s_t < C$ where $C$ is the length of $\beta$. This means for only countably many $t\in[-\epsilon,\epsilon]$, $s_t > 0$. So, there are only countably many nondegenerate $N_t$ for $t\in[-\epsilon,\epsilon]$. Hence, for all other $t$, $N_t$ is degenerate. This means there exist uncountably many $t\in[-\epsilon,\epsilon]$, where $\Gamma_t$ bounds a unique area minimizing disk. Since $\Gamma_0$ is arbitrary, this proves $\mathcal{A} '$ is dense in $\mathcal{A}$.
**Claim 2:** $\mathcal{A}'$ is generic in $\mathcal{A}$.
We will prove that $\mathcal{A} '$ is countable intersection of open dense subsets of $\mathcal{A}$. Then the result will follow by Baire category theorem.
Since the space of continuous maps from circle to boundary of $M$, $C^0(S^1,\partial M)$, is complete with supremum metric, then the closure of $\mathcal{A}$ in $C^0(S^1,\partial M)$, $\bar{\mathcal{A}}\subset C^0(S^1,\partial M)$, is also complete.
Now, we will define a sequence of open dense subsets $U^i\subset \mathcal{A}$ such that their intersection will give us $\mathcal{A} '$. Let $\Gamma\in \mathcal{A}$ be a simple closed curve in $\partial M$. As in the Claim 1, let $N(\Gamma)\subset \partial M$ be a neighborhood of $\Gamma$ in $\partial M$, which is an open annulus. Then, define an open neighborhood $U_\Gamma$ of $\Gamma$ in $\mathcal{A}$, such that $U_\Gamma = \{\alpha \in \mathcal{A} \ | \ \alpha(S^1)\subset N(\Gamma), \
\alpha \mbox{ is homotopic to } \Gamma\}$. Clearly, $\mathcal{A}= \bigcup_{\Gamma\in \mathcal{A}}
U_\Gamma$. Now, define a finite curve $\beta_\Gamma$ as in Claim 1, which intersects transversely all the area minimizing disks bounding the curves in $U_\Gamma$.
Now, for any $\alpha \in U_\Gamma$, by Lemma 3.1, there exists a canonical region $N_\alpha$ in $M$ (which can be degenerate if $\alpha$ bounds a unique area minimizing disk). Let $I_{\alpha,\Gamma}
= N_\alpha \cap \beta_\Gamma$. Then let $s_{\alpha,\Gamma}$ be the length of $I_{\alpha,\Gamma}$ ($s_{\alpha,\Gamma}$ is $0$ if $N_\alpha$ degenerate). Hence, for every element $\alpha$ in $U_\Gamma$, we assign a real number $s_{\alpha,\Gamma} \geq 0$.
Now, we define the sequence of open dense subsets in $U_\Gamma$. Let $U^i_\Gamma = \{\alpha\in
U_\Gamma \ | \ s_{\alpha,\Gamma} < 1/i \ \}$. We claim that $U^i_\Gamma$ is an open subset of $U_\Gamma$ and $\mathcal{A}$. Let $\alpha\in U^i_\Gamma$, and let $s_{\alpha,\Gamma} = \lambda <
1/i$. So, the interval $I_{\alpha,\Gamma}\subset \beta_\Gamma$ has length $\lambda$. Let $I '
\subset \beta_\Gamma$ be an interval containing $I_{\alpha,\Gamma}$ in its interior, and has length less than $1/i$. By the proof of Claim 1, we can find two simple closed curves $\alpha^+, \alpha^-
\in U_\Gamma$ with the following properties.
- $\alpha^\pm$ are disjoint from $\alpha$,
- $\alpha^\pm$ are lying in opposite sides of $\alpha$ in $\partial M$,
- $\alpha^\pm$ bounds a unique area minimizing disk $D_{\alpha^\pm}$,
- $D_{\alpha^\pm} \cap \beta_\Gamma \subset I '$.
The existence of such curves is clear from the proof of Claim 1, as if one takes any foliation $\{\alpha_t\}$ of a small neighborhood of $\alpha$ in $\partial M$, there are uncountably many curves in the family bounding a unique area minimizing disk, and one can choose sufficiently close pair of curves to $\alpha$, to ensure the conditions above.
After finding $\alpha^\pm$, consider the open annulus $F_\alpha$ in $\partial M$ bounded by $\alpha^+$ and $\alpha^-$. Let $V_\alpha = \{ \gamma\in U_\Gamma \ | \ \gamma(S^1)\subset F_\alpha
, \ \gamma \mbox{ is homotopic to } \alpha \}$. Clearly, $V_\alpha$ is an open subset of $U_\Gamma$. If we can show $V_\alpha\subset U^i_\Gamma$, then this proves $U^i_\Gamma$ is open for any $i$ and any $\Gamma\in \mathcal{A}$.
Let $\gamma\in V_\alpha$ be any curve, and $N_\gamma$ be its canonical neighborhood given by Lemma 3.1. Since $\gamma(S^1)\subset F_\alpha$, $\alpha^+$ and $\alpha^-$ lie in opposite sides of $\gamma$ in $\partial M$. This means $D_{\alpha^+}$ and $D_{\alpha^-}$ lie in opposite sides of $N_\gamma$. By choice of $\alpha^\pm$, this implies $N_\gamma \cap \beta_\Gamma= I_{\gamma,\Gamma}
\subset I '$. So, the length $s_{\gamma,\Gamma}$ is less than $1/i$. This implies $\gamma\in
U^i_\Gamma$, and so $V_\alpha\subset U^i_\Gamma$. Hence, $U^i_\Gamma$ is open in $U_\Gamma$ and $\mathcal{A}$.
Now, we can define the sequence of open dense subsets. Let $U^i = \bigcup_{\Gamma\in \mathcal{A}} U^i_\Gamma$ be an open subset of $\mathcal{A}$. Since, the elements in $\mathcal{A} '$ represent the curves bounding a unique area minimizing disk, for any $\alpha\in \mathcal{A} '$, and for any $\Gamma\in \mathcal{A}$, $s_{\alpha,\Gamma} = 0$. This means $\mathcal{A}'\subset U^i$ for any $i$. By Claim 1, $U^i$ is open dense in $\mathcal{A}$ for any $i>0$.
As we mention at the beginning of the proof, since the space of continuous maps from circle to boundary of $M$, $C^0(S^1,\partial M)$ is complete with supremum metric, then the closure $\bar{\mathcal{A}}$ of $\mathcal{A}$ in $C^0(S^1,\partial M)$ is also complete metric space. Since $\mathcal{A}'$ is dense in $\mathcal{A}$, it is also dense in $\bar{\mathcal{A}}$. As $\mathcal{A}$ is open in $C^0(S^1,\partial M)$, this implies $U^i$ is a sequence of open dense subsets of $\bar{\mathcal{A}}$. On the other hand, since $U_1 \supseteq U_2 \supseteq ... \supseteq U_n
\supseteq ...$ and $ \bigcap_{i=1}^\infty U_i = \mathcal{A}'$, $\mathcal{A}'$ is generic in $\mathcal{A}$.
Notice that we use the homology condition just to make sure that $D^+\cup D^-$ is a separating sphere in $M$, and hence to define the canonical region between them in Lemma 3.1. So, if we replace $H_2(M,{\mathbf{Z}})=0$ condition with irreducibility of $3$-manifold (any embedded $2$-sphere bounds a $3$-ball in $M$), the same proof for Lemma 3.1 and Theorem 3.2 would go through. In other words, Theorem 3.2 is also true for compact, irreducible, orientable, mean convex $3$-manifolds.
Generic Uniqueness for Area Minimizing Surfaces
===============================================
In this section, we will prove the generic uniqueness result for $H$-extreme curves in the absolutely area minimizing case. The technique is basically same with area minimizing disk case. First, we will prove an analogous version of Lemma 2.1 \[MY2, Theorem 6\] for absolutely area minimizing surfaces. However, the analogous version of Lemma 2.1 is not true in general for global version. Hence, we will prove it for a local version which suffices for our purposes. See Remark 4.1.
Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Let $A$ be an annulus in $\partial M$ whose core $\Gamma$ is nullhomologous in $M$. If $\Gamma_1$ and $\Gamma_2$ are two disjoint simple closed curves in $A$ which are homotopic to $\Gamma$ in $A$, then any absolutely area minimizing surfaces $\Sigma_1$ and $\Sigma_2$ in $M$ with $\partial \Sigma_i = \Gamma_i$ are disjoint, too. Moreover, if $\Sigma$ and $\Sigma '$ are two absolutely area minimizing surfaces in $M$ where $\partial \Sigma = \partial \Sigma ' = \Gamma$, then they must be disjoint, too.
Let $M$ be a mean convex $3$-manifold, and $A$ is an annulus in $\partial M$ whose core $\Gamma$ is nullhomologous in $M$. Let $\Gamma_1$ and $\Gamma_2$ are two disjoint simple closed curves in $A$ which are homotopic to $\Gamma$ in $A$. Let $\Sigma_1$ and $\Sigma_2$ be absolutely area minimizing surfaces in $M$ with $\partial \Sigma_i = \Gamma_i$. We want to show that $\Sigma_1$ and $\Sigma_2$ are disjoint.
Assume on the contrary that $\Sigma_1\cap\Sigma_2 \neq \emptyset$. Now, let $\widehat{N}$ be the convex hull of $A$ in $M$. Then, by maximum principle, $\Sigma_1$ and $\Sigma_2$ are in $\widehat{N}$. Moreover, as $\Gamma_1$ separates the annulus $A$, then $\Sigma_1$ is separating in $\widehat{N}$. Similarly, $\Sigma_2$ is separating, too. Now, if $\Sigma_1\cap\Sigma_2 = \gamma$ where $\gamma$ is a collection of closed curves, then $\Sigma_1$ separates $\Sigma_2$ into two subsurfaces $S^1_1$ and $S^1_2$ where $\partial S^1_1 = \gamma$ and $\partial S^1_2 = \gamma\cup
\Gamma_1$. Similarly, $\Sigma_2$ separates $\Sigma_1$ into two subsurfaces $S^2_1$ and $S^2_2$ where $\partial S^2_1 = \gamma$ and $\partial S^2_2 = \gamma\cup \Gamma_2$. Now, we will use the Meeks-Yau exchange roundoff trick to get a contradiction [@MY2].
As $\Sigma_1$ and $\Sigma_2$ are absolutely area minimizing surfaces in $M$, $|S^1_1| = |S^2_1|$ where $ | S |$ is the area of $S$. Now define a new surface by swaping the subsurfaces $S^1_1$ and $S^2_1$. In other words, let $T_1 = (\Sigma_1 - S^1_1) \cup S^2_1$. As $T_1$ and $\Sigma_1$ have same area, then $T_1$ is also absolutely area minimizing surface. However, $\gamma$ is a folding curve in $T_1$ as in [@MY2]. This is a contradiction (One can also argue with the regularity of the absolutely area minimizing surfaces [@Fe]). Hence, this shows that $\Sigma_1$ and $\Sigma_2$ in $M$ with $\partial \Sigma_i = \Gamma_i$ are disjoint absolutely area minimizing surfaces in $M$.
Now, we will consider same boundary case. Let $A$ and $\Gamma$ be as in the statement of the theorem. Let $\Sigma$ and $\Sigma '$ be two absolutely area minimizing surfaces where $\partial
\Sigma = \partial \Sigma ' = \Gamma$. Let $\widehat{N}$ be as above. Then, $\Sigma_1$ and $\Sigma_2$ are separating in $\widehat{N}$. As in the previous paragraph, $\Sigma_1$ and $\Sigma_2$ separates each other, and by swaping argument again, we get a contradiction. The proof follows.
The techniques for Lemma 2.1 (or \[MY2, Theorem 6\]) is not working for an analogous theorem in absolutely area minimizing surfaces case in general. In other words, if we just assume $\Gamma_1\cap\Gamma_2=\emptyset$, and not require them to be in the annulus $A$, then the techniques of the above lemma do not apply. This is because if, for example, $\Gamma_1$ or $\Gamma_2$ are not separating in $\partial M$, then the intersection of absolutely area minimizing surfaces $\Sigma_1$ and $\Sigma_2$ might contain a nonseparating curve $\gamma$ in one of the surfaces, say $\Sigma_1$. Hence, we cannot make any surgery there because $\gamma$ may not bound a subsurface in $\Sigma_1$. So, we went to a local version of this theorem (which is enough for our purposes) by restricting $\partial M$ to a small subannnulus $A$ in $\partial M$ to make sure that each essential curve is separating in $A$, and we can make surgery in the intersection of surfaces.
Now, we will give a generalization of Lemma 3.1 in absolutely area minimizing surface case.
Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Let $\Gamma$ be a nullhomologous (in $M$) simple closed curve in $\partial M$. Then either there is a unique absolutely area minimizing surface $\Sigma$ in $M$ with $\partial \Sigma = \Gamma$, or there are uniquely defined two canonical extremal absolutely area minimizing surfaces $\Sigma^+$ and $\Sigma^-$ in $M$ with $\partial \Sigma^\pm = \Gamma$, and any other absolutely area minimizing surface in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $\Sigma^+$ and $\Sigma^-$ in $M$.
Let $M$ be a mean convex $3$-manifold and let $\Gamma\subset\partial M$ be a nullhomologous simple closed curve. Take a small neighborhood $A$ of $\Gamma$ in $\partial M$, which will be a thin annulus where $\Gamma$ is the core. $\Gamma$ separates the annulus $A$ into two parts, say $A^+$ and $A^-$ by giving a local orientation. Define a sequence of pairwise disjoint simple closed curves $\{\Gamma_i^+\} \subset A^+ \subset \partial M$ such that $\lim\Gamma_i^+ =
\Gamma$. Now, by Lemma 2.2, for any curve $\Gamma_i^+$, there exist an embedded absolutely area minimizing surface $\Sigma_i^+$ with $\partial \Sigma_i^+ = \Gamma_i^+$. This defines a sequence of absolutely area minimizing surfaces $\{ \Sigma_i^+\}$ in $M$. By [@Fe], there exists a subsequence $\{\Sigma_{i_j}^+\}$ converging to an absolutely area minimizing surface $\Sigma^+$ with $\partial \Sigma^+ = \Gamma$. Similarly, by defining a similar sequence $\{\Gamma_i^-\}$ in $A^-$ and similar construction, an absolutely area minimizing surface $\Sigma^-$ with $\partial
\Sigma^- = \Gamma$ can be defined.
Now, we claim these absolutely area minimizing surfaces $\Sigma^+$ and $\Sigma^-$ are canonical, depending only on $\Gamma$ and $M$, and independent of the choice of the sequence $\{\Gamma_i\}$ and $\{ \Sigma_i\}$. Let $\{\gamma_i^+\}$ be another sequence of simple closed curves in $A^+$. Assume that there exists another absolutely area minimizing surface $S^+$ with $\partial S^+ = \Gamma$ and $S^+$ is a limit of the sequence of absolutely area minimizing surfaces $S_i^+$ with $\partial S_i^+ = \gamma_i^+ \subset A^+$. As $\Sigma^+$ and $S^+$ are absolutely area minimizing surfaces with same boundary $\Gamma$, they are disjoint by Lemma 4.1. Then one of them is [*above*]{} the other one (in the positive side of the other one in the local orientation). If $\Sigma^+$ is above $S^+$, then $\Sigma^+$ between the sequence $S_i^+$ and $S^+$. This is because, all $S_i^+$ are disjoint and above $S^+$ as $\partial S_i^+ = \gamma_i$ are disjoint and above $\Gamma$. Similarly, $\Sigma^+$ is below $S_i$ for any $i$, as $\partial \Sigma^+ = \Gamma$ is below the curves $\gamma_i^+\subset A^+$. Now, since $\Sigma^+$ is between the sequence $\{S_{i_j}^+\}$ and its limit $S^+$, and $S^+$ and $\Sigma^+$ are disjoint, $\Sigma^+$ will be a barrier for the sequence $\{S_{i_j}^+\}$, and so they cannot limit on $S^+$. This is a contradiction. Similarly, $\Sigma^+$ cannot be below $S^+$, so they must be same. Hence, $\Sigma^+$ and $\Sigma^-$ are canonical absolutely area minimizing surfaces for $\Gamma$.
Now, we will show that any absolutely area minimizing surface in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $\Sigma^+$ and $\Sigma^-$ in $M$, i.e. $\partial N \supseteq \Sigma^+ \cup\Sigma^-$ ($H_2(M,{\mathbf{Z}})=0$). Let $T$ be any absolutely area minimizing surface with boundary $\Gamma$. By Lemma 4.1, $T$ is disjoint from $\Sigma^+$ and $\Sigma^-$. Hence, if $T$ is not in $N$, then it must be completely outside of $N$. So, $T$ is either above $\Sigma^+$ or below $\Sigma^-$. Assume $T$ is above $\Sigma^+$. However, $\Sigma^+ = \lim
\Sigma_{i_j}^+$ and $\Gamma_{i_j}^+ \rightarrow \Gamma$ from above. Moreover, again by Lemma 4.1, $T$ must be disjoint from $\Sigma_{i_j}^+$. Hence, $T$ is a barrier between the subsequence $\Sigma_{i_j}^+$ and its limit $\Sigma$. Like in previous paragraph, this is a contradiction. Similarly, the same is true for $\Sigma^-$. Hence, any absolutely area minimizing surface in $M$ with boundary $\Gamma$ must belong to the canonical region $N$ bounded by $\Sigma^+$ and $\Sigma^-$ in $M$. This also shows that if $\Sigma^+ =\Sigma^-$, then there exists a unique absolutely area minimizing surface in $M$ with boundary $\Gamma$.
Now, we can prove the generic uniqueness result for absolutely area minimizing surfaces.
Let $M$ be a compact, orientable, mean convex $3$-manifold with $H_2(M,{\mathbf{Z}})=0$. Then for a generic nullhomologous (in $M$) simple closed curve $\Gamma$ in $\partial M$, there exists a unique absolutely area minimizing surface $\Sigma$ in $M$ with $\partial \Sigma = \Gamma$. In other words, let $\mathcal{A}$ be the space of nullhomologous (in $M$) simple closed curves in $\partial M$ and let $\mathcal{A}'\subset \mathcal{A}$ be the subspace containing the curves bounding a unique absolutely area minimizing surface in $M$. Then, $\mathcal{A}'$ is generic in $\mathcal{A}$, i.e. $\mathcal{A}'$ is countable intersection of open dense subsets.
The idea is basically same with Theorem 3.2. We will imitate the same proof in this context. Again, we will prove this theorem in 2 steps.
**Claim 1:** $\mathcal{A}'$ is dense in $\mathcal{A}$ as a subspace of $C^0(S^1,\partial M)$ with the supremum metric.
Let $\mathcal{A}$ be the space of nullhomologous simple closed curves in $\partial M$. We parametrize this space with $C^0$ parametrizations, and use supremum metric, i.e. $\mathcal{A}= \{\alpha\in
C^0(S^1,\partial M)\ | \ \alpha(S^1) \mbox{ is an embedding, and nullhomologous in } M\}$.
Now, let $\Gamma_0\in \mathcal{A}$ be a nullhomologous simple closed curve in $\partial M$. As in the proof of Theorem 3.2, let $N(\Gamma_0)$ be an annulus in $\partial M$ and Let $\Gamma:[-\epsilon,\epsilon]\rightarrow \mathcal{A}$ foliates $N(\Gamma)$ with simple closed curves $\Gamma_t$.
By Lemma 4.2, for any $\Gamma_t$ either there exists a unique absolutely area minimizing surface $\Sigma_t$ in $M$, or there is a canonical region $N_t$ in $M$ between the canonical area minimizing surfaces $\Sigma_t^+$ and $\Sigma_t^-$. As in the proof of Theorem 3.2, if $\Gamma_t$ bounds a unique absolutely area minimizing surface $\Sigma_t$ in $M$, define $N_t=\Sigma_t$ as a degenerate canonical neighborhood for $\Gamma_t$. Clearly, degenerate neighborhood $N_t$ means $\Gamma_t$ bounds a unique absolutely area minimizing surface, and nondegenerate neighborhood $N_s$ means that $\Gamma_s$ bounds more than one absolutely area minimizing surface. Note that by Lemma 4.1, all canonical neighborhoods in the collection are pairwise disjoint.
Like before, let $\widehat{N} = \bigcup_{t\in [-\epsilon,\epsilon]} N_t$. Let $p^+$ be a point in $\Sigma_\epsilon^+$ and $p^-$ be a point in $\Sigma_{-\epsilon}^-$. Let $\beta$ be a finite curve from $p^+$ to $p^-$ intersecting transversely all the canonical neighborhoods in the collection $\widehat{N}$. For each $t\in[-\epsilon,\epsilon]$, assign a real number $s_t$ to be the length of $I_t = \beta \cap N_t$. Clearly if $N_t$ is nondegenerate ($\Gamma_t$ bounds more than one absolutely area minimizing surface), then $s_t > 0$. Then, $\sum_{t\in[-\epsilon,\epsilon]} s_t <
C$ where $C$ is the length of $\beta$. This means for only countably many $t\in[-\epsilon,\epsilon]$, $s_t > 0$. So, there are only countably many nondegenerate $N_t$ for $t\in[-\epsilon,\epsilon]$. Hence, for all other $t$, $N_t$ is degenerate. This means there exist uncountably many $t\in[-\epsilon,\epsilon]$, where $\Gamma_t$ bounds a unique absolutely area minimizing surface. Since $\Gamma_0$ is arbitrary, this proves $\mathcal{A} '$ is dense in $\mathcal{A}$.
**Claim 2:** $\mathcal{A}'$ is generic in $\mathcal{A}$.
Let $\mathcal{A}$ be as in the proof of Theorem 3.2. Again, we will define a sequence of open dense subsets $U^i\subset \mathcal{A}$ such that their intersection will give us $\mathcal{A} '$. Let $\Gamma\in \mathcal{A}$ be a simple closed curve in $\partial M$. As in the Claim 1, let $N(\Gamma)\subset
\partial M$ be a neighborhood of $\Gamma$ in $\partial M$, which is an open annulus. Then, define an open neighborhood $U_\Gamma$ of $\Gamma$ in $\mathcal{A}$, such that $U_\Gamma = \{\alpha \in
\mathcal{A} \ | \ \alpha(S^1)\subset N(\Gamma), \ \alpha \mbox{ is homotopic to } \Gamma\}$. Clearly, $\mathcal{A}= \bigcup_{\Gamma\in \mathcal{A}} U_\Gamma$. Now, define a finite curve $\beta_\Gamma$ as in Claim 1, which intersects all the absolutely area minimizing surfaces bounding the curves in $U_\Gamma$.
Now, for any $\alpha \in U_\Gamma$, by Lemma 4.2, there exists a canonical region $N_\alpha$ in $M$. Let $I_{\alpha,\Gamma} = N_\alpha \cap \beta_\Gamma$. Then let $s_{\alpha,\Gamma}$ be the length of $I_{\alpha,\Gamma}$ ($s_{\alpha,\Gamma}$ is $0$ if $N_\alpha$ degenerate). Now, we define the sequence of open dense subsets in $U_\Gamma$.
Let $U^i_\Gamma = \{\alpha\in U_\Gamma \ | \ s_{\alpha,\Gamma} < 1/i \ \}$. We claim that $U^i_\Gamma$ is an open subset of $U_\Gamma$ and $\mathcal{A}$. Let $\alpha\in U^i_\Gamma$, and let $s_{\alpha,\Gamma} = \lambda < 1/i$. So, the interval $I_{\alpha,\Gamma}\subset \beta_\Gamma$ has length $\lambda$. Let $I ' \subset \beta_\Gamma$ be an interval containing $I_{\alpha,\Gamma}$ in its interior, and has length less than $1/i$. Now, let $\alpha^+$ and $\alpha^-$ be as in the proof of Theorem 3.2. Consider the open annulus $F_\alpha$ in $\partial M$ bounded by $\alpha^+$ and $\alpha^-$. Let $V_\alpha = \{ \gamma\in U_\Gamma \ | \ \gamma(S^1)\subset F_\alpha , \ \gamma
\mbox{ is homotopic to } \alpha \}$. Clearly, $V_\alpha$ is an open subset of $U_\Gamma$. If we can show $V_\alpha\subset U^i_\Gamma$, then this proves $U^i_\Gamma$ is open for any $i$ and any $\Gamma\in \mathcal{A}$.
Let $\gamma\in V_\alpha$ be any curve, and $N_\gamma$ be its canonical neighborhood given by Lemma 4.2. Since $\gamma(S^1)\subset F_\alpha$, $\alpha^+$ and $\alpha^-$ lie in opposite sides of $\gamma$ in $\partial M$. This means $\Sigma_{\alpha^+}$ and $\Sigma_{\alpha^-}$ lie in opposite sides of $N_\gamma$. By choice of $\alpha^\pm$, this implies $N_\gamma \cap \beta_\Gamma=
I_{\gamma,\Gamma} \subset I '$. So, the length $s_{\gamma,\Gamma}$ is less than $1/i$. This implies $\gamma\in U^i_\Gamma$, and so $V_\alpha\subset U^i_\Gamma$. Hence, $U^i_\Gamma$ is open in $U_\Gamma$ and $\mathcal{A}$. The remaining part of the proof is just like Theorem 3.2.
Concluding Remarks
==================
In this paper, we showed that for a generic nullhomotopic, simple closed curve in the boundary of a mean convex $3$-manifold $M$, there exists a unique area minimizing disk in $M$. We also prove a similar theorem for absolutely area minimizing surfaces. In many sense, the techniques used in this paper are purely topological and simple. They are quite original and can be applied to many similar settings.
Note that all the results of this paper for compact $3$-manifolds with mean convex boundary. For the noncompact case, like in [@MY3] and [@HS], with additional condition of being homogeneously regular on the manifold $M$, all the results of this paper will go through easily by using the analogous theorems from the same references.
There have been many embeddedness and uniqueness results for the Plateau problem. In extreme and $H$-extreme curve case, there have been many embeddedness results like [@TT], [@AS], [@MY1]. There are also “strong uniqueness” results for $H$-extreme curves like [@MY3], [@Li], [@Wh3]. However, those results do not say anything about the number of area minimizing disks bounded by an $H$-extreme curve. In those papers, authors gave a dichotomy that either an $H$-extreme curve bounds more than one minimal disk, or the only minimal surface bounded by that curve is an area minimizing disk. One should not combine this result with ours in a wrong way. Our result tells that a generic $H$-extreme curve bounds a unique area minimizing disk. However, bounding a unique area minimizing disk does not prohibit to bound other minimal surfaces. So, it is not true that for a generic $H$-extreme curve, the only minimal surface bounded by that curve is an area minimizing disk.
On the other hand, generic uniqueness for area minimizing disks and generic uniqueness for absolutely area minimizing surfaces might sound contradicting at the first glance. This is because if we have an absolutely area minimizing surface (which is not a disk) in $M$, we can construct two different area minimizing disks in different sides of the surface. There are two points to consider here. The first obvious thing is that an absolutely area minimizing surface might be a disk. The other less obvious fact is that having two different disks in different sides of the surface does not mean that the curve has more than one area minimizing disk. This is because they are area minimizing disks in that part of $M$, not the whole $M$. The area minimizing disk in $M$ could be completely different than the others, and it still can be a unique area minimizing disk in $M$ bounded by that curve.
Another important point here is that these techniques may not work for surfaces which are area minimizing in a fixed topological class. If they are not absolutely area minimizing in the homology class, or area minimizing disk, then Lemma 2.1 and its local generalization Lemma 4.1 are not true in general. One should keep in mind that two [*just*]{} minimal surfaces with same extreme boundary curve can intersect in a certain way, but two area minimizing disks, or two absolutely area minimizing surfaces must stay disjoint because of area constraints (intersection implies area reduction). In those lemmas, we are essentially using Meeks-Yau exchange roundoff trick, and a surgery argument. However, two surfaces which are area minimizing in a fixed topological class may not give a surface in the same topological class after surgery. Hence, the key point in our technique (disjointness for the summation argument) fails in this case. However, as we pointed out in the introduction, White [@Wh1] already gave a strong generic uniqueness result for this case in any dimension and codimension with some smoothness condition.
[MSY]{}
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[^1]: The author is partially supported by EU-FP7 Grant IRG-226062 and TUBITAK Grant 107T642
|
---
abstract: 'This paper proposes the architecture of partial sum generator for constituent codes based polar code decoder. Constituent codes based polar code decoder has the advantage of low latency. However, no purposefully designed partial sum generator design exists that can yield desired timing for the decoder. We first derive the mathematical presentation with the partial sums set $\bm{\beta^c}$ which is corresponding to each constituent codes. From this, we concoct a shift-register based partial sum generator. Next, the overall architecture and design details are described, and the overhead compared with conventional partial sum generator is evaluated. Finally, the implementation results with both ASIC and FPGA technology and relevant discussions are presented.'
author:
-
bibliography:
- 'IEEEabrv.bib'
title: An Efficient Partial Sums Generator for Constituent Code based Successive Cancellation Decoding of Polar Codes
---
Introduction {#Introduction}
============
Recently, polar code [@arikan2009channel] has received increasing attentions because it is the first code which provably achieves the channel capacity. Its low-complexity encoding and decoding schemes make it very promising for practical application. There are three widely known algorithms for polar codes decoding. E. Arikan in [@arikan2009channel] presents a successive cancellation (SC) algorithm which can successively accomplish decoding with recursive cancellation. I. Tal [@tal2011list] makes the SC algorithm more competitive by exploring more paths among the codewords tree; this method is referred as list successive cancellation (LSC). Also, N. Hussami et al. in [@hussami2009performance] shows that the belief propagation (BP) can be applied as decoding algorithm.
Although many efforts have been made for BP decoder [@xu2015xj], [@yuan2013architecture] and [@lin2016high], the BP decoding still suffers from the problem of high computing complexity. Thus, SC and LSC attract more studies especially on their hardware architecture [@leroux2013semi] [@zhang2013low] [@yuan2014low] [@che2015overlapped] [@balatsoukas2014hardware] [@balatsoukas2014llr]. SC decoding is based on the feedback, which is also called partial sum, from decoded codewords. A partial sum generator (PSG) is needed for each SC decoder. The partial sum needs to be calculated at the same clock cycle when the codewords are determined. Thus, the calculation of partial sum is on the critical path of the decoding and can affect the maximum frequency of the decoder. Some works have been done for a good PSG design. C. Leroux [@leroux2013semi] proposed an indicator function based PSG (IF-PSG). C. Zhang [@zhang2013low] proposed a PSG with feedback part (FB-PSG). J. Lin [@lin2015hybrid] proposed a hybrid PSG for LSC. G. Berhault proposed a shift-register-based PSG (SR-PSG) [@berhault2013partial] [@berhault2015partialsum], which is able to increase the timing performance and reduce the hardware complexity. Y. Fan [@fan2014efficient] proposed a similar architecture with SR-PSG however with higher level simplification.
Both SC and LSC suffer from the long latency problem. The constituent code based decoding has been studied recently since it is capable of significantly reducing decoding latency [@alamdar2011simplified] [@sarkis2014fast] [@che2016tc]. All the aforementioned PSGs are capable of increasing the timing performance of SC decoder. However, none of them has considered the constituent codes based decoding. Since introducing the concept of constituent codes into decoding processing can significantly reduce the latency, it is reasonable and necessary to design a constituent-codes-compatible PSG. In this paper, we propose an efficient PSG for constituent code based SC decoding, and this is the first architecture of PSG for constituent code based SC decoder. First, we derive the mathematical presentation for constituent based PSG. This derivation is based on the SR-PSG for conventional SC decoder. Next, the overall hardware architecture and design details are proposed. The timing and hardware complexity are evaluated as well. Finally, the implementation result are presented. This architecture is implemented with both VLSI and FPGA technology. The relevant discussions are also mentioned as well.
This paper is organized as follows. The relative background is reviewed in section \[Background\]. In following, the proposed design including the mathematical derivation are described in section \[Proposed Design\]. After that, the implementation results and reverent discussions are presented in section \[Implementation results and discussions\]. Finally, this paper is concluded in section \[Conclusion\].
Background {#Background}
==========
Polar Code {#Polar Code}
----------
As introduced by E. Arikan [@arikan2009channel], we can construct polar code by successively performing channel polarization. Fig. \[encoder\] shows an example of the construction of 8-bit polar code. Mathematically, polar codes are linear block codes of length $n = 2^m$. The coded codeword ${\bm{x}}\triangleq {(x_1,x_2,\cdots,x_n)}$ is computed by $\bm{x}=\bm{u}\bm{G}$ where $\bm{G=F^{\otimes m}}$, and $\bm{F^{\otimes m}}$ is the $m$-th Kronecker power of $\bm{F} =
\begin{bmatrix}
1&0\\
1&1
\end{bmatrix}
$. Each row of $G$ is corresponding to an equivalent polarizing channel. For an $(n,k)$ polar code, $k$ bits that carry source information in $\bm{u}$ are called information bits. They are transmitted via the most $k$ reliable channels. While the rest $n-k$ bits, called frozen bits, are set to zeros and are placed at the least $n-k$ reliable channels.
Polar codes can be decoded by recursively applying successive cancellation to estimate $\hat{u}_i$ from the channel output $y_{0}^{n-1}$ and the previously estimated bits $\hat{u}_{0}^{i-1}$. This method is referred as successive cancellation (SC) decoding. Actually, SC decoding can be regarded as a binary tree traversal as described in Fig. \[SC\_tree\]. The number of bits of one node in stage $m (m = 0,1,2...log_2n)$ is equal to $2^m$. $\bm{\alpha}$ stands for the soft reliability value, typically is log-likelihood ratio (LLR). Each left and right child nodes can calculate the LLR for current node via $f$ and $g$ functions, respectively [@leroux2013semi]. However, in order to solve $g$ function, a feedback $\bm{{\beta}_l}$ from left child of the same parent node is needed. This kind of feedback is called partial sum. At stage 0, $\beta$ of a frozen node is always zero, and for information bit its value is calculated by threshold detection of the soft reliability according to $$\label{hard_decision}
\beta=h(\alpha)=
\left
\{
\begin{array}{c}
0,~if~\alpha \geqslant 0 \\
1,~otherwise \\
\end{array}
\right.$$ At intermediate stages, $\bm{\beta}$ can be recursively calculated by $$\label{feedback}
\beta[i] =
\left
\{
\begin{array}{ll}
\beta_{l}[i] \oplus \beta_{r}[i]~if~i\leq~N^{m}/2 \\
\beta_{r}[i-N^{m}/2]~otherwise \\
\end{array}
\right.$$
Constituent codes based SC decoding {#Constituent codes based SC decoding}
-----------------------------------
SC decoding generally suffers from the high latency due to its inherent serial property. The processing of obtaining the partial sum from each node significantly constrains the decoding speed. Thus, in order to reduce the latency caused by partial sum calculation, constituent code based SC decoding has been proposed [@alamdar2011simplified], [@sarkis2014fast]. By finding some certain patterns in the source code, some part of the codeword and their corresponding partial sums can be estimated immediately without traversal. This method significantly reduces the partial-sum-constrained latency. $\mathcal{N}^0$, $\mathcal{N}^1$, $\mathcal{N}^{SPC}$ and $\mathcal{N}^{REP}$ are the four commonest constituent code.
$\mathcal{N}^0$ and $\mathcal{N}^1$ only contain either frozen bits or information bits, respectively. For $\mathcal{N}^0$ codes, we can set the corresponding partial sums to $0$ immediately. For $\mathcal{N}^1$ node, the partial sums can be directly determined via threshold detection Eq. (\[hard\_decision\]). $\mathcal{N}^{SPC}$ and $\mathcal{N}^{REP}$ contain both frozen bits and information bits. In the $\mathcal{N}^{SPC}$ codes, only the first bit is frozen. It makes the length $n$ constituent codes as a rate $(n-1)/n$ single parity check (SPC) code. This kind of code can be decoded by performing parity check with the least reliable bit. Typically it is the one with the minimum absolute value of LLR. In the $\mathcal{N}^{REP}$ codes, only the last bit is information bit. In this case, all the corresponding partial sums should be the same since they all are the reflection of the last information bit. Thus, the decoding algorithm starts by summing all input LLRs and the partial sums are calculated by performing the hard detection to the final summary. Fig. \[2tree\] shows an example of how constituent code can simplify the SC decoding tree. According to T. Che’s implementation of constituent code based SC decoder [@che2016tc], the latency of length $n$ constituent code can be reduced from $2n-2$ to 1, 1, $log_2n+1$ and $log_2n$ for $\mathcal{N}^0$, $\mathcal{N}^1$, $\mathcal{N}^{SPC}$ and $\mathcal{N}^{REP}$ codes, respectively. In order to further optimize the performance constituent codes based decoder, a specific designed PSG for it is very necessary.
Shift-register-based partial sums generator {#Shift-register-based partial sums generator}
-------------------------------------------
![The architecture of SR-PSG[]{data-label="SR_PSG"}](SR_PSG.eps){width="3in"}
Among all the aforementioned PSGs design, shift-register-based PSG (SR-PSG) has a better performance in terms of both the timing and hardware complexity. For length $n$ polar code decoder, it consists of $n$ registers and some other simple combination logic. Along with the estimation of each $\hat{u}_i$, the registers perform shift calculation and the partial sums can be obtained from their corresponding register. Its architecture is illustrated in Fig. \[SR\_PSG\]. This architecture is built according to the following rule: $$\label{SR_PSG_e}
\left
\{
\begin{array}{l}
R_0~\Leftarrow~\hat{u}_i\cdot c_{i,0} \\
R_k~\Leftarrow~R_{k-1}\oplus (\hat{u}_i\cdot c_{i,k}) ,~if~k \geqslant 0 \\
\end{array}
\right.$$ where $\cdot$ and $\oplus$ stand for $and$ and $exclusive$-$or$ operation, respectively. In Fig. \[SR\_PSG\], $R_k$ means the $k$th register, $\hat{u}_i$ means the $i$th estimated bit. $\beta_{i,j}$ means the $j$th partial sum in stage $i$. $c_{i,k}$ means the $i$th row and $k$th column in the generate matrix $G$. The matrix generation unit is able to generate $c_{i,k}$ with very simple logic. The SC decoder consists of many basic computation parts called processing unit (PU). Each partial sum needs to be feed into the corresponding PU. The shift register based architecture can guarantee that all partial sum required by a PU are all generated in the same register, which can avoid any extra routing logic in the circuit.
Such architecture is able to receive the estimated bit and update the corresponding partial sum by every valid cycle, which is highly consistent with SC decoding processing. However, this architecture is not suitable for constituent codes based SC decoder since some partial sums are obtained directly instead of calculating from estimated bits. Thus, a PSG for constituent codes based SC decoder should have the capability to generate the new partial sums from either the directly got intermediate partial sums or the estimated bits, and to maintain the coherence of them.
Proposed Design {#Proposed Design}
===============
In this section, we first derive the mathematical presentation of constituent code based partial sum from Eq. (\[SR\_PSG\_e\]). Then, the overall hardware architecture and subsequent design details are presented.
Mathematical Presentation {#Mathematical Presentation}
-------------------------
For a length $n$ constituent code, its corresponding estimated bits and partial sums are denoted as $\hat{u}_{i-n+1}^c~\ldots~\hat{u_i}^c$ and $\beta_0^c\ldots \beta_{n-1}^c$, respectively. All the $\bm{\beta^c}$ are obtained at the same time. For those bits that do not belong to any constituent codes, we still have to calculate their corresponding partial sums accroding to Eq. (\[SR\_PSG\_e\]). Thus, if we still want to keep consistency between directly calculated intermediate partial sums and the one-by-one-estimated bits, we need to derive the mathematical presentation with $\bm{\beta^c}$ from Eq. (\[SR\_PSG\_e\]).
For $k~\geqslant~n$ and $k\in[a\cdot n,(a+1)\cdot n-1], a~=~1,2,\ldots$, according to Eq. (\[SR\_PSG\_e\]), we have $$\begin{aligned}
\ R_k &= R_{k-1}\oplus (\hat{u}_i^c\cdot c_{i,k})\\
&= R_{k-2}\oplus (\hat{u}_{i-1}^c\cdot c_{i-1,k-1})\oplus (\hat{u}_i^c\cdot c_{i,k})\\
&\cdots\\
&=R_{k-n}\oplus\\
&\left(
\left[
\begin{array}{lll}
\hat{u}_{i-n+1}^c,&\cdots,&\hat{u}_i^c
\end{array}
\right]
\left[
\begin{array}{c}
c_{i-n+1,k-n+1}\\
\vdots\\
c_{i,k}
\end{array}
\right]
\right).
\label{klargerthann_1}
\end{aligned}$$ As we know, $c_{i,k}$ is the element of generation matrix $G$ which is the Kronecker power of $\bm{F} =
\begin{bmatrix}
1&0\\
1&1
\end{bmatrix}
$. Combine this property with our observation on the matrix, we conduct the following rule which is also noted in Fig. \[observation\]. $$\left[
\begin{array}{c}
c_{i-n+1,k-n+1}\\
\vdots\\
c_{i,k}
\end{array}
\right]
=
\left[
\begin{array}{c}
c_{i-n+1,(a+1)\cdot n - (k~mod~n)-1}\\
\vdots\\
c_{i,(a+1)\cdot n - (k~mod~n)-1}
\end{array}
\right].
\label{shift_g}$$
According to the definition of generation matrix and concept of constituent code, when $c_{i,k}~=~0$, the right part of Eq. (\[shift\_g\]) is equal to a all $\bm{zero}$ vector, and when $c_{i,k}~=~1$ the right part of Eq. (\[shift\_g\]) is equal to the $(n-(k~mod~n)-1)$th column in the generation matrix for length $n$ polar code. According to the definition of partial sum and Eq. (\[feedback\]), we get $$[\hat{u}_{i-n+1}^c,\cdots,\hat{u}_i^c]\cdot[c_{i-n+1,p(k)},\cdots,c_{i,p(k)}]^T=\beta_{p(k)}
\label{get_partial_2}$$ where $p(k) = (n-(k~mod~n)-1)$.
Now we apply the above observation back to Eq (\[klargerthann\_1\]). We define the vector $\bm{R_a}=[R_{a\cdot n},\cdots,R_{a\cdot n + n-1}]$ and $\bm{c_{i,a}} = [c_{i,a\cdot n},\cdots,c_{i,a\cdot n + n-1}]$ for $k\in[a\cdot n,(a+1)\cdot n-1], a~=~1,2,\ldots$. We also define the vectors $\bm{\hat{u}^c}=[\hat{u}_{i-n+1}^c,\cdots,\hat{u}_i^c]$ and $\bm{\hat{\beta}^c}=[\beta_{n-1}^c,\ldots,\beta_0^c]$. Then, we have $$\begin{aligned}
\bm{R_a} &=[R_{a\cdot n},\cdots,R_{a\cdot n + n-1}] \\\\
&=[R_{(a-1)\cdot n},\cdots,R_{a\cdot n -1}]\oplus \\
&\left(
[\hat{u}_{i-n+1}^c,\cdots,\hat{u}_i^c]
\left[
\begin{array}{ccc}
c_{i-n+1,a\cdot n-n+1} &\cdots &c_{i-n+1,a\cdot n} \\
\vdots &\ddots &\vdots \\
c_{i,a\cdot n} &\cdots &c_{i,a\cdot n+n-1}
\end{array}
\right]
\right)\\
&=[R_{(a-1)\cdot n},\cdots,R_{a\cdot n -1}]\oplus \\
&\left(
\bm{\hat{u}^c}
\left[
\begin{array}{ccc}
c_{i-n+1,p(a\cdot n)} &\cdots &c_{i-n+1,p(a\cdot n+n-1)} \\
\vdots &\ddots &\vdots \\
c_{i,p(a\cdot n)} &\cdots &c_{i,p(a\cdot n+n-1)}
\end{array}
\right]
\right)\\
=&
\left
\{
\begin{array}{ll}
\bm{0},if~\bm{c_{i,a}}~=~\bm{0} \\
\bm{R_{a-1}}\oplus\cdot\bm{\beta^c},if~\bm{c_{i,a}}~=~\bm{1}\\
\end{array}
\right.
\label{klargerthann_2}
\end{aligned}$$
For the consistent with Eq. (\[SR\_PSG\_e\]), we rewrite Eq. (\[klargerthann\_2\]) as follow: $$\bm{R_a}=\bm{R_{a-1}}\oplus(\bm{\beta^c} \& \bm{c_{i,a}})
\label{klargerthann_3}$$ where $\&$ stands for the bit-wise $and$ operation.
For $0~\leqslant~k<~n$, similar to Eq. (\[klargerthann\_1\]), we have $$R_k =[\hat{u}_{i-k}^c,\cdots,\hat{u}_i^c]\cdot[c_{i-k,0},\cdots,c_{i,k}]^T
\label{ksmallerthann_1}$$ According to the definition of $G$ and constituent codes, we can conduct that for any length $n$ constituent codes, the first $n$ columns of its corresponding rows in $G$ should also be a generation matrix $G_n$ for length $n$ polar code. As described in Fig. \[diagonal\_shift\], the diagonal cycle shift is same as each correspond column, and consider the $G_n$ is a lower triangular matrix, we get $$\begin{aligned}
&[c_{i-n+1,k+1},\cdots,c_{i-k-1,n-1},c_{i-k,0},\cdots,c_{i,k}]^T\\
&=[c_{i-n+1,n-1-k},\cdots,c_{i,n-1-k}]^T\\
&=[0,\cdots,0,c_{i-k,n-1-k},\cdots,c_{i,n-1-k}]^T
\end{aligned}
\label{cycle_shift}$$ Thus, Eq. (\[ksmallerthann\_1\]) can be rewritten as: $$\begin{aligned}
R_k
&=[\hat{u}_{i-k+1}^c,\cdots,\hat{u}_i^c]\cdot[c_{i-k,0},\cdots,c_{i,k}]^T\\
&=[\hat{u}_{i-n+1}^c,\cdots,\hat{u}_i^c]\cdot[0,\cdots,0,c_{i-k,0},\cdots,c_{i,k}]^T\\
&=[\hat{u}_{i-n+1}^c,\cdots,\hat{u}_i^c]\cdot[c_{i-n+1,n-1-k},\cdots,c_{i,n-1-k}]^T\\
&=\beta_{n-k-1}^c
\label{ksmallerthann_2}
\end{aligned}$$ Thus, combining Eq. (\[ksmallerthann\_2\]) and Eq. (\[klargerthann\_3\]), we derive the mathematical presentation for partial sum of constituent based polar decoder as follow: $$\label{psg_cbpc}
\bm{R_a} =
\left
\{
\begin{array}{ll}
\bm{\beta^c},~if~a~=~0 \\
\bm{R_{a-1}}\oplus(\bm{\beta^c} \& \bm{c_{i,a}}), if~a\geqslant~1. \\
\end{array}
\right.$$
Proposed architecture {#Proposed architecture}
---------------------
![Overall architecture of SR-CB-PSG[]{data-label="SR_CB_PSG"}](SR_CB_PSG.eps){width="3.5in"}
According to Eq (\[psg\_cbpc\]), the shift-register constituent-code based partial sum generator (SR-CB-PSG) is proposed as in Fig. \[SR\_CB\_PSG\]. Compared with Fig. \[SR\_PSG\], there are three differences. The first difference is the input. For SR-PSG, only current estimated bit is sent into, which means the input is only from the $PU$ from stage $0$. However, for SR-CB-PSG, the inputs are from $PU$s of any stage, depending on the length of constituent code. Thus, a multiplexing networking is needed to route all the inputs values to the right registers. The second difference is the shift function. According to Eq (\[psg\_cbpc\]), instead of just shifting by one bit, the shifter should have the capability to shift $n$-bit where $n$ is the length of constituent code. According to the definition of constituent code, $n$ should be the any power of $2$. Thus, A specific design $(2^m-1)$-bit shifter is proposed. The control signals for both the muxing networking and shifter are from the $Control~Signal~Generator (CSG)$ with simple logic. The last difference is matrix generation unit. For each constituent code, its corresponding $c_{i,j}$ should be the $i$th row of the generation matrix, where $i$ is the index of the last bit in the constituent code. Due to the irregularity of the constituent code, it’s unnecessary to build an online generator for that. Thus, a pre-calculated ROM is placed. It is a trade-off between design complexity and hardware resource. It can be replaced by a re-configurable memory device like RAM for flexibility.
![ (a) PU tree of SC decoder, (b) PUs and their corresponding register, and (c) architecture of multiplexing network []{data-label="mux_networking"}](mux_networking.eps){width="3.5in"}
Fig. \[mux\_networking\] shows an example of partial sum routing for 8 bit constituent code based polar code. We can see each register has specific corresponding PU from each stage. They need the multiplexing networking to route the partial sums to the each right register. For length $n$ polar code, there are $log_2n$ stages in the decoder and $n/2$ registers in the SR-CB-PSG. If the multiplexing networking is built from the basic 2-bit MUX, each register is assigned an identical MUXs networking made by $(log_2n~-~1)$ MUXs. All the networkings share the same control signal. According to its architecture, the control signals are the direct binary mapping of its stage index. In total, $n/2\cdot(log_2n~-~1) $ MUXs are needed. Since the multiplexer networking needs to wait each PU finish computing to get the valid inputs, it is on the critical path of the decoder. Thus, it causes additional $\lceil log_2(log_2n)\rceil \cdot \bigtriangleup(MUX)$ delay, where $\bigtriangleup(MUX)$ is the delay for a single MUX.
![An example of $(2^m-1)$ shifter for 16-bit polar code decoder[]{data-label="shifter"}](shifter.eps){width="3in"}
For the $(2^m-1)$ shifter, we proposed a barrel-shifter-based architecture. For length $n$ polar code, $m\leqslant (log_2n-1)$. The shifter performs logic right shift. For $k<n$, where $k$ is the index of the register and $n$ is the length of the current constituent code, zeros are added to the left. For $k\geqslant n$, we do shift. Those behaviors satisfy the first and second in Eq (\[psg\_cbpc\]).
Fig. \[shifter\] shows an example of $(2^m-1)$ shifter for 16-bit polar code decoder. All the MUXs in the same row can shall the same control signal. Those signals are generated by a $k~to~2^k~decoder$, where $k=\lceil log_2(log_2n) \rceil$ for length $n$ polar code. For length $n$ polar code, there are $(n/2-1)\cdot(log_2n~-~1) $ MUXs are needed for the shifter. Since the shifter can start shift data without waiting $PU$ to finish computing, it is not on the critical path. Thus, it should not deteriorate the timing performance of the decoder at all.
Implementation results and discussions {#Implementation results and discussions}
======================================
To the best of our knowledge, the proposed design is the first PSG design especially design for constituent codes based SC decoder. Thus, there is no reference design we can directly compare with. In this section, we list all the results we have and presents some relevant discussions.
Critical Path
------------------------------ -----------------------------------------------------------------------------------------------
SR-PSG[@berhault2013partial] $ \bigtriangleup(AND) + \bigtriangleup(XOR) $
Proposed $\lceil log(logN)\rceil \cdot \bigtriangleup(MUX) +\bigtriangleup(AND) + \bigtriangleup(XOR)$
: Critical Path Comparison[]{data-label="Critical Path Comparison"}
Table \[Critical Path Comparison\] shows the critical path comparison between proposed PSG and the PSG in [@berhault2013partial]. We can tell the delay overhead comes from the muxing network. Ideally, the maximum frequency of constituent codes based SC decoder is lower than that of conventional SC decoder. However, after taking the latency reduction into account, as shown in Table \[latency\_reduction\], constituent codes based SC decoder is able to achieve much higher throughput. The conventional SC decoder is referred from [@yuan2014low] which is the lowest latency conventional SC decoder to the best of out knowledge.
---------------------------------------- ------ ------ ------ ------ ------
0.2 0.35 0.5 0.65 0.8
latency of conventional [@yuan2014low]
latency of constituent code based 263 298 266 200 160
reduction($\%$) 65.7 61.1 65.3 73.9 79.1
---------------------------------------- ------ ------ ------ ------ ------
: Decoder Latency comparison for length=1024 polar code[]{data-label="latency_reduction"}
Table \[Estimated resource consumption \] shows the resource consumption estimation of proposed SR-CB-PSG for length $n$ polar code decoder and the comparison with other two conventional PSG. The most resource consumption part is the $MUX$ since it used in both multiplexer networking and shifter. The estimation for the ROM size is based on the average calculation since the decoding latency changes along with the code rate.
proposed [@berhault2013partial] [@zhang2013low]
---------- ---------------------- ------------------------ -----------------
DFF $n/2$ $n$ $(n^2-4)/12$
MUX $(n-1)\cdot(logn-1)$ - $n-2$
XOR $n/2-1$ $n-2$ $n/2-1$
AND $n/2$ $n/2$ -
ROM(bit) $n^2/10$(average) - -
: Resource comparison []{data-label="Estimated resource consumption "}
The proposed design can be targeted on either ASIC or FPGA. We synthesized both with Nangate FreePDK 45nm process and on Xilinx Kintex-7 FPGA KC705 Evaluation board. Table \[hardware resource of SR-CB-PSG\] shows the hardware resource of SR-CB-PSG for 1024 code length polar code decoder on both of them.
nangate 45nm
-- ---------------------- --------------------- ----------------
slice LUTs slice REGs area
$1569(\textless1\%)$ $512(\textless1\%)$ $16333\mu m^2$
: hardware resource of SR-CB-PSG for 1024 code length polar code decoder []{data-label="hardware resource of SR-CB-PSG"}
Noticeably, the architecture we discussed in this paper is based on the consideration for the worst case, which is that the maximum length of constituent codes could be $n/2$. However, for practical application, the maximum length of constituent is fix for certain code rate and usually cannot approach $n/2$. For those case, the logic of both the multiplexer networking and shifter could be even simpler, which will result in a better timing and silicon area performance.
Conclusion {#Conclusion}
==========
This paper proposed an efficient PSG hardware design for constituent code based SC decoder. Conventional PSG is not compatible with the constituent code based SC decoder. This is because that the conventional one is only capable of taking estimated bit one by one but the constituent code based decoder is generated the intermediate partial sum directly. To solve this problem, we first derive the mathematical presentation for constituent code based PSG from the SR-PSG for conventional SC decoder. Then, the overall hardware architecture and design details are proposed. Finally, the implementation result with both VLSI and FPGA technology are presented, and the relevant discussions are carried out.
|
---
abstract: 'We consider a position-dependent quantum walk on ${\bf Z}$. In particular, we derive a detection method for edge defects by embedded eigenvalues of its time evolution operator. In the present paper, an edge defect is a set $ \{ y-1 ,y\} $ for $y\in {\bf Z}$ on which the coin operator is an anti-diagonal matrix. In fact, under some suitable assumptions, the existence of a finite number of edge defects is equivalent to the existence of embedded eigenvalues of the time evolution operator. In view of applications, by checking the spectrum, we can detect the existence of disconnecting edge (in the sense of edge defects above) on the line without directly watching it.'
address:
- 'Graduate School of Science and Engineering, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577, Japan'
- 'Graduate School Educational Promotion Center, Yokohama National University, Tokiwadai 79-7, Hodogaya-ku, Yokohama 240-8501, Japan'
author:
- Hisashi MORIOKA
- Etsuo SEGAWA
title: Detection of edge defects by embedded eigenvalues of quantum walks
---
Introduction
============
Quantum walks have been studied in various kinds of research fields (see [@Am], [@Sh], [@Ven] et al. and its references). Recently, there is an abundance of studies on position-dependent quantum walks in view of the spectral theory of unitary operators. Some results of the weak limit theorem for position-dependent quantum walks were proved by Konno-Luczak-Segawa [@KLS], Endo-Konno [@EK1] and Endo et al. [@EK2]. In view of the scattering theory, the wave operators associated with the time evolution operator were introduced by Suzuki [@Su] under the short-range type condition, as well as the asymptotic velocity of the quantum walker and the weak limit theorem were considered as applications. We also mention about Richard-Suzuki-Tiedra de Aldecoa [@RST]. A Mourre theory for unitary operators is given and its application to the spectral theory of the quantum walk is derived.
In some models of quantum walks, localization occurs depending on its initial states, and eigenvalues of the time evolution operator have a crucial role in the localization. If $U$ is a unitary time evolution operator for one-dimensional, two-state quantum walks, eigenvalues and eigenspaces are defined as follows. If there exists a non-trivial solution $\psi \in \ell^2 ({\bf Z} ; {\bf C}^2 ) $ to the equation $U\psi = e^{i\theta} \psi $ for $ \theta \in [0 , 2\pi )$, we call $ e^{i\theta} $ an eigenvalue of $U$. Thus the associated eigenspace $\mathcal{E} (\theta )$ is a subspace of $ \ell^2 ({\bf Z} ; {\bf C}^2 )$. As has been shown by Cantero et al. [@CGM], and Suzuki [@Su], if the initial state has an overlap with $\mathcal{E} (\theta )$ i.e. the initial state is not in $\mathcal{E} (\theta )^{\perp}$ in the sense of $\ell^2 ({\bf Z} ; {\bf C}^2 )$, the localization occurs in the associated quantum walk. Examples of localizations with one-defect model are in Cantero et al. [@CGM], Konno-Luczak-Segawa [@KLS] and Fuda-Funakawa-Suzuki [@FuFuSu]. More generally, we can see a similar result for localizations for quantum walks on graphs (see Segawa-Suzuki [@SS]).
In this paper, we consider an approach of *detection of edge defects* by using embedded eigenvalues of the time evolution operator of the one-dimensional, two-state quantum walk. The rigorous meaning of edge defects will be defined below. Let $ \mathcal{H} = \ell^2 ({\bf Z} ; {\bf C}^2 )$ be the space of states. The unitary operator $U$ is given by $$(U \psi )(x)= P(x+1) \psi (x+1) +Q (x-1) \psi (x-1) , \quad x\in {\bf Z} ,$$ for every $\psi \in \mathcal{H} $ and $$P(x)= \left[ \begin{array}{cc}
a (x) & b (x) \\ 0 & 0 \end{array} \right] , \quad Q(x)= \left[ \begin{array}{cc}
0 & 0 \\ c (x) & d (x) \end{array} \right] .$$ Here we assume $C(x) := P(x)+Q(x) \in U(2)$ for every $x\in {\bf Z} $ and $U$ is rewritten by $U=SC$ where $S$ is the shift operator defined by $$(S\psi )(x)= \left[ \begin{array}{c} \psi^{(0)} (x+1) \\ \psi ^{(1)} (x-1) \end{array} \right] , \quad \psi \in \mathcal{H} , \quad x\in {\bf Z} .$$ Taking an initial state $\psi_0 \in \mathcal{H}$, we put $\psi (t, \cdot ) := U^t \psi_0$ for $t\in \{ 0,1,2, \ldots \} $. Since the operator $U$ depends on the position, we call this discrete time evolution *one dimensional position-dependent quantum walk*. Thus we call $C$ the *coin operator* of the operator $U$. The corresponding position-independent quantum walk is given by $U_0 = SC_0 $ where $C_0 := P_0 + Q_0 \in U(2)$ and $$P_0 = \left[ \begin{array}{cc}
a_0 & b_0 \\ 0 & 0 \end{array} \right] , \quad Q_0 = \left[ \begin{array}{cc}
0 & 0 \\ c_0 & d_0 \end{array} \right] .$$ We adopt the representation of $C_0$ which is introduced in [@RST]. Precisely, we put $ a_0 = pe^{i\alpha} $, $b_0 = q e^{i\beta} $, $c_0 = -q e^{- i ( \beta - \gamma )} $ and $d_0 = p e^{-i ( \alpha - \gamma )} $ for $ \alpha , \beta , \gamma \in [0,2\pi ) $ and $p,q\in [0,1]$ with $p^2 + q^2 =1$ : $$C_0 = e^{i\gamma /2} \left[ \begin{array}{cc}
pe^{i(\alpha - \gamma /2 )} & q e^{i (\beta - \gamma /2)} \\ -q e^{-i (\beta - \gamma /2 )} & p e^{-i (\alpha - \gamma /2)} \end{array} \right] .
\label{S1_def_C0}$$ Throughout of the paper, we assume that there exist constants $\rho , M>0$ such that $$\| C(x) - C_0 \| _{\infty } \leq M e^{-\rho \langle x\rangle } , \quad x\in {\bf Z} ,
\label{S1_eq_exp}$$ where $\| \cdot \|_{\infty } $ is the norm of $2\times 2$-matrices defined by $$\| A \| _{\infty} = \max _{1 \leq j,k \leq 2 } | a_{jk} | , \quad A= [a_{jk} ]_{1\leq j,k \leq 2} ,$$ and $ \langle x \rangle = \sqrt{1+x^2 }$.
In the present paper, we consider the existence or the non-existence of *edge defects* on ${\bf Z}$. Here we define edge defects as follows.
We call the set ${\bf e}_y = \{ y-1 ,y \} $ for $y\in {\bf Z}$ an edge defect if $C(x)=C_1$ for $x\in {\bf e}_y$ where $$C_1 = e^{ i \gamma ' /2} \left[ \begin{array}{cc}
0 & e^{i (\beta ' - \gamma ' /2)} \\ -e^{-i ( \beta ' - \gamma ' /2)} & 0 \end{array} \right] ,
\label{S1_def_C1}$$ for $ \beta ' , \gamma ' \in [0,2\pi )$.
\[S1\_def\_edefect\]
Let us make a remark on Definition \[S1\_def\_edefect\] in view of applications. If the edge defect occurs, then there is a disconnection between $ \{y-1,y \} $ in the network by the definition. So in this paper we propose a detection way of the existence of a disconnecting part without directly watching it. Turning our mind to quantum search algorithms driven by quantum walks, we notice that the quantum coins at the target vertices are also perfect reflection operators. Then it is possible to interpret that the setting of the edge defect is an [*infinite system’s*]{} analogue of quantum search algorithms whose target vertices are e.g., $ \{0,1 \} $ ; in this “algorithm", we can find how the defects occurs at the targets checking the spectrum of this system (see Figs. \[fig\_distvertex\]-\[fig\_edge\] in §5).
Under the assumption (\[S1\_eq\_exp\]), we show that one can detect the existence of edge defects by that of eigenvalues of $U$ embedded in the interior of the continuous spectrum $\sigma _{ess} (U)$. The first result of the present paper is as follows.
Let $p\in (0,1]$. We assume that there is no edge defect i.e. there exists a constant $\delta >0$ such that $|a(x)| \geq \delta $ for all $x\in {\bf Z} $. Moreover, suppose that $C$ and $C_0$ satisfy the condition (\[S1\_eq\_exp\]). Then the continuous spectrum of $U$ is $\sigma_{ess} (U) = \{ e^{i\theta } \ ; \ \theta \in J_{\gamma} \} $ where $J_{\gamma} = J_{\gamma ,1} \cup J_{\gamma ,2} $ with $$\begin{gathered}
\begin{split}
& J_{\gamma ,1} = [ \arccos p + \gamma /2 , \pi - \arccos p + \gamma /2 ] , \\
& J_{\gamma ,2} = [ \pi + \arccos p + \gamma /2 , 2\pi - \arccos p + \gamma /2 ] .
\end{split}\end{gathered}$$ Moreover, there is no eigenvalue in $\sigma_{ess} (U) \setminus \mathcal{T} $ where $\mathcal{T} = \{ e^{i\theta} \in \sigma_{ess} (U) \ ; \ \theta \in J_{\gamma , \mathcal{T}} \} $ with $$\begin{gathered}
J_{\gamma , \mathcal{T}} = \left\{
\begin{split}
& \arccos p + \gamma /2 , \ \pi - \arccos p + \gamma /2 , \\
& \pi + \arccos p + \gamma /2 , \ 2\pi - \arccos p + \gamma /2
\end{split}
\right\} . \end{gathered}$$ \[S1\_mainthm1\]
If there are some edge defects, the operator $U$ is given as follows. Let $C_1 $ be defined by (\[S1\_def\_C1\]). For a positive integer $N>0$, we take $y_1 , \cdots , y_N \in {\bf Z} $, and put $${\bf e} = \bigcup _{j=1}^N {\bf e}_{y_j} , \quad {\bf e} _{y_j} = \{ y_j -1 , y_j \} .$$ For any subset $A\subset {\bf Z} $, let the operator $F_A$ on $\mathcal{H}$ be defined by $ (F_A \psi )(x)= \psi (x)$ for $x\in A$ and $( F_A \psi )(x)=0 $ for $x \in {\bf Z} \setminus A$. Then we put $$C = \sum _{j=1}^N F_{{\bf e}_{y_j}} C_1 + (1-F_{{\bf e}} ) C_2 = F_{{\bf e}} C_1 + (1-F_{{\bf e}} ) C_2 ,
\label{S1_eq_CC}$$ where the coin operator $C_2$ given by $$C_2 (x) = \left[ \begin{array}{cc}
a_2 (x) & b_2 (x) \\ c_2 (x) & d_2 (x) \end{array} \right] \in U (2) , \quad x\in {\bf Z} ,$$ satisfies the assumption (\[S1\_eq\_exp\]) and there exists a constant $\delta >0$ such that $|a_2 (x) | \geq \delta $ for all $x\in {\bf Z}$. In this case, the situation of $U$ and $U_0$ is same as Theorem \[S1\_mainthm2\] in ${\bf Z}\setminus {\bf e}$. However, there exists an embedded eigenvalue as follows.
Let $p\in (0,1]$ and $C$ be given by (\[S1\_eq\_CC\]).\
(1) The continuous spectrum of $U$ is $\sigma_{ess} (U)=\{ e^{i\theta} \ ; \ \theta \in J_{\gamma} \} $.\
(2) For any $\gamma ' \in [0,2\pi )$, we have $\pm i e^{i\gamma ' /2} \in \sigma_p (U)$, and we can take associated eigenfunctions $\Psi _{\pm} \in \mathcal{H}$ such that $\mathrm{supp} \Psi _{\pm} \subset {\bf e}$.\
(3) If $ ( \gamma ' + \pi )/2 \in J_{\gamma} \setminus J_{\gamma , \mathcal{T} }$, we have $ \pm i e^{i \gamma ' /2} \in \sigma_p (U) \cap ( \sigma_{ess} (U) \setminus \mathcal{T} )$. Any associated eigenfunctions $\Psi_{\pm}$ vanish in $\{x\in {\bf Z} \ ; \ x>x^* \, \text{or} \ x<x_* \} $ where $x^* = \max \{ x\in {\bf e} \} $ and $x_* = \min \{ x\in {\bf e} \} $. \[S1\_mainthm2\]
As a consequence of Theorems \[S1\_mainthm1\] and \[S1\_mainthm2\], we can state the conclusion of this paper.
Let $p\in (0,1]$ and $( \gamma ' + \pi )/2 \in J_{\gamma} \setminus J_{\gamma , \mathcal{T}}$. Suppose $C$ is given by (\[S1\_eq\_CC\]). There is no edge defect i.e. $ {\bf e} = \emptyset $ if and only if $U$ has no eigenvalue in $\sigma_{ess} (U)\setminus \mathcal{T}$. \[S1\_cor\_main\]
Theorems \[S1\_mainthm1\] and \[S1\_mainthm2\] are analogues of the Rellich type uniqueness theorem for the Helmholtz equation $(-\Delta -\lambda )u=0$ on the Euclidean space. Originally it was introduced by Rellich [@Re] and Vekoua [@Vek]. This theorem has been generalized to a broad class of partial differential equations, since it plays important roles in the spectral theory ([@Tr], [@Li1], [@Li2], [@Ho], [@Mu] and [@RaTa]). Recently, this theorem was generalized for the discrete Schrödinger operator on perturbed periodic graphs ([@IsMo], [@Ves] and [@AIM]). Note that the Rellich type uniqueness theorem holds in a Banach space larger than $L^2$-space or $\ell^2$-space. However, it is sufficient to prove in $\ell^2 ({\bf Z} ; {\bf C}^2 )$ for our purpose of the paper. For the proof, we use a Paley-Wiener theorem and the theory of complex variable.
The plan of this paper is as follows. In §2, we recall basic properties of spectra of unitary operators. The proof of Theorem \[S1\_mainthm1\] is given in §3. The precise construction of embedded eigenvalues and the associated eigenfunctions are given in §4. We summarize our arguments in §5, using some numerical examples.
Throughout of this paper, we use the following basic notations. We denote the flat torus by ${\bf T} = {\bf R} / ( 2\pi {\bf Z} )$ and the complex torus by ${\bf T} _{{\bf C}} = {\bf C} /( 2\pi {\bf Z} )$. For any $s\in {\bf R}$, we put $\langle s \rangle = \sqrt{ 1+s^2 }$. The unit circle on the complex plane ${\bf C}$ is denoted by $S^1 $.
Continuous spectrum
====================
Spectral decomposition of unitary operators
-------------------------------------------
Here let us recall some general properties of spectra of unitary operators. Let $ \mathcal{H} $ be a Hilbert space. We denote by $ (\cdot , \cdot ) _{\mathcal{H}} $ the inner product of $ \mathcal{H} $ and by $\| \cdot \| _{\mathcal{H}} $ the associated norm.
Let $U$ be a unitary operator on $ \mathcal{H} $. It is well-known that there exists a spectral decomposition $E_U ( \theta )$ for $ \theta \in {\bf R}$ such that $$U= \int_0^{2\pi} e^{i\theta} dE_U ( \theta ) ,$$ where $E_U ( \theta )$ is extended to be zero for $\theta \in (-\infty , 0)$ and to be $1$ for $\theta \in [ 2\pi ,\infty )$. It is well-known that $\sigma (U) \subset S^1 $. Since $E_U (\theta )$ is a measure on ${\bf R}$, applying Radon-Nikodým theorem, it provides the orthogonal decomposition of $ \mathcal{H} $ associated with $U$ as $$\mathcal{H} = \mathcal{H}_p (U) \oplus \mathcal{H}_{sc} (U) \oplus \mathcal{H} _{ac} (U) ,$$ where $ \mathcal{H}_p (U ) $ is spanned by eigenfunctions of $U$, $ \mathcal{H}_{sc} (U) $ and $\mathcal{H}_{ac} (U)$ are orthogonal projections on the pure point, the singular continuous and the absolutely continuous subspace of $ \mathcal{H}$, respectively. Then we put $$\sigma_p (U) = \text{the set of eigenvalues of } U \text{ in } \mathcal{H} ,$$ $$\sigma_{sc} (U)= \sigma ( U | _{\mathcal{H} _{sc} (U)} ) , \quad \sigma_{ac} (U)= \sigma (U | _{\mathcal{H}_{ac} (U)} ) ,$$ and we call them the point spectrum, the singular continuous spectrum and the absolutely continuous spectrum of $U$, respectively.
We also define the discrete spectrum and the essential spectrum of $U$. The discrete spectrum $\sigma_d (U)$ is the set of isolated eigenvalues of $U$ with finite multiplicities. The essential spectrum $\sigma_{ess} (U)$ is defined by $ \sigma_{ess} (U) = \sigma (U) \setminus \sigma_d (U)$. Then if $\lambda \in \sigma_{ess} (U)$, $\lambda $ is either an eigenvalue of infinite multiplicity or an accumulation point of $\sigma (U)$.
As in the case of self-adjoint operators, the essential spectrum of $U$ is characterized by singular sequences as follows.
We have $e^{i \theta } \in \sigma_{ess} (U)$ for $ \theta \in [0,2\pi )$ if and only if there exists a sequence $\{ \psi _n \} _{n=1}^{\infty} $ in $ \mathcal{H} $ such that $\| \psi_n \| _{\mathcal{H} } =1$, $\psi_n \to 0$ weakly in $\mathcal{H}$ and $ \| (U-e^{i\theta} )\psi_n \| _{\mathcal{H}} \to 0$ as $n\to \infty $. \[S2\_lem\_essspec\]
Proof. Suppose $ e^{i\theta} \in \sigma_{ess} (U)$. When $e^{i\theta} $ is an eigenvalue of infinite multiplicities, we can take an orthonormal basis $\{ \psi_n \}_{n=1}^{\infty} $ in $ \mathrm{Ker} (U-e^{i\theta} )$. When $e^{i\theta} $ is an accumulation point of $\sigma (U)$, we can take a sequence $ \{ \theta_n \} _{n=1}^{\infty} $ such that $e^{i\theta_n} \in \sigma (U)$ and $\theta_n \to \theta$. We take sufficiently small $\epsilon _n >0$ so that $I_n = ( \theta_n - \epsilon_n , \theta_n + \epsilon_n )$ satisfies $I_n \cap I_m = \emptyset $ for $m\not= n$. By choosing $\psi_n \in \mathrm{Ran}( E_U ( I_n ))$ with $\| \psi_n \| _{\mathcal{H}} =1$, we have an orthonormal basis $\{ \psi_n \} _{n=1}^{\infty} $. Moreover, we obtain $$\| (U-e^{i\theta} ) \psi_n \|^2 _{\mathcal{H}} = \int_{I_n} |e^{is} - e^{i\theta} |^2 d( E_U (s) \psi_n , \psi_n )_{\mathcal{H}} \leq C\epsilon_n^2 \to 0 .$$
Suppose that there exists a sequence $\{ \psi_n \} _{n=1}^{\infty}$ such that $\psi_n$ satisfies the condition in the statement of the lemma. If $e^{i\theta} \not\in \sigma (U)$, there exists a constant $\delta >0$ such that $ E_U (( \theta - \delta , \theta + \delta ))=0$ and $ \| (U-e^{i\theta} ) \psi \| _{\mathcal{H}} \geq \delta $ for any $\psi \in \mathcal{H} $. This is a contradiction. If $e^{i \theta } \in \sigma_d (U)$, there exists a constant $ \epsilon >0$ such that $ E_U (( \theta - \epsilon , \theta + \epsilon ))= E_U ( \{ \theta \} )$ for $e^{i\theta} \not= 1$ or $E_U ((-\epsilon , \epsilon )) + E_U ((2\pi - \epsilon , 2\pi + \epsilon ))=E_U (\{ 0\} ) + E_U (\{ 2\pi \} )$ for $e^{i\theta} =1$. In the following, we shall prove the case $e^{i\theta} \not= 1$. For $e^{i\theta} =1$, the proof is similar.
We can take an orthonormal basis $\{ \phi_j \} _{j=1}^m $ of $\mathrm{Ker} (U-e^{i\theta} )$ for a positive integer $m$. Applying the Gram-Schmidt orthonormalization to $\{ \phi_j \} _{j=1}^m \cup \{ \psi _k \} _{k=1}^{\infty }$, we put the resulting sequence $\{ \phi '_j \} _{j=1}^{\infty } $. Note that $\phi'_j = \phi_j $ for $j=1 ,\cdots , m$. Hence we have $\| (U-e^{i\theta} ) \phi'_j \| _{\mathcal{H}} \to 0 $ as $j\to \infty $. On the other hand, we have $$\| (U-e^{i\theta} ) \phi '_j \|^2_{\mathcal{H}} = \int _{|s-\theta | \geq \epsilon } |e ^{i (s-\theta )} -1 |^2 d( E_U (s) \phi'_j , \phi '_j )_{\mathcal{H}} \geq \epsilon ^2 ,$$ for $j>m$. This is a contradiction.
As a consequence, we can see that compact perturbations of $U$ do not change its essential spectrum.
Let $U'$ and $U$ be unitary operators on $\mathcal{H}$. If $U'-U$ is compact on $\mathcal{H}$, we have $\sigma_{ess} (U') = \sigma_{ess} (U)$.
\[S2\_lem\_weyltype\]
Proof. Let $e^{i\theta} \in \sigma_{ess} (U)$. In view of Lemma \[S2\_lem\_essspec\], there exists a sequence $\{ \psi_n \} _{n=1}^{\infty} $ in $\mathcal{H}$ such that $\| \psi_n \| _{\mathcal{H}} =1$, $\psi_n \to 0$ weakly in $\mathcal{H}$ and $\| (U-e^{i\theta } )\psi_n \| _{\mathcal{H}} \to 0$ as $n\to \infty $. Since $U' -U$ is compact, we have $(U'-U )\psi_n \to 0$ in $\mathcal{H}$. Then we have $$\| (U' -e^{i\theta} )\psi_n \| _{\mathcal{H}} \leq \| (U-e^{i\theta } )\psi_n \| _{\mathcal{H}} + \| (U' -U) \psi_n \| _{\mathcal{H}} \to 0.$$ Applying Lemma \[S2\_lem\_essspec\] to $U'$, we obtain $e^{i\theta} \in \sigma_{ess} (U')$. This implies $ \sigma_{ess} (U) \subset \sigma_{ess} (U') $. We can prove $\sigma_{ess} (U' ) \subset \sigma_{ess} (U) $ by the same way.
Essential spectrum
------------------
We turn to the quantum walk. In the following, the notations $U$ and $U_0$ are used in order to represent the unitary operators of time evolution for the quantum walk, and $\mathcal{H} = \ell^2 ({\bf Z} ; {\bf C}^2 )$. Let $ \mathcal{F} : \mathcal{H} \to \widehat{\mathcal{H}} := L^2 ({\bf T} ; {\bf C}^2 )$ be the unitary operator defined by $$(\mathcal{F} \psi )(\xi ) = \left[ \begin{array}{c}
\widehat{\psi }^{(0)} (\xi ) \\ \widehat{\psi }^{(1)} (\xi )
\end{array} \right] ,\quad \widehat{\psi }^{(j)} (\xi )= \frac{1}{\sqrt{2\pi }} \sum _{x\in {\bf Z}} e^{-ix\xi } \psi ^{(j)} (x) ,$$ for $\xi \in {\bf T}$, $j=0,1$, and every $ \psi \in \mathcal{H} $. Then the adjoint operator $ \mathcal{F}^* : \widehat{\mathcal{H}} \to \mathcal{H} $ is given by $$(\mathcal{F}^* \widehat{\phi} )(x ) = \left[ \begin{array}{c}
\phi ^{(0)} (x ) \\ \phi ^{(1)} (x )
\end{array} \right] ,\quad \phi^{(j)} (x) = \frac{1}{\sqrt{2\pi}} \int _{{\bf T}} e^{ix\xi} \widehat{\phi}^{(j)} (\xi ) d\xi ,$$ for $x\in {\bf Z}$, $j=0,1$, and every $\widehat{\phi} \in \widehat{\mathcal{H}} $.
Letting $$\widehat{U}_0 = \mathcal{F} U_0 \mathcal{F}^* = \mathcal{F} SC_0 \mathcal{F}^*,$$ we have that $\widehat{U}_0 $ is the operator of multiplication by the unitary matrix $$\widehat{U}_0 (\xi ) = \left[ \begin{array}{cc}
a_0 e^{i\xi} & b_0 e^{i\xi} \\ c_0 e^{-i\xi} & d_0 e^{-i\xi} \end{array} \right] .
\label{S2_eq_U0hat}$$ In view of (\[S1\_def\_C0\]), we have $$\widehat{U}_0 (\xi ) = e^{i\gamma /2} \left[ \begin{array}{cc}
p e^{i(\alpha - \gamma /2 )} e^{i\xi} & q e^{i(\beta - \gamma /2)} e^{i\xi} \\ -q e^{-i(\beta -\gamma /2)} e^{-i\xi} & p e^{-i( \alpha - \gamma /2 )} e^{-i\xi} \end{array} \right] .
\label{S2_eq_U0hat2}$$ Moreover, we obtain for any $ \lambda \in {\bf C}$ $$\mathrm{det} (\widehat{U}_0 (\xi ) -\lambda ) = \lambda^2 -2 \lambda pe^{i\gamma /2} \cos \left( \xi + \alpha - \frac{\gamma}{2} \right) + e^{i\gamma } .
\label{S2_eq_det}$$ In view of (\[S2\_eq\_det\]), we can see the following fact. For the proof, see Lemma 4.1 in [@RST].
\(1) If $p=0$, we have $ \sigma (U_0 ) = \sigma_p (U_0 ) = \{ \pm i e^{i\gamma /2} \} $.\
(2) If $p\in (0,1)$, we have $\sigma (U_0 )= \sigma_{ac} (U_0 ) = \{ e^{i \theta} \ ; \ \theta \in J_{\gamma} \}$.\
(3) If $p=1$, we have $\sigma (U_0 )= \sigma_{ac} (U_0 )= S^1 $. \[S2\_lem\_specU0\]
In view of the assumption (\[S1\_eq\_exp\]), the operator $U-U_0 $ is compact on $\mathcal{H} $. Applying Lemma \[S2\_lem\_weyltype\], we obtain the following lemma.
\(1) If $p\in (0,1) $, we have $\sigma_{ess} (U)=\sigma_{ess} (U_0 ) = \{ e^{i\theta} \ ; \ \theta \in J_{\gamma} \}$.\
(2) If $p=1$, we have $\sigma_{ess} (U)=\sigma_{ess} (U_0 ) = S^1 $.
\[S2\_lem\_essspecUU0\]
Absence of embedded eigenvalues
===============================
Thresholds
----------
Let $$\begin{gathered}
M ( \theta ) = \{ \xi \in {\bf T} \ ; \ p(\xi , \theta )=0 \} , \label{S3_def_fermi} \\
M _{reg} ( \theta ) = \{ \xi \in {\bf T} \ ; \ p(\xi , \theta )=0 , \partial _{\xi} p(\xi ,\theta ) \not= 0 \} , \label{S3_def_fermi_reg} \\
M_{sng} (\theta ) = \{ \xi \in {\bf T} \ ; \ p(\xi , \theta )=0 , \partial _{\xi} p (\xi , \theta )=0 \} , \label{S3_def_fermising}\end{gathered}$$ where $p(\xi ,\theta )= \mathrm{det} ( \widehat{U}_0 (\xi ) - e^{i\theta } )$. Note that $p(\xi ,\theta )$ is a trigonometric polynomial in $\xi $ (see (\[S2\_eq\_det\])).
Suppose $p\in (0,1]$. If $ \theta \in J_{\gamma} \setminus J_{\gamma ,\mathcal{T}} $, we have $M(\theta )= M_{reg} (\theta )$ and $M_{sng} (\theta )= \emptyset$. If $ \theta \in J_{\gamma ,\mathcal{T}} $, we have $M(\theta )= M_{sng} (\theta )$ and $M_{reg} (\theta )= \emptyset$. \[S3\_lem\_singular\]
Proof. Note that $$\partial_{\xi} p (\xi , \theta )= 2p e^{i \gamma /2} e^{i\theta} \sin \left( \xi + \alpha -\frac{\gamma}{2} \right) .$$ Then $ \partial_{\xi} p (\xi , \theta )=0$ if and only if $ \xi + \alpha - \gamma /2 =0$ modulo $\pi $. If $p(\xi , \theta )=\partial_{\xi} p (\xi , \theta ) =0$, we have that $e^{i\theta} $ must be equal to one of the following values : $$e^{i \gamma /2} \left( p \pm i \sqrt{ 1-p^2 } \right) , \quad e^{i \gamma /2} \left( - p \pm i \sqrt{ 1-p^2 } \right) .$$ The lemma follows from these observations.
Absence of embedded eigenvalues {#section_proofmain1}
-------------------------------
In §\[section\_proofmain1\], we prove Theorem \[S1\_mainthm1\]. For the proof, we suppose that there exists an eigenvalue in $ \sigma_p (U) \cap ( \sigma_{ess} (U) \setminus \mathcal{T} ) $ and we show a contradiction.
Let us recall the assumptions which we adopt in §\[section\_proofmain1\] :
1. $p \in (0,1]$ and there exists a constant $\delta >0$ such that $|a(x)| \geq \delta $ for all $x\in {\bf Z} $.
2. There exist constants $\rho ,M>0$ such that $\| C(x)-C_0 \| _{\infty} \leq Me^{-\rho \langle x\rangle }$ for any $x\in {\bf Z} $.
We assume $e^{i\theta} \in \sigma_p (U) \cap (\sigma_{ess} (U) \setminus \mathcal{T} )$ and let $\psi \in \mathcal{H}$ be the associated eigenfunction. Putting $f=-(U-U_0 )\psi \in \mathcal{H}$, the equation $(U-e^{i\theta } )\psi =0$ is rewritten as $$(U_0 -e^{i\theta} )\psi =f \quad \text{on} \quad {\bf Z} .$$ In view of the assumption (2), we have $e^{r \langle \cdot \rangle} f \in \mathcal{H}$ for any $r\in (0,\rho )$. Passing to the Fourier series, we have $$(\widehat{U}_0 (\xi )-e^{i\theta} ) \widehat{\psi} = \widehat{f} \quad \text{on} \quad {\bf T}.
\label{S3_eq_eigen_torus}$$ Moreover, we multiply the equation (\[S3\_eq\_eigen\_torus\]) by the cofactor matrix of $ \widehat{U}_0 (\xi )-e^{i\theta}$. Note that each component of the cofactor matrix is trigonometric polynomials. Then the matrix $ \widehat{U}_0 (\xi )-e^{i\theta}$ is diagonalized and it is sufficient to consider the equation of the form $$p(\xi , \theta ) \widehat{u} = \widehat{g} \quad \text{on} \quad {\bf T} ,
\label{S3_eq_eigen_torus2}$$ where $\widehat{u} , \widehat{g} \in L^2 ({\bf T})$.
Here we need a Paley-Wiener type theorem. The following one is Theorem 6.1 in [@Ves].
Let $k_0 >0$ be a constant. For a function $\phi \in \ell^2 ({\bf Z} )$, $e^{k\langle \cdot \rangle} \phi \in \ell^2 ({\bf Z} )$ for any $k\in (0,k_0 )$ if and only if the function $\widehat{\phi}$ extends to analytic function in $\{ z\in {\bf T}_{{\bf C}} \ ; \ | \mathrm{Im} \, z| < k_0 / (2\pi ) \}$. \[S3\_thm\_PW\]
As a direct consequence, we have the following fact.
The function $\widehat{g}$ in (\[S3\_eq\_eigen\_torus2\]) extends to an analytic function in $\{ z\in {\bf T}_{{\bf C}} \ ; \ | \mathrm{Im} \, z| < \rho / (2\pi ) \}$. \[S3\_lem\_PW\]
Proof. Since we have $e^{r \langle \cdot \rangle} f \in \mathcal{H}$ for any $r\in (0, \rho )$, we apply Theorem \[S3\_thm\_PW\] to $f$ so that $\widehat{f}$ is analytic in $\{ z\in {\bf T}_{{\bf C}} \ ; \ | \mathrm{Im} \, z| < \rho / (2\pi ) \}$. Each component of the cofactor matrix is trigonometric polynomials. Then $\widehat{g}$ is also analytic in $\{ z\in {\bf T}_{{\bf C}} \ ; \ | \mathrm{Im} \, z| < \rho / (2\pi ) \}$.
Next we discuss about the regularity of $\widehat{u}$.
Let $\widehat{u} \in L^2 ({\bf T})$ satisfy the equation (\[S3\_eq\_eigen\_torus2\]). Then $\widehat{u} \in C^{\infty} ({\bf T} )$. In particular, we have $\widehat{g} (\xi (\theta )) =0$ for $\xi (\theta )\in M(\theta )$.\
\[S3\_lem\_smoothness\]
Proof. We take $\xi (\theta ) \in M(\theta )$. Note that $M(\theta )= M_{reg} (\theta )$ from $e^{i\theta} \in \sigma_p (U) \cap ( \sigma_{ess} (U) \setminus \mathcal{T} )$. Let $\chi \in C^{\infty} ({\bf T} )$ satisfy $\chi (\xi (\theta )) =1$ with small support. In view of $\xi (\theta )\in M_{reg} (\theta )$, we have $ \partial _{\xi} p (\xi (\theta ),\theta ) \not= 0$. Thus we can make the change of variable $$\eta = \cos \left( \xi + \alpha - \frac{\gamma}{2} \right) - \cos \left( \xi (\theta ) + \alpha - \frac{\gamma}{2} \right) ,$$ in a small neighborhood of $\xi (\theta )$. Letting $ \widehat{u} _{\chi} = \chi \widehat{u} $ and $\widehat{g} _{\chi} = \chi \widehat{g} $, we rewrite the equation (\[S3\_eq\_eigen\_torus2\]) as $$\eta \widehat{u} _{\chi} = -\frac{1}{2p} e^{-i (\theta + \gamma /2)} \widehat{g} _{\chi} \quad \text{on} \quad {\bf T} .
\label{S3_eq_smoothness1}$$
Now let us define the Fourier transformation by $$\widetilde{u _{\chi}} (t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-it\eta} \widehat{u} _{\chi} (\eta ) d\eta , \quad t \in {\bf R} .$$ We define $\widetilde{g _{\chi}} (t)$ by the same way. Then the equation (\[S3\_eq\_smoothness1\]) is reduced to the differential equation $$\partial_t \widetilde{u_{\chi}} = \frac{i}{2p} e^{-i( \theta + \gamma /2)} \widetilde{g_{\chi}} .
\label{S3_eq_smoothness2}$$ Integrating this equation, we have $$\widetilde{u_{\chi} } (t)= \frac{i}{2p} e^{-i(\theta + \gamma /2)} \int_0^t \widetilde{g_{\chi}} (s) ds + \widetilde{u_{\chi} } (0) .$$ In view of Lemma \[S3\_lem\_PW\], $\widehat{g}_{\chi} $ is smooth. Hence $\widetilde{g_{\chi}} $ is rapidly decreasing at infinity. From $\widehat{u}_{\chi} \in L^2 ({\bf T} )$, we have $\widetilde{u_{\chi}} (t)\to 0$ as $|t| \to \infty $. Then the limit $$\lim _{t\to \infty} \widetilde{u_{\chi}} (t)= \frac{i}{2p} e^{-i (\theta + \gamma /2)} \int_0^{\infty} \widetilde{g_{\chi}} (s) \, ds + \widetilde{u_{\chi}} (0) ,$$ exists and we obtain $$\widetilde{u_{\chi}} (0)= - \frac{i}{2p} e^{-i (\theta +\gamma /2)} \int_0^{\infty} \widetilde{g_{\chi} } (s) \, ds .$$ Therefore, $\widetilde{u_{\chi}} $ is represented by the rapidly decreasing function $$\widetilde{u_{\chi}} (t)=- \frac{i}{2p} e^{-i ( \theta + \gamma /2)} \int_t^{\infty} \widetilde{g_{\chi}} (s) \, ds , \quad t \geq 0 .
\label{S3_eq_smoothness3}$$ Similarly, we have as $t\to - \infty$ $$\lim _{t\to -\infty} \widetilde{u_{\chi}} (t)=- \frac{i}{2p} e^{ -i ( \theta + \gamma /2)} \int_{-\infty}^0 \widetilde{g_{\chi}} (s) \, ds + \widetilde{u_{\chi}} (0) ,$$ and $$\widetilde{u_{\chi}} (0) = \frac{i}{2p} e^{-i (\theta + \gamma /2)} \int_{-\infty}^0 \widetilde{g_{\chi}} (s) \, ds .$$ Hence we obtain $$\widetilde{u_{\chi}} (t)= \frac{i}{2p} e^{-i ( \theta + \gamma /2)} \int^t_{-\infty} \widetilde{g_{\chi}} (s) \, ds , \quad t \leq 0 .
\label{S3_eq_smoothness4}$$ Then $\widetilde{u_{\chi}} (t)$ is rapidly decreasing as $|t| \to \infty $ and this implies that $\widehat{u}_{\chi} \in C^{\infty} ({\bf T} )$. Obviously, $\widehat{u}$ is smooth outside any small neighborhood of $\xi (\theta )$. Then we have $\widehat{u} \in C^{\infty} ({\bf T} )$. It follows from the equation (\[S3\_eq\_eigen\_torus2\]) that $\widehat{g} $ vanishes at $\xi (\theta )$.
The meromorphic function $ \widehat{g} (z) /p(z, \theta )$ is analytic in $\{ z\in {\bf T} _{{\bf C}} \ ; \ | \mathrm{Im} \, z| < \rho / (2\pi ) \} $.
\[S3\_lem\_uextention\]
Proof. If $ p(z,\theta )=0$ for $e^{i\theta} \in \sigma_{ess} (U)\setminus \mathcal{T} $, we have $$\cos \left( z+\alpha - \frac{\gamma}{2} \right) = \frac{1}{p} \cos \left( \theta -\frac{\gamma}{2} \right) .$$ This implies $\mathrm{Im} \, z =0$ if $p(z,\theta )=0$ for $e^{i \theta } \in \sigma_{ess} (U) \setminus \mathcal{T} $. Therefore, in order to show the analyticity of $\widehat{g} (z) /p(z,\theta )$, it is sufficient to consider a neighborhood of $\xi (\theta ) \in M(\theta )$. We expand $p(z, \theta )$ and $ \widehat{g} (z) $ into Taylor series at $\xi ( \theta ) \in M(\theta )$ : $$p(z,\theta )= \sum _{n=0}^{\infty} p_n (z-\xi (\theta ))^n , \quad \widehat{g} (z)= \sum _{n=0}^{\infty} g_n ( z-\xi (\theta ))^n ,$$ for $p_n , g_n \in {\bf C} $. In view of $M(\theta )= M_{reg} (\theta )$, we have $p_0 =0$ and $p_1 \not= 0$. Then Lemma \[S3\_lem\_smoothness\] implies $g_0 =0 $ and $\widehat{g} (z) / p(z, \theta )$ is analytic in a neighborhood of $\xi (\theta )$. The Lemma follows from Lemma \[S3\_lem\_PW\].
In the next step, we show that the eigenfunction $\psi$ decays super-exponentially as $|x| \to \infty $.
For any $k>0$, we have $e^{k \langle \cdot \rangle } \psi \in \mathcal{H} $. \[S3\_lem\_superexp\]
Proof. It follows from Lemma \[S3\_lem\_uextention\] that the function $$u(x ) := \frac{1}{\sqrt{2\pi}} \int _{{\bf T}} e^{ix\xi} \widehat{u} (\xi ) \, d\xi ,$$ satisfies $e^{r\langle \cdot \rangle} u \in \ell^2 ({\bf Z} )$ for $r\in (0,\rho )$ so that $e^{r \langle \cdot \rangle} \psi \in \mathcal{H} $. The assumption (2) implies that the function $f=(U-U_0 )\psi $ satisfies $e^{2r \langle \cdot \rangle } f \in \mathcal{H} $ for any $r\in (0,\rho )$. Repeating the arguments in the proofs of Lemmas \[S3\_lem\_PW\]-\[S3\_lem\_uextention\], we can see $ e^{2r \langle \cdot \rangle } \psi \in \mathcal{H} $. We can repeat this procedure any number of times. Therefore, we have $e^{mr \langle \cdot \rangle} \psi \in \mathcal{H}$ for any $m>0$.
*Proof of Theorem \[S1\_mainthm1\].* Plugging Lemmas \[S3\_lem\_PW\]-\[S3\_lem\_superexp\], the eigenfunction $\psi $ satisfies $e^{k\langle \cdot \rangle} \psi \in \mathcal{H}$ for any $k>0$. The equation $(U-e^{i\theta} )\psi =0$ is rewritten as $$\begin{gathered}
a(x+1) \psi ^{(0)} (x+1) + b(x+1) \psi^{(1)} (x+1)=e^{i\theta} \psi ^{(0)} (x) , \label{S3_eq_eigen11} \\
c(x-1) \psi ^{(0)} (x-1) + d(x-1) \psi ^{(1)} (x-1) = e^{i\theta} \psi ^{(1)} (x) . \label{S3_eq_eigen12} \end{gathered}$$ Recalling the assumptions (1) and (2), we put $$K_1 = \max \left\{ 1 , \, \sup _{x\in {\bf Z}} \| C(x)\| _{\infty} \right\} , \quad K_2 = \max \left\{ 1, \delta^{-1} \right\} .$$ From the equations (\[S3\_eq\_eigen11\]) and (\[S3\_eq\_eigen12\]), we have $$\begin{gathered}
\begin{split}
a(x) \psi ^{(0)} (x) = & \, \left( -e^{-i\theta} b(x) c(x-1) + e^{i\theta} \right) \psi ^{(0)} (x-1) \\
& \, - e^{-i\theta} b(x)d(x-1) \psi^{(1)} (x-1) ,
\end{split}\end{gathered}$$ and then $$| \psi ^{(0)} (x) | \leq 2 K_1^2 K_2 \left( | \psi ^{(0)} (x-1) | + | \psi ^{(1)} (x-1) | \right) .$$ Repeating the same estimate on the right-hand side, we can see for any $y >0$ that $$| \psi ^{(0)} (x) | \leq 2^{2y-1} K_1^{2y} K_2^y \left( | \psi ^{(0)} (x-y) | + | \psi ^{(1)} (x-y) | \right) .$$ In view of Lemma \[S3\_lem\_superexp\], we obtain $$| \psi ^{(0)} (x) | \leq 2^{2y} K_1^{2y} K_2^y e^{-k\langle x-y \rangle } ,$$ for any $k>0$. Taking a sufficiently large $k$ and tending $y\to \infty$, we see $|\psi^{ (0)} (x)|=0$. Since $x\in {\bf Z} $ is arbitrary, $\psi^{(0)} $ vanishes on ${\bf Z} $.
Let us go back the equation (\[S3\_eq\_eigen12\]). The equation is rewritten as $$d(x-1) \psi^{(1)} (x-1) = e^{i\theta} \psi ^{(0)} (x) ,$$ so that $$|\psi^{(1)} (x)| \leq K_1 |\psi^{(1)} (x-1)| \leq \cdots \leq K_1^y |\psi^{(1)} (x-y)| ,$$ for any $y>0$. Hence we also have $$|\psi^{(1)} (x)|\leq K_1^y e^{-k \langle x-y \rangle} ,$$ for any $k>0$. Taking a sufficiently large $k>0$ and tending $y\to \infty$, we obtain $\psi ^{(1)} (x)=0$ for any $x\in {\bf Z} $.
Existence of embedded eigenvalues
=================================
Finite support of eigenfunctions
--------------------------------
In this section, we turn to the coin operator $C$ given by (\[S1\_eq\_CC\]). Since $C(x) -C_0$ satisfies the assumption (\[S1\_eq\_exp\]), Lemma \[S2\_lem\_essspecUU0\] also holds for this case i.e. $\sigma_{ess} (U)= \sigma_{ac} (U_0 )$. The set of thresholds $\mathcal{T}$ is also defined by the same manner of Theorem \[S1\_mainthm1\]. Thus the assertion (1) of Theorem \[S1\_mainthm2\] holds. On the other hand, the assertion of Theorem \[S1\_mainthm1\] does not hold for this case. However, we can prove the assertion (3) of Therem \[S1\_mainthm2\] which is weaker than Theorem \[S1\_mainthm1\].
*Proof of (3) of Theorem \[S1\_mainthm2\].* We can apply Lemmas \[S3\_lem\_PW\]-\[S3\_lem\_superexp\] to $U$. Then we have $e^{k\langle \cdot \rangle} \psi \in \mathcal{H} $ for any $k>0$. Since we have $a(x)= p e^{i\alpha} \not= 0 $ for $x<x_*$, we can use the estimate which is derived in the proof of Theorem \[S1\_mainthm1\]. Then we have $\psi =0$ for $x<x_*$. In view of the equations (\[S3\_eq\_eigen11\]) and (\[S3\_eq\_eigen12\]), we have $$\begin{gathered}
\begin{split}
d(x) \psi ^{(1)} (x) = & \, -e^{i\theta} a(x+1) c(x) \psi^{(0)} (x+1) \\
& \, + \left( e^{i\theta} - e^{-i\theta} b(x+1)c(x) \right) \psi ^{(1)} (x+1) .
\end{split}\end{gathered}$$ Note that $d(x)= pe^{i\alpha} e^{i\gamma} \not= 0$ for $x>x^*$. Then we have $$| \psi^{(1)} (x) | \leq 2^{2y-1} K_1^{2y} K_2^y e^{-k \langle x+y \rangle} ,$$ for any large $k>0$ and $y>0$. We obtain $\psi ^{(0)} (x)=0$ for $x>x^*$ tending $y\to \infty $. From the equation (\[S3\_eq\_eigen11\]), we have $$|\psi ^{(0)} (x)|\leq K_1^y | \psi ^{(0)} (x+y)| \leq K_1^y e^{-k \langle x+y\rangle} ,$$ for any large $k>0$ and $y>0$. Hence we also obtain $\psi^{(1)} (x)=0$ for $x>x^*$ tending $y \to \infty $.
Embedded eigenvalues
--------------------
In order to construct eigenfunctions precisely, we consider the auxiliary operator $U_1 = SC_1$. Note that $\sigma (U_1 )= \sigma_p (U_1 )= \{ \pm i e^{i\gamma '/2} \} $ (see Lemma \[S2\_lem\_specU0\]).
Let $\delta (x) = \delta _{x0}$ for $x\in {\bf Z}$. Then the function $$\psi _{\pm} (x)= \frac{1}{\sqrt{2} } \left[ \begin{array}{c} \mp i e^{i( \beta ' -\gamma '/2 )} \delta (x+1) \\ \delta (x) \end{array} \right] , \quad \beta ' , \gamma ' \in [0,2\pi ),
\label{S4_eq_eigenfunction}$$ are normalized eigenfunctions of $U_1$ with eigenvalues $\pm ie^{i\gamma '/2}$, respectively. \[S4\_lem\_supportef\]
Proof. The equation $(U_1 - (\pm ie^{i\gamma '/2} )) \psi _{\pm} =0$ is equivalent to $$\left[ \begin{array}{cc}
\mp i e^{i\gamma '/2} & e^{i \beta '} e^{i\xi} \\ -e^{-i\beta '} e^{i\gamma '} e^{-i\xi} & \mp i e^{i\gamma '/2} \end{array} \right] \left[ \begin{array}{c} \widehat{\psi} _{\pm}^{(0)} (\xi ) \\ \widehat{ \psi }_{\pm}^{(1)} (\xi ) \end{array} \right] =0 , \quad \xi \in {\bf T} .$$ By a direct computation, we have $$\left[ \begin{array}{c} \widehat{\psi} _{\pm}^{(0)} (\xi ) \\ \widehat{ \psi }_{\pm}^{(1)} (\xi ) \end{array} \right] = s(\xi ) \left[ \begin{array}{c}
\mp i e^{ i(\beta ' -\gamma '/2)} e^{i\xi} \\ 1 \end{array} \right] ,$$ for any scalar functions $s(\xi )$. Taking $s(\xi )= (2 \sqrt{\pi} )^{-1}$, we obtain the lemma.
The operator of translation $T_y$ for $y\in {\bf Z}$ is defined by $$(T_y \psi )(x)= \psi (x-y) , \quad x\in {\bf Z} ,
\label{S4_def_translation}$$ for $\psi \in \mathcal{H}$. Obviously, $T_y \psi _{\pm} $ are also eigenfunctions of $U_1$ with eigenvalues $\pm i e^{i\gamma '/2} $, respectively. Moreover, we have $\mathrm{supp} T_y \psi^{(0)} _{\pm} = \{ y-1 \}$ and $ \mathrm{supp} T_y \psi^{(1)} _{\pm} = \{ y \} $.
*Proof of (2) of Theorem \[S1\_mainthm2\].* We put $$\Psi _{\pm} = \kappa_1 T_{y_1} \psi _{\pm} + \cdots + \kappa_N T_{y_N} \psi _{\pm } ,$$ for any $ \kappa_1 , \cdots , \kappa_N \in {\bf C} $, where $\psi _{\pm} $ is given by (\[S4\_eq\_eigenfunction\]). Then we have $ \mathrm{supp} \Psi^{(0)}_{\pm} = \{ y_1 -1 , \cdots , y_N -1 \} $ and $\mathrm{supp} \Psi_{\pm}^{(1)} =\{ y_1 , \cdots , y_N \} $. Since we have $(F_{{\bf e}_{y_j}} C) \big| _{{\bf e}_{y_j}} = C_1 $ for each $j=1,\cdots ,N$, $\Psi_{\pm} $ satisfies the equation $U\Psi_{\pm} = \pm i e^{i\gamma '/2} \Psi _{\pm} $. Then $ \pm i e^{i\gamma '/2} \in \sigma_p (U)$ for any $\gamma ' \in [0,2\pi )$.
In view of the assertion (3) of Theorem \[S1\_mainthm2\], if $ \pm i e^{i\gamma '/2} \in \sigma_p (U) \cap (\sigma_{ess} (U)\setminus \mathcal{T} )$, associated eigenfunctions vanish for $x>x^* $ and $x<x_*$.
Summary and discussion
======================
Finally, we summarize our results of the present paper as a conclusive remark by using typical numerical examples. We consider two typical cases. We put $ {\bf e} = {\bf e}_0 \cup {\bf e}_1 = \{ -1 ,0,1 \} $. Let $U_v = SC_v$ and $U_e = SC_e$ be defined by $$\begin{gathered}
C_v = F_{{\bf e}} \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] + (1-F_{{\bf e}} ) \left[ \begin{array}{cc} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{array} \right] , \label{S5_def_cv} \\
C_e = F_{{\bf e}} \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right] + (1-F_{{\bf e}} ) \left[ \begin{array}{cc} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{array} \right] . \label{S5_def_ce}\end{gathered}$$ For $U_e$, ${\bf e}_0 $ and ${\bf e}_1 $ are edge defects. On the other hand, $U_v$ does not have edge defects but are perturbed on ${\bf e}$. From Lemma \[S2\_lem\_essspecUU0\], we have $\sigma_{ess} (U_v)= \sigma_{ess} (U_e) = \{ e^{i\theta} \ ; \ \theta \in J \}$ with $$J= [ \pi /4 , 3\pi /4] \cup [ 5\pi /4 , 7\pi /4 ] .$$
Taking the initial state $ \psi_0$ given by $$\psi_0 (x) = \left[ \begin{array}{c} 1/ \sqrt{6} \\ i/\sqrt{6} \end{array} \right] , \ x\in {\bf e} , \quad \psi _0 \big| _{{\bf Z} \setminus {\bf e}} =0 ,$$ we put $\psi_v (t,\cdot ):=U_v^t \psi_0 $ and $\psi_e (t,\cdot ) := U_e^t \psi_0 $ for $t \geq 0$. Then we compute the probability $P_* (X_t = x)=| \psi_* (t,x)|^2 $ where $*=v$ or $e$ and $X_t$ is the position of the quantum walker at time $t$. For the numerical results at $t=100$, see Figures \[fig\_distvertex\] and \[fig\_distedge\]. Localization occurs near $x=0$ for both of $P_v (X_t =x)$ and $P_e (X_t =x)$. Here localization means $ \limsup _{t\to \infty} P_* (X_t =x) >0$ for some $x\in {\bf Z} $. Thus we cannot detect edge defects by the existence of localization.
![The distribution of $P_e ( X_t = x)$ at $t=100$.[]{data-label="fig_distedge"}](Dist_vertex.png){width="60mm"}
![The distribution of $P_e ( X_t = x)$ at $t=100$.[]{data-label="fig_distedge"}](Dist_edge.png){width="60mm"}
![The distribution of $\sigma (U_e) $.[]{data-label="fig_edge"}](Spec_vertex.png){width="55mm"}
![The distribution of $\sigma (U_e) $.[]{data-label="fig_edge"}](Spec_edge.png){width="55mm"}
If the initial state $ \psi_0 $ has an overlap with an eigenvector of $U_*$, then localization occurs (see [@SS]). For the locations of $\sigma (U_v )$ and $ \sigma ( U_e )$, see Figures \[fig\_vertex\] and \[fig\_edge\]. $ \sigma _{ess} (U_* )$ is approximated by eigenvalues of the finite rank operator $ U _* \big |_{\{ -60 \leq x \leq 60 \} } $. The operator $U_v$ has discrete eigenvalues. On the other hand, $U_e$ has eigenvalues $ \pm i$ which are embedded in the interior of $ \sigma_{ess} (U_e )$. Localizations of $U_v$ and $U_e$ occur due to eigenvectors of these eigenvalues. Thus the existence of edge defects is distinguished by the location of eigenvalues. Precisely, if there exist eigenvalues embedded in the interior of the continuous spectrum, there are some edge defects.
These examples are typical situations to which our main results are applicable (see Theorems \[S1\_mainthm1\] and \[S1\_mainthm2\] and Corollary \[S1\_cor\_main\]).
[99]{}
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|
---
abstract: '[We present an empirical study of the first passage time (${\mathrm{FPT}}$) of order book prices needed to observe a prescribed price change $\Delta$, the time to fill (${\mathrm{TTF}}$) for executed limit orders and the time to cancel (${\mathrm{TTC}}$) for canceled ones in a double auction market. We find that the distribution of all three quantities decays asymptotically as a power law, but that of ${\mathrm{FPT}}$ has significantly fatter tails than that of ${\mathrm{TTF}}$. Thus a simple first passage time model cannot account for the observed ${\mathrm{TTF}}$ of limit orders. We propose that the origin of this difference is the presence of cancellations. We outline a simple model, which assumes that prices are characterized by the empirically observed distribution of the first passage time and orders are canceled randomly with lifetimes that are asymptotically power law distributed with an exponent $\lambda_{\mathrm{LT}}$. In spite of the simplifying assumptions of the model, the inclusion of cancellations is enough to account for the above observations and enables one to estimate characteristics of the cancellation strategies from empirical data.]{}'
author:
- Zoltán Eisler
- János Kertész
- Fabrizio Lillo
- 'Rosario N. Mantegna'
bibliography:
- 'execution9.bib'
title: Diffusive behavior and the modeling of characteristic times in limit order executions
---
Introduction
============
Understanding the market microstructure is crucial for both theoretical and practical purposes [@biais.jfm2005]. On double auction markets the limit order book contains most of the information about the market microstructure and price discovery. Recently there has been considerable effort to investigate limit order book dynamics. Empirical studies [@biais.jf1995; @handa.jf1996; @harris.jfqa1996; @kavajecz.jf1999; @sandas.rfs2001; @maslov.pa2001; @challet.pa2001; @lo.limitorder; @potters.pa2003; @hollifield.res2004; @farmer.qf2004; @zovko.farmer; @mike.empirical2; @weber.qf2005; @hollifield.gains; @ponzi.ph2006] have been devoted to the search for the key determinants of price formation, the trading process and market organization. A large number of papers have focused on modeling the limit order book with [@parlour.rfs1998; @daniels.prl2003; @foucault.flow; @foucault.liquidity; @rosu.dynamic] or without [@glosten.jf1004; @chakravarty.jfi1995; @seppi.rfs1997; @luckock.qf2003] dynamics. Market microstructure studies consider a large number of aspects of the price discovery mechanism and these studies can greatly contribute to the success of the modeling of financial markets. The market mechanism, along with the complex interactions among market participants results in the emergence of a collective action of continuous price formation. Some of the studies have used an agent based modeling approach. Examples are market models described in terms of agents interacting through an order book based on simple rules [@chiarella.iori; @licalzi.qf2003] and models where the assumptions about the trading strategies are kept as minimal as possible [@daniels.prl2003; @mike.empirical2]. One of the most striking findings was that even if trends and investor strategies are neglected, purely random trading may be adequate to describe certain basic properties of the order book [@zovko.farmer].
Most of the above papers focus on limit order executions, and very few deal with cancellations, even though the frequency of the two outcomes is comparable [@lo.limitorder]. The uncertainty of execution represents a primary source of risk [@chakrabarty.competing]. Another major risk factor is adverse selection, also known as “pick-off” risk. This risk is associated with the waiting time until order execution. During this period those with excess information can take advantage of the liquidity provided by the limit orders of less informed traders, and hence it is important to accurately quantify these waiting times. Lo et al. [@lo.limitorder] apply survival analysis to limit order data, and they find that the time between order placement and execution is very sensitive to the limit price, but not to the volume of the order. They also investigate the dependence on further explanatory variables such as the bid-ask spread and the volatility. The dynamics of the limit order book has also been investigated by using a joint model of executions and cancelations in a framework of competing risks[^1]. Within this approach Hollifield et al. [@hollifield.gains], by using observations on order submissions and execution and cancellation histories, estimate both the distribution of traders’ unobserved valuations for the stock and latent trader arrival rates. Chakrabarty et al. [@chakrabarty.competing] show that executions are more sensitive to price variation and less to volume variation than cancellations. This last work also analyzes the relationship between execution time and market depth.
In this paper we aim to go a step further, and combine the framework of competing risks with random walk theory. In particular, we analyze the difference observed between the time to fill a limit order, which is the time one had to wait before a limit order was executed, and the first passage time [@feller], i.e., the time elapsed between an initial instant and the time when the transaction price crosses a given predefined threshold. In addition, the largest difference between our approach and most previous studies (e.g., Refs. [@lo.limitorder; @chakrabarty.competing]) is that while those placed more emphasis on the typical values of execution and cancellation times, we will concentrate on the accurate description of the rare events, and the related asymptotic tail behavior of the distributions.
We observe that for a fixed price change the first passage time distributions of transaction price, best bid and best ask are quite well described asymptotically by the theoretical form expected for a Markov process with symmetric jump length distribution (including Brownian motion) [@feller; @chechkin.JPA2003]. The empirical time to fill of executed orders is smaller than the first passage time. We attribute this difference to canceled and expired orders. We propose a simple competing risks model, where limit orders are removed from the order book when either of two events happens: (i) when they are executed, this is modeled as the first time when the transaction price reaches the limit price, (ii) or when they are canceled, the time horizon of cancellations is modeled as a random process that is independent from price changes. In this framework we are able to predict constraints about the tail behavior of the time to fill and time to cancel probability densities. Our model also allows us to estimate the distribution of the time horizons of the placed limit orders. We show that the assumption of independence between the price changes and order cancellations, while it is a large simplification compared to real data, does not affect our conclusions significantly.
The paper is organized as follows. In Section \[sec:book\] we describe the investigated market and the variables of interest. In Section \[sec:fpt\] we study the first passage time and in Section \[sec:ttfttc\] the time to fill and the time to cancel. Section \[sec:model\] describes a simple limit order model and Section \[sec:predictions\] is devoted to testing the model empirically. Section \[sec:moredelta\] extends the result to limit orders placed inside the spread. Section \[sec:conclusions\] discusses the validity of the assumptions and summarizes the results. [Finally, in the Appendices we show that the results are unchanged if time is measured in transactions.]{} Then we present a critical discussion of the fitting procedure we used to estimate the tail bahavior of the time to fill and time to cancel distributions.
The dataset {#sec:book}
===========
The empirical analysis presented in this study is based on the trading data of the electronic market (SETS) of London Stock Exchange (LSE) during the year $2002$. These data can be purchased directly from the London Stock Exchange. We investigate $5$ highly liquid stocks, AstraZeneca (AZN), GlaxoSmithKline (GSK), Lloyds TSB Group (LLOY), Shell (SHEL), and Vodafone (VOD). Opening times of LSE are divided into three periods. The intervals 7:50–8:00 and 16:30–16:35 are called the opening and the closing auction, respectively. These follow different rules and thus also observe different statistical properties than the rest of the trading. Therefore we discarded limit orders placed during these times, and focused only on the periods of continuous double auction during 8:00–16:30. We also removed limit orders that were placed during 8:00–16:30 but were canceled (or expired) during the opening/closing auctions. We measure time intervals in trading time, i.e., we discard the time between the closing and the opening of the next day.[^2] Finally, whenever we refer to prices we exclude all transactions that were executed on the SEAQ market[^3] and not in the limit order book.
We denote the best bid price[^4] by $b(t)$, the best ask price by $a(t)$ and the bid-ask spread is $s(t)=a(t)-b(t)$. Except for very special cases, there are already other limit orders waiting inside the book when one wants to place a new one. Let $b(t)-\Delta$ denote the price of a new buy limit order, and $a(t)+\Delta$ the price of a new sell limit order. Orders placed exactly at the existing best price correspond to $\Delta = 0$, orders placed inside the spread have $\Delta < 0$, while $\Delta > 0$ means orders placed “inside the book”. It is possible to have so called crossing orders with such large negative values of $\Delta$ that they cross the spread, i.e., $\Delta < b(t)-a(t)$. These orders can be partially or fully executed immediately by limit orders from the other side of the book. Since a trader would place a crossing limit order to execute (at least part of) it immediately, we will not consider them as limit orders in our analysis.
Any limit order which was not executed can be canceled at any time by the trader who placed it. The order can also have a predetermined validity after which it is automatically removed from the book, this is called expiry. We will not distinguish between these mechanisms and we will call both of them cancellation. Throughout the paper we will use ticks as units of price and all logarithms are $10$-base.
The first passage time {#sec:fpt}
======================
Let the latest transaction price of an asset at time $t_0=0$ be $S_0$. The first passage time [@feller] of price through a prescribed level $S_0+\Delta$ with some fixed $\Delta > 0$ is defined as the time $t$ of the first transaction when $S(t)\geq S_0+\Delta$. Similarly we can determine the first time after $t_0=0$ when the transaction price was below or equal to $S_0-\Delta$ and we will consider this time as another, independent observation of $t$. We will call the distribution of the quantity $t$ the first passage time distribution to a distance $\Delta$, and denote it by $P_{\mathrm{FPT};\Delta}(t)$.
Such first passage processes have been studied extensively [@redner]. For simplicity we will restrict ourselves to driftless processes. This is justified, because in real data for time horizons $t$ of up to a day the drift of the prices is negligible. This means that the ratio $|\mu|\sqrt{t}/\sigma$ is small (it is always less than $10^{-1}$ in our dataset), where $\mu$ is the mean price change over unit time, and $\sigma$ is the standard deviation of price changes during a unit time (i.e., the volatility). Throughout the paper we use real time[^5].
For the following analysis of empirical data, it is useful to review the first passage time distribution for Brownian motion without drift. This is can be written as [@feller] $$P_{\mathrm{FPT};\Delta}(t) = \frac{\Delta}{\sqrt{2\pi\sigma^2}}t^{-3/2}
\exp\left(-\frac{\Delta^2}{2\sigma^2t}\right),
\label{eq:fpt_brownian_full}$$ which is the fully asymmetric $1/2$-stable distribution. For any fixed $\Delta$ the asymptotics for long times is $$P_{\mathrm{FPT};\Delta}(t)\propto t^{-3/2}.
\label{eq:fpt_brownian}$$
A recent study [@chechkin.JPA2003] has clarified that this asymptotic behavior is valid not only for Brownian motion but also for any Markov process with symmetric jump length distribution.[^6] Of course, real price changes are not described by continuous values, and transactions and order submissions are also separated by finite waiting times, which a continuous time random walk formalism could take into account [@scalas.minireview; @Montero]. However, in this paper we are interested in time intervals much longer than these waiting times, so the discrete aspects of the dynamics are negligible. Thus, we will model prices as if they varied continuously in time.
Let us now investigate empirically the first passage time behavior. The first passage time distribution for the transaction price, bid and ask when $\Delta = 1$ tick is shown in Fig. \[fig:Lfpt\] for the stock GSK. The distribution is obtained by sampling the first passage time at each second. One can see that there are no significant differences in the behavior of the three prices. Qualitatively, the distribution is similar to Eq. , and the long time asymptotic of real data seems to decay approximately as $t^{-3/2}$. For times shorter than $1$ minute the curves significantly deviate both from the power law behavior and from the prediction of Eq. . We choose to fit the first passage time distribution with the function $$P_{\mathrm{FPT};\Delta}(t) = \frac{Ct^{-\lambda_\mathrm{FPT}}}
{1+[t/T_\mathrm{FPT}(\Delta)]^{-\lambda_\mathrm{FPT}+\lambda'_\mathrm{FPT}}}.
\label{eq:pfptfit}$$
This form, that we will use to fit also the other distributions introduced below, is characterized by two power law regimes. Normalization conditions of Eq. imply that $\lambda_\mathrm{FPT}>1$ and $\lambda'_\mathrm{FPT}<1$. For $t\ll T_\mathrm{FPT}(\Delta)$ it is $P_{\mathrm{FPT};\Delta}(t)\propto t^{-\lambda'_\mathrm{FPT}}$, whereas for $t\gg
T_\mathrm{FPT}(\Delta)$ it is $P_{\mathrm{FPT};\Delta}(t)\propto
t^{-\lambda_\mathrm{FPT}}$. We will discuss the motivations for choosing this form in Section IV and in the Appendix.
Table \[tab:fpt\] contains the fitted parameters $\lambda_\mathrm{FPT}$, $\lambda'_\mathrm{FPT}$, and $T_\mathrm{FPT}(\Delta)$ for $\Delta=1,\dots,4$ ticks. The difference between the actual values of $\lambda_\mathrm{FPT}$ and $3/2$ from Eq. is small. Systematic deviations due to clustered volatility or the fluctuations of trading activity could not be identified. [For example, the asymptotic shape of the distribution does not change, even if time is measured in transactions instead of seconds (see Appendix \[app:ttime\]).]{}
The observation that $\lambda_\mathrm{FPT} < 2$ implies that the theoretical mean and standard deviation of the first passage time distribution are infinite. Thus one should be careful with the interpretation of means calculated from finite samples. Throughout the paper we will rely on the determination of quantiles (e.g., the median) instead, which are always well-defined regardless of the shape of the distribution.
The inset of Fig. \[fig:Lfpt\] shows the median first passage time as a function of $\Delta$ for the five investigated stocks. The behavior is not exactly quadratic ($\Delta^2$) as one would expect from Eq. . If prices followed a Brownian motion, the $q$-th quantile (${T_q}$) of the first passage time distribution would be $$T_q=\frac{\Delta^2}{2\sigma^2[\mathrm{erfc}^{-1}(q)]},
\label{eq:fpttq}$$ where the median ($\med{{\mathrm{FPT}}}$) corresponds to $q=0.5$. In reality, the power law behavior with $\Delta$ is less evident, as shown by the inset of Fig. \[fig:Lfpt\]. Assuming a behavior $\med{{\mathrm{FPT}}}\propto \Delta^{\eta}$ would require an exponent varying between $1.5$ and $1.8$ depending on the specific stock and the precise range of $\Delta$ used for the estimation of $\eta$. A similar deviation from the prediction of Brownian motion was reported in Ref. [@simonsen.optimal] in the analysis of closure index values sampled at a daily time horizon.
There are many differences between real prices and Brownian motion, and the above non-quadratic behavior can come from any of them: the non-Gaussian distribution of returns, the superdiffusivity of price, perhaps both or none. We have performed a series of shuffling experiments and preliminary results support the conclusion that the main role is played by the deviation from Gaussianity. This non-Gaussianity is well documented in the literature down to the scale of single transactions [@farmer.qf2004]. A similar effect was seen for Levy flights, whose increments are also very broadly distributed, and their value of $\eta$ can be different from $2$, and it is related to the index of the corresponding Levy distribution [@seshadri.pnas1982].
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ --------
stock
$\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$
AZN $1.50$ $0.14$ $58$ $1.50$ $0.22$ $140$ $1.50$ $0.18$ $240$ $1.49$ $0.11$ $350$
GSK $1.52$ $0.16$ $62$ $1.52$ $0.18$ $230$ $1.50$ $-0.02$ $390$ $1.48$ $-0.21$ $520$
LLOY $1.54$ $0.22$ $85$ $1.55$ $0.20$ $280$ $1.53$ $0.01$ $460$ $1.51$ $-0.12$ $630$
SHEL $1.52$ $0.20$ $83$ $1.53$ $0.27$ $160$ $1.51$ $0.02$ $360$ $1.51$ $0.00$ $450$
VOD $1.57$ $0.43$ $150$ $1.54$ $-0.19$ $450$ $1.49$ $-0.69$ $720$ $1.51$ $-0.66$ $1500$
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ --------
![First passage time distributions for the price, bid and ask quotes of GlaxoSmithKline (GSK), distance $\Delta = 1$ tick. The dotted line is the first passage time distribution for Brownian motion with volatility $\sigma=1/7$ penny$\times$ sec$^{-1/2}$. The thick solid line is a fit with Eq. as given in Table \[tab:fpt\]. The inset shows the median first passage time as a function of $\Delta$.[]{data-label="fig:Lfpt"}](Lfpt){width="240pt"}
Time to fill, time to cancel {#sec:ttfttc}
============================
For an executed order the time elapsed between its placement and its complete execution is called *time to fill*. Orders are often not executed in a single transaction, thus one can also define *time to first fill*, which is the time from order placement to the first transaction this order participates in. Finally, for canceled orders one can define the *time to cancel* which is the time between order placement and cancellation. The distribution of these three quantities will be in the following denoted by $ P_\mathrm{TTF}(t)$, $P_\mathrm{TTFF}(t)$, and $ P_\mathrm{TTC}(t)$, respectively.
Properties of the distributions
-------------------------------
As a first characteristic of the order book, we investigate the distribution of time to fill and time to cancel for the stocks in our dataset. Fig. \[fig:GttfGSK\] shows these distributions for GlaxoSmithKline (GSK) for different values of $\Delta$. Similarly to the first passage time, we fitted the empirical density with the function $$P_{\mathrm{TTF};\Delta}(t) =
\frac{C't^{-\lambda_\mathrm{TTF}}}{1+[t/T_\mathrm{TTF}(\Delta)]^{-\lambda_\mathrm{TTF}+
\lambda'_\mathrm{TTF}}}.
\label{eq:pttffit}$$
This form , which we used to fit the FPT in the previous Section, is different from the more familiar generalized Gamma distribution used in Ref. [@lo.limitorder]. The reason for our choice is that we concentrate on the tail behavior of time distributions. According to our measurements the FPT, TTF and TTC distributions have fat tails, which can be well described by power laws. The generalized Gamma function has too slow convergence to a power law to describe the observed tails in the time range of our investigations. A detailed discussion of this problem is provided in Appendix \[app:fit\].
We also emphasize that in the present study we do not intend to discuss in detail the behavior on short time scales. We assume that this regime is simply characterized by the exponent $\lambda'$ only to perform a quick and efficient fit. This choice will have no direct relevance to our main conclusions, which always apply to the tails of the distribution.
Nevertheless, in addition to the very good fit at large times the above formula gives for some cases an overall good description also at short times. Table \[tab:ttf\] shows the results for all five stocks. We find that $\lambda_\mathrm{TTF}$, which gives the asymptotic behavior of the distribution, ranges between $1.8$ and $2.2$ for up to $\Delta = 4$ ticks. This is greater than the value Ref. [@challet.pa2001] found for NASDAQ. The exponent $\lambda'_\mathrm{TTF}$ varies between $-0.4$ and $0.6$. Finally $T_\mathrm{TTF}$ typically grows with $\Delta$, as orders placed deeper into the book are executed later. We will return to this observation in Section \[sec:delta\]. For $\Delta > 4$ the small number of limit orders in our sample does not allow us to make reliable estimates for the shape of the distribution. Fig. \[fig:GttfGSK\] also gives a comparison of four further stocks (AZN, LLOY, SHEL and VOD) to show that our findings are quite general. The distribution of time to first fill is indistinguishable from time to fill.
For time to cancel one finds a similarly robust behavior, also shown in Fig. \[fig:GttfGSK\]. Its distribution is again well fitted by the form $$P_{\mathrm{TTC};\Delta}(t) =
\frac{C''t^{-\lambda_\mathrm{TTC}}}{1+[t/T_\mathrm{TTC}(\Delta)]^{-\lambda_\mathrm{TTC}+
\lambda'_\mathrm{TTC}}},
\label{eq:pttcfit}$$ where the long time asymptotics has an exponent $\lambda_\mathrm{TTC}$ ranging between $1.9$ and $2.4$. Unlike the case of $\lambda_\mathrm{TTF}$, the measured values of of $\lambda_\mathrm{TTC}$ are in agreement with those measured in Ref. [@challet.pa2001] for NASDAQ. All results concerning the time to cancel are given in Table \[tab:ttc\].
[As for the FPT, for both TTF and TTC the asymptotic power law behavior and the value of exponents is preserved if time is measured in transactions, see Appendix \[app:ttime\].]{}
{width="220pt"}{width="220pt"}
{width="220pt"}{width="220pt"}
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
stock
$\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$
AZN $2.0$ $-0.0$ $65$ $1.9$ $0.0$ $100$ $1.8$ $-0.0$ $120$ $1.9$ $0.0$ $200$
GSK $1.9$ $-0.2$ $68$ $1.9$ $-0.2$ $150$ $1.8$ $-0.4$ $190$ $1.8$ $-0.3$ $320$
LLOY $2.0$ $-0.1$ $85$ $1.9$ $-0.1$ $160$ $1.9$ $-0.2$ $240$ $1.9$ $-0.2$ $350$
SHEL $1.9$ $-0.1$ $77$ $1.9$ $-0.2$ $110$ $1.9$ $0.0$ $270$ $1.8$ $-0.1$ $250$
VOD $1.8$ $-0.4$ $190$ $1.8$ $-0.5$ $490$ $1.8$ $-0.4$ $980$ – – –
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
------- ----------- ------------ ------- ----------- ------------ -------- ----------- ------------ -------- ----------- ------------ --------
stock
$\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$
AZN $2.2$ $0.6$ $87$ $2.2$ $0.6$ $90$ $2.2$ $0.6$ $85$ $2.2$ $0.7$ $100$
GSK $2.2$ $0.5$ $110$ $2.0$ $0.5$ $90$ $1.9$ $0.5$ $94$ $1.9$ $0.6$ $170$
LLOY $2.3$ $0.5$ $130$ $2.2$ $0.4$ $140$ $2.0$ $0.4$ $120$ $2.0$ $0.5$ $250$
SHEL $2.4$ $1.1$ $150$ $2.3$ $1.1$ $140$ $2.3$ $1.1$ $68$ $2.2$ $1.0$ $56$
VOD $2.0$ $0.9$ $300$ $2.1$ $0.8$ $1000$ $2.2$ $0.6$ $1500$ $1.9$ $0.5$ $1000$
------- ----------- ------------ ------- ----------- ------------ -------- ----------- ------------ -------- ----------- ------------ --------
Comparison of characteristic times
----------------------------------
The emphasis of this paper is on the interplay between order execution, order cancellation and the first passage properties of price. To understand this relationship, consider the following argument proposed in Ref. [@lo.limitorder]. Imagine that there are no cancellations. Let a buy order be placed at the price $b_0-\Delta$, when the current best bid is at $b_0$ (the argument goes similarly for sell orders). How much time does it take until this order is executed? It is certain that the order cannot be executed before the best bid decreases to $b_0-\Delta$, because until then there will always be more favorable offers in the book. On the other hand, once the price decreases to $b_0-\Delta-\epsilon$ where $\epsilon$ is the tick size of the stock, it is certain, that all possible offers at the price $b_0-\Delta$ have been exhausted, including ours. Therefore both time to fill and time to first fill for any order placed at a distance $\Delta$ from the best offer is greater than the first passage time of price to a distance $\Delta$, and less than that to $\Delta+\epsilon$. Since this is true for every individual order, one expects the following inequality for the distribution functions of characteristic times: $$\begin{aligned}
\int_0^t
P_{\mathrm{FPT};\Delta}(t')dt' \geq \int_0^t P_{\mathrm{TTF};\Delta}(t')dt'
\geq \nonumber \\ \int_0^t P_{\mathrm{TTFF};\Delta}(t')dt' \geq \int_0^t
P_{\mathrm{FPT};\Delta+\epsilon}(t')dt'.\end{aligned}$$ Using the empirical distributions above, a straightforward calculation yields $$\lambda_\mathrm{FPT}=\lambda_\mathrm{TTFF}=\lambda_\mathrm{TTF},
\label{eq:lambdawrong}$$ which is in clear disagreement with the data, where pronouncedly $\lambda_\mathrm{FPT}<\lambda_\mathrm{TTF}\approx
\lambda_\mathrm{TTFF}$. This inequality for the tail exponents means that one finds less orders with very long time to (first) fill than expected. The resolution of this apparent contradiction is that cancellations have to be taken into account: Orders which would have to wait too long before being executed are often canceled and thus removed from the statistic. The measurement of the cancellation time distribution suffers from the same bias. The observed distribution of time to cancel does not characterize how traders would actually cancel their orders, because here the executed orders are missing from the statistics.
In Section \[sec:model\] we will present a simple model that gives insight into the features pointed out so far. However, before doing so, we would like to present one further point concerning the empirical data.
The role of entry depth {#sec:delta}
-----------------------
How do order execution times change as a function of the entry depth $\Delta$? Similarly to first passage times, the empirical distributions found for time to fill/cancel have a slowly decaying tail such that the means might diverge. Therefore, in the following we will use the medians of all quantities as a measure of their typical value.
In Fig. \[fig:GlifedistGSK\] we show that the median of time to fill is empirically well described by $$\med{{\mathrm{TTF}}} \propto \Delta^{1.4},$$ which is quite different from the $\med{{\mathrm{TTF}}} \propto \Delta^2$ expected naively from Eq. and a Brownian motion assumption, and also from the $\Delta^{1.5-1.8}$ behavior observed for the first passage time. We show that cancellations play an important role in these discrepancies.
Let us make a surrogate experiment with the data of the stock GSK. We select all filled orders, and from the time of their placement we calculate the [*first*]{} time when the transaction price becomes equal to or better than the price of the order. If one plots the median of this quantity versus the $\Delta$ of the orders, the resulting curve is indistinguishable from the median of time to fill \[Fig. \[fig:GlifedistGSK\](left), curve labeled as “TTF/FPT filled ord”\]. Thus the exponent $1.4$ does not come from the difference between order executions and first passage times.
In another surrogate experiment we keep the time of order placements, but shuffle the $\Delta$ values between orders. This way we destroy correlations between volatility and order placement. We record the corresponding first passage times. The resulting curve is labeled as “FPT shuff. all ord”. This new curve now agrees with the first passage time of price \[curve “FPT, price (book only)”\] when $\Delta > 8$ ticks, which corresponds to a median time of about $1-2$ hours. The origin of the anomalous $\Delta$-dependence is, at least in the large $\Delta$ case, therefore the presence of cancellations. The explanation of other contributions requires more involved arguments which are beyond the scope of this paper.
The dependence of median time to cancel on the entry depth $\Delta$ has a less clear functional form, as shown by Fig. \[fig:GcancdistCOMPARISON\]. While $\med{{\mathrm{TTC}}}$ appears to be a monotonically increasing function of $\Delta$, the curves for the different stocks show only a qualitative similarity. One of the reasons may be that different cancellation mechanisms are treated together.
{width="220pt"}{width="220pt"}
![The dependence of median time to cancel on the entry depth $\Delta$ of the limit order. The curves have an increasing tendency and they are qualitatively similar across stocks. However, they do not follow any obvious functional form.[]{data-label="fig:GcancdistCOMPARISON"}](GcancdistCOMPARISON){width="240pt"}
A simple model of the characteristic times {#sec:model}
==========================================
The problem of the interplay between time to fill and time to cancel is an example for competing risks [@bernoulli.risk; @hollifield.gains]. In this framework mutually exclusive events are considered in time [@bedford.competing; @hollifield.gains]: in our case after its placement a limit order is either executed or canceled. Each of these events has its own probability distribution for the time when it will occur, but only the earliest one of the events is observed. In this section we present a simple joint model [^7] of limit order placement and cancellation that is of this type. We will see that the model gives predictions that can be tested against real data. Moreover, it also gives indications on the statistical properties of a quantity that is directly unobservable: the “lifetime” an agent is willing to wait for a limit order to be executed.
We make the following assumptions:
1. We consider one “representative agent” [@kirman.representative]. At time $t=0$ the agent places a single buy[^8] limit order at a $\Delta > 0$ distance from the current best offer. (A generalization to $\Delta \leq 0$ is given in Section \[sec:moredelta\].) We treat all the other market participants on an aggregate level.
2. The agent is not willing to wait indefinitely for the order to be executed. Instead, at the time of placement the agent also decides about a cancellation (or more appropriately expiration) time $t'$ for the order. This is a value drawn randomly from the distribution ${P}_{{\mathrm{LT}};\Delta}(t')$. We will call this function the *lifetime distribution*. If the order is not executed until $t'$, then the order is canceled. The agent has no additional cancellation strategy. This assumption is very restrictive (cf. Ref. [@mike.empirical2]), but as Section \[sec:clock\] will show, it does not affect our results significantly.
3. The market is very liquid and tick sizes are small. As a consequence,
1. before its execution, the effect of the agent’s limit order on the evolution of the market price is negligible. This point neglects that traders reveal private information about their valuation of the stock by placing limit orders.
2. the interval between the time when the best bid reaches the order price and when the agent’s order is executed is negligible. We also assume that such immediate execution is independent of the volume of the agent’s order. A simple way to motivate that the volume present at a given price does not strongly affect execution times is to measure the typical ratio between time to fill and time to first fill as a function of the volume of the order. For at least $75\%$ of the orders of any volume this is close to $1$. The only exceptions can be very large orders with $\Delta = 1$. Here the price reaches the order quickly, but it takes about $20\%$ longer to execute it completely (see also Ref. [@lo.limitorder]). Moreover, for real limit orders the median time to fill does not depend too strongly on the volume of the order, except for very large volumes, see Fig. \[fig:Glifevol\].
In our study we included SHEL and VOD which are known to have large tick/price ratios, so Assumption 3 would be invalid. Contrary to our expectations, we did not find any indication of anomalies like in other studies [@eisler.sizematters; @eisler.liquidity; @mike.empirical2; @bouchaud.relation], and the model proved useful for these stocks as well.
{width="220pt"}{width="220pt"}
Under our assumptions one can write a joint density function that describes both the price diffusion process and cancellations. The probability $P_\Delta (t,t')$ that the price reaches an order placed at a distance $\Delta>0$ from the current best offer at time $t$ (and then it can be executed immediately), and that the agent decides to cancel the order a time $t'$ can be written as a product of two independent distributions: $$P_\Delta (t,t')=P_{\mathrm{FPT};\Delta}(t) P_{{\mathrm{LT}};\Delta}(t').
\label{eq:pprod}$$ For each limit order values of $t$ and $t'$ are drawn from $P$. The limit order is executed if $t<t'$ or it is canceled if $t>t'$. The two cases are illustrated in detail in Fig. \[fig:model\].
{height="180pt"}
The predictions of the model {#sec:predictions}
============================
Competing risk models are often estimated by the procedure introduced by Kaplan and Meier [@kaplan.meier]. This is a statistically consistent, non-parametric method to estimate the marginal distributions ${P_\mathrm{FPT}}$ and ${P_\mathrm{LT}}$ from ${P_\mathrm{TTF}}$ and ${P_\mathrm{TTC}}$ under the assumption that execution and cancellation are independent as we already assumed in writing Eq. . We will now calculate these estimates in another, but strictly equivalent analytical way.
Let us denote distribution functions as follows: $$P_{X;\Delta}(>t)=\int_t^\infty P_{X;\Delta}(\tau)d\tau,$$ where $X$ can be any process introduced above (FPT, LT, TTF, TTFF, TTC). We will omit the lower index $\Delta$ for brevity. Let us first express the previously introduced quantities in terms of the joint probability $ P_\Delta (t,t')$ and via Eq. . For executed orders $t<t'$, thus the distribution of time to fill is given by $$\begin{aligned}
P_\mathrm{TTF}(t)=\frac{ P_\mathrm{FPT}(t) P_{\mathrm{LT}}(>t)}{\int_0^\infty
P_\mathrm{FPT}(\tau) P_{\mathrm{LT}}(>\tau)d\tau}= \nonumber \\ \mathcal N[ P_\mathrm{FPT}(t)
P_{\mathrm{LT}}(>t)].\label{eq:pf}\end{aligned}$$ We introduced the operator $\mathcal N[\cdot]$, which normalizes a function to an integral of $1$. Symmetrically for time to cancel $t<t'$: $$\begin{aligned}
P_\mathrm{TTC}(t)=\frac{ P_\mathrm{FPT}(>t) P_{\mathrm{LT}}(t)}{\int_0^\infty P_\mathrm{FPT}(>\tau) P_{\mathrm{LT}}(\tau)d\tau}=\nonumber \\ \mathcal N[P_\mathrm{FPT}(>t) P_{\mathrm{LT}}(t)].
\label{eq:pc}\end{aligned}$$
As and are two equations with only one unknown function, namely the lifetime distribution $ P_{\mathrm{LT}}(t)$, one can calculate that from, e.g., Eq. , and then see if the solution is consistent with Eq. . We can express from Eq. , that $$P_{\mathrm{LT}}(>t) \propto \frac{ P_\mathrm{TTF}(t)}{P_\mathrm{FPT}(t)}
\label{eq:pbar}$$ and thus $$P_{\mathrm{LT}}(t)=-\frac{d}{dt} P_{\mathrm{LT}}(>t)=-\mathcal N\left[\frac{d}{dt} \frac{P_\mathrm{TTF}(t)}{P_\mathrm{FPT}(t)}\right].
\label{eq:pcp1}$$ It is also possible to estimate the same quantity directly from Eq. : $$P_{\mathrm{LT}}(t)=\mathcal N\left[\frac{
P_\mathrm{TTC}(t)}{ P_\mathrm{FPT}(>t)}\right].
\label{eq:pcp2}$$ Let us eliminate the lifetime distribution, and substitute the large $t$ asymptotic power law behavior of all probabilities. After simple calculations one finds that $$\lambda_\mathrm{TTF}=\lambda_\mathrm{TTC}. \\
\label{eq:lambdaright1}$$ Then we substitute this result back into Eq. to find that the lifetime distribution also has to decay asymptotically as a power law: $$P_{\mathrm{LT}}(t)\propto
t^{-\lambda_{\mathrm{LT}}},
\label{eq:pclock}$$ with $$\lambda_{\mathrm{LT}}=\lambda_\mathrm{TTF}-\lambda_\mathrm{FPT}+1=\lambda_\mathrm{TTC}-\lambda_\mathrm{FPT}+1. \label{eq:lambdaright2}$$
Eq. is in good agreement with the results of Section \[sec:ttfttc\], where $\lambda_\mathrm{TTF}=1.8-2.2$, and $\lambda_\mathrm{TTC}=1.9-2.4$. This is a clear improvement compared to Eq. . The introduction of the simplest possible cancellation model gives a good prediction for the difference between the exponents describing the asymptotics of the first passage time and time to fill.
Moreover, one can now observe the hidden distribution of lifetimes. By substituting the typical values into Eq. , one gets $\lambda_{\mathrm{LT}}\approx 1.6$. In comparison, a paper by Borland and Bouchaud [@Borland] describes a GARCH-like model obtained by introducing a distribution of traders’ investment horizons and the model reproduces empirical values of volatility correlations for $\lambda_{\mathrm{LT}}=1.15$, which is not far from our estimate. More recently it has been shown [@Lillo06] that the limit order price probability distribution is consistent with the solution of an utility maximization problem in which the limit order lifetime is power law distributed with an exponent $\lambda_{\mathrm{LT}}\simeq 1.75$. The origin of the power law distribution of limit order lifetimes is not clear. Unfortunately the data do not allow us to separate individual traders. Therefore we do not know whether such a result arises from the broad distribution of the time horizons of each trader, or simply a distribution of traders with different investment strategies. Based on an empirical investigation at the broker level, in Ref. [@Lillo06] it is argued that heterogeneity of investors could be the determinant of the power law lifetime distribution. Notice, however, two points: (i) We are not speaking about how long the investors *hold* the stock. Instead, $ P_{\mathrm{LT}}$ is the distribution of *how long investors are willing to wait* for their limit orders to be executed and before they cancel or revise their offers. (ii) None of the limit orders we are discussing here are truly long-term. Even the orders with relatively long lifetime spend at most a few days in the book.
An extension to $\Delta \leq 0$ {#sec:moredelta}
===============================
So far we only considered orders with prices which were worse than the best offer at the time of their placement, i.e., $\Delta > 0$. However, this group only accounts for less than half of the actual limit orders. Measurements for $\Delta \leq 0$ orders give the surprising result that these execution times are described by statistics very similar to those for $\Delta > 0$. One example stock (GSK) is shown in Fig. \[fig:GttfGSKnegative\](left). The results of our fitting procedure performed with Eq. are given in Table \[tab:ttfneg\] for all five stocks.
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
stock
$\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$
AZN $2.2$ $0.6$ $110$ $2.2$ $1.0$ $230$ $2.1$ $1.2$ $320$
GSK $2.2$ $0.5$ $110$ $2.1$ $1.1$ $180$ $2.2$ $1.3$ $410$
LLOY $2.2$ $0.5$ $120$ $2.1$ $1.0$ $150$ $2.0$ $1.2$ $220$
SHEL $2.2$ $0.4$ $110$ $2.1$ $1.0$ $120$ $2.1$ $1.0$ $140$
VOD $2.1$ $0.5$ $160$ $2.0$ $1.1$ $130$ – – –
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
stock
$\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$ $\lambda$ $\lambda'$ $T$
AZN $2.3$ $0.6$ $130$ $2.0$ $0.7$ $90$ $1.9$ $0.8$ $72$
GSK $2.1$ $0.6$ $130$ $1.9$ $0.7$ $90$ $1.8$ $0.8$ $50$
LLOY $2.2$ $0.5$ $120$ $1.9$ $0.7$ $70$ $1.8$ $0.9$ $85$
SHEL $2.3$ $1.0$ $220$ $2.2$ $1.0$ $130$ $2.0$ $1.0$ $160$
VOD $2.0$ $0.7$ $200$ $1.8$ $0.8$ $120$ – – –
------- ----------- ------------ ------- ----------- ------------ ------- ----------- ------------ -------
According to our model, these orders should have been executed within a negligible time of their placement. While this is true for a number of them, certainly not for all. Let us assume that we are placing a new buy limit order. If our order has $\Delta = 0$, then it will be among the best offers at the time of its placement. If our order has $\Delta < 0$, then it becomes the single best offer in the book, and hence it will trade with certainty if the next event is a buy market order. Why can our order still take a long time before being executed? The answer is naturally that before our order is executed, a new buy limit order may enter the book. If this new order has $\Delta < 0$ (where $\Delta$ now has to be measured from our order), it means that it has an even better price than our order and it will gain priority of execution. On the other hand, our order now effectively has $\Delta > 0$, and the original model can be applied.
In order to test such a hypothesis, we carried out the following calculation. For the sake of simplicity, we will consider the time to first fill instead of time to fill. Section \[sec:model\] argued that for the majority of orders the difference between the two is negligible. From the time of its placement, we tracked every single at least partially filled $\Delta \leq 0$ order until the time it was first filled. We defined the reduced entry depth ($\Delta'$) and the reduced time to first fill ($\mathrm{TTFF}'$) for these orders as follows
1. For orders, where from their placement to their first fill there were no even more favorable orders both placed and then at least partially filled, $\Delta'=0$ and $\mathrm{TTFF}'=\mathrm{TTFF}$.
2. For orders where after their placement but before their first fill there was at least one new, more favorable order introduced with $\Delta_\mathrm{new}<0$ and then this new order was at least partially filled, we selected the first of such new orders placed after the original one and set $\Delta'=-\Delta_\mathrm{new}$. Thus, $\Delta'$ is the new position of the original order, after the new one was placed. $\mathrm{TTFF}'$ is defined as the time to first fill of our order measured from the placement of this new order.
The typical distribution of $\mathrm{TTFF}'$ for different groups in $\Delta'$ is shown in Fig. \[fig:GttfGSKnegative\](right). For orders with $\Delta'=0$ this is – except for here uninteresting very short times – well described by a stretched exponential distribution $P_\mathrm{TTFF'}(t)=\frac{1}{25}\exp\left[-\left(\frac{t}{6}\right)^{1/2}\right]$. These are the orders, where there was no better offer made, and hence their execution times were purely determined by the incoming market orders. The distribution is very close to the distribution of the times between two consecutive transactions of the stock \[see Fig. \[fig:GttfGSKnegative\](right)\].
For orders with $\Delta'>0$, one recovers the results of the previous sections, and the distribution of reduced time to first fill asymptotically decays as a power law with a power close to $2.0$. Eqs. and are expected to be valid for orders with $\Delta < 0$ and $\Delta' > 0$ as well, given that we use them in terms of $\Delta'$ and $\mathrm{TTFF}'$.
As a summary, time to first fill for orders with $\Delta \leq 0$ is a two-component process. If there is no better order placed before the first fill, then time to first fill is basically identical to the waiting time distribution between opposite market orders. If there is a better offer submitted, then the order effectively becomes $\Delta > 0$, and the diffusion approximation applies. As this latter process has a much fatter tail than the former one, long waiting times and the tail exponent of the joint process are again dominated by a first passage process.
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Discussion {#sec:conclusions}
==========
Lifetime distribution {#sec:clock}
---------------------
Before discussing the results let us analyze the most important simplifying assumption of our model, namely the independence of the lifetime of the order from the evolution of price. This would mean that traders decide about an expiry time of their limit orders at the time of their placement, and then do not cancel them earlier, which resembles the random cancellation process as introduced in Ref. [@zovko.farmer]. In order to see the relevance of our assumption one should calculate the cross-correlation coefficient of first passage times and the lifetime process. However, as mentioned in Section \[sec:model\] we are limited by the fact that the lifetime is hidden. It is not possible to calculate cross-correlations between time to fill and time to cancel either, because for the same order one cannot observe both variables. This issue is related to the identifiability problem of competing risks [@bedford.competing].
We suggest the following approach to resolve the above issue: Let us consider canceled orders only. There one can observe the values of the lifetime, because they were realized as an actual time to cancel. Moreover, our model assumed, that the order would have been executed at the first passage time (the time of the first transaction at the order’s or a better price). Now it is possible to quantify cross-correlations between these two quantities, but one has to keep in mind three points. (Note that we will consider orders with $\Delta = 1$ to have the largest possible sample.)
1. For very short times the price dynamics is dominated by bid-ask bounce, and other non-diffusive processes [@Roll84]. Our model is not valid in this regime, because rapid order executions are not governed by a first passage process. Hence we discard all orders which were canceled within $L=4$ minutes of their placement.
2. In order to avoid problems arising from the possible non-existence of the moments of the distributions, we choose to evaluate Spearman’s rank-correlation coefficient[^9] ($\rho$), instead of Pearson’s correlation coefficient. The quantity $\rho$ has further favorable statistical properties, for example it is not very sensitive to extreme events.
3. As we can only consider canceled orders, we know that ${\mathrm{FPT}}>{\mathrm{LT}}$. This constraint alone, and regardless of the choice of correlation measure, will cause strong positive correlations between the two quantities. Even if ${\mathrm{FPT}}$ and ${\mathrm{LT}}$ are independent, the conditional joint distribution reads $$\begin{aligned}
P({\mathrm{FPT}}= t ,{\mathrm{LT}}= t'\vert {\mathrm{FPT}}> {\mathrm{LT}}) = \nonumber \\ \mathcal N[\Theta(t-t'){P_\mathrm{FPT}}(t){P_\mathrm{LT}}(t')],\end{aligned}$$ where $\Theta$ is the Heaviside step function. Due to our restricted observations this is clearly not a product of two independent densities.
Instead, a more convenient null hypothesis is to measure the correlations between ${\mathrm{FPT}}/{\mathrm{LT}}$ and ${\mathrm{LT}}$. $L=4$ min was chosen such that for $\Delta = 1$ the distribution of the first passage time is well described by the power law $$\begin{aligned}
P_{\mathrm{FPT}}(t\vert t>L)\sim\frac{\lambda_{\mathrm{FPT}}-1}{L^{\lambda_{\mathrm{FPT}}-1}}t^{-\lambda_{\mathrm{FPT}}}.
\label{eq:pttl}\end{aligned}$$ If ${\mathrm{FPT}}$ and ${\mathrm{LT}}$ are independent, then $$\begin{aligned}
P\left({\mathrm{FPT}}/{\mathrm{LT}}=x , {\mathrm{LT}}=t'\vert {\mathrm{FPT}}>{\mathrm{LT}}\right)=\nonumber \\ \mathcal N[\Theta(x-1){P_\mathrm{FPT}}(xt'){P_\mathrm{LT}}(t')]= \nonumber \\
\mathcal N[\Theta(x-1)x^{-\lambda_{\mathrm{FPT}}}] \times \mathcal N[{P_\mathrm{FPT}}(t'){P_\mathrm{LT}}(t')].\end{aligned}$$ Eq. was used for the second equality. The final result is a product form in functions of $x$ and of $t'$, which means that ${\mathrm{FPT}}/{\mathrm{LT}}$ is independent from ${\mathrm{LT}}$, given that we restrict ourselves to ${\mathrm{FPT}}>{\mathrm{LT}}$. Remember that the only assumption for this result is that first passage times are asymptotically power law distributed, which seems to hold very well in our data down to $L\approx 4$ min.
We calculated Spearman’s rank correlations between ${\mathrm{FPT}}/{\mathrm{LT}}$ and ${\mathrm{LT}}$ in our restricted sample for various stocks, this we will denote by $\rho_\mathrm{res}$. Results are summarized in Table \[tab:indep\]. One finds negative correlation between the two quantities at all usual significance levels.[^10] This means that those limit orders that would have been executed later were canceled earlier, i.e., that traders update their decision on when to cancel a limit order by tracking the price path. This is in line with the results of Ref. [@mike.empirical2]. To prove that this value of $\rho$ truly comes from correlations, we generated surrogate datasets by randomizing the pairs ${\mathrm{FPT}}/{\mathrm{LT}}$ and ${\mathrm{LT}}$ while keeping the constraint ${\mathrm{FPT}}>{\mathrm{LT}}$. According to Table \[tab:indep\] this completely destroys the correlations between ${\mathrm{FPT}}/{\mathrm{LT}}$ and ${\mathrm{LT}}$, $\rho_\mathrm{surr}=0$.
It is important to remember that this value of $\rho_\mathrm{res}$ is not the actual correlation coefficient between the first passage time and the lifetime process. To quantify the true value of cross-correlations, we introduce $\rho_\mathrm{true}$ which is Spearman’s rank-correlation coefficient between ${\mathrm{LT}}$ and ${\mathrm{FPT}}$. While this cannot be measured directly, there is a procedure to estimate it from a known value of $\rho_\mathrm{res}$ based on Monte Carlo simulation. Let us assume that ${\mathrm{FPT}}$ and ${\mathrm{LT}}$ are adequately described by power law distributions with the known tail exponents. We model the cross-correlation between the two processes by copulas (see Ref. [@frees.understanding]). Morgenstern’s copula reads $$\begin{aligned}
P(>t, >t')={P_\mathrm{FPT}}(>t){P_\mathrm{LT}}(>t')\nonumber \\ \left\{1+3\rho_\mathrm{true}[1-{P_\mathrm{FPT}}(>t)][1-{P_\mathrm{LT}}(>t')]\right\},\end{aligned}$$ with some $-1/3<\rho_\mathrm{true}<1/3$, while Frank’s copula assumes $$P(>t,>t')=\frac{1}{\alpha}\ln\left[1+\frac{(e^{\alpha {P_\mathrm{FPT}}(>t)}-1)(e^{\alpha {P_\mathrm{LT}}(>t')}-1)}{e^\alpha-1}\right],$$ with some $-\infty < \alpha < \infty$. Here $P(>t, >t')=\int_t^\infty d\tau\int_{t'}^\infty d\tau' P(\tau, \tau')$ which is the joint distribution function.
Monte Carlo measurements based on random pairs from these copulas suggest a nearly linear relationship between the true and the restricted correlation coefficients. With the substitution of the typical values of $\lambda_{\mathrm{FPT}}$ and $\lambda_{\mathrm{LT}}$ one finds that $$\rho_\mathrm{true}=r\times\rho_\mathrm{res},$$ where $r \approx 1.66$ for Morgenstern’s and $r \approx 1.55$ for Frank’s copula. The resulting estimates are given in Table \[tab:indep\]. Naturally, the shuffled surrogate datasets yield $\rho_\mathrm{true}=\rho_\mathrm{res}=0$.
These calculations have shown that there is a strong negative correlation between the first passage time and the lifetime of an order in agreement with Ref. [@mike.empirical2] but contrary to our model assumption $2$ and Eq. . So the key question is: How much does the presence of this correlation affect the predictions of our model? We performed a series of Monte Carlo simulations of the execution and cancellation processes by using the empirically observed value of tail exponents and cross correlations (Table \[tab:indep\]). We found that for a fixed value of $\lambda_{\mathrm{FPT}}$ and $\lambda_{\mathrm{LT}}$ the introduction of such correlations increases the values of $\lambda_{\mathrm{TTF}}$ and $\lambda_{\mathrm{TTC}}$ by about $0.1$, which is comparable to the error bars of our estimates, and the power law behavior is well preserved. Moreover, the central part of our arguments, Eq. , remains valid. Thus the presence of a dynamic cancellation strategy does not significantly affect the validity of our model.
stock $\rho_\mathrm{res}$ $\rho_\mathrm{surr}$ $\rho_\mathrm{true}$ (Morg.) $\rho_\mathrm{true}$ (Frank) number of points
------- --------------------- ---------------------- ------------------------------ ------------------------------ ------------------
AZN $-0.12 \pm 0.02$ $-0.001 \pm 0.001$ $-0.19 \pm 0.03$ $-0.18 \pm 0.03$ $3277$
GSK $-0.13 \pm 0.01$ $0.000 \pm 0.001$ $-0.21 \pm 0.02$ $-0.20 \pm 0.02$ $8573$
LLOY $-0.10 \pm 0.01$ $0.000 \pm 0.001$ $-0.16 \pm 0.02$ $-0.15 \pm 0.02$ $8201$
SHEL $-0.13 \pm 0.02$ $-0.001 \pm 0.001$ $-0.21 \pm 0.03$ $-0.19 \pm 0.03$ $2791$
VOD $-0.134 \pm 0.008$ $0.000 \pm 0.001$ $-0.22 \pm 0.01$ $-0.21 \pm 0.01$ $16392$
Conclusions
-----------
In this paper we focused on the tails of the distributions of characteristic times in the limit order book. Our empirical observations, based on five highly liquid stocks on the London Stock Exchange, underline the importance of cancellations when comparing the first passage time to the time to execute an order. We found that the distributions follow asymptotically power laws for the first passage time, the time to (first) fill and time to cancel. The differences between the statistical properties of these characteristic times are informative of the interdependence of order executions and cancellations. These observations are quite robust and can be seen as “stylized facts” characterizing the order book.
[We did not find significant difference between the behavior of buy and sell orders, in contrast with Refs. [@lo.limitorder; @cho.execution] for US markets, but in accord with Ref. [@hollifield.res2004] for the case of Ericsson stock traded at the Stockholm Stock Exchange. We are therefore not able to conclude whether the symmetric behavior we observe in the London Stock Exchange is common to most markets or specific to some of them or to certain time periods.]{}
In addition to the empirical findings summarized in Tables \[tab:fpt\], \[tab:ttf\] and \[tab:ttc\] we introduced a model, where order execution times are related to the first passage time of price, and orders are canceled randomly with lifetimes that are asymptotically power law distributed. This can be considered as the simplest possible model to take cancellations into account. In this framework we showed that the characteristic exponents of the asymptotic power law behavior of the first passage time, the time to (first) fill and time to cancel are related to each other by simple rules which are in agreement with our empirical observations. These results are in contrast with another study (the NASDAQ data investigated in Ref. [@challet.pa2001]). Therefore further investigations are needed to clarify whether or not our findings are market specific.
[The observed heterogeneity of cancellation times may be driven by traders having different time horizons or by traders following different cancellation strategies in different market environments. Methods that can discriminate between these mechanisms represent a major objective for future research.]{}
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank two anonymous referees for useful comments and suggestions. ZE is grateful to Jean-Philippe Bouchaud for discussions of the order book, to Michele Tumminello for advice on bootstrapping, and to Ingve Simonsen for help with the measurement of first passage times. The hospitality of l’Ecole de Physique des Houches and Capital Fund Management is also thankfully acknowledged. This work was supported by COST–STSM–P10–917 and OTKA T049238. FL and RNM acknowledge support from MIUR research project “Dinamica di altissima frequenza nei mercati finanziari” and NEST-DYSONET 12911 EU project.
Results in transaction time {#app:ttime}
===========================
[The typical time between transactions strongly depends on market conditions and it is very far from strictly stationary. This fact, also closely related to volatility clustering, could influence the distribution of first passage times, time to fill and time to cancel. Many recent studies measure time in transactions in order to remove fluctuations in trading activity. In order to better understand the role of activity fluctuations, we repeated our calculations in transaction time, but we did not find any changes that affect the conclusions of our paper. Fig. \[fig:GfptGSK1tr\] shows comparisons between real time and transaction time for the probability distributions of ${\mathrm{FPT}}$, ${\mathrm{TTF}}$ and ${\mathrm{TTC}}$ (for the stock GSK, $\Delta = 1$ tick). The short time regime is quite different, while for long times the fluctuations in trading activity are less relevant, and all the distributions remain power laws asymptotically. The changes in the values of the tail exponents are also small. The bottom right panel of Fig. \[fig:GfptGSK1tr\] compares $P_\mathrm{FPT}$, $P_\mathrm{TTF}$ and $P_\mathrm{TTF}$ in transaction time. Our arguments still hold, as $\lambda_\mathrm{FPT} < \lambda_\mathrm{TTF} \approx \lambda_\mathrm{TTC}$.]{}
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{width="220pt"}{width="220pt"}
Fitting functions for the distributions of time to fill and time to cancel {#app:fit}
==========================================================================
In this Appendix we present a critical discussion regarding our decision to fit the empirical distributions of FPT, TTF and TTC with the functional form presented in Eq. . In our preliminary investigations, we fitted the distribution of FPT, TTF and TTC with two different distributions. The first one was the one we consider throughout the paper, i.e., $$P_Z(t) =
\frac{C't^{-\lambda}}{1+(t/T)^{-\lambda+
\lambda'}}.
\label{eqz}$$ The second one was the generalized Gamma distribution $$P_G(t)=\frac{\lambda|p|\kappa^\kappa(\lambda
t)^{p\kappa-1}\exp[-(\lambda t)^p\kappa]}{\Gamma(\kappa)},
\label{gammadist}$$ which has been used in some of the existing studies on TTF (e.g. Ref. [@lo.limitorder]). Another common form, the Weibull distribution, is a special case of Eq. for $\kappa=1$. Our empirical analysis shows that the Weibull distribution fits the data poorly and it will not be considered in this Appendix. For large values of $t$ the density of Eq. behaves as $$P_Z(t)\sim\frac{1}{t^\lambda}~$$ The asymptotic behavior of $P_G(t)$ depends on the sign of the parameter $p$. If $p<0$ (as for the investigated data) it can be written as $$P_G(t)\sim\frac{\exp(-c/t^{|p|})}{t^{1+|p|\kappa}}$$ where $c$ is a constant. Thus the generalized Gamma distribution, similarly to Eq. , is consistent with a power law tail, although it is modulated by an exponential function which becomes less and less important as $t\to\infty$. In order to estimate the optimal parameters of the distributions we used a Maximum Likelihood Estimator (MLE). For illustrative purposes, here we consider the case of TTF for AZN and $\Delta=0$ but the results are similar for other stocks, other values of $\Delta$ and for both TTF and TTC.
{width="220pt"}{width="220pt"}
Fig. \[fig:Lapp\](left) shows the distribution of TTF for AZN and $\Delta=0$ together with fits by Eqs. and . Both $P_Z$ and $P_G$ give a good fit both in the tail and in the body of the distribution. One finds that $P_G$ has a slightly larger likelihood ${\cal L}$ than $P_Z$. Since the two distributions have the same number of parameters (degrees of freedom) the likelihoods can be compared directly. However if one computes the tail exponents of the distribution from the fitted parameters one finds a puzzling result. The tail exponent obtained from the generalized Gamma distribution fit is $4.5$, whereas the tail exponent obtained from the $P_Z$ fit is $2.2$. Such a difference in the exponent should be detectable in data. Still, Fig. \[fig:Lapp\](left) shows that both distributions fit the tail reasonably well. The reason of this contradiction is shown in the inset of Fig. \[fig:Lapp\](left). This plots the local tail exponent of the generalized Gamma distribution, given by $d[\log
P_G(t)]/d[\log t]$, as a function of $t$. The local exponent of the generalized Gamma distribution converges extremely slowly to the asymptotic value $4.5$ and in the range of the TTF from $10^3$ to $10^4$ the local exponent is between $2$ and $3$, which is approximately consistent with the values obtained from $P_Z$.
As we have repeatedly stated, in this paper we are interested in the tail behavior of the distribution of the time to fill and time to cancel. The analysis summarized in Fig. \[fig:Lapp\](left) shows that the parameters estimated from a fit to a generalized Gamma distribution are not suitable to estimate the tail exponent of the distribution, or at least not in the regime of TTF and TTC values that can be explored within our dataset. In other words, even if the generalized Gamma distribution gives a (slightly) better fit in terms of likelihood, it is hard to estimate the tail exponent from the fitted parameters due to the slow convergence of the local exponent. On the contrary, the parameters estimated from the fit with functional form of Eq. give a better estimate of the tail exponent. In order to support this claim, we estimate independently the tail exponent by using the Hill estimator [@hill; @embrechts]. In Fig. \[fig:Lapp\](right) we show the Hill plot of the time to fill of AZN with $\Delta=0$. It is clear that the Hill estimator converges to a value which is much closer to $2.2$ (as in the $P_Z$ distribution) than to $4.5$ (as in the $P_G$ distribution).
In conclusion our analysis shows that, although the generalized Gamma distribution gives a slightly better *overall* fit of time to fill and time to cancel than our proposed form \[Eq. \], the parameters obtained from the fit of $P_G$ suggest an unrealistic value of the tail exponent. On the contrary, our function $P_Z$ allows us to both fit the data reasonably well *and* to obtain values of the tail exponent which are consistent with the Hill estimator.
[^1]: The notion of competing risks applies to problems where one deals with several “risks”, i.e., random events, of which only the first one can be observed [@bedford.competing]. For example, limit orders are either executed or canceled and both events can be modeled by some random process. If an order gets canceled, one can no longer directly observe what time it would have been eventually executed, and vice versa. Thus it is not possible to independently estimate either process without a bias, if one simply ignores information from the other one.
[^2]: In our analyses, we removed the data of trading on September 20, 2002. This is because on that day very unusual trading patterns were observed, including an anomalous behavior of the bid-ask spread.
[^3]: Many studies refer to this colloquially as the “upstairs” market.
[^4]: [In most of the literature the logarithm of the price is modeled, while throughout the paper we intentionally use price itself. Our study is concerned with very small price changes on the order of the spread, when there is little difference between the two approaches. In our case it is important to keep bare prices, as stocks have a finite tick size (minimal price change). Taking bare prices enables us to classify the orders into discrete categories by price difference. The size of ticks depends on the stock, the possible values are $1/4$, $1/2$ or $1$ penny.]{}
[^5]: [We repeated the statistical analysis with transaction time and observed a similar power law decay of the first passage time for large times. The value of the power law exponent turns out to be different for real time analysis and transaction time analysis. See Appendix \[app:ttime\] for details.]{}
[^6]: This result is consistent with the Sparre-Andersen theorem [@redner]. Alternative descriptions obtained for the asymptotic time dependence of the FPT of Lévy flights which were hypothesizing a dependence of the distribution exponent from the index of the Lévy distribution have missed the fact that the method of images, which is extremely powerful in Gaussian diffusion, fails for Lévy flight processes [@chechkin.JPA2003]. The behavior is of course more complex in the case of Lévy random processes described by using a subordination scheme. In these cases the asymptotic behavior of first passage time depends on the complete properties of the subordination procedure [@sokolov].
[^7]: Ref. [@bouchaud.relation] shows that similar arguments give a very good approximation for the average shape of the order book.
[^8]: Note that throughout the paper we use the language of buy orders, but analogous definitions can be given for sell orders. All measurements include both buy and sell orders.
[^9]: This is defined by first, for both quantities separately, replacing each observation by its rank in the sample (i.e., assigning $1$ to the largest observation of first passage time, $2$ to the second largest, etc., and then repeating the procedure for lifetimes). Then the usual cross-correlation coefficient is calculated for the ranks [@lee.statistics].
[^10]: The error bars were estimated by the bootstrapping procedure suggested in Ref. [@schmid.bootstrapping] (for more details see Refs. therein).
|
---
abstract: 'We present the results of experimental and numerical study of the distribution of the reflection coefficient $P(R)$ and the distributions of the imaginary $P(v)$ and the real $P(u)$ parts of the Wigner’s reaction $K$ matrix for irregular fully connected hexagon networks (graphs) in the presence of strong absorption. In the experiment we used microwave networks, which were built of coaxial cables and attenuators connected by joints. In the numerical calculations experimental networks were described by quantum fully connected hexagon graphs. The presence of absorption introduced by attenuators was modelled by optical potentials. The distribution of the reflection coefficient $P(R)$ and the distributions of the reaction $K$ matrix were obtained from the measurements and numerical calculations of the scattering matrix $S$ of the networks and graphs, respectively. We show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory (RMT).'
address: |
$^1$Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland\
$^2$University of Hradec Králové, Hradec Králové, Czech Republic\
$^3$Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha, Czech Republic
author:
- 'Micha[ł]{} [Ł]{}awniczak$^1$, Oleh Hul$^1$, Szymon Bauch$^1$, Petr Šeba$^{2,3}$, and Leszek Sirko$^1$'
date: 'April 16, 2008'
title: '**Experimental and numerical investigation of the reflection coefficient and the distributions of Wigner’s reaction matrix for irregular graphs with absorption**'
---
Quantum graphs of connected one-dimensional wires were introduced more than sixty years ago by Pauling [@Pauling]. Next the same idea was used by Kuhn [@Kuhn] to describe organic molecules by free electron models. Quantum graphs can be considered as idealizations of physical networks in the limit where the lengths of the wires are much bigger than their widths, i.e. assuming that the propagating waves remain in a single transversal mode. Among the systems modelled by quantum graphs one can find e.g., electromagnetic optical waveguides [@Flesia; @Mitra], mesoscopic systems [@Imry; @Kowal] , quantum wires [@Ivchenko; @Sanchez] and excitation of fractons in fractal structures [@Avishai; @Nakayama]. Recently it has been shown that quantum graphs are excellent paradigms of quantum chaos [@Kottossmilansky; @Kottos; @Prlkottos; @Zyczkowski; @Kus; @Tanner; @Kottosphyse; @Kottosphysa; @Gaspard; @Blumel; @Hul2004]. More complicated and thus more realistic systems - microwave networks with moderate absorption strength $\gamma =2\pi \Gamma /\Delta \leq 7.1$, where $\Gamma$ is the absorption width and $\Delta$ is the mean level spacing, have been experimentally investigated in [@Hul2005; @Hul2007]. Other interesting open objects - quantum graphs with leads - have been analyzed in details in [@Kottosphyse; @Kottosphysa]. However, the properties of networks and graphs with strong absorption have not been studied experimentally neither numerically so far. Therefore, in this paper we study experimentally and numerically the distribution of the reflection coefficient $P(R)$ and the distributions of the Wigner’s reaction matrix [@Akguc2001] (in the literature often called $K$ matrix [@Fyodorov2004]) for networks (graphs) with time reversal symmetry ($\beta=1$) in the presence of strong absorption.
In the case of a single channel antenna experiment the $K$ matrix is related to the scattering matrix $S$ by the following relation $$\label{Eq.1} S=\frac{1-iK}{1+iK}.$$ Eq. (1) holds for the systems with absorption but without direct processes [@Fyodorov2004]. It is important to mention that the function $Z=iK$ has a direct physical meaning of the electric impedance that has been recently measured in the microwave cavity experiment [@Anlage2005]. In the one channel case the $S$ matrix can be parameterized as $$\label{Eq.2} S=\sqrt{R}e^{i\theta},$$ where $R$ is the reflection coefficient and $\theta$ the phase.
Properties of the statistical distributions of the $S$ matrix with direct processes and imperfect coupling have been studied theoretically in several important papers [@Lopez1981; @Doron1992; @Brouwer1995; @Savin2001; @Fyodorov2003; @Fyodorov2005]. Recently the distribution of the $S$ matrix has been also measured experimentally for chaotic microwave cavities with absorption [@Kuhl2005]. The distribution $P(R)$ of the reflection coefficient $R$ and the distributions of the imaginary $P(v)$ and the real $P(u)$ parts of the Wigner’s reaction $K$ matrix are theoretically known for any dimensionless absorption strength $\gamma $ [@Fyodorov2004; @Savin2005]. In the case of time reversal systems (symmetry index $\beta=1$) $P(R)$ has been studied experimentally by Méndez-Sánchez et al. [@Sanchez2003]. The distributions $P(v)$ and $P(u)$ have been studied for chaotic microwave cavities in [@Anlage2005; @Anlage2006] and for microwave networks for moderate absorption strength $\gamma \leq 7.1$ in [@Hul2005; @Hul2007]. For systems without time reversal symmetry ($\beta=2$) and a single perfectly coupled channel $P(R)$ was calculated by Beenakker and Brouwer [@Beenakker2001] while the exact formulas for the distributions $P(v)$ and $P(u)$ were given by Fyodorov and Savin [@Fyodorov2004].
In the experiment quantum graphs can be simulated by microwave networks. The analogy between quantum graphs and microwave networks is based upon the equivalency of the Schrödinger equation describing the quantum system and the telegraph equation describing the microwave circuit [@Hul2004].
A general microwave network consists of $N$ vertices connected by bonds e.g., coaxial cables. A coaxial cable consists of an inner conductor of radius $r_1$ surrounded by a concentric conductor of inner radius $r_2$. The space between the inner and the outer conductors is filled with a homogeneous material having a dielectric constant $\varepsilon$. For a frequency $\nu$ below the onset of the next TE$_{11}$ mode only the fundamental TEM mode can propagate inside a coaxial cable. (This mode is in the literature often called a Lecher wave.) The cut-off frequency of the TE$_{11}$ mode is $\nu_{c} \simeq \frac{c}{\pi (r_1+r_2)
\sqrt{\varepsilon}} = 32.9$ GHz [@Jones], where $r_1$ = 0.05 cm is the inner wire radius of the coaxial cable (SMA-RG402), while $r_2$ = 0.15 cm is the inner radius of the surrounding conductor, and $\varepsilon \simeq 2.08$ is the Teflon dielectric constant [@Breeden1967; @Savytskyy2001].
From the experimental point of view absorption of the networks can be changed by the change of the bonds’ (cables’) lengths [@Hul2004] or more effectively by the application of microwave attenuators [@Hul2005; @Hul2007]. In the numerical calculations weak absorption inside the cables can be described with the help of complex wave vector [@Hul2004]. We will show that strong absorption inside an attenuator can be described by a simple optical potential. The corresponding mathematical theory has been developed in [@Ex1].
The distribution $P(R)$ of the reflection coefficient $R$ and the distributions of the imaginary and real parts of the Wigner’s reaction matrix $K$ for microwave networks with absorption were found using the impedance approach [@Anlage2005; @Anlage2006; @Hul2007]. In this approach the real and imaginary parts of the normalized impedance $Z$ $$\label{Equation22} Z = \frac{\textrm{Re } Z_{n}+i(\textrm{Im }
Z_{n}-\textrm{Im } Z_{r})}{\textrm{Re } Z_{r}}$$ of a chaotic microwave system are measured, with $Z_{n(r)}=Z_0(1+S_{n(r)})/(1-S_{n(r)})$ being the network (radiation) impedance expressed by the network (radiation) scattering matrix $S_{n(r)}$ and $Z_0$ is the characteristic impedance of the transmission line. The radiation impedance $Z_{r}$ is the impedance seen at the output of the coupling structure for the same coupling geometry, but with the vertices of the network removed to infinity. The Wigner’s reaction matrix $K$ can be expressed by the normalized impedance as $K=-iZ$. The scattering matrix $S$ of a network for the perfect coupling case (no direct processes present) required for the calculation of the reflection coefficient $R$ (see Eq. (2)) can be finally extracted from the formula $S=(1-Z)/(1+Z)$.
Figure 1(a) shows the experimental setup for measuring the single-channel scattering matrix $S_{n}$ of fully connected hexagon microwave networks necessary for finding of the impedance $Z_{n}$. We used Hewlett-Packard 8720A microwave vector network analyzer to measure the scattering matrix $S_n$ of the networks in the frequency window: 7.5–11.5 GHz. The networks were connected to the vector network analyzer through a lead - a HP 85131-60012 flexible microwave cable - connected to a 6-joint vertex. The other five vertices of the networks were connected by 5-joints. Each bond of the network presented in Fig. 1(a) contains a microwave attenuator.
The radiation impedance $Z_{r}$ was found experimentally by measuring the scattering matrix $S_{r}$ of the 6-joint connector with five joints terminated by 50 $\Omega $ loads (see Figure 1(b)).
The experimentally measured fully connected hexagon networks were described in numerical calculations by quantum fully connected hexagon graphs with one lead attached to the 6-joint vertex. In the calculations attenuators (absorbers) were modelled by optical potentials [@Ex1]. To be explicit we suppose that the fully connected hexagon graph $\Upsilon$ with one coupled antenna is described in the Hilbert space $L^2(\Upsilon):= \bigoplus_{(j,n)}
L^2(0,\ell_{jn})\bigoplus L^2(0,\infty)$, where $\ell_{jn}$ stays for the lengths of the bond connecting the vertices $j$ and $n$ and the halfline $(0,\infty)$ describes the attached antenna.
We define the Schrödinger operator $H$ by $$\label{graph SO}
H{\psi_{jn} := \, -\psi''_{jn}+ U_{jn}\psi_{jn}},$$
with $\psi_{jn}\in L^2(0,\ell_{jn})$ for the bonds and
$$\label{graph ant}
H{\psi_{0n} := \, -\psi_{0n}''},$$
with $\psi_{0n}\in L^2(0,\infty)$ describing the wave function of the antenna connected to the vertex $n$ (note that the “infinite” vertex of the antenna has index 0) .
At the vertices the wave functions are linked together with the boundary values
$$\label{boundary values}
\psi_{jn}(j):=\lim_{x\to 0+} \psi_{jn}(x)\,, \quad \psi'_{jn}(j):\lim_{x\to 0+} \psi'_{jn}(x)\,,$$
satisfying boundary conditions $\,\psi_{jn}(j)=\psi_{jm}(j)=:\psi_j$ for all $n,m$ describing connected vertices, and
$$\label{delta}
\sum_{n\in\nu(j)} \psi'_{jn}(j)= 0.$$
The optical potentials $U_{jn}$ are purely imaginary and describe the absorber inserted between the vertices $(j,n)$.
Since the graph $\Upsilon$ is infinite (due to the attached antenna) we can look for solutions of the equation
$$\label{local SO}
H\psi= k^2\psi,$$
referring to the continuous spectrum, where $k$ is the wave vector. For microwaves propagating inside a lossless bond with a dielectric constant $\varepsilon$ the wave vector $k=2\pi
\varepsilon \nu/c $, where $\nu $ and $c$ denote the frequency of a microwave field and the speed of light in the vacuum, respectively. To solve this equation we used the graph duality principle [@Ex1]. According to this principle we need to solve the equation $-f''+U_{jn}f=k^2f$ on $[0,\ell_{jn}]$ satisfying the normalized Dirichlet boundary conditions
$$u_{jn}(\ell_{jn})= 1\!-\!(u_{jn})'(\ell_{jn})=0\,, \;\; v_{jn}(0)1\!-\!(v_{jn})'(0)=0\,.$$
The Wronskian of this solution is naturally equal to $W_{jn}-v_{jn}(\ell_{jn}) =u_{jn}(0)$. Then according to [@Ex1] the corresponding boundary values (\[boundary values\]) satisfy the equation
$$\label{discrete delta}
\sum_{n} {\psi_n\over W_{jn}}\,-\, \left(\, \sum_{n\in\nu(j)}
{(v_{jn})'(\ell_{jn}) \over W_{jn}}\, \right)\psi_j\,=\, 0\,.$$
Conversely, any solution $\psi_j$ of the system (\[discrete delta\]) determines a solution of (\[local SO\]) by
$$\begin{aligned}
\psi_{jn}(x)= {\psi_n\over W_{jn}}\,u_{jn}(x) -\,{\psi_j\over
W_{jn}}\,v_{jn}(x) \;\;
& {\rm if} & n = 1,..,6\,, \label{reconstruction i} \\
\psi_{jn}(x)= -\,{\psi_j\over W_{jn}}\,v_{jn}(x) \;\; & {\rm if} &
n=0\,. \label{reconstruction b}\end{aligned}$$
As already mentioned the microwave attenuators are modelled by optical potentials localized inside the inserted component. It is well known that any smooth and localized potential can be easily approximated by a sequence of delta potentials inside the support of the potential - see [@demkov] for details. We will use this fact and express the optical potential as a sum of $N$ delta-potentials with imaginary coupling constants: $U(x)=ib\sum_{r=1}^{N} \delta(x-(r-1)l_b/(N-1))$. The delta-potentials were equally spaced inside the length $l_{b}$ of the absorbing element (attenuator). By changing the number $N$ and the strength $b$ of delta-potentials we were able to vary absorbing properties as well as reflective properties of attenuators. We used $N=10$ delta-potentials with $b=0.028$ $m^{-1}$ for simulation of the 1 dB attenuators and $N=12$ delta-potentials with $b=0.045$ $m^{-1}$ for the 2 dB attenuators. In both cases the length of the attenuator was $l_{b}=2.65$ cm. Furthermore, in the numerical calculations of the scattering matrices $S_n$ of the graphs the weak absorption inside the microwave cables was taken into account by replacing the real wave vector $k$ by the complex vector $k+ia\sqrt{k}$ [@Goubau], where the absorption coefficient was assumed to be $a=0.009$ $m^{-1/2}$ [@Hul2004].
In order to find the distribution $P(R)$ of the reflection coefficient $R$ and the distributions of the imaginary and real parts of the K matrix we measured the scattering matrix $S_n$ of $88$ and $74$ network configurations containing in each bond a single 1 dB and 2 dB microwave SMA attenuator, respectively. The total optical lengths of the microwave networks containing 1 dB attenuators, including joints and attenuators, varied from 574 cm to 656 cm. For the networks with 2 dB attenuators the optical lengths varied from 554 cm to 636 cm. To avoid degeneracy of eigenvalues of the networks the lengths of the bonds were chosen as incommensurable.
In Figure 2 the modulus $|S_n|$ and the phase $\theta $ of the scattering matrix $S_n$ of the microwave networks with $\gamma =
19.9 \textrm{ and } 47.9$, respectively, are presented in the frequency range 7.5 - 9 GHz. The measurements were done for two networks containing 1 dB and 2 dB attenuators, respectively. Their total “optical" lengths including joints and attenuators were 574 cm and 554 cm, respectively.
For systems with time reversal symmetry ($\beta=1$), the explicit analytic expression for the distribution $P(R)$ of the reflection coefficient $R$ is given by [@Savin2005]
$$\label{Equation23}
P(R)=\frac{2}{(1-R)^2}P_0\Bigl(\frac{1+R}{1-R}\Bigr).$$
The probability distribution $P_0(x)$ is given by the expression
$$P_0(x)=-\frac{dW(x)}{dx},$$
where the integrated probability distribution $W(x)$ is expressed by the formula [@Savin2005]
$$W(x)=\frac{x+1}{4\pi}\Bigl[f_{1}(w)g_{2}(w)+f_{2}(w)g_{1}(w)+h_{1}(w)j_{2}(w)+h_{2}(w)j_{1}(w)\Bigr]_{w=(x-1)/2}.$$
The functions $f_{1},g_{1},h_{1},j_{1}$ are defined as follows
$$f_{1}(w)=\int_{w}^{\infty}dt\frac{\sqrt{t\mid t-w\mid}e^{-\gamma
t/2}}{(1+t)^{3/2}}\Bigl[1-e^{-\gamma}+\frac{1}{t}\Bigr],$$
$$g_{1}(w)=\int_{w}^{\infty}dt\frac{e^{-\gamma t/2}}{\sqrt{t\mid
t-w\mid}(1+t)^{3/2}},$$
$$h_{1}(w)=\int_{w}^{\infty}dt\frac{\sqrt{\mid t-w\mid} e^{-\gamma
t/2}}{\sqrt{t(1+t)}}\Bigl[\gamma+(1-e^{-\gamma})(\gamma
t-2)\Bigr],$$
$$j_{1}(w)=\int_{w}^{\infty}dt\frac{e^{-\gamma t/2}}{\sqrt{t\mid
t-w\mid} (1+t)^{1/2}}.$$
Their counterparts with the index 2 are given by the same expressions but the integration is performed in the interval $t \in
[0, w]$ instead of $[w, \infty)$.
The distributions of the imaginary and the real parts $P(v)$ and $P(u)$ of the $K$ matrix [@Fyodorov2004] can be also expressed by the probability distribution $P_0(x)$: $$\label{Equation34} P(v)=\frac{\sqrt{2}}{\pi
v^{3/2}}\int^{\infty}_{0}dqP_0\Bigl[q^2+\frac{1}{2}\Bigl(v+\frac{1}{v}\Bigr)\Bigr],$$ and $$\label{Equation35} P(u)=\frac{1}{2\pi
\sqrt{u^{2}+1}}\int^{\infty}_{0}dqP_0\Bigl[\frac{\sqrt{u^2+1}}{2}\Bigl(q+\frac{1}{q}\Bigr)\Bigr],$$ where $-v=\textrm{Im} \, K<0$ and $u=\textrm{Re} \,K$ are, respectively, the imaginary and real parts of the $K$ matrix.
Figure 3 shows the experimental distributions $P(R)$ (squares) of the reflection coefficient $R$ for two mean values of the parameter $\bar{\gamma }$, viz., 19.3 and 47.7. The distribution for $\bar{\gamma } = 19.3$ is obtained by averaging over 88 realizations of the microwave networks containing 1 dB attenuators. The distribution for $\bar{\gamma } = 47.7$ is obtained by averaging over 74 realizations of the microwave networks containing 2 dB attenuators. The experimental values of the $\gamma$ parameter were estimated for each realization of the network by adjusting the theoretical mean reflection coefficient $\langle R \rangle _{th}$ to the experimental one $\langle R
\rangle=\langle SS^{\dag}\rangle $, where $$\label{Equation36} \langle R \rangle _{th} = \int _0^1dRRP(R).$$
Figure 3 also presents the corresponding distributions $P(R)$ (solid and dashed lines, respectively) evaluated from Eq. (\[Equation23\]). A good overall agreement of the experimental distributions $P(R)$ with their theoretical counterparts is seen.
Figure 4 shows the numerically evaluated distributions $P(R)$ (circles) of the reflection coefficient $R$ for the graphs at $\bar{\gamma } = 19.3 \textrm{ and } 47.7$ compared to the theoretical ones evaluated from the formula Eq. (\[Equation23\]). The numerical distributions are the result of averaging over 162 and 214 realizations of the graphs with optical potentials simulating 1 dB and 2 dB attenuators, respectively. The numerical values of $\gamma$ parameter were also estimated by adjusting the theoretical mean reflection coefficient to the numerical one. The agreement between the numerical results for $\bar{\gamma } = 47.7$ and the theoretical ones (dashed line) is good. However, for $\bar{\gamma } = 19.3$ for $R < 0.15$ some discrepancies between the numerical results and the theoretical ones (solid line) are visible.
In Figure 5 the experimental distribution $P(v)$ of the imaginary part of the $K$ matrix is shown for the two mean values of the parameter $\bar{\gamma } = 19.3 \textrm{ and } 47.7$, respectively. The distribution is the result of averaging over 88 and 74 realizations of the networks with the attenuators 1 dB and 2 dB, respectively. The experimental results in Figure 5 are in general in good agreement with the theoretical ones. However, both experimental distributions are slightly higher than the theoretical ones in the vicinity of their maxima.
Results of the numerical calculations of the distributions $P(v)$ are shown in Figure 6 for two mean values of the parameter $\bar{\gamma }=19.3 \textrm{ and } 47.7$, respectively. They are compared to the theoretical ones evaluated from the formula Eq. (\[Equation34\]). Figure 6 shows also a good agreement between the numerical and theoretical results, which confirms usefulness of the optical potential approach in describing the microwave attenuators.
Measurements of the distribution $P(u)$ of the real part of the Wigner’s reaction matrix give an additional test of the consistency of the $\gamma$ evaluation. In Figure 7 we show this distribution obtained for two values of $\bar{\gamma }=19.3
\textrm{ and } 47.7$, respectively, compared to the theoretical ones evaluated from the formula Eq. (\[Equation35\]). Also here we observe good overall agreement between the experimental and theoretical results. However, Figure 7 shows that for the networks with 2 dB attenuators the theoretical distribution is in the middle ($-0.1<u<0.1$) slightly higher than the experimental one. According to the definition of the $K$ matrix (see Eq. (1)) such a behavior of the experimental distribution $P(u)$ suggests deficiency of small values of $|\textrm{Im} S|$, whose origin is not known. Moreover, the experimental distribution $P(u)$ obtained for the networks assembled with 1 dB attenuators is slightly asymmetric for $|u|>0.5$.
In Figure 8 the comparison of the numerical distribution $P(u)$ obtained for two values of $\bar{\gamma } = 19.3 \textrm{ and }
47.7$, respectively, to the theoretical one evaluated from the formula Eq. (\[Equation35\]) is presented. In this case we see a good overall agreement between the numerical and theoretical results.
In spite of the above mentioned discrepancies which appeared mainly in the case of the experimental distribution $P(u)$ the overall good agreement between the experimental and theoretical results justifies *a posteriori* the chosen procedure of calculating the experimental $\gamma$. The same is true also for the numerical simulations.
In summary, we measured and calculated numerically the distribution of the reflection coefficient $P(R)$ and the distributions of the imaginary $P(v)$ and the real $P(u)$ parts of the Wigner’s reaction $K$ matrix for irregular fully connected hexagon networks and graphs in the presence of strong absorption. In the case of the microwave networks consisting of SMA cables and attenuators the application of attenuators allowed for effective change of absorption in the graphs. In the numerical calculations absorption in an attenuator was modelled by an optical potential. We showed that in the case of the time reversal symmetry ($\beta=1$) the experimental and numerical results for $P(R)$, $P(v)$ and $P(u)$ are in good overall agreement with the theoretical predictions. The agreement of the numerical and theoretical results strongly confirms the usefulness of the optical potential approach in the description of the microwave attenuators.
Acknowledgments. This work was partially supported by Polish Ministry of Science and Higher Education grant No. N202 099 31/0746 and by the Ministry of Education, Youth and Sports of the Czech Republic within the project LC06002.
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|
---
address: |
Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, MI\
48109, USA
author:
- 'Szu-yuan Chen, Anatoly Maksimchuk & Donald Umstadter'
nocite:
- '[@Thomson]'
- '[@Mourou]'
- '[@Vachaspati; @Brown; @Sarachik; @Castillo; @Esarey; @Hartemann; @Hartemann2]'
- '[@Sarachik; @Castillo; @Esarey]'
- '[@Esarey]'
- '[@Esarey2; @Esarey3]'
- '[@Meyer; @Basov; @Malka; @Englert; @Bula]'
- '[@Brunel]'
- '[@Malka]'
- '[@Chen]'
- '[@Chen]'
- '[@Sarachik; @Castillo; @Esarey]'
- '[@Esarey]'
- '[@Castillo]'
- '[@LeBlanc]'
- '[@Esarey]'
- '[@Esarey]'
- '[@Thomson]'
- '[@Mourou]'
- '[@Vachaspati]'
- '[@Brown]'
- '[@Sarachik]'
- '[@Castillo]'
- '[@Esarey]'
- '[@Hartemann]'
- '[@Hartmann2]'
- '[@Esarey2]'
- '[@Esarey3]'
- '[@Meyer]'
- '[@Basov]'
- '[@Malka]'
- '[@Englert]'
- '[@Bula]'
- '[@Brunel]'
- '[@Chen]'
- '[@LeBlanc]'
title: Experimental observation of nonlinear Thomson scattering
---
[**A century ago, J. J. Thomson$^{1}$ showed that the scattering of low-intensity light by electrons was a linear process (i.e., the scattered light frequency was identical to that of the incident light) and that light’s magnetic field played no role. Today, with the recent invention of ultra-high-peak-power lasers$^{2}$ it is now possible to create a sufficient photon density to study Thomson scattering in the relativistic regime. With increasing light intensity, electrons quiver during the scattering process with increasing velocity, approaching the speed of light when the laser intensity approaches 10$^{18}$ W/cm$^{2}$. In this limit, the effect of light’s magnetic field on electron motion should become comparable to that of its electric field, and the electron mass should increase because of the relativistic correction. Consequently, electrons in such high fields are predicted to quiver nonlinearly, moving in figure-eight patterns, rather than in straight lines, and thus to radiate photons at harmonics of the frequency of the incident laser light$^{3-9}$, with each harmonic having its own unique angular distribution$^{5-7}$. In this letter, we report the first ever direct experimental confirmation of these predictions, a topic that has previously been referred to as nonlinear Thomson scattering$%
^{7}$. Extension of these results to coherent relativistic harmonic generation$^{10,11}$ may eventually lead to novel table-top x-ray sources.**]{}
In this experiment, we used a laser system that produces 400-fs-duration laser pulses at 1.053-$\mu $m wavelength with a maximum peak power of 4 TW. The 50-mm diameter laser beam was focused with an f/3.3 parabolic mirror onto the front edge of a supersonic helium gas jet. The focal spot is consisted of a 7-$\mu $m FWHM Gaussian spot (containing 60 $\%$ of the total energy) and a large ($>$ 100 $\mu $m) dim spot. The helium gas was fully ionized by the foot of the laser pulse. A half-wave plate was used to rotate the axis of linear polarization of the laser beam in order to vary the azimuthal angle ($\phi $) of observation. We define $\theta =0^{\circ }$ as along the direction opposite to that of the laser propagation and $\phi
=0^{\circ }$ as along the axis of linear polarization. In a linearly polarized laser field, electrons move in a figure-eight trajectory lying in the plane defined by the axis of linear polarization and the direction of beam propagation.
While the observation of harmonics in laser-plasma (or electron beam) interactions has been made by several groups$^{12-16}$, that alone is insufficient to unambiguously identify nonlinear Thomson scattering and its underlying dynamics. Several other mechanisms might generate continuum or harmonics under our experimental conditions, and, therefore, need to be isolated and discriminated from the signal generated by nonlinear Thomson scattering: (1) continuum generated from self-phase modulation of laser beam in gas, (2) harmonics generated from atomic nonlinear susceptibility of gas or, especially, from the ionization process$^{17}$, (3) continuum generated from (a) (relativistic) self-phase modulation of laser pulse in the plasma, or from (b) electron-electron bremsstrahlung and electron-ion bremsstrahlung, and (4) harmonics generated from the interaction of laser pulses with a transverse electron-density gradient$^{14}$.
The main focal spot of the laser pulse undergoes relativistic-ponderomotive self-channeling when high laser power and gas density are used$%
^{18}$. Side imaging ($\theta $ = 90$^{\circ }$) of the 1st harmonic light (at the laser frequency) from nonlinear Thomson scattering shows that the laser channel has a diameter of $<$10 $\mu $m FWHM. However, interferograms $^{18}$ show that the diameter of the plasma column is about 100-200 $\mu $m, which is created by the wings with intensities $>10^{15}$ W/cm$^{2}$ (the ionization threshold). Therefore, the light generated from laser-gas interaction should be observed to originate from the entire region of plasma, rather than from the narrow laser channel. Results of side imaging ($\theta $ = 90$^{\circ }$ and arbitrary $\phi $) of the 2nd and 3rd harmonics using a matching interference filter (10 nm bandwidth) show that the signal is emitted only from the narrow laser channel. In addition, the images of the harmonics have spatial distributions similar to the images of the 1st harmonic light, and their profiles vary in the same way as the laser power and gas density are changed. This rules out the possibility that the harmonic signal observed in the side images is a result of laser-gas interaction ((1) and (2)).
According to theory$^{5-7}$, the harmonic signal generated from nonlinear Thomson scattering should have two important features: (1) it is linearly proportional to the electron density because it is an incoherent single electron process (the harmonics generated from a collection of electrons interfere with each other destructively, leaving only an incoherent signal, which is equal to the single-electron result multiplied by the total number of electrons which radiate), and (2) it increases roughly as $I^{n}$, where $n$ is the harmonic number, and gradually saturates when $a_{0}$ is on the order of unity$%
^{7} $, where $a_{0}=eE/m_{0}\omega _{0}c=8.5\times 10^{-10}\lambda $\[$\mu $m\]$I^{1/2}$\[W/cm$^{2}$\] is the normalized vector potential, $E$ is the amplitude of laser electric field, and $I=cE^{2}/8\pi $ is the laser intensity. These are characteristically different from the behavior of any other mechanisms. For instance, bremsstrahlung radiation should be proportional to the square of gas density ($N_{e}\cdot N_{e}$ or $N_{e}\cdot
N_{i}$). In this experiment, the intensity of the harmonic signal was determined from the peak intensity or the average intensity of the images of harmonics, when it was plotted as a function of the observing angle, gas density and laser power. Both showed the same variations. Figure \[powden\] shows the variation of the 2nd harmonic signal as a function of laser power and plasma (electron) density. The experimental results show a reasonable fit with the theoretical predictions. The 1st and 3rd harmonics show the same match with the theory.
Although the above two observations are consistent with nonlinear Thomson scattering as the source of the harmonic signal, the observation of the unique angular patterns is necessary in order to prove that the detailed dynamics of nonlinear Thomson scattering are indeed the same as the theoretical prediction. Figure\[plon2f90\](a) shows the $\phi $ -dependence of the 2nd harmonic signal at $\theta =$ 90$^{\circ }$. The experimental results match qualitatively with the theoretical prediction, both having a quadrupole-type radiation pattern, which is characteristically different from the dipole pattern for other mechanisms (1)-(4), and linear Thomson scattering. Other measurements such as the $\phi $-dependence of the 2nd harmonic light at $\theta =$ 51$^{\circ }$ (an “anti-dipole” pattern), shown in Fig.\[plon2f90\](b), and the $\phi $-dependence of the 3rd harmonic light at $\theta =$ 90$^{\circ }$ (a “butterfly” pattern), shown in Fig.\[plon3f90\], were also made, all showing reasonable matches between the experimental data and the theoretical predictions. Such angular radiation patterns directly prove that electrons do indeed oscillate with figure-eight trajectories in an intense (relativistic) laser field. The angular pattern of the 1st harmonic light (linear component) of nonlinear Thomson scattering is also included in Fig.\[plon2f90\](b) for comparison.
Measurements of the spectra of the harmonics show that each of the spectra of 2nd and 3rd harmonics contains a peak at roughly the harmonic wavelength and a red-shifted broader peak, as shown in Fig.\[spectra\]. The red-shifted broader peaks are believed to be part of the harmonic spectra generated by nonlinear Thomson scattering, because they vary in amplitude proportionally with the corresponding unshifted harmonic signals when the gas density and the laser power are changed. It was expected that the spectra of harmonics should be broadened tremendously for electrons in a high-fluid-velocity plasma wave$^{6}$. A fast-phase-velocity electron plasma wave (with a maximum fluid velocity of as large as $\sim $0.2 $c$, where $c$ is the speed of light in vacuum) excited by stimulated Raman forward scattering$^{19}$ was observed in this experiment at high laser power and gas density. However, the fact that the spectral distribution of the harmonics was not observed to change significantly with variation of gas density and laser power, when the plasma wave amplitude was, indicates that such spectral structure has nothing to do with the collective drift motion of electrons in the plasma waves. Although the angular radiation patterns of the harmonics could also be affected by such a 0.2 $c$ fluid-velocity oscillation, the changes are not significant enough (compare the solid and dash lines in Fig. \[plon2f90\](a)) to be identified from the experimental data$^{7}$. In other words, all measurements done in this experiment match qualitatively with the prediction of incoherent nonlinear Thomson scattering of electrons without drift motion; the results appear not to be affected by the existence of plasma waves, probably due to destructive coherent interference. The absolute scattering efficiency is measured to be $8\times
10^{-4}$ and $1\times 10^{-4}$ photons per electron per pulse for the 2nd and 3rd harmonics (including both the unshifted and red-shifted spectral components), respectively, at $\theta =90^{\circ }$, $\phi =50^{\circ }$, for an angle of collection of $7\times 10^{-3}$ steradians. These numbers match reasonably well with the theoretical predictions for incoherent nonlinear Thomson scattering, which are $8\times 10^{-4}$ and $5\times
10^{-4}$, respectively.
In summary, the results reported here confirm for the first time several predictions of relativistic electrodynamic theory, which were formulated forty years ago, coincident with the invention of the laser. As predicted $^{7}$, a century-old fundamental “constant,” the Thomson cross-section, is now shown to depend on the strength of light.
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[**Acknowledgements**]{} This work was supported by U. S. National Science Foundation and the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy. The authors would also like to thank G. Mourou, R. Wagner and X.-F. Wang for their useful discussions.
|
---
abstract: 'During mammalian embryo development, reprogramming of DNA methylation plays important roles in the erasure of parental epigenetic memory and the establishment of naïve pluripogent cells. Multiple enzymes that regulate the processes of methylation and demethylation work together to shape the pattern of genome-scale DNA methylation and guid the process of cell differentiation. Recent availability of methylome information from single-cell whole genome bisulfite sequencing (scBS-seq) provides an opportunity to study DNA methylation dynamics in the whole genome in individual cells, which reveal the heterogeneous methylation distributions of enhancers in embryo stem cells (ESCs). In this study, we developed a computational model of enhancer methylation inheritance to study the dynamics of genome-scale DNA methylation reprogramming during exit from pluripotency. The model enables us to track genome-scale DNA methylation reprogramming at single-cell level during the embryo development process, and reproduce the DNA methylation heterogeneity reported by scBS-seq. Model simulations show that DNA methylation heterogeneity is an intrinsic property driven by cell division along the development process, and the collaboration between neighboring enhancers is required for heterogeneous methylation. Our study suggest that the mechanism of genome-scale oscillation proposed by Rulands et al. (2018) might not necessary to the DNA methylation during exit from pluripotency.'
author:
- 'YUSONG YE$^1$,ZHUOQIN YANG$^1$,and JINZHI LEI$^{2}$[^1]'
title: DNA methylation heterogeneity induced by collaborations between enhancers
---
1. [School of Mathematics and Systems Science and LMIB, Beihang University, Beijing 100191, China]{}
2. [Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084, China]{}
[2.0]{}
**Keywords:** DNA methylation, embryo development, heterogeneity, genome-scale oscillation, stochastic simulation
Introduction
============
In mammalian development, reprogramming of DNA methylation (5-methylcytosine) patterns play a crucial role in defining cell fate. Upon fertilization, DNA methylation marks represent an epigenetic barriers that restrict mammalian development, and hence need to be restored and subsequently rebuilt with the commitment to particular cell fates[@seisenberger2013reprogramming]. The segregation of cell lineages give rise to different somatic tissues associated with tissue-specific DNA methylation patterns [@styblo2000comparative; @Greenberg:2019hw; @Hon:2013jr]. The genome-wide DNA methylation reprogramming events coincide with the changes in concentrations of DNA methyltranferases (DNMTs) and the enzymes that initiate the removal of DNA methylation (ten-eleven-translocation family proteins, TETs)[@seisenberger2013reprogramming; @Greenberg:2019hw]. The recent maturation of single-cell sequencing technologies has enable us to observe a variety of sequencing information at individual cells level, such as the genome, transcriptome, and epigenome [@Stuart:2019iv]. It becomes a challenge issue in computational biological to develop single-cell based computational model that can help us to better understand the process of DNA methylation pattern formation as well as cell fate decision during early embryo development.
The availability of methyleome information from single-cell whole genome bisulfite sequencing (scBS-seq) provides an opportunity to study DNA methylation patterns in the whole genome in individual cells[@Farlik:2015bw; @rulands2018genome; @Smallwood:2014kn]. A recent study applied scBS-seq to embryo stem cells (ESCs) cultured under naïve (two chemical inhibitors (“2i”) of MEK1/2 and GSK3$\alpha$/$\beta$) and primed (“serum”) conditions to explore DNA methylation dynamics in cells undergoing a biological transition[@rulands2018genome], primed ESCs had increased variance at several genomic annotations associated with active enhancer elements, including H3K4me1 and H3K27ac sites and low methylated regions (LMRs). Analysis of scBS-seq data shown that individual primed ESCs have average DNA methylation levels varying between 17% and 86% at enhancers, while naïve ESCs showed minimal cell-to-cell variability, and DNA methylation heterogeneity was resolved upon differentiation to embryoid bodies[@rulands2018genome]. In [@rulands2018genome], it was proposed that the DNA methylation heterogeneity is associated with coherent, genome-scale oscillations in DNA methylation, and amplitude is dependent on the CpG density. Moreover, a mathematical model of delay differential equation with autocatalytic *de novo* methylation was proposed to show that global oscillations may emerge from the biochemistry of methylation turnover due to Hopf bifurcation with increasing values of the time delay, and a Kuramoto model was applied to describe the global heterogeneous coupling of CpGs via DNMT3a/b binding. Nevertheless, genome-scale oscillations in DNA methylation is a very strong assumption, which may imply global oscillations in transcriptions of most genes, and is not supported by the experimental data of DNA methylation dynamics during transition from naïve to primed pluripotency *in vitro* (detailed below). Hence, we asked how the transitions of DNA methylation heterogeneity should be explained through a simple mechanism?
DNA methylation and chromatin dynamics have been modeled quantitatively in various genomic contexts of biological significance[@Berry:2017km; @haerter2013collaboration; @Huang:2017jr; @Sneppen:2011db; @song2017collaborations]. The collaboration between neighboring CpGs was highlight in recent studies[@haerter2013collaboration; @song2017collaborations], which play essential roles in the formation of global patterns and the genome-scale transitions of DNA methylation. The collaboration may directly come from the binding of DNMT3a/b to neighboring CpGs[@rulands2018genome], or indirectly through the interaction with methylations in the histones H3K9[@Lehnertz:2003eb] and H3K36[@Weinberg:2019gv]. In additional to the *de novo* methylation and demethylation, dilution and maintenance of of methylated marks during DNA replication may also contribute to the cell-to-cell variance of DNA methylation.
Here, motivated by the scBS-seq data of heterogeneous methylation at genomic annotations associated with active enhancer elements, we developed a model of DNA methylation, considering the stochastic dynamics methylation levels of enhancers over cell divisions and the collaboration between neighboring enhancers, to investigate the transition of DNA methylation from naïve to primed ESCs. The model focus at the random inheritance of DNA methylations during cell cycling, and can automatically reproduce the DNA methylation heterogeneity on enhancers during embryonic development. Our results suggest that the mechanism of genome-scale oscillation proposed by [@rulands2018genome] may not required for the observed heterogeneity during exit from pluripotency.
Results
=======
Transition of DNA methylation patterns from naïve to primed ESCs
----------------------------------------------------------------
We analyzed the scBS-seq data separately for ESCs cultured under naive (‘‘2i’’) and primed (‘‘serum’’) conditions[@rulands2018genome]. Similar to the analysis in [@rulands2018genome], taking published H3K4me1 chromatin immunoprecipitation sequencing (ChIP-seq) data form primed ESCs[@creyghton2010histone] as a definition of enhancer elements, the methylation levels of enhancers in primed ESCs increase comparing with naïve ESCs (Fig. \[fig:1\]A). Here, the methylation level of an enhancer is defined as the average level of all CpG sites contained in the enhancer. For each CpG site, we assigned a value $0$ for unmethylated, $0.5$ for half-methylated, and $1$ for full methylated, hence the methylation level of an enhancer takes value from the interval $[0, 1]$ (or from $0\%$ to $100\%$ methylated). Moreover, we calculated the distribution of methylation levels of all enhancers in individual cells. Primed ESCs shown higher cell-to-cell variability at the distribution patterns than the naïve ESCs (Fig. \[fig:1\]B & C). We also analyzed the parallel scM&T sequencing of *in vivo* epiblast cells at E4.5, E5.5, and E6.5[@rulands2018genome], which shown an increase in the methylation level in enhancers from E4.5 to E5.5 (Fig. \[fig:1\]D). We note that there are a few cells shown low methylation levels at E5.5, however all cells have high methylation at E6.5, which suggest a transition dynamics of methylation levels (Fig. \[fig:1\]D).
To quantify the heterogeneity of DNA methylations among different cells, we proposed a definition of heterogeneity index based on the methylation levels of enhancers in each cell. Assuming that there are $n$ cells, and $p_i$ ($1\leq i \leq n$) the distribution of all enhancer methylation levels of the $i$’th cell, we defined the heterogeneity index ($H$) as the average of Kullback-Leibler divergence between any two cells. Mathematically, the heterogeneity index is formulated as $$\label{eq:HI}
H = \dfrac{1}{n(n-1)}\sum^n_{i,j=1}KL(p_i || p_j),$$ where $KL(p_i||p_j)$ means the Kullback-Leibler divergence between the distributions of enhancer methylation levels for the two cells $i$ and $j$, $$KL(p_i ||p_j) = \int_{0}^{1} p_i(x) \log \dfrac{p_i(x)}{p_j(x)} dx.$$
We calculated the heterogeneity index based on the above data from naïve and primed ESCs, and the cells at E4.5, E5.5, and E6.5 mice embryo. The is no significant changes in the heterogeneity of naïve ESCs in comparing with the primed ESCs (Fig. \[fig:1\]E). For the mice embryo cells, the heterogeneity index increases from E4.5 to E5.5, and decreases from E6.6 to E6.5 (Fig. \[fig:1\]E).
![**DNA methylation transition in embryo cells.** (A). DNA methylation variance in naïve and primed ESCs compared the enhancer elements defined using published H3K4me1 ChIP-seq data[@creyghton2010histone], and the average methylation levels in individual cells. (B). Distributions of enhancer methylation levels in two cells from the naïve condition. (C). Distributions of enhancer methylation levels in three cells from the primed condition. (D). DNA methylation variance of epiblast cells at E4.5, E5.5, and E6.5 of mice embryo, and the average methylation levels in individual cells. (E). Heterogeneity indexes of cells under different conditions. (F). Dynamics of average DNA methylation from $0$ to $8$ h in the “2i release” experiments. Here the methylation levels were calculated from the average over the enhancers at ch1. All data were obtained from [@rulands2018genome]. []{data-label="fig:1"}](Ye_Fig_1.eps){width="14cm"}
To validate the genome-scale DNA methylation oscillations in [@rulands2018genome], we analyzed the same data from an *in vitro* “2i release” model in which cells were transferred from naïve 2i to primed serum culture conditions. The average methylation levels of enhancers from the first chromosome (ch1) were calculated to obtain the dynamics from $0$ to $8$ h after 2i release. The results do not show significant oscillations in the methylation level (Fig. \[fig:1\]F). This findings suggest that the assumption of genome-scale DNA methylation oscillations might not necessary to explain the transition of methylation and heterogeneity from naïve to primed ESCs.
Stochastic dynamics of enhancer methylation levels
--------------------------------------------------
The dynamics of DNA methylation/demethylation have been modeled quantitatively with exquisite details at biochemistry of single CpG sites[@haerter2013collaboration; @song2017collaborations; @Lei:2018ev; @Sontag:2006iw]. The methylation state of a single CpG site is often random due to the stochastic biochemical reactions. Nevertheless, the average methylation level of CpG sites associated with a genomic segment is more predictable. During embryo development, the most significant changes in DNA methylation occur during DNA replication when the 5-methylcytosine marks are dilute to two daughter strains and are restored through enzymes DNMT1 and nuclear protein 95 (NP95 or UHRF1). Correlating global DNA methylation with replication timing repli-seq data shown that late-replicating regions did not have lower DNA methylation than early-replicating regions[@rulands2018genome]. Thus, while we omit the details dynamics between DNA replications, we can represent the methylation level of an enhancer by the average methylation level at late-replication stage of each cell cycle.
### Formulation
To consider the dynamics of enhancer methylation levels, assuming that there are $N$ enhancers in a chromatin, and letting $\beta_i^t$ the methylation level of the $i$’th enhancer at cycle $t$ ($0\leq \beta_i^t \leq 1$), we only need to formulate the dynamics of the states $$\vec{\beta}^t = (\beta_1^t, \beta_2^t, \cdots, \beta_N^t)$$ over cell cycles $t$. During cell cycling, the methylation states update as a consequence of the regulations through enzymes DNMT1, DNMT3a/b and TETs, which leads to the following iteration $$\label{eq:1}
T: \left( \beta^t_1, \beta^t_2,\beta^t_3, ..., \beta^t_N \right) \xrightarrow{\mbox{cell cycle}}\left( \beta^{t+1}_1, \beta^{t+1}_2,\beta^{t+1}_3, ..., \beta^{t+1}_N \right).$$ Hence, while we omit the biochemistry details, the stochastic dynamics of enhancer methylation levels can be formulated as an iteration $$\label{eq:m}
\vec{\beta}^{t+1} = T(\vec{\beta}^t)$$ for each cell cycle. Here, $\vec{\beta}^{t}$ represents the methylation state of a cell before cell division, and $\vec{\beta}^{t+1}$ is the state of one daughter cell after cell division.
We note that the iteration Eq. is usually a random map. Given the state of a mother cells, the methylation state of the daughter cell is a random valuable whose probability density is dependent on the state of the mother cell. Hence, to formulate the iteration map, we need to write down the conditional probability density function $$\mathrm{Prob}(\vec{\beta}^{t+1} = \vec{x}| \vec{\beta}^t),$$ the probability of $\vec{\beta}^{t+1}$ given the state of the mother cell $\vec{\beta}^t$. While the probability of each enhancer, given the state of the mother cell, is independent to each other, we have $$\label{eq:4}
\mathrm{Prob}(\vec{\beta}^{t+1} = \vec{x}| \vec{\beta}^t) = \prod_{i=1}^N \mathrm{Prob}(\beta_i^{t+1} = x_i | \vec{\beta}^t).$$
The probability density $\mathrm{Prob}(\beta_i^{t+1} = x_i | \vec{\beta}^t)$ is usually not known and may depend on the biochemistry details of methylation/demethylation. Nevertheless, it is possible to write down the phenomena formulation if we overlook the detail process. If there are $m_i$ CpGs in the $i$’th enhancer, each CpG has a probability $p_i^t$ to be methylated (here the superscript $t$ specified the dependence with the time $t$), and $1-p_i^t$ to be unmethylated after cell division (here we omitted the state of half-methylation), the probability to have $k_i$ methylated CpGs is given by a binomial distribution $C_{m_i}^{k_i}(p_i^t)^{k_i} (1-p_i^t)^{m_i-k_i}$ with the parameters $p_i^t$. To extend the probability to a more general situation, the binomial distribution can be generalized to a beta-binomial distribution through two shape parameters $a_i^t$, $b_i^t$. Moreover, while we are only interested at the probability of the methylation level $x_i = k_i/m_i$, we replaced the beta-binomial distribution with the beta distribution, and hence $$\label{eq:5}
\mathrm{Prob}(\beta_i^{t+1} = x_i | \vec{\beta}^t) = \dfrac{x^{a_i^t-1} (1-x_i)^{b_i^t-1}}{B(a_i^t, b_i^t)},\quad B(a, b) = \dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},$$ where $\Gamma(\cdot)$ is the gamma function, $a_i^t$ and $b_i^t$ are shape parameters depending on $\vec{\beta}^t$. Eqs. \[eq:4\]–\[eq:5\] together define the conditional probability $\mathrm{Prof}(\vec{\beta}^{t+1} = \vec{x} | \vec{\beta})$. The beta distribution is one of few common “named” distributions that give probability $1$ to a finite interval. As the shape parameters $a$ and $b$ vary, the beta distribution can take different shapes, include strictly decreasing ($a \leq 1, b > 1$), strictly increasing ($a > 1, b \leq 1$), U-shaped ($a < 1, b < 1$), or unimodal ($a> 1, b>1$). Thus, the dependences of parameters $a_i^t, b_i^t$ on the state $\vec{\beta}^t$ are important to define the shape of the distribution function.
Now, to define the dependences $a_i^t(\vec{\beta}^t)$ and $b_i^t(\vec{\beta}^t)$, assuming that there are functions $\phi_i^t(\vec{\beta}^t)$ and $\eta_i^t$ so that average of $\beta_i^{t+1}$, given $\vec{\beta}^t$, is $$\left.\langle \beta_i^{t+1}\rangle\right|_{\vec{\beta}^t} = \phi_i^t(\vec{\beta}^t),$$ and the variance $$\left.\mathrm{Var}(\beta_i^{t+1})\right|_{\vec{\beta}^t} = \dfrac{1}{ 1 + \eta_i^t} \left(1-\phi_i^t(\vec{\beta}^t)\right) \phi_i^t(\vec{\beta}^t),$$ than[^2] $$\label{eq:ab}
a_i = \eta_i^t \phi_i^t(\vec{\beta}^t),\quad b_i^t = \eta_i^t \left(1 - \phi_i^t(\vec{\beta}^t)\right).$$ Hence, we only need to identify the functions $\phi_i(\vec{\beta}^t)$ and $\eta_i^t$, and the parameters $a_i^t$ and $b_i^t$ can be defined accordingly. Specifically, we take $\eta_i^t$ as the number of CpGs in the $i$’th enhancer, *i.e.*, $$\eta_i^t = m_i.$$ This simple assumption means the inverse proportion of the variance of enhancer methylation levels with the number of CpGs.
To define the function $\phi_i^t$, we assumed the methylation level of daughter cell depends on that of the mother cell through three components: basal methylation level, autocatalytic effect, and collaboration between enhancers. Hence, the function $\phi_i^t$ was formulated as $$\label{eq:6}
\phi_i^t(\vec{\beta}^t)=H_{[0,1]}(z),\quad z=\underbrace{\mu_0}_{\mathrm{basal}}+\underbrace{\mu_1\dfrac{(\beta^t_i)^n}{(\beta^t_i)^n+v}}_{\mathrm{autocatalysis}}+\underbrace{\dfrac{\alpha}{2L+1}\sum\limits_{|j-i|\leq L}({\beta^{t}_j }-\beta^t_i)}_{\mathrm{collaborations}},$$ where $$H_{[0,1]}(z) = \left\{
\begin{array}{ll}
0,&\quad z<0,\\
z,&\quad 0\leq z\leq 1\\
1,&\quad z>1.
\end{array}\right.$$ Here $\mu_0, \mu_1, n, v, \alpha, L$ are parameters, with $\mu_0$ the basal level, $\mu_1$ the coefficient for autocatalysis, $n$ the Hill coefficient, $v$ the parameter for the autocatalytic efficiency, $\alpha$ the coefficient for long distance collaboration, and $L$ defines the range of collaboration between enhancers. The autocatalytic efficiency $v$ usually depends on the activity of *de novo* methylation/demethylation regulated by DNMT3a/b and TETs, and hence can change with time $t$ during embryo development. Here, we always have $0\leq \phi_i^t \leq1$ due to the function $H_{[0,1]}(z)$.
The third term in Eq. \[eq:6\] shows the collaboration between neighboring enhancers. Here, we assumed the coherent collective behaviors of DNA methylation/demethylation when the average coupling through enzymes binding is sufficiently strong so that the nearby enhancers tend to the same trends of either methylation or demethylation. The similar mechanism was introduced previously to reproduce the long distance correlation of DNA methylation between CpGs[@song2017collaborations]. Here the collaboration effect is limited by cooperative range $L$ and the coefficient $\alpha$.
### Numerical scheme and parameters
To model the methylation dynamics following developmental process with the above iteration Eq. \[eq:1\], we first initialized the methylation level of $N$ enhancers $\vec{\beta}^0 = (\beta_1^0, \beta_2^0, \cdots, \beta_N^0)$ (here $t=0$). Here, each enhancer $i$ associates with an integer $m_i$ for the number of CpGs. Next, at each step, we calculated the shape parameters $a_i^t$ and $b_i^t$ using Eqs. \[eq:ab\]–\[eq:6\]. Finally, for each enhancer $i$, generated a random number in according to the beta distribution Eq. \[eq:5\], and set $t=t+1$. We repeated the above scheme to generate the dynamics of methylations in each enhancer following multiple cell cycles.
![[**Number of CpG sites and the initial methylation level.**]{} (A). Distribution of number of CpG sites in enhancers and the fitting curve (Eq. ). (B). Distributions of methylation at a naïve ESC.[]{data-label="fig:7"}](Ye_Fig_2.eps){width="14cm"}
In model simulations, we took $\mu_0 = 0.01, \mu_1 = 0.9, n = 3, \alpha = 0.5, L=50$, and varied the autocatalysis ability $v$ to represent different culture conditions ($v = 0.1, 0.04, 0.01$ for situations of high, mediate, and low level DNA methylation, respectively). Moreover, to mimic an enhancers in a chromosome, we performed simulations with $N=1000$ enhancers, the CpG numbers for each enhancer were taken following an exponential distribution $$\label{eq:fit}
\mathrm{Prob}(m_i=m)=\frac{e^{(-m/3)}}{1.6}.$$ in according to the statistics from mouse genome(Fig. \[fig:7\]A). In simulations, we can refer the distribution of methylation levels from a naïve as the initial condition(Fig. \[fig:7\]B).
Transition of DNA methylation heterogeneity during the development process
--------------------------------------------------------------------------
To verify the proposed model, we varied the parameter $v$ ($v = 0.1, 0.04, 0.01$) to mimic different conditions. For each value $v$, we initialized a cell with an initial state of a naïve cell and ran the model simulation for 15 cell cycles in order to mimic the transition from naïve to primed condition. The simulated distribution of enhancers methylation levels at each cell cycle were calculated, and are shown by Fig. \[fig:3\]A-C. The enhancers methylation distributions depend on the parameter $v$: when $v=0.1$, the enhancers were homogeneous with low level methylation; when $v=0.04$, the cells shown obvious methylation heterogeneity, the methylation levels transfer from low to high along with cell cycling; when $v=0.01$, most enhancers change to high level methylations in a few cycles. When $v=0.04$, we have low, mediate, or high methylation levels in the enhancers at different cycles (Fig. \[fig:3\]D). These results are in agree with experimental observations of ESCs cultured under serum condition, and hence the model can be used to mimic the DNA methylation heterogeneity of ESCs during exist from pluripotency.
To further examine the transition dynamics of methylation heterogeneity in ESCs, we set $v=0.04$ and initialized a population of cells according to the methylation distribution at Fig. \[fig:7\]B. Next, we performed the simulation scheme to mimic a development process of 48 h. In simulations, each cell divides with a probability of $0.3$ per h, and collected a subpopulation of cells to calculate the average methylation level and the heterogeneity index every 3 h. Simulations shown that the average methylation level increased from $0$ to $48$ h, but shown obvious diversity from $12$ to $30$ h (Fig. \[fig:3\]E). The heterogeneity index shown non-monotonous with the development process, firstly increase from $0$ to $27$ h to reach a high level of $0.12$, and then decrease to a low level heterogeneity at 48h (Fig. \[fig:3\]F). In embryo development, DNA methylation heterogeneity in ESCs increased from naïve to primed, and the heterogeneity was resolved upon differentiation to embryoid bodies [@rulands2018genome]. These results shown similar dynamics in both experiments and our model simulations.
![**DNA methylation heterogeneity from model simulation.** (A-C). Distributions (violin plots) of methylation levels in enhancers obtained from model simulation with $v=0.1$ (A), $v=0.04$ (B), and $v=0.01$ (C). (D). Histogram methylation levels in enhancers for cycles 0, 5, and 14. (E). Transition of DNA methylation from low to high level. Each dot represent the average methylation level in a cell. (F). Evolution of the heterogeneity index (HI). The HI were calculated from the cells shown by (E). Here $v=0.04$ in (E)-(F).[]{data-label="fig:3"}](Ye_Fig_3.eps){width="14cm"}
Collaboration and methylation heterogeneity
-------------------------------------------
The proposed model includes collaboration between neighboring enhancers so that there are coherent collective behaviors during enzyme binding. To investigate the effects of neighboring collaboration, we varied the parameters $\alpha$ and $L$ and examined the changes in methylation heterogeneity. Here, the parameter $\alpha$ measures the collective strength, and $L$ gives the regions of neighboring collaboration. Simulations shown that the heterogeneity increased with $\alpha$ for different values of $L$, and increased with $L$ for a large value $\alpha$ (Fig. \[fig:4\]A).
To further examine how collaborations may affect the DNA methylation dynamics, we varied the parameters $\alpha$ and $L$ and calculated the evolution dynamics over a period of $48$ h. When $\alpha = 0.5$ and $L=10$, the cells shown highly heterogeneous during the intermediate transition region (Fig. \[fig:4\]B). When either $L$ or $\alpha$ decreases, the cells shown less heterogeneity (Fig. \[fig:4\]C-D). In particular, when $\alpha=0$, which represents the situation without collaboration, all cells shown similar DNA methylation dynamics during simulation from $0$ to $48$ h, with the average methylation increases from a low level to an intermediate level of $50$% (Fig. \[fig:4\]E). These results suggest that the collaboration between neighboring enhancers is required to produce DNA methylation heterogeneity.
![[**Collaboration and methylation heterogeneity.**]{} (A). Heterogeneity index with different values of the parameter $\alpha$ and $L$. (B-D) Transition dynamics of the average methylation level with varied parameter values of $\alpha$ and $L$.[]{data-label="fig:4"}](Ye_Fig_4.eps){width="12cm"}
Discussion
==========
Reprogramming of DNA methylation plays important roles in mammalian early embryo development. ESCs show heterogeneous methylation distributions under primed conditions. To understand the mechanism of methylation heterogeneity, previous studies suggest a mechanism of genome-scale oscillations in DNA methylation. Nevertheless, experiment data did not support the assumption of genome-scale oscillations. Here, we proposed a computational model for the stochastic transitions of enhancer methylations during cell cycling. The model combines random distribution of methylation marks in DNA replication, autocatalysis of DNA methylation due to the binding of DNMT3a/b, and the collaboration between neighboring enhancers during the reconstruction of methylation marks. The proposed model can nicely explain the transition of methylation level and heterogeneous methylation distributions. Model simulations shown that the proper values of the autocatalysis is important for the heterogeneity between different cells, and increasing the collaboration between neighboring enhancers can promote the heterogeneity. Our model suggest that methylation heterogeneity is a nature consequence of stochastic transition of DNA methylation between cell cycles and the collaboration between CpGs, however the assumption of genome-scale oscillations might not necessary for the observed heterogeneous methylation distributions.
The proposed model mainly considers the dynamics of enhancer methylation levels in a cell, and omits the biochemical reactions involved in methylation or demethylation, which may be regulated by various enzymes. In the model, we assumed a beta distribution that connects the methylation level in daughter cells with those of the mother cell. The beta distribution can take different forms based on the shape parameters that are defined by the state of mother cells. Thus, the proposed model framework mainly focus at the general effect of methylation state transition between cell cycles, while omit the detail biochemical reactions. On the other hand, despite the complex biochemical reactions, they mainly affect the methylation levels through the based methylation/demethylation processes and the collaboration between CpGs, and hence may end up to the function $\phi$ in the model. Hence, the proposed model provides a general framework to sum up different level biochemical reactions model for DNA methylation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the National Natural Science Foundation of China (91730301, 11831015, and 11372017).
Disclosure of Potential Conflicts of Interest {#disclosure-of-potential-conflicts-of-interest .unnumbered}
=============================================
No potential conflicts of interest were disclosed.
[18]{} natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix
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[^1]: Corresponding author, Email: jzlei@tsinghua.edu.cn
[^2]: Here we note that $$\left.\langle \beta_i^{t+1}\rangle\right|_{\vec{\beta}^t} = \dfrac{a_i^t}{a_i^t + b_i^t},$$ and $$\left.\mathrm{Var}(\beta_i^{t+1})\right|_{\vec{\beta}^t} = \dfrac{a_i^t\, b_i^t}{ (a_i^t + b_i^t) (1 + a_i^t + b_i^t)}.$$
|
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abstract: |
Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to non-trivial permutation symmetries (beyond the conventional paradigm of symmetric–antisymmetric states).
The present work obtains (a) selection rules for additional non-separable (dependent on spins and coordinates) $k$-body interactions: the Young diagrams, associated with the initial and the final states of a transition, can differ by relocation of no more than $k$ boxes between their rows; and (b) correlation rules: eigenstate-averaged local correlations of $k$ particles vanish if $k$ exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry — in 1929, Dirac identified those with characters of the symmetric group — relating them to experimentally observable correlations of several particles.
The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.
author:
- 'Vladimir A. Yurovsky'
title: 'Permutation symmetry in spinor quantum gases: selection rules, conservation laws, and correlations'
---
Selection rules constrain possible transitions between states of quantum systems [@dirac; @landau]. They allow the prediction of essential properties of physical systems based on their symmetries, without expensive calculations. Like every other symmetry, permutation symmetry leads to conservation laws, which were identified by Dirac[@dirac1929] (see also[@dirac]). This symmetry has been used in the Yang-Gaudin model [@yang1967; @*sutherland1968] and has gained increasing attention due to recent progress in the control of many-body states of cold atoms [@fang2011; @daily2012; @harshman2014].
The Pauli exclusion principle (see the review [@kaplan2013] and references therein) states that a many-body wavefunction changes its sign on permutation of two identical fermions and remains unchanged on permutation of two identical bosons. At first glance, this fixes the permutation properties of each system and leave no room for selection rules. However, the symmetric group ${\mathcal{S}}_N$ of permutations of $N$ symbols has also multidimensional, non-Abelian, irreducible representations (irreps), when a permutation operator ${\mathcal{P}}$ transforms the wavefunction into a superposition of several wavefunctions in the representation (see [@hamermesh; @kaplan; @pauncz_symmetric]). In physical systems, such wavefunctions can appear where a many-body Hamiltonian $\hat{H}=\hat{H}_{\mathrm{spat}}+\hat{H}_{\mathrm{spin}}$ is a sum of a spin-independent $\hat{H}_{\mathrm{spat}}$ and coordinate-independent $\hat{H}_{\mathrm{spin}}$, and each of $\hat{H}_{\mathrm{spat}}$ and $\hat{H}_{\mathrm{spin}}$ is permutation-invariant. For example, $\hat{H}_{\mathrm{spat}}$ can represent particles with spin-independent interactions, and $\hat{H}_{\mathrm{spin}}$ can describe an interaction with a homogeneous magnetic field. The spatial and spin eigenfunctions of $\hat{H}_{\mathrm{spat}}$ and $\hat{H}_{\mathrm{spin}}$, respectively, form multidimensional irreps of the symmetric group. The total wavefunction is a sum of products of the spin and spatial functions and satisfies the exclusion principle. Hamiltonians and wavefunctions of this type appear in the spin-free quantum chemistry [@pauncz_symmetric]. They can also describe spinor quantum gases, which are extensively studied starting from the first experiments [@myatt1997; @stamper1998] and the classical theoretical investigations [@ho1998; @*ohmi1998] (see book [@pitaevskii], reviews [@stamper2013; @*guan2013], and references therein). Such gases, containing atoms in several states (hyperfine or magnetic), can demonstrate a variety of non-trivial symmetries (see [@wu2003; @*wu2006] and references therein). A general Hamiltonian of a spinor gas [@ho1998] contains spin-dependent interactions. However, if atoms have closed electron shells and nuclear spins (e.g., ${}^{87}$Sr [@boyd2006; @*desalvo2010; @*tey2010] and ${}^{173}$Yb [@fukuhara2007], used in experiments), the interactions will be spin-independent with a good accuracy due to weak interaction of nuclear magnetic moments. Spin-independent interactions between the atoms can also be provided by magnetic, optical, or microwave Feshbach resonances (see [@stamper2013; @*guan2013] and references therein). In these cases, the Hamiltonian can be separated to spin-independent and coordinate-independent parts. Instead of coordinates and spins, other two kinds degrees of freedom can be considered, e.g., electronic and spin ones [@gorshkov2010].
If $\hat{H}_{\mathrm{spin}}$ is independent of the spin components, the gas becomes to be $SU(M)$-symmetric [@honerkamp2004; @gorshkov2010; @cazalilla2009], where $M=2s+1$ is the multiplicity and $s$ is the spin of the atom. This symmetry has been recently observed in experiments [@zhang2014; @*scazza2014]. States of $SU(M)$-symmetric systems are classified according to the Young diagrams $\lambda=[\lambda_1,\ldots,\lambda_M]$ — sets of $M$ non-negative non-increasing integers $\lambda_m$ that sum to $N$ \[they are pictured as $M$ rows of $\lambda_m$ boxes, see e.g. Fig. \[Fig\_sel\]\]. Transformations in the spin space couple functions within irrep of $SU(M)$. Functions in different irreps, associated with the same Young diagram, are coupled by permutations of particles, forming irreps of the symmetric group. A set of all states associated with the Young diagram $\lambda$ will be referred to here as a $\lambda$-multiplet. In generic, non-$SU(M)$ invariant systems with coordinate-independent $\hat{H}_{\mathrm{spin}}$, only the permutation symmetry survives. If $s=1/2$, the Young diagram is unambiguously determined by the total spin of the many-body system $S$ as $\lambda_1-\lambda_2=2S$. If $s>1/2$, the irreps of both groups contain contributions with different total spins [@landau; @kaplan].
Every permutation ${\mathcal{P}}$ commutes with the Hamiltonian and, therefore, is an integral of motion [@dirac; @dirac1929]. However, permutations do not commute with each other. The commuting integrals of motion [@dirac; @dirac1929] are the character operators $\hat{\chi}(C_N)=\sum_{{\mathcal{P}}\in C_N}{\mathcal{P}}/g(C_N)$. Here the sum is over all $g(C_N)$ permutations ${\mathcal{P}}$ of $N$ particles in a conjugate class $C_N$ (two permutations ${\mathcal{P}}$ and ${\mathcal{P}}'$ are conjugate if there exist a permutation ${\mathcal{Q}}$ such that ${\mathcal{P}}'={\mathcal{Q}}^{-1} {\mathcal{P}}{\mathcal{Q}}$, see [@hamermesh; @kaplan; @pauncz_symmetric]). The operator $\hat{\chi}$ for transpositions (permutations of two particles) was also used [@fang2011] for the classification of states of a Bose-Fermi mixture.
#### Wavefunctions
The spin $\Xi^{[\lambda]}_{t l}$ and spatial $\Phi^{[\lambda]}_{t n}$ eigenfunctions form irreps of the symmetric group, associated with the Young diagram $\lambda$, and are transformed by a permutation ${\mathcal{P}}$ as [@hamermesh; @kaplan; @pauncz_symmetric] $
{\mathcal{P}} \Xi^{[\lambda]}_{t l}=\sum_{t'} D_{t' t}^{[\lambda]}({\mathcal{P}})\Xi^{[\lambda]}_{t' l}
$, $
{\mathcal{P}} \Phi^{[\lambda]}_{t n}=\mathrm{sig}({\mathcal{P}})\sum_{t'} D_{t' t}^{[\lambda]}({\mathcal{P}})\Phi^{[\lambda]}_{t' n}
$, where the standard Young tableaux $t$ and $t'$ of the shape $\lambda$ label the functions within irreps, and $D_{t' t}^{[\lambda]}({\mathcal{P}})$ are the Young orthogonal matrices (see [@kaplan; @pauncz_symmetric]). The factor $\mathrm{sig}({\mathcal{P}})$ is the permutation parity for fermions and $\mathrm{sig}({\mathcal{P}})\equiv 1$ for bosons. For fermions $D_{t' t}^{[\tilde{\lambda}]}({\mathcal{P}})\equiv\mathrm{sig}({\mathcal{P}})D_{t' t}^{[\lambda]}({\mathcal{P}})$ are matrices of the conjugate representation with $\tilde{\lambda}$ obtained from $\lambda$ by changing rows and columns. The functions belonging to the same irrep can be considered as components of a vector (or pseudovector) of the same dimension $f_\lambda$ as the representation. Each permutation corresponds then to a rotation \[represented by the matrix $D_{t' t}^{[\lambda]}({\mathcal{P}})$\] of the vectors. The total wavefunction $$\Psi^{[\lambda]}_{n l}=f_\lambda^{-1/2}\sum_{t}
\Phi^{[\lambda]}_{t n}\Xi^{[\lambda]}_{t l} ,
\label{Psilamnl}$$ being a scalar product of the vectors of the spin and spatial wavefunctions, is then scalar (or pseudoscalar) and is transformed as ${\mathcal{P}}\Psi^{[\lambda]}_{n l}=\mathrm{sig}({\mathcal{P}})\Psi^{[\lambda]}_{n l}$, in the agreement to the exclusion principle. Different irreps, associated with the same Young diagram, are labeled by $n$ and $l$ for the spatial and spin functions, respectively.
Each particle ($j$) can occupy one of the spin states $|m(j)\rangle$, $1\leq m\leq M=2s+1$. The quantum number $m$ can also denote internal states of composite particles, e.g., hyperfine states of atoms. In the last case, the even (odd) number $M$ of internal states corresponds to the integer (half-integer) spin, with no relation to the permutation symmetry of the total wavefunction. The many-body spin eigenfunctions are expressed as sums of configurations [[@supplement]]{}, $$\Xi^{[\lambda]}_{t l}=\sum_{\{N\},r} B^{[\lambda]}_{l \{N\} r}
\sum_{{\mathcal{P}}}D_{t r}^{[\lambda]}({\mathcal{P}})
\prod_{j=1}^{N}|m_j({\mathcal{P}}j)\rangle .
\label{SpinFun}$$ Here the configurations correspond to the different occupations $N_m$ of the states $|m\rangle$, such that $m_j=m$ for $\sum_{i=1}^{m-1}N_i< j\leq\sum_{i=1}^{m}N_i$. The spatial wavefunction can be represented in a similar form [[@supplement]]{}, like the configuration-interaction method in quantum chemistry (see [@pauncz_symmetric]). The total wavefunction [(\[Psilamnl\])]{} cannot be represented as a product of the states of individual particles. It is therefore a wavefunction of a many-body entangled state.
The spin state occupations $N_m$ in the wavefunction [(\[SpinFun\])]{} are restricted by the associated Young diagram. For $s=1/2$-particles, the total spin $S=\lambda_1-N/2$. Its projection $S_z$ is related to the occupations $N_{\uparrow\downarrow}$ of the spin up/down states as $S_z=N_{\uparrow}-N/2=N/2-N_{\downarrow}$, leading to $N_{\uparrow\downarrow}\leq \lambda_1$, since $-S\leq S_z\leq S$. Similar restrictions are obtained in the general case of $s>1/2$ [@supplement]. They are: the spin-state occupations $N_i$ cannot exceed the first row length, $N_i\leq \lambda_1$ (obtained in [@kaplan]); if occupations of $m$ states ($1\ldots m$ for definiteness) are equal to lengths of the first $m$ rows, $N_i=\lambda_i (1\leq i \leq m)$, occupations of other states cannot exceed the next row length, $N_i\leq\lambda_{m+1} (m+1\leq i\leq N$). This demonstrates the physical meaning of the Young diagrams. These restrictions are valid for spatial functions as well; for fermions the spatial state occupations are restricted by row lengths of the conjugate Young diagram $\tilde{\lambda}$, which are equal to the column lengths of $\lambda$.
#### Selection rules
If an interaction depends on spins or coordinates only, it can couple only the states [(\[Psilamnl\])]{} associated with the same Young diagram, due to orthogonality of the spatial or spin functions, respectively. A nonseparable spin- and coordinate-dependent interaction of $k$ particles $j_1,\ldots,j_k$, $$\begin{aligned}
\hat{W}_k(\{j\})=\sum_{\{m\},\{m'\}}\langle\{m'\}|\hat{W}(\mathbf{r}_{j_1},\ldots,\mathbf{r}_{j_k})|\{m\}\rangle
\nonumber
\\
\times
\prod_{i=1}^k|m'_i(j_i)\rangle\langle m_i(j_i)| ,
\label{Wk}
$$ can couple only the states if their Young diagrams, $\lambda$ and $\lambda'$, differ by relocation of no more than $k$ boxes between their rows [[@supplement]]{}. These selection rules (see Fig. \[Fig\_sel\]) can be expressed as $$\sum_{m=1}^M |\lambda_m-\lambda'_m|\leq 2k ,
\label{SelRule}$$ while both diagrams have to satisfy the standard relations, $\lambda_{m+1}\leq\lambda_m$, $\lambda_m\geq 0$, and $\sum_{m=1}^M\lambda_m=N$. For $s=1/2$ and $k=1$, we have $|\lambda'_1-\lambda_1|\leq 2$, or $|S'-S|\leq 1$. It agrees to the conventional selection rule for dipole transitions. Although many-body states of higher-spin particles generally do not have a defined total spin [@landau; @kaplan], a maximal spin $$S=(s+1)N-\sum_{m=1}^M m\lambda_m
\label{MaxSpin}$$ can be introduced [[@supplement]]{}. However, in this case the selection rule restricts $2s$ parameters and cannot be expressed in therms of $S$ alone. For example, for $s=1$ and $k=1$ there are only 6 allowed transitions \[see Figs. \[Fig\_sel\](a) and \[Fig\_sel\_en\_cor\](a)\], although the allowed number of $\lambda$-multiplets with given $S$ is of order of $N$.
![Selection rules [(\[SelRule\])]{}. (a) Six states, which can be coupled to a given state (associated with the central Young diagram) by a one-body interaction $\hat{W}_1$ \[[Eq. (\[Wk\])]{}\] for $s=1$. (b) An example of the coupling by a two-body interaction $\hat{W}_2$ \[[Eq. (\[Wk\])]{}\]. The dashed boxes are relocated. \[Fig\_sel\]](sel_N12k1_2.eps){width="3.4in"}
![ (a) Allowed transitions between $\lambda=[(N-\lambda_2+S)/2,\lambda_2,(N-\lambda_2-S)/2]$-multiplets (red arrows) due to a one-body interaction $W_1$ [(\[Wk\])]{} for $N=18$ bosons with the spin $s=1$. The average energies of the multiplets \[circles, see [Eq. (\[Eav0\])]{}\] for bosons are proportional to the average local two-body correlations [(\[rhotilklam\])]{}. The dashed lines connect the points with the same maximal spin $S$. (b) The $3$- and $4$-body average local correlations for the same multiplets (black and red, respectively). (c) The $3$-body average local correlations for the $N=18$ fermions with the spin $s=1$ as functions of $\lambda_3$ given the maximal spin $S$ (denoted by numbers). \[Fig\_sel\_en\_cor\]](correl234_3.eps){width="3.4in"}
#### Correlation rules
The probabilities of finding the given distances $\mathbf{R}_i$ between $k$ particles or the given differences $\mathbf{q}_i$ between their momenta, the $k$-body spatial $\bar{\rho}_k(\{\mathbf{R}\})$ or momentum $\bar{g}_k(\{\mathbf{q}\})$ correlations, respectively, are the expectation values of the operators $$\begin{aligned}
\hat{\rho}_k(\{\mathbf{R}\})=\prod_{i=2}^k
\delta(\mathbf{r}_1-\mathbf{r}_i-\mathbf{R}_{i-1})
\label{hatrhok}
\\
\hat{g}_k(\{\mathbf{q}\})=\prod_{i=2}^k
\delta(\mathbf{p}_1-\mathbf{p}_i-\mathbf{q}_{i-1}) .
\label{hatgk}\end{aligned}$$ Here $\mathbf{r}_i$ and $\mathbf{p}_i$ are, respectively, $D$-dimensional coordinates and momenta (in physical applications, $D$ can be either $1$, $2$, or $3$), and for $D>1$ the $\delta$-functions in [(\[hatrhok\])]{} are properly renormalized. The local correlations, probabilities of finding $k$ particles in the same point (or with the same momenta) are determined by $ \hat{\rho}_k(\{0\})$ (or $ \hat{g}_k(\{0\})$). Their eigenstate expectation values, $\langle \Psi^{[\lambda]}_{n l}|\hat{\rho}_k(\{0\})|\Psi^{[\lambda]}_{n l}\rangle$ and $\langle \Psi^{[\lambda]}_{n l}|\hat{g}_k(\{0\})|\Psi^{[\lambda]}_{n l}\rangle$, vanish if the correlation order $k$ exceeds the first row length in the Young diagram for the spatial wavefunction — $\lambda_1$ for bosons or $\tilde{\lambda}_1$ for fermions (which is equal to the number of rows in the Young diagram $\lambda$ for the spin wavefunction)[@supplement]. For fermions, these restrictions are stricter than the ones provided by the Pauli principle, which states that $k$ cannot exceed the number of different spin states (this number can be greater than the number of rows).
#### Correlations and characters
The $\lambda$-multiplet-average of a $k$-body spin-independent operator $\hat{F}_k$ is expressed as [[@supplement]]{}, $$\begin{aligned}
&& \bar{F}^{[\lambda]}_k\equiv\frac{1}
{\tilde{\mathcal{N}}_{\lambda}}
\sum_n \langle \Psi^{[\lambda]}_{n l}|\hat{F}_k|\Psi^{[\lambda]}_{n l}\rangle
\nonumber
\\
&& =\frac{f_{\lambda}}{\tilde{\mathcal{N}}_{\lambda}}
\sum_{C_N}\mathrm{sig}(C_N)g(C_N)\tilde{\chi}_{\lambda}(C_N)
\langle F_k\rangle_{C_N}
\label{Fkexp}
\end{aligned}$$ where $\tilde{\mathcal{N}}_{\lambda}$ is the total number of the spatial wavefunctions, associated with the Young diagram $\lambda$. The multiplet-dependence is given by the normalized characters $\tilde{\chi}_{\lambda}(C_N)$. The factors $\langle F_k\rangle_{C_N}$ [[@supplement]]{} are independent of the multiplet. If $\hat{F}_k$ is the coordinate-dependent Hamiltonian $\hat{H}_{\mathrm{spat}}$ and each spatial orbital is occupied only by one particle, [Eq. (\[Fkexp\])]{} is reduced to the average multiplet energy, obtained in [@heitler1927].
The local spatial (or momentum) correlations are determined by $\langle \rho_k(\{0\})\rangle_{C_k}$ (or $\langle g_k(\{0\})\rangle_{C_k}$), which become independent of the conjugate class $C_k$ [[@supplement]]{} if each spatial orbital is occupied only by one particle. In this case $\tilde{\mathcal{N}}_{\lambda}=f_{\lambda}$. Then the multiplet dependence of the average local correlations, $\bar{\rho}^{[\lambda]}_k(\{0\})=\tilde{\rho}^{[\lambda]}_k\langle \rho_k(\{0\})\rangle$ and $\bar{g}^{[\lambda]}_k(\{0\})=\tilde{\rho}^{[\lambda]}_k\langle g_k(\{0\})\rangle$, is given by the universal factor $$\tilde{\rho}^{[\lambda]}_k=
\sum_{C_k}\mathrm{sig}(C_k)g(C_k)\tilde{\chi}_\lambda (C_k) .
\label{rhotilklam}$$ Thus, the integrals of motion $\tilde{\chi}_\lambda (C_k)$, corresponding to the permutation symmetry, are related to quantities $\bar{\rho}^{[\lambda]}_k(\{0\})$ and $\bar{g}^{[\lambda]}_k(\{0\})$, which can be measured in experiments.
In a system with zero-range two-body interactions, $V(\mathbf{r}'-\mathbf{r})=V N(N-1)\hat{\rho}_2(\{0\})/2$, the average energy of the $\lambda$-multiplet, counted from the multiplet-independent energy of non-interacting particles, is $$\bar{E}_{\lambda}=V\frac{N(N-1)}{2}[1\pm \tilde{\chi}_\lambda (\{2\})]\langle \rho_2(\{0\})\rangle .
\label{Eav0}$$ Here, the sign $+/-$ is taken for bosons/fermions and $\{2\}$ is the conjugate class of the transpositions [[@supplement]]{}. The energy attains its maximum for bosons and minimum for fermions at $\lambda=[N]$, when the normalized character for transpositions attains its maximum $\tilde{\chi}_\lambda (\{2\})=1$ [@supplement]. In this state (belonging to a one-dimensional irrep) the total spin is defined and has the maximal allowed value $N s$. The minimal average energy for bosons and the maximal one for fermions correspond to $\lambda=[(\lambda_M+1)^k,\lambda_M^{M-k}]$, where $\tilde{\chi}_\lambda (\{2\})$ attains its minimum $[(N+k)\lambda_M+(M-k+1)k-M N]/[N(N-1)]$ [@supplement]. (Here $\lambda_M$ and $k$ are, respectively, the quotient and remainder of the division of $N$ by $M$.) This $\lambda$-multiplet corresponds to the minimum $S=(M-k)k/2$ of the maximal spin. If $N$ is a multiple of $M$, it has the defined total spin $S=0$.
These general properties are confirmed for particular values of $s$ using the explicit expressions [@supplement] obtained with the characters [@lassalle2008]. For $s=1/2$, the energy $\bar{E}_{\lambda}=V[N(N-1)/2\pm (N^2/4-N+S^2+S)]\langle \rho_2(\{0\})\rangle$ is a monotonic function of the total spin $S$. For $s=1$, $\bar{E}_{\lambda}=V[N(N-1)/2 \pm(N^2-6N+S^2+4S+3\lambda_2^2-2N\lambda_2)/4]\langle \rho_2(\{0\})\rangle$ is a sum of quadratic functions of the maximal spin $S$ and $\lambda_2$. Its dependence of $S$ may be non-monotonic \[see Fig. \[Fig\_sel\_en\_cor\](a)\]. The multiplet-dependencies of the $3$- and $4$-body correlations ($\tilde{\rho}^{[\lambda]}_3$ and $\tilde{\rho}^{[\lambda]}_4$) are shown in Fig. \[Fig\_sel\_en\_cor\](b,c). For fermions, $3$-body correlations vanish for two-row Young diagrams \[$\lambda_3=0$, see Fig. \[Fig\_sel\_en\_cor\](c)\], in agreement with the correlation rules. The averages are independent of the particle spin, which only restricts the number of the Young diagram rows.
![A scheme of population of $\lambda$-multiplets using spatially-homogeneous spin-changing pulses $\hat{W}_{\text{hom}}(t)$ (blue dashed arrows) and spatially-inhomogeneous spin-conserving pulse $\hat{W}_{\text{inh}}(\mathbf{r},t)$ (red solid arrows). Shapes of the Young diagram boxes denote numbers of occupied atomic spin states. \[Fig\_trans\]](popYoung.eps){width="3.4in"}
#### Possible realization
The states, associated with various Young diagrams, may be selectively populated using two types of pulses (see Fig. \[Fig\_trans\]). A spatially-homogeneous spin-changing pulse $\hat{W}_{\text{hom}}(t)=\sum_{m\neq m'}W_{m m'}(t)|m\rangle\langle m'|$ changes the spin states of atoms but, being coordinate-independent, does not change the Young diagram associated with the many-body state. $\pi/2$ pulses of this type are generally used in experiments with spinor cold gases [@matthews1998; @sagi2010]. A spatially-inhomogeneous spin-conserving pulse $\hat{W}_{\text{inh}}(\mathbf{r},t)=\sum_{m}W_{m}(\mathbf{r},t)|m\rangle\langle m|$ is a one-body interaction of the form [(\[Wk\])]{}. It can relocate one box in the Young diagram, according to the selection rules, but does not change the spin states of the atoms. If all atoms are initially formed in the same spin states, the many-body spin wavefunction is associated with the one-row Young diagram $[N]$. A pulse of the type $\hat{W}_{\text{hom}}(t)$ can transfer each atom to a superposition of two spin states. For fermions, all local correlations vanish in this state, since its spatial wavefunction is associated with the one-column Young diagram. Then a pulse of the type $\hat{W}_{\text{inh}}(\mathbf{r},t)$ can lead to the spin wavefunction associated with a two-row Young diagram $[N-1,1]$, depleting the $[N]$ state. The depletion could be detected in a Ramsey experiment (like [@sagi2010; @sagi2010a]) by applying the second pulse $\hat{W}_{\text{hom}}(t)$. Further pulses of the type $\hat{W}_{\text{inh}}(\mathbf{r},t)$ can provide only one- and two-row Young diagrams, since only two atomic spin states are occupied. For fermions, only two-body local correlations do not vanish in these many-body states. A population of the third atomic spin state by $\hat{W}_{\text{hom}}(t)$ does not change the vanishing correlations, but allows to provide three-row Young diagrams using $\hat{W}_{\text{inh}}(\mathbf{r},t)$. States, associated with arbitrary Young diagrams can be populated in this way, and, for fermions, the number of rows can be tested by non-vanishing correlations. More comprehensive information on the populated states can be provided by the correlation values, since they are related to the characters \[see [(\[rhotilklam\])]{}\] and the Young diagram is unambiguously related to the values of all characters [@dirac; @dirac1929]. Then the correlations can allow to analyze coherent or statistical mixtures of various $\lambda$-multiplets.
The selection and correlation rules are applicable to any system with two kinds of separable degrees of freedom, e.g. to a spinor gas with spin-independent interactions in arbitrary trap potentials. The simple relation [(\[rhotilklam\])]{} between correlations and characters is obtained for the single occupations of spatial orbitals. This can be realized, for example, with cold atoms in a $D$-dimensional optical lattice [@bloch2008; @*yukalov2009; @*svistunov] in the unit-filling Mott (or fermionic band)-insulator regime, when each lattice site is occupied by one atom. In this regime, the spatial correlations do not demonstrate a substantial dependence on the spin state [[@supplement]]{} (indeed, in any state, each site is occupied by one atom). The momentum correlations oscillate as functions of each component of $\mathbf{q}_j$ with the maximal values (at $\mathbf{q}_j=0$) [[@supplement]]{} $$\bar{g}_k(\{0\})=\tilde{\rho}^{[\lambda]}_k f_k(\{0\}) .
\label{bargkzero}$$ Here the probability of finding differences $\mathbf{q}_{j}$ between momenta of $k$ non-interacting particles $
f_k(\{\mathbf{q}\})=\int d^D p |\tilde{w}(\mathbf{p})|^2
\prod_{i=2}^k |\tilde{w}(\mathbf{p}+\mathbf{q}_{i-1})|^2
$ is the convolution of the momentum distributions $|\tilde{w}(\mathbf{p})|^2$. The correlations and momentum distributions in [(\[bargkzero\])]{} can be measured in experiments, and the factor $\tilde{\rho}^{[\lambda]}_k$ is a linear combination of the characters [(\[rhotilklam\])]{}.
#### Conclusions
Rather abstract mathematical constructs — Young diagrams and characters of the symmetric group — have a physical meaning. Young diagrams classify many-body states of systems with separable spin and spatial degrees of freedom. For such state, a maximal spin [(\[MaxSpin\])]{}, occupations of one-body states, and non-vanishing correlations are determined by row lengths and number of rows in the associated Young diagram. A transition due to a nonseparable $k$-body interaction cannot move more than $k$ boxes between the Young diagram rows \[see selection rules [(\[SelRule\])]{}\]. The characters — integrals of motion, corresponding to permutation symmetry — are related to correlations of several particles in the coordinate or momentum space \[see [(\[Fkexp\])]{} and [(\[rhotilklam\])]{}\], which can be measured in experiments. This demonstrates that the characters have a physical meaning, similarly to the integrals of motion corresponding to many other symmetries.
This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The author gratefully acknowledge useful conversations with A. Ben-Reuven, N. Davidson, I. G. Kaplan, M. Olshanii, R. Pugatch, A. Simoni, and B. Svistunov.
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[**Supplemental material for: Permutation symmetry in spinor quantum gases: selection rules, conservation laws, and correlations**]{}
Vladimir A. Yurovsky
Numbers of equations in the Supplemental material are started from S. References to equations in the Letter do not contain S.
Spin wavefunctions {#SI_SpinWF}
==================
Suppose that $N$ particles, labeled by $j$, occupy $M$ orthonormal one-body states $|m(j)\rangle$, $1\leq m\leq M$. The basic functions of irreducible representations (irreps) of the symmetric group ${\mathcal{S}}_N$ can be expressed as (see [@kaplan; @pauncz_symmetric]), $$|\{N\},\lambda,t,r\rangle=\left( \frac{f_{\lambda}}{N!}\right)^{1/2}
\sum_{{\mathcal{P}}}D_{t r}^{[\lambda]}({\mathcal{P}})\Xi_{\{N\}{\mathcal{P}}} ,
\label{Nlamtr}$$ where $D_{t r}^{[\lambda]}({\mathcal{P}})$ are the Young orthogonal matrices, $\lambda$ is a Young diagram, associated with the irrep, $t$, $r$ are the standard Young tableaux of the shape $\lambda$, and ${\mathcal{P}}$ are permutations of $N$ symbols. A permutation ${\mathcal{P}}$ transforms the functions [(\[Nlamtr\])]{} as $${\mathcal{P}}|\{N\},\lambda,t,r\rangle =\sum_{t'}D_{t' t}^{[\lambda]}({\mathcal{P}})
|\{N\},\lambda,t',r\rangle.$$ Thus $t$ labels the basic functions of the representation and $r$ labels different representations, associated with the same $\lambda$. The irrep dimension $f_{\lambda}$ is equal to the number of standard Young tableaux of the shape $\lambda$ [@kaplan], $$f_{\lambda}=\frac{N!\prod_{m<m'}(\lambda_m-m-\lambda_{m'}+m')}
{\prod_{m=1}^M (\lambda_m+M-m)!} .$$ The permutted states $$\Xi_{\{N\}{\mathcal{P}}}=\prod_{j=1}^N |m_j({\mathcal{P}}j)\rangle
\label{XiNP}$$ are determined by the set $\{N\}$ of occupations $N_m$ of the states $|m\rangle$. The one-body states are arranged in non-decreasing order of $m$, such that $m_j=m$ for $\sum_{i=1}^{m-1}N_i< j\leq\sum_{i=1}^{m}N_i$ and $\sum_{m=1}^{M}N_m=N$. The number of possible sets $\{N\}$, the number $\mathcal{N}(N,M)$ of distributions of $N$ identical particles to $M$ distinct states, is calculated in combinatorics [@bogart] as $$\mathcal{N}(N,M)=\frac{(N+M-1)!}{N!(M-1)!} .$$ If all states have single occupation ($N_m=1$), all $m_j$ are different, all the functions [(\[Nlamtr\])]{} are orthogonal, $$\langle \{N\},\lambda',t',r'|\{N\},\lambda,t,r\rangle =
\delta_{\lambda \lambda'} \delta_{t t'} \delta_{r r'},$$ and $r$ can be any of the $f_{\lambda}$ standard Young tableaux. However, if multiple occupations of the states $|m\rangle$ are allowed, the permutted states [(\[XiNP\])]{} become non-orthogonal $$\langle \Xi_{\{N'\}{\mathcal{P'}}}|\Xi_{\{N\}{\mathcal{P}}}\rangle=\delta_{\{N\} \{N'\}}
\sum_{{\mathcal{P}}^{\{N\}}}\delta_{{\mathcal{P'}},{\mathcal{P}}{\mathcal{P}}^{\{N\}}} ,$$ where $${\mathcal{P}}^{\{N\}}=\prod_{m=1}^M{\mathcal{P}}^{(m)}
\label{PN}$$ and ${\mathcal{P}}^{(m)}$ can be either the identity permutation or any permutation of $\sum_{i=1}^{m-1}N_i< j\leq\sum_{i=1}^{m}N_i$, leaving all other $j$ unchanged.
This leads to non-orthogonality of the basic functions [(\[Nlamtr\])]{} in different irreps, $$\begin{aligned}
\langle \{N'\},\lambda',t',r'&|&\{N\},\lambda,t,r\rangle =
\delta_{\{N\} \{N'\}}\frac{\sqrt{f_{\lambda'}f_{\lambda}}}{N!}
\nonumber
\\
&&\times
\sum_{{\mathcal{P}},{\mathcal{P}}^{\{N\}}}D_{t' r'}^{[\lambda']}({\mathcal{P}})
D_{t r}^{[\lambda]}({\mathcal{P}}{\mathcal{P}}^{\{N\}})
\nonumber
\\
&& = \delta_{\{N\} \{N'\}}\delta_{\lambda \lambda'} \delta_{t t'}
\tilde{D}_{r' r}^{[\lambda]}(\{N\}),
\phantom{qqqq}
\label{Nlamtrover}
$$ where $$\tilde{D}_{r' r}^{[\lambda]}(\{N\})=
\sum_{{\mathcal{P}}^{\{N\}}}D_{r' r}^{[\lambda]}({\mathcal{P}}^{\{N\}}) .
\label{tildeD}$$ The above derivation uses the following properties of the Young orthogonal matrices (see [@kaplan; @pauncz_symmetric]), $$\begin{aligned}
D_{r t}^{[\lambda]}({\mathcal{P}}{\mathcal{Q}})=
\sum_{t'} D_{r t'}^{[\lambda]}({\mathcal{P}}) D_{t' t}^{[\lambda]}({\mathcal{Q}})
\label{ProdYoung}
\\
\sum_{{\mathcal{P}}} D_{t' r'}^{[\lambda']}({\mathcal{P}})D_{t r}^{[\lambda]}({\mathcal{P}})
= \frac{N!}{f_{\lambda}}\delta_{\lambda \lambda'} \delta_{t t'} \delta_{r r'}
\label{OrthYoung}\end{aligned}$$ The matrix $\tilde{D}_{r' r}^{[\lambda]}$ is symmetric, since $D_{r r'}^{[\lambda]}({\mathcal{P}})=D_{r' r}^{[\lambda]}({\mathcal{P}}^{-1})$ for orthogonal matrices and $({\mathcal{P}}^{\{N\}})^{-1}$ belongs to the subgroup $\{{\mathcal{P}}^{\{N\}}\}$. Therefore it has $f_{\lambda}$ orthogonal normalized eigenvectors $d_{\nu r}^{[\lambda]}(\{N\})$, corresponding to eigenvalues $\delta_{\nu}^{[\lambda]}(\{N\})$, and can be represented as $$\tilde{D}_{r' r}^{[\lambda]}=\sum_{\nu}d_{\nu r'}^{[\lambda]}
\delta_{\nu}^{[\lambda]}d_{\nu r}^{[\lambda]} .
\label{Devecval}$$ A similar representation exists for the square of this matrix, $$(\tilde{D}^{[\lambda]})_{r' r}^2=\sum_{\nu}d_{\nu r'}^{[\lambda]}
(\delta_{\nu}^{[\lambda]})^2 d_{\nu r}^{[\lambda]} .
\label{D2evecval}$$ At the same time, using [Eq. (\[ProdYoung\])]{} we get, $$\begin{aligned}
(\tilde{D}^{[\lambda]})_{r' r}^2=\sum_{t,{\mathcal{P}}^{\{N\}},\tilde{{\mathcal{P}}}^{\{N\}}}D_{r' t}^{[\lambda]}({\mathcal{P}}^{\{N\}})
D_{t r}^{[\lambda]}(\tilde{{\mathcal{P}}}^{\{N\}})
\nonumber
\\
=\{N\}!\tilde{D}_{r' r}^{[\lambda]} ,
$$ since the product ${\mathcal{P}}^{\{N\}}\tilde{{\mathcal{P}}}^{\{N\}}$ belongs to the subgroup $\{{\mathcal{P}}^{\{N\}}\}$. Here $\{N\}!\equiv\prod_{m=1}^N N_m!$. Then, Eqs. [(\[Devecval\])]{} and [(\[D2evecval\])]{} lead to the equality $(\delta_{\nu}^{[\lambda]})^2=\{N\}!\delta_{\nu}^{[\lambda]}$. This means that $\delta_{\nu}^{[\lambda]}$ can have the value of either $0$ or $\{N\}!$. Finally, [Eq. (\[Nlamtrover\])]{} allows to prove that the functions $$|\{N\},\lambda,t,\nu\rangle = (\{N\}!)^{-1/2}
\sum_r d_{\nu r}^{[\lambda]}|\{N\},\lambda,t,r\rangle ,
\label{Nlamtnu}$$ with $\nu$ corresponding to $\delta_{\nu}^{[\lambda]}>0$, are normalized and orthogonal for different $\nu$.
Thus the number of irreps $\tilde{f}_{\lambda}(\{N\})$, associated with the same $\lambda$, is equal to the number of non-zero eigenvalues $\delta_{\nu}^{[\lambda]}$. Functions in these irreps form a complete basic, since Eqs. [(\[Nlamtr\])]{}, [(\[XiNP\])]{}, [(\[Nlamtrover\])]{}, [(\[ProdYoung\])]{}, [(\[OrthYoung\])]{}, and [(\[Devecval\])]{} lead to $$\begin{aligned}
\sum_{\lambda,t}\sum_{\nu}{}'
\langle \Xi_{\{N\}{\mathcal{P}}'}|\{N\},\lambda,t,\nu\rangle
\langle \{N\},\lambda,t,\nu|\Xi_{\{N\}{\mathcal{P}}}\rangle
\nonumber
\\
=\sum_{{\mathcal{P}}^{\{N\}}}\delta_{{\mathcal{P'}},{\mathcal{P}}{\mathcal{P}}^{\{N\}}} ,
\phantom{qqqqq}
\label{ResIdent}
$$ where $\sum_{\nu}'$ means the summation over all $\nu$ with $\delta_{\nu}^{[\lambda]}>0$. Equation [(\[ResIdent\])]{} is nothing but the resolution of identity, as $\Xi_{\{N\}{\mathcal{P}}{\mathcal{P}}^{\{N\}}}$ is equal to $\Xi_{\{N\}{\mathcal{P}}}$ for each ${\mathcal{P}}^{\{N\}}$.
For example, if $M=2$ (as for $s=1/2$ particles) only one irrep is associated with each $\lambda$ (see [@yurovsky2013], where explicit expressions are derived for the wavefunctions in this case).
Consider a permutation-symmetric Hamiltonian $\hat{H}_{\mathrm{spin}}={\mathcal{P}}^{-1}\hat{H}_{\mathrm{spin}}{\mathcal{P}}$. Using Eqs. [(\[Nlamtr\])]{}, [(\[ProdYoung\])]{}, and [(\[OrthYoung\])]{}, we get
$$\begin{aligned}
\langle \{N'\},\lambda',t',r'|\hat{H}_{\mathrm{spin}}|\{N\},\lambda,t,r\rangle
&=&\frac{\sqrt{f_{\lambda'}f_{\lambda}}}{N!}\sum_{{\mathcal{P}},{\mathcal{P}}'}
D_{t' r'}^{[\lambda']}({\mathcal{P}}')D_{t r}^{[\lambda]}({\mathcal{P}})
\langle \Xi_{\{N'\}{\mathcal{P}}^{-1}{\mathcal{P}}'}|{\mathcal{P}}^{-1}\hat{H}_{\mathrm{spin}}{\mathcal{P}}|\Xi_{\{N\}{\mathcal{E}}}\rangle
\nonumber
\\
&=&\frac{\sqrt{f_{\lambda'}f_{\lambda}}}{N!}\sum_{{\mathcal{P}},{\mathcal{Q}}}
D_{t' r'}^{[\lambda']}({\mathcal{P}}{\mathcal{Q}})D_{t r}^{[\lambda]}({\mathcal{P}})
\langle \Xi_{\{N'\}{\mathcal{Q}}}|\hat{H}_{\mathrm{spin}}|\Xi_{\{N\}{\mathcal{E}}}\rangle
\nonumber
\\
&=&\delta_{\lambda \lambda'} \delta_{t t'} \sum_{{\mathcal{Q}}}D_{r r'}^{[\lambda]}({\mathcal{Q}})\langle \Xi_{\{N'\}{\mathcal{Q}}}|\hat{H}_{\mathrm{spin}}|\Xi_{\{N\}{\mathcal{E}}}\rangle ,
\label{Hspinmatr}
\end{aligned}$$
where ${\mathcal{E}}$ is the identity permutation. This means that the coupling of the states [(\[Nlamtnu\])]{} is diagonal in $\lambda$ and $t$, $\langle \{N'\},\lambda',t',\nu'|\hat{H}_{\mathrm{spin}}|\{N\},\lambda,t,\nu\rangle \sim \delta_{\lambda \lambda'} \delta_{t t'}$, and independent of $t$ (indeed, it is a general group-theoretical property of irrep basic functions, see [@kaplan]). Then the eigenfunctions of $\hat{H}_{\mathrm{spin}}$ can be expanded as $$ \Xi^{[\lambda]}_{t l}=\sum_{\{N\}}\sum_{\nu}{}' A^{[\lambda]}_{l \{N\} \nu}
|\{N\},\lambda,t,\nu\rangle ,
\label{Xilamtl_Nlamtnu}$$ where the coefficients $A^{[\lambda]}_{l \{N\} \nu}$ form eigenvectors of the Hamiltonian matrix, $$\begin{aligned}
\sum_{\{N'\}}\sum_{\nu'}{}'
\langle \{N\},\lambda,t,\nu|\hat{H}_{\mathrm{spin}}|\{N'\},\lambda,t,\nu'\rangle
A^{[\lambda]}_{l \{N'\} \nu'}
\nonumber
\\
=E^{[\lambda]}_{l}A^{[\lambda]}_{l \{N\} \nu} .
\phantom{qqqqq}
\label{HAeqEA}
$$ Due to hermiticity of the matrix, its eigenvectors from a complete and orthonormal basic set, $$\begin{aligned}
\sum_l A^{[\lambda]}_{l \{N'\} \nu'} A^{[\lambda]}_{l \{N\} \nu}=
\delta_{\{N\}\{N'\}} \delta_{\nu \nu'}
\label{Acomp}
\\
\sum_{\{N\}}\sum_{\nu}{}' A^{[\lambda]}_{l' \{N\} \nu} A^{[\lambda]}_{l \{N\} \nu}=
\delta_{l l'} .
\label{Aorth}\end{aligned}$$ Finally, using [Eq. (\[Nlamtnu\])]{}, one obtains [Eq. ([2]{})]{} with $$B^{[\lambda]}_{l \{N\} r}= \left( \frac{f_\lambda}{N! \{N\}!}\right) ^{1/2}
\sum_{\nu}{}' d_{\nu r}^{[\lambda]}A^{[\lambda]}_{l \{N\} \nu} .
\label{Blamlnr}$$
Spatial wavefunctions {#SI_SpatWF}
=====================
Like the spin wavefunctions, the spatial ones can be expressed in terms of basic functions of the symmetric group irreps, $$\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}=
\left( \frac{f_{\lambda}}{N!}\right)^{1/2}
\sum_{{\mathcal{P}}}\mathrm{sig}({\mathcal{P}})D_{t r}^{[\lambda]}({\mathcal{P}})
\prod_{j=1}^{N} \varphi_{m_j}(\mathbf{r}_{{\mathcal{P}}j}) .
\label{tilPhilamNtr}$$ Here the factor $\mathrm{sig}({\mathcal{P}})$ is the permutation parity for fermions and $\mathrm{sig}({\mathcal{P}})\equiv 1$ for bosons. For fermions, it provides the conjugate representation with the matrices $D_{\tilde{t} \tilde{r}}^{[\tilde{\lambda]}}({\mathcal{P}})=\mathrm{sig}({\mathcal{P}})D_{t r}^{[\lambda]}({\mathcal{P}})$, where $\tilde{\lambda}$ is obtained from $\lambda$ by changing rows and columns. (In the following, the notation $\tilde{\lambda}$ will be used for bosons as well, meaning $\tilde{\lambda}=\lambda$.) The proper orthonormal one-body spatial orbitals $\varphi_{m}(\mathbf{r})$ ($1\leq m\leq M_{\mathrm{spat}}$) depend on the $D$-dimensional coordinate $\mathbf{r}$ (in concrete applications, $D$ can be either $1$, $2$, or $3$). The quantum numbers $m_j$ are determined by the set of occupations $\{\tilde{N}\}$ in the same way as in the case of the spin functions. Eigenfunctions of the permutation-symmetric Hamiltonian $\hat{H}_{\mathrm{spat}}$, $$\Phi^{[\lambda]}_{t n}=\sum_{\{\tilde{N}\},r}
\left(\{\tilde{N}\}!\right) ^{-1/2}\sum_{\nu}{}'
A^{[\lambda]}_{n \{\tilde{N}\} \nu}
d_{\nu r}^{[\tilde{\lambda}]}(\{\tilde{N}\})
\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r} ,
\label{Philamtn}$$ are linear combinations of the basic functions [(\[tilPhilamNtr\])]{}, where the coefficients $A^{[\lambda]}_{n \{\tilde{N}\} \nu}$ are solutions of the eigenproblem of the form of [Eq. (\[HAeqEA\])]{}, $$\begin{aligned}
\sum_{\{\tilde{N}'\},r,r'}\sum_{\nu'}{}'
\left( \{\tilde{N}\}!\{\tilde{N}'\}!\right) ^{-1/2}
&&d_{\nu r}^{[\tilde{\lambda}]}(\{\tilde{N}\})
\nonumber
\\
\times
\langle \tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}|\hat{H}_{\mathrm{spat}}|\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}'\} t r'}\rangle
&&d_{\nu' r'}^{[\tilde{\lambda}]}(\{\tilde{N}'\})
A^{[\lambda]}_{n \{\tilde{N}'\} \nu'}
\nonumber
\\
&&=E^{[\lambda]}_{n}A^{[\lambda]}_{n \{\tilde{N}\} \nu} .
\phantom{qqqqq}
\label{HspatAeqEA}
$$ Here the Hamiltonian matrix $$\begin{aligned}
&&\langle \tilde{\Phi}^{[\lambda]}_{\{\tilde{N}'\} t r'}|\hat{H}_{\mathrm{spat}}|\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}\rangle=
\sum_{{\mathcal{Q}}}D_{r' r}^{[\tilde{\lambda}]}({\mathcal{Q}})
\nonumber
\\
&&\times
\int d^{DN} r \prod_{j'=1}^N
\varphi^*_{m'_{{\mathcal{Q}}j'}}(\mathbf{r}_{j'})\hat{H}_{\mathrm{spat}}
\prod_{j=1}^N \varphi_{m_{j}}(\mathbf{r}_{j})
\label{Hspatmatr}
$$ is derived like [Eq. (\[Hspinmatr\])]{}.
Although [Eq. (\[Philamtn\])]{} involves only a finite number of the orbitals, the spatial wavefunction can be approximated with the required accuracy if the number of the orbitals is sufficiently large.
The maximal state occupations {#SI_MaxOccup}
=============================
Since the permutations ${\mathcal{P}}^{\{N\}}$ [(\[PN\])]{} do not affect the functions $\Xi_{\{N\}{\mathcal{P}}}$ [(\[XiNP\])]{}, the wavefunctions [(\[Nlamtr\])]{} can be expressed as, $$\begin{aligned}
|\{N\}&,&\lambda,t,r\rangle=\frac{1}{\{N\}!}\left( \frac{f_{\lambda}}{N!}\right)^{1/2}
\sum_{{\mathcal{P}},{\mathcal{P}}^{\{N\}}}D_{t r}^{[\lambda]}({\mathcal{P}})\Xi_{\{N\}{\mathcal{P}}{\mathcal{P}}^{\{N\}}}
\nonumber
\\
&&=\frac{1}{\{N\}!}\left( \frac{f_{\lambda}}{N!}\right)^{1/2}
\sum_{{\mathcal{P}},r_M}D_{t r_M}^{[\lambda]}({\mathcal{P}})\Xi_{\{N\}{\mathcal{P}}}
\nonumber
\\
&&\times
\sum_{{\mathcal{P}}^{(M)},r_{M-1}}D_{r_M r_{M-1}}^{[\lambda]}({\mathcal{P}}^{(M)})
\cdots \sum_{{\mathcal{P}}^{(1)}}D_{r_1 r}^{[\lambda]}({\mathcal{P}}^{(1)}) .
\phantom{qq}
$$ As ${\mathcal{P}}^{(1)}$ are elements of the subgroup ${\mathcal{S}}_{N_1}$ of permutations of $N_1$ first symbols, a reduction to subgroup (see [@kaplan]) can be used, $D_{r_1 r}^{[\lambda]}({\mathcal{P}}^{(1)})=D_{\bar{r}_1 \bar{r}}^{[\bar{\lambda}]}({\mathcal{P}}^{(1)})$, where the Young tableaux $\bar{r}_1$ and $\bar{r}$, corresponding to the same Young diagram, $\bar{\lambda}$, are obtained by the removal of the symbols $N_1+1\ldots N$ from the tableaux $r_1$ and $r$, respectively. ($D_{r_1 r}^{[\lambda]}({\mathcal{P}}^{(1)})=0$ if $\bar{r}_1$ and $\bar{r}$ correspond to different Young diagrams or if the symbols $N_1+1\ldots N$ are placed differently in $r_1$ and $r$.) Let us introduce the notation $[0]$ for the Young tableau of the proper shape in which the symbols are arranged by rows in the sequence of natural numbers. Taking into account that $D_{[0] [0]}^{[N_1]}({\mathcal{P}}^{(1)})=1$ as the Young diagram $[N_1]$, having one row of length $N_1$, corresponds to the identity representation, we get (using [Eq. (\[OrthYoung\])]{}), $$\begin{aligned}
\sum_{{\mathcal{P}}^{(1)}} D_{r_1 r}^{[\lambda]}({\mathcal{P}}^{(1)})
&=& \sum_{{\mathcal{P}}^{(1)}} D_{\bar{r}_1 \bar{r}}^{[\bar{\lambda}]}({\mathcal{P}}^{(1)})
D_{[0] [0]}^{[N_1]}({\mathcal{P}}^{(1)})
\nonumber
\\
&=&N_1! \delta_{\bar{\lambda} [N_1]} \delta_{\bar{r}_1 [0]} \delta_{\bar{r} [0]} .
\label{sumP1}
$$ This means that $N_1$ cannot exceed $\lambda_1$ (another proof of this statement is given in [@kaplan]). Besides, $r_1=r$, since, according to [Eq. (\[sumP1\])]{} $\bar{r}_1=\bar{r}$ and the remaining parts of $r_1$ and $r$ coincide, as mentioned above.
Consider now the case of $N_1=\lambda_1$. Each permutation ${\mathcal{P}}^{(2)}$ can be represented as a product of elementary transpositions ${\mathcal{P}}_{j,j+1}$ (see [@kaplan; @pauncz_symmetric]) of symbols $N_1< j < N_1+N_2$. All these transpositions do not affect the first row of $r$ occupied by the first $N_1$ symbols. Therefore, the Young orthogonal matrix $D_{r_2 r_1}^{[\lambda]}({\mathcal{P}}^{(2)})=D_{r_2 r}^{[\lambda]}({\mathcal{P}}^{(2)})$ will have non-zero elements only if the first row of $r_2$ is occupied by the first $N_1$ symbols. Further, as the Young orthogonal matrix for an elementary transposition depends only on the distance between the permutted symbols (see [@kaplan; @pauncz_symmetric]), $D_{r_2 r}^{[\lambda]}({\mathcal{P}}^{(2)})=D_{r''_2 r''}^{[\lambda'']}({\mathcal{P}}^{(2)})$, where $r''$ and $r''_2$, obtained by removal of the first row from the tableaux $r$ and $r_2$, respectively, correspond to the same Young diagram $[\lambda'']$. The same argumentation as for ${\mathcal{P}}^{(1)}$ leads then to the conclusion that $N_2\leq \lambda_2$, the symbols $N_1+1,\ldots,N_1+N_2$ occupy the second row of $r$ in the sequence of natural numbers, and $r_2=r$. Repeating this for $D_{r_i r_{i-1}}^{[\lambda]}({\mathcal{P}}^{(i)})$ with $3\leq i\leq m$, one gets that if $N_i=\lambda_i$ for all $1\leq i\leq m-1$ then $N_m\leq \lambda_m$, $r_m=r$, and symbols $1\leq j\leq\sum_{i=1}^{m}N_i$ occupy first $m$ rows of $r$ in the sequence of natural numbers. Therefore, there is only one irrep for $N_m=\lambda_m$ ($1\leq m\leq M$), and its label is $r=[0]$.
The maximal spin and boundaries of characters {#SI_MaxSpin}
=============================================
Let us attribute the spin projection $s_z=s+1-m$ to the state $|m\rangle$. The functions of the irrep considered in the previous section have the maximal possible occupation $N_1=\lambda_1$ of the state with the maximal spin projection $s$, and the occupation of the state $|m\rangle$ does not exceed occupations of the states $|m'\rangle$ with higher projections. Therefore, the functions have the maximal possible projection of the total spin. This projection $$S=\sum_{m=1}^M (s+1-m)\lambda_m
=\frac{M+1}{2}N-\sum_{m=1}^M m\lambda_m
\label{MaxSpinS}$$ can be considered as the maximal total spin, corresponding to the Young diagram $\lambda$. Irreps of $SU(M)$, associated with the Young diagram, can be decomposed into irreps of $R(3)$, having a defined spin. Examples of such decomposition are presented in [@kaplan]. Equation [(\[MaxSpinS\])]{} agrees with the maximal spin appearing in these examples.
The maximal spin [(\[MaxSpinS\])]{} attains its maximum value $N(M-1)/2=N s$ at the one-row Young diagram $\lambda=[N]$. Indeed, for any other Young diagram $[N-\sum_{m=2}^M \lambda_m,\lambda_2,\ldots,\lambda_M]$ the maximal spin will be $$S=\frac{M-1}{2}N-\sum_{m=2}^M (m-1)\lambda_m \leq \frac{M-1}{2}N
\label{proofmaxS}$$ The one-row Young diagram is associated with a one-dimensional irrep. The basic function is symmetric over all permutations and is an eigenfunction of the total spin.
The normalized character for the conjugate class of transpositions can be expressed as [@lassalle2008] $$\begin{aligned}
\tilde{\chi}_\lambda (\{2\})=\frac{1}{N(N-1)}
\left[ \sum_{m=1}^M(\lambda_m^2-2 m \lambda_m)+N\right]
\nonumber
\\
=\frac{1}{N(N-1)}
\left( \sum_{m=1}^M\lambda_m^2+2S-M N\right) .
\phantom{qq}
\label{chartran}
$$ It attains its maximum $\tilde{\chi}_{[N]} (\{2\})=1$ at the one-row Young diagram. Indeed, for any other Young diagram $$\begin{aligned}
\sum_{m=1}^M\lambda_m^2= N^2
-2\sum_{m=2}^M \lambda_m\left( N-\sum_{m'=2}^M \lambda_{m'}\right)
\\*
-\sum_{m\neq m'}\lambda_m \lambda_{m'}
\leq N^2 .
$$ Taking into account [Eq. (\[proofmaxS\])]{}, we get $\tilde{\chi}_\lambda (\{2\})\leq 1$.
The zero value of the maximal spin [(\[MaxSpinS\])]{} is reached if $N$ is a multiple of $M$ and $\lambda=[(N/M)^M]$ has $M$ rows of the equal length $N/M$. If $N$ is not a multiple of $M$ ($N=M \lambda_M+k$, $k<M$), the minimum value of the maximal spin $(M-k)k/2$ is attained at the Young diagram $\lambda=[(\lambda_M+1)^k,\lambda_M^{(M-k)}]$. Indeed, for any Young diagram $\lambda'$ the row length can be represented as $\lambda'_m=\lambda_M+1+\Delta\lambda_m$ if $m\leq k$ and $\lambda'_m=\lambda_M+\Delta\lambda_m$ if $m> k$ with $\sum_{m=1}^M \Delta\lambda_m=0$. The change of the second term in [Eq. (\[MaxSpinS\])]{} can be then expressed as $$\begin{aligned}
\sum_{m=1}^M m \Delta\lambda_m=\frac{M+1}{2}\sum_{m=1}^M \Delta\lambda_m
\nonumber
\\
+\sum_{m=1}^{M/2} \left( m-\frac{M+1}{2}\right) (\Delta\lambda_m-\Delta\lambda_{M+1-m})
$$ It is non-positive, since $m-(M+1)/2\leq 0$ and rows of Young diagrams have non-increasing lengths. As a result, we get $S\geq (M-k)k/2$ for any Young diagram.
The first sum in the normalized character [(\[chartran\])]{} can be expressed as $$\begin{aligned}
\sum_{m=1}^M\lambda_m^2=(N+k)\lambda_M+k+2\sum_{m=1}^k\Delta\lambda_m
+\sum_{m=1}^M\Delta\lambda_m^2
\nonumber
\\
\geq (N+k)\lambda_M+k
$$ since $\sum_{m=1}^k\Delta\lambda_m\geq 0$ (otherwise, $\lambda'_m$ will not form a non-increasing sequence). Therefore, $$\tilde{\chi}_\lambda (\{2\})\geq \frac{(N+k)\lambda_M+(M-k+1)k-M N}{N(N-1)}$$ and the minimum is attained at $\lambda=[(\lambda_M+1)^k,\lambda_M^{M-k}]$.
The boundaries for characters are used for calculation of boundaries for energies.
Selection rules {#SI_sel}
===============
Since the total wavefunctions are symmetric (or antisymmetric) over permutations, matrix elements of the $k$-body non-separable interaction $\hat{W}_k(\{j\})$ \[see [Eq. ([3]{})]{}\] are independent of the choice of the interacting particles $j_1,\ldots,j_k$, and, without loss of generality, we can consider the matrix element
$$\begin{aligned}
&&\langle\Psi^{[\lambda']}_{n' l'}|\hat{W}_k(N-k+1,\ldots,N)|\Psi^{[\lambda]}_{n l}\rangle=\frac{N!}{f_\lambda f_{\lambda'}}
\sum_{t,t',\{N\},\{N'\},r,r'}
B^{[\lambda']}_{l' \{N'\} r'}
B^{[\lambda]}_{l \{N\} r}
\nonumber
\\
&&
\times\langle\Phi^{[\lambda']}_{t' n'}|
\langle\{N'\},\lambda',t',r'|\hat{W}_k(N-k+1,\ldots,N)
|\{N\},\lambda,t,r\rangle |\Phi^{[\lambda]}_{t n}\rangle .\end{aligned}$$
It is expressed, using Eqs. ([1]{}), [(\[Xilamtl\_Nlamtnu\])]{}, [(\[Nlamtnu\])]{}, and [(\[Blamlnr\])]{}, in terms of wavefunctions $|\{N\},\lambda,t,r\rangle$, which do not take into account interactions of spins. These wavefunctions keep the Young diagrams of the total wavefunctions, since the Hamiltonian $\hat{H}_{\mathrm{spin}}$ is diagonal in $\lambda$ (see [Eq. (\[Hspinmatr\])]{}). Then the matrix element is expressed in terms of one-body spin states using Eqs. [(\[Nlamtr\])]{}, [(\[XiNP\])]{}, and [Eq. ([3]{})]{} $$\begin{aligned}
\langle\{N'\},\lambda',t',r'|\hat{W}_k(N-k+1,\ldots,N)
|\{N\},\lambda,t,r\rangle=\frac{\sqrt{f_\lambda f_{\lambda'}}}{N!}
\sum_{{\mathcal{P}},{\mathcal{P}}'}D_{r' t'}^{[\lambda']}({\mathcal{P}}')
D_{r t}^{[\lambda]}({\mathcal{P}})
\nonumber
\\
\times\prod_{j=1}^{N-k}\delta_{m'_{{\mathcal{P}}'j}m_{{\mathcal{P}}j}}
\langle m'_{{\mathcal{P}}'(N-k+1)}\ldots m'_{{\mathcal{P}}'N}|\hat{W}(\mathbf{r}_{N-k+1},\ldots,\mathbf{r}_N)|m_{{\mathcal{P}}(N-k+1)}\ldots m_{{\mathcal{P}}N}\rangle .
\label{Wk_spin_me}\end{aligned}$$ The Kronecker $\delta$-symbols here remain invariant if we replace ${\mathcal{P}}'$ by ${\mathcal{P}}'{\mathcal{Q}}$ and ${\mathcal{P}}$ by ${\mathcal{P}}{\mathcal{Q}}$, where ${\mathcal{Q}}$ is any permutation of the first $N-k$ symbols, and the matrix element of $\hat{W}(\mathbf{r}_{N-k+1},\ldots,\mathbf{r}_N)$ is independent of these symbols. Therefore, we can average [Eq. (\[Wk\_spin\_me\])]{} over the permutations ${\mathcal{Q}}$, replacing the product of the Young matrices by $$\frac{1}{(N-k)!}\sum_{{\mathcal{Q}}}D_{r' t'}^{[\lambda']}({\mathcal{P}}'{\mathcal{Q}})
D_{r t}^{[\lambda]}({\mathcal{P}}{\mathcal{Q}})=\frac{1}{(N-k)!}
\sum_{t'',t'''}D_{r' t'''}^{[\lambda']}({\mathcal{P}}')
D_{r t''}^{[\lambda]}({\mathcal{P}})\sum_{{\mathcal{Q}}}
D_{\bar{t}''' \bar{t}'}^{[\bar{\lambda}']}({\mathcal{Q}})
D_{\bar{t}'' \bar{t}}^{[\bar{\lambda}]}({\mathcal{Q}}) ,$$
where [Eq. (\[ProdYoung\])]{} and the reduction to subgroup (see [@kaplan]) are used. The Young tableaux $\bar{t}'''$, $\bar{t}'$, $\bar{t}''$, and $\bar{t}$ are obtained by removal of the symbols $N-k+1,\ldots,N$ from the tableaux $t'''$,$t'$, $t''$, and $t$, respectively. Finally, the summation over ${\mathcal{Q}}$ using [Eq. (\[OrthYoung\])]{} leads to $$\langle\Psi^{[\lambda']}_{n' l'}|\hat{W}_k(N-k+1,\ldots,N)|\Psi^{[\lambda]}_{n l}\rangle\sim \delta_{\bar{\lambda}' \bar{\lambda}} .$$ The Young diagrams $\bar{\lambda}'$ and $\bar{\lambda}$ are obtained by removing of $k$ boxes from the diagrams $\lambda'$ and $\lambda$, respectively. Therefore, $\lambda$ and $\lambda'$ can be different by the relocation of no more than $k$ boxes between their rows.
Correlation rules
=================
Using [Eq. ([1]{})]{}, orthogonality of spin wavefunctions, and [Eq. (\[Philamtn\])]{}, the expectation value of local spatial correlations [Eq. ([6]{})]{} can be expressed in terms of wavefunctions $\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}$ of non-interacting atoms, $$\begin{aligned}
\langle \Psi^{[\lambda]}_{n l}|\rho_k(\{0\})|\Psi^{[\lambda]}_{n l}\rangle=
\sum_{\{\tilde{N}\},r,\{\tilde{N'}\},r'}
\left(\{\tilde{N}\}!\{\tilde{N'}\}!\right) ^{-1/2}
\nonumber
\\
\times
\sum_{\nu,\nu'}{}'
A^{[\lambda]}_{n \{\tilde{N}\} \nu}A^{[\lambda]}_{n \{\tilde{N'}\} \nu'}
d_{\nu r}^{[\tilde{\lambda}]}(\{\tilde{N}\})
d_{\nu' r'}^{[\tilde{\lambda}]}(\{\tilde{N'}\})
\nonumber
\\
\times
\frac{1}{f_{\lambda}}\sum_{t}\langle\tilde{\Phi}^{[\lambda]}_{\{\tilde{N'}\} t r'}|\rho_k(\{0\})|\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}\rangle .
\phantom{qqqqqq}\end{aligned}$$ Equation [(\[tilPhilamNtr\])]{} allows us to express the matrix element over $\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}$ (the last line of the equation above) in terms of the one-body spatial orbitals $\varphi_{m}(\mathbf{r})$, $$\begin{aligned}
\frac{1}{N!}\sum_{{\mathcal{P}},{\mathcal{P}}',t}D_{r' t}^{[\tilde{\lambda}]}({\mathcal{P}}')
D_{r t}^{[\tilde{\lambda}]}({\mathcal{P}})\prod_{j=k+1}^{N}\delta_{m'_{{\mathcal{P}}'j}m_{{\mathcal{P}}j}}
\nonumber
\\
\times
\int d^{D} r \prod_{j'=1}^k \varphi^*_{m'_{{\mathcal{P'}}j'}}(\mathbf{r})
\prod_{j=1}^k \varphi_{m_{{\mathcal{P}}j}}(\mathbf{r}).
\label{rhok0tilPhi}\end{aligned}$$ The spatial orbitals of the correlating atoms are taken in the same point. Therefore $$\prod_{j'=1}^k \varphi^*_{m'_{{\mathcal{P'}}j'}}(\mathbf{r})=\prod_{j'=1}^k \varphi^*_{m'_{{\mathcal{P'}}{\mathcal{Q}}^{-1}j'}}(\mathbf{r}) \quad ({\mathcal{Q}}\in S_k)$$ and [Eq. (\[rhok0tilPhi\])]{} is invariant over permutations ${\mathcal{Q}}\in S_k$ of $j'$. Denoting ${\mathcal{R}}={\mathcal{P'}}{\mathcal{Q}}^{-1}$, averaging over permutations ${\mathcal{Q}}$, and using [Eq. (\[ProdYoung\])]{}, [Eq. (\[rhok0tilPhi\])]{} can be transformed to $$\begin{aligned}
\frac{1}{N!}\sum_{{\mathcal{P}},{\mathcal{R}},t,t'}D_{r' t'}^{[\tilde{\lambda}]}({\mathcal{R}})
D_{r t}^{[\tilde{\lambda}]}({\mathcal{P}})\prod_{j=k+1}^{N}\delta_{m'_{{\mathcal{R}}j}m_{{\mathcal{P}}j}}
\\
\times
\int d^{D} r \prod_{j'=1}^k \varphi^*_{m'_{{\mathcal{R}}j'}}(\mathbf{r})
\prod_{j=1}^k \varphi_{m_{{\mathcal{P}}j}}(\mathbf{r})
\frac{1}{k!}\sum_{{\mathcal{Q}}\in S_k} D_{t' t}^{[\tilde{\lambda}]}({\mathcal{Q}}) .\end{aligned}$$ The last sum in this expression can be transformed in the same way as in [Eq. (\[sumP1\])]{} $$\sum_{{\mathcal{Q}}\in S_k} D_{t' t}^{[\tilde{\lambda}]}({\mathcal{Q}})
=k! \delta_{\bar{\tilde{\lambda}} [k]} \delta_{\bar{t'} [0]} \delta_{\bar{t} [0]} ,$$ where the Young tableaux $\bar{t'}$ and $\bar{t}$ (obtained by the removal of the symbols $k+1\ldots N$ from the tableaux $t'$ and $t$, respectively) correspond to the same Young diagram $\bar{\tilde{\lambda}}$. The first Kronecker $\delta$ symbol in the above identity zeroes if $\tilde{\lambda}_1<k$. As a result, the expectation values of local spatial correlations vanish if $k>\tilde{\lambda}_1$. Transforming the spatial wavefunctions to the momentum representation, we arrive at the same result for local momentum correlations.
Multiplet averages of expectation values {#SI_MultAv}
========================================
Equation ([1]{}) and orthogonality of the spin functions lead to the following average of a spin-independent operator $\hat{F}_k$ over a $\lambda$-multiplet, $$\begin{aligned}
\bar{F}^{[\lambda]}_k\equiv\frac{1}
{\tilde{\mathcal{N}}_{\lambda}(N,M_{\mathrm{spat}})}
\sum_n \langle \Psi^{[\lambda]}_{n l}|\hat{F}_k|\Psi^{[\lambda]}_{n l}\rangle
\nonumber
\\
=\frac{1}{f_{\lambda}
\tilde{\mathcal{N}}_{\lambda}(N,M_{\mathrm{spat}})}\sum_{t,n}
\langle\Phi^{[\lambda]}_{t n}|\hat{F}_k|\Phi^{[\lambda]}_{t n}\rangle ,
\label{Fkexps}
$$ where $$\tilde{\mathcal{N}}_{\lambda}(N,M_{\mathrm{spat}})=\sum_{\{\tilde{N}\}} \tilde{f}_{\tilde{\lambda}}(\{\tilde{N}\})$$ is the total number of the spatial wavefunctions, associated with the Young diagram $\lambda$. The average [(\[Fkexps\])]{} is independent of the spin quantum numbers $l$. Since the total wavefunctions are symmetric (or antisymmetric) over permutations, we can suppose, without loss of generality, that the operator $\hat{F}_k$ acts to $\mathbf{r}_1,\ldots,\mathbf{r}_k$. Then equations [(\[Philamtn\])]{}, [(\[Acomp\])]{}, [(\[tilPhilamNtr\])]{}, and [(\[ProdYoung\])]{} lead to
$$ \bar{F}^{[\lambda]}_k=\frac{1}
{ N! \tilde{\mathcal{N}}_{\lambda}(N,M_{\mathrm{spat}})}
\sum_{\{\tilde{N}\},r,r'}\frac{1}{\{\tilde{N}\}!}\sum_{\nu}{}' d_{\nu r}^{[\tilde{\lambda}]}
d_{\nu r'}^{[\tilde{\lambda}]}
\sum_{{\mathcal{P}},{\mathcal{R}}}D_{r' r}^{[\tilde{\lambda}]}({\mathcal{R}})
\prod_{j=k+1}^{N}\delta_{m_{{\mathcal{R}}{\mathcal{P}}j}m_{{\mathcal{P}}j}}
\int d^{Dk} r
\prod_{j'=1}^k \varphi^*_{m_{{\mathcal{R}}{\mathcal{P}}j'}}(\mathbf{r}_{j'})\hat{F}_k
\prod_{j=1}^k \varphi_{m_{{\mathcal{P}}j}}(\mathbf{r}_{j}).
\label{barFkPR}
$$
The summand in [Eq. (\[barFkPR\])]{} is the same for all $(N-k)!$ permutations ${\mathcal{P}}$ corresponding to the given set $\{j\}$ of the symbols $j_i={\mathcal{P}}i$ with $1\leq i\leq k$. Except of this, Eqs. [(\[Devecval\])]{} and [(\[tildeD\])]{} lead to $\sum'_{\nu} d_{\nu r}^{[\tilde{\lambda}]}
d_{\nu r'}^{[\tilde{\lambda}]}=
\sum_{\nu} d_{\nu r}^{[\tilde{\lambda}]}\delta_{\nu}^{[\tilde{\lambda}]}
d_{\nu r'}^{[\tilde{\lambda}]}/\{\tilde{N}\}!
=\sum_{{\mathcal{P}}^{\{\tilde{N}\}}}D_{r r'}^{[\tilde{\lambda}]}({\mathcal{P}}^{\{\tilde{N}\}})/\{\tilde{N}\}!$. Using [Eq. (\[ProdYoung\])]{} and taking into account that $m_{{\mathcal{P}}^{\{\tilde{N}\}}j}=m_j$ for each $j$, one gets $$\begin{aligned}
\bar{F}^{[\lambda]}_k=\frac{(N-k)!}
{N! \tilde{\mathcal{N}}_{\lambda}(N,M_{\mathrm{spat}})}\sum_{\{\tilde{N}\}}\frac{1}{\{\tilde{N}\}!}\sum_{\{j\}}{}'
\sum_{{\mathcal{R}}}\sum_r
D_{r r}^{[\tilde{\lambda}]}({\mathcal{R}})
\nonumber
\\
\times
\prod_{j'\notin\{j\}}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\int d^{Dk} r
\prod_{i'=1}^k \varphi^*_{m_{{\mathcal{R}}j_{i'}}}(\mathbf{r}_{i'})\hat{F}_k
\prod_{i=1}^k \varphi_{m_{j_i}}(\mathbf{r}_{i}),
\label{barFkQP}
$$ where $\sum'_{\{j\}}$ denotes summation over all $j_i$ ($1\leq i\leq k$) such that $j_i\neq j_{i'}$. The summation over $r$ here leads to the trace of the Young matrix — the character $\chi_{\lambda}$, $$\sum_t D_{t t}^{[\lambda]}({\mathcal{Q}})\equiv
\chi_{\lambda}({\mathcal{Q}})=f_{\lambda}\tilde{\chi}_{\lambda}({\mathcal{Q}}) .$$ The characters, as well as the normalized characters $\tilde{\chi}_{\lambda}$, are the same for all permutation in a given conjugate class $C_N$ (see [@hamermesh; @kaplan; @pauncz_symmetric]). As a result, we get [Eq. ([8]{})]{} with $$\begin{aligned}
\langle F_k\rangle_{C_N}=\frac{(N-k)!}
{N!g(C_N)}
\sum_{\{\tilde{N}\}}\frac{1}{\{\tilde{N}\}!}
\sum_{\{j\}}{}'
\sum_{{\mathcal{R}}\in C_N}
\prod_{j'\notin\{j\}}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\nonumber
\\
\times
\int d^{Dk} r
\prod_{i'=1}^k \varphi^*_{m_{{\mathcal{R}}j_{i'}}}(\mathbf{r}_{i'})\hat{F}_k
\prod_{i=1}^k \varphi_{m_{j_i}}(\mathbf{r}_{i}).
\phantom{qqqqq}
\label{FkavCk}
$$
The conjugate classes of the symmetric group ${\mathcal{S}}_k$ are characterized by the cyclic structure of the permutations. All permutations in the class $C_k=\{k^{\nu_k} \ldots 2^{\nu_2}\}$ have $\nu_l$ cycles of length $l$. This class notation omits the number of cycles of the length one, i.e. the number of symbols which are not affected by the permutations in the class. This number is determined by the condition $\sum_{l=1}^k l \nu_l=k$. Thus, the same notation can be used for classes $C_k$ and $C_N$ of the groups ${\mathcal{S}}_k$ and ${\mathcal{S}}_N$, respectively. The number of elements in the class $C_N$ of the group ${\mathcal{S}}_N$ is expressed as [@kaplan; @pauncz_symmetric] $$ g_{C_N}=\frac{N!}{\prod_{l=1}^N \nu_l! l^{\nu_l}} ,
\label{gclass}$$ and the permutations in this class have the parity $$\mathrm{sig}(C_N)=\prod_{l=1}^{[N/2]} (-1)^{\nu_{2l}} ,$$ where $[x]$ is the integer part of $x$.
If each spatial orbital is occupied only by one particle, $\tilde{N}_m=1$, the set of permutations $\{{\mathcal{P}}^{\{\tilde{N}\}}\}$ includes only the identity permutation. In this case, the Kronecker $\delta$-symbols in [Eq. (\[FkavCk\])]{} select only the permutations ${\mathcal{R}}$ of $k$ symbols $j_i$. All such permutations belong to the conjugate classes $C_k$ of the subgroup ${\mathcal{S}}_k$, i.e. $\sum_{l=2}^k l \nu_l\leq k$. Therefore, [Eq. ([8]{})]{} will include the summation over these classes only. It should be stressed that even in this case, $\tilde{\chi}_{\lambda}(C_k)$ are characters in irreps of ${\mathcal{S}}_N$, i.e. $\sum_{m=1}^M \lambda_m=N$. Equation [(\[FkavCk\])]{} can be simplified in this case by a substitution of ${\mathcal{R}}j_{i}=j_{{\mathcal{P}}i}$, where ${\mathcal{P}}$ are permutations of $k$ symbols, $$\begin{aligned}
\langle F_k\rangle_{C_k}=\frac{(N-k)!}
{N!g(C_k)}
\sum_{\{j\}}{}'
\sum_{{\mathcal{P}}\in C_k}
\int d^{Dk} r
\prod_{i'=1}^k \varphi^*_{m_{j_{{\mathcal{P}}i'}}}(\mathbf{r}_{i'})\hat{F}_k
\nonumber
\\*
\times
\prod_{i=1}^k \varphi_{m_{j_i}}(\mathbf{r}_{i}).
\phantom{qqqqqq}
\label{FkavCk1}
$$
Multiplet-averaged correlations {#SI_corr}
===============================
Consider the operator given by [Eq. ([6]{})]{}. Its expectation values are the $k$-body spatial correlations $\bar{\rho}_k(\{\mathbf{R}\})$. In this case, [Eq. (\[FkavCk\])]{} leads to
$$\langle \rho_k(\{\mathbf{R}\})\rangle_{C_N}=\frac{(N-k)!}{N!g(C_N)}
\sum_{\{\tilde{N}\}}\frac{1}{\{\tilde{N}\}!}
\sum_{\{j\}}{}'
\sum_{{\mathcal{R}}\in C_N}\prod_{j'\notin\{j\}}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\int d^{D}r
\prod_{i=1}^k \varphi^*_{m_{{\mathcal{R}}j_{i}}}(\mathbf{r}-\mathbf{R}_{i-1}) \varphi_{m_{j_i}}(\mathbf{r}-\mathbf{R}_{i-1})$$ with $\mathbf{R}_0=0$. If each spatial orbital is occupied only by one particle, [Eq. (\[FkavCk1\])]{} gives us $$\langle \rho_k(\{\mathbf{R}\})\rangle_{C_k}=
\frac{(N-k)!}{N!g(C_k)}
\sum_{\{j\}}{}'\sum_{{\mathcal{P}}\in C_k}
\int d^{D}r
\prod_{i=1}^k \varphi^*_{m_{j_i}}(\mathbf{r}-\mathbf{R}_{({\mathcal{P}}i)-1}) \varphi_{m_{j_i}}(\mathbf{r}-\mathbf{R}_{i-1}) .
\label{rhokavCk}$$ For the local correlations, with all $\mathbf{R}_{i}=0$, [Eq. (\[rhokavCk\])]{} becomes independent of the conjugate class $C_k$ since the integrand therein becomes independent of the permutation ${\mathcal{P}}$.
The $k$-body momentum correlations $\bar{g}_k(\{\mathbf{q}\})$ are the expectation values of the operator given by [Eq. ([7]{})]{}. In the coordinate representation, it becomes an integral operator with the kernel, $$(2\pi)^{-D(k-1)}\delta\left( \sum_{j=1}^k(\mathbf{r}'_j-\mathbf{r}_j)\right)
\exp\left( i \sum_{j=2}^k \mathbf{q}_{j-1}(\mathbf{r}'_j-\mathbf{r}_j)\right)
\prod_{j=k+1}^N \delta(\mathbf{r}_j-\mathbf{r}'_j) .$$ In this case, [Eq. (\[FkavCk\])]{} leads to $$\begin{aligned}
\langle g_k(\{\mathbf{q}\})\rangle_{C_N}=
\frac{(N-k)!}{(2\pi)^{D(k-1)}N!g(C_N)}&&
\sum_{\{\tilde{N}\}}\frac{1}{\{\tilde{N}\}!}
\sum_{\{j\}}{}'
\sum_{{\mathcal{R}}\in C_N}\prod_{j'\notin\{j\}}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\nonumber
\\
&&\times
\int d^{Dk}rd^{Dk}r'
\delta\left( \sum_{i=1}^k(\mathbf{r}'_i-\mathbf{r}_i)\right)
\exp\left( i \sum_{i=2}^k \mathbf{q}_{i-1}(\mathbf{r}'_{i}-\mathbf{r}_i)\right)
\prod_{i=1}^k \varphi^*_{m_{{\mathcal{R}}j_{i}}}(\mathbf{r}'_i)\varphi_{m_{j_i}}(\mathbf{r}_i) .\end{aligned}$$ If each spatial orbital is occupied only by one particle, [Eq. (\[FkavCk1\])]{} gives us $$\langle g_k(\{\mathbf{q}\})\rangle_{C_k}=
\frac{(N-k)!}{(2\pi)^{D(k-1)}N!g(C_k)}
\sum_{\{j\}}{}'\sum_{{\mathcal{P}}\in C_k}
\int d^{Dk}r d^{Dk}r'
\delta\left( \sum_{i=1}^k(\mathbf{r}'_i-\mathbf{r}_i)\right)
\exp\left( i \sum_{i=2}^k \mathbf{q}_{i-1}(\mathbf{r}'_{{\mathcal{P}}i}-\mathbf{r}_i)\right)
\prod_{i=1}^k \varphi^*_{m_{j_i}}(\mathbf{r}'_i)\varphi_{m_{j_i}}(\mathbf{r}_i) .
\label{gkavCk}$$
This expression becomes independent of the conjugate class $C_k$ if all $\mathbf{q}_j=0$.
Optical lattice {#SI_latt}
===============
Consider $N$ cold atoms in a $D$-dimensional optical lattice [@bloch2008; @*yukalov2009; @*svistunov]. Spin-independent interactions between the atoms lead to the Hamiltonian $$\hat{H}=\sum_{j=1}^N \left[ -\frac{1}{2} \nabla_j^2 +U_{\mathrm{latt}}(\mathbf{r}_j)+
U(\mathbf{r}_j)\right] +\sum_{j<j'} V(\mathbf{r}_j-\mathbf{r}_{j'})
\label{Hlatt}$$ (using units with the mass of the atom and the Plank constant $\hbar$ are equal to $1$) Here, $\mathbf{r}_j$ is a $D$-dimensional coordinate of the $j$th atom and $\nabla_j$ is the $D$-dimensional gradient. The periodic lattice potential $U_{\mathrm{latt}}(\mathbf{r})$ has $D$ primitive vectors $\mathbf{a}_l$ ($1\leq l\leq D$). The trap potential $U(\mathbf{r})$ is flat on the scale of the lattice period.
In the case of a deep lattice, the spatial wavefunctions can be expanded in terms of the lowest-band Wannier functions $w(\mathbf{r}-\mathbf{T}(\mathbf{n}))$, where $\mathbf{T}(\mathbf{n})=\sum_{l=1}^D n_l \mathbf{a}_l$ is the lattice vector, and $\mathbf{n}\equiv(n_1,\ldots,n_D)$ is a $D$-dimensional integer vector. The matrix elements of the terms in $\hat{H}$ will be $$\begin{aligned}
J_{\mathbf{n}}&=&-\int d^D r w(\mathbf{r}-\mathbf{T}(\mathbf{n}))
\left( -\frac{1}{2} \nabla^2 +U_{\mathrm{latt}}(\mathbf{r})\right)w(\mathbf{r})
\nonumber
\\
V_{\mathbf{n}}&=&\int d^D r d^D r'
| w(\mathbf{r}'-\mathbf{T}(\mathbf{n}))w(\mathbf{r})|^2 V(\mathbf{r}'-\mathbf{r})
\label{LattMatEl}
\\
U_{\mathbf{n}}&=&U(\mathbf{T}(\mathbf{n}))
\nonumber\end{aligned}$$ For a zero-range interaction, the exchange interaction will be equal to $V_{\mathbf{n}}$ too. In the unit-filling Mott-insulator regime ($V_0\gg J_{\mathbf{n}}$ for $\mathbf{n}\neq 0$) each lattice site is occupied by one atom. (In experiments, this regime can be realized in the vicinity of the minimum of $U(\mathbf{r})$.) Due to the single occupation of each Wannier state, we have $\tilde{D}^{[\lambda]}_{r' r}=\delta_{r' r}$, $\{\tilde{N}\}!=1$, and $d_{\nu r}^{[\lambda]}=\delta_{\nu r}$. Then, using [(\[tilPhilamNtr\])]{} and [(\[Philamtn\])]{} with $\varphi(\mathbf{r})=w(\mathbf{r}-\mathbf{T}(\mathbf{n}))$, the spatial wavefunction can be expressed as $$\Phi^{[\lambda]}_{t n}=\sqrt{\frac{f_{\lambda}}{N!}}\sum_{r}
A^{[\lambda]}_{n r}
\sum_{{\mathcal{P}}}D_{t r}^{[\lambda]}({\mathcal{P}}) \mathrm{sig}({\mathcal{P}})
\prod_{j=1}^N w(\mathbf{r}_{{\mathcal{P}}j}-\mathbf{T}(\mathbf{n}_j)) ,
\label{Philatt}$$ where $\mathbf{n}_j$ correspond to the occupied sites. The coefficients $A^{[\lambda]}_{n r}$ are determined by the eigenproblem [(\[HspatAeqEA\])]{}, with $\hat{H}_{\mathrm{spat}}=\hat{H}$ of [Eq. (\[Hlatt\])]{}. Using orthogonality of the Wannier functions, the matrix elements [(\[LattMatEl\])]{}, and neglecting the hopping ($J_{\mathbf{n}}$ with $\mathbf{n}\neq 0$), we get the Hamiltonian matrix [(\[Hspatmatr\])]{} $$\langle \tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r}|\hat{H}|\tilde{\Phi}^{[\lambda]}_{\{\tilde{N}\} t r'}\rangle=
(-N J_0+\sum_j U_{\mathbf{n}_j})\delta_{r r'}+V^{[\lambda]}_{r r'}$$ with $$V^{[\lambda]}_{r r'}=\sum_{j<j'}V_{\mathbf{n}_j-\mathbf{n}_{j'}}
[\delta_{r r'}\pm D_{r r'}^{[\lambda]}({\mathcal{P}}_{j j'})]$$ The resulting eigenvalue equation for the coefficients $A^{[\lambda]}_{n r}$ and the state energies $E^{[\lambda]}_{n}$ has the form, $$E^{[\lambda]}_{n} A^{[\lambda]}_{n r}=
\sum_{r'}V^{[\lambda]}_{r r'}A^{[\lambda]}_{n r'} .$$ The energies are counted from $-N J_0+\sum_j U_{\mathbf{n}_j}$. They form $\lambda$-multiplets with the average energies [Eq. ([10]{})]{}. In the case of equal lengths of the primitive vectors $\mathbf{a}_i$, taking into account only near-neighbor interactions, $V_{\mathrm{near}}=V_{\mathbf{n}}$ with $|\mathbf{n}|=1$, we get $$\bar{E}_{\lambda}=\frac{N N_{\mathrm{near}}V_{\mathrm{near}}}{2}
\left( 1\pm \frac{N(N-1)}{2}\tilde{\chi}_{\lambda}(\{2\})\right) .$$ Here $N_{\mathrm{near}}$ is the number of neighboring sites for each site.
In order to evaluate correlations, let us use the harmonic oscillator approximation for the Wannier functions, $$w(\mathbf{r})=\pi^{-D/4}a_{HO}^{-D/2}\exp(-\mathbf{r}^2/(2a_{HO}^2)),
\label{WanHarmOsc}$$ where $a_{HO}$ is the range of the harmonic potential, approximating the lattice potential in the vicinities of its minima.
For the spatial correlations, [Eq. (\[rhokavCk\])]{} takes the form
$$\langle \rho_k(\{\mathbf{R}\})\rangle_{C_k}=
\frac{(N-k)!}{N!g(C_k)}
\sum_{{\mathcal{P}}\in C_k}\sum_{\{j\}}{}'
\int d^{D}r
\prod_{i=1}^k
w(\mathbf{r}-\mathbf{T}(\mathbf{n}_{j_i})-\mathbf{R}_{({\mathcal{P}}i)-1}) w(\mathbf{r}-\mathbf{T}(\mathbf{n}_{j_i})-\mathbf{R}_{i-1}) .
$$ If ${\mathcal{P}}={\mathcal{E}}$, the sum over $\{j\}$ can contain terms with $\mathbf{T}(\mathbf{n}_{j_i})=-\mathbf{R}_{i-1}$, which are not exponentially small. However, if $C_k\neq \{\}$, even these terms become exponentially small, as ${\mathcal{P}}i\neq i$ for at less two $i$, and the integral, calculated with the functions [(\[WanHarmOsc\])]{}, will be proportional to $$\exp\left(-\frac{1}{2a_{HO}^2}\left[ \sum_{i=1}^k \left( \Delta \mathbf{R}_i-\frac{1}{k}\sum_{i'=1}^k\Delta \mathbf{R}_{i'}\right)^2
+\frac{1}{2 k}\left(\sum_{i=1}^k\Delta \mathbf{R}_i\right)^2\right] \right) .$$ Here $\Delta \mathbf{R}_i=\mathbf{R}_{({\mathcal{P}}i)-1}-\mathbf{R}_{i-1}$.
For the momentum correlations, [Eq. (\[gkavCk\])]{} leads to $$\begin{aligned}
\langle g_k(\{\mathbf{q}\})\rangle_{C_k}=
\frac{(N-k)!}{(2\pi)^{D(k-1)}N!g(C_k)}
\sum_{{\mathcal{P}}\in C_k}\sum_{\{j\}}{}'
\exp\left( i\sum_{i=1}^k \mathbf{T}(\mathbf{n}_{j_i})[\mathbf{q}_{i-1}-\mathbf{q}_{({\mathcal{P}}i)-1}]\right)
\nonumber
\\
\times
\int d^{Dk}rd^{Dk}r'
\delta\left( \sum_{i=1}^k(\mathbf{r}'_i-\mathbf{r}_i)\right)\prod_{i=1}^k
\exp\left( i \mathbf{q}_{i-1}(\mathbf{r}_i-\mathbf{r}'_{i})\right) w(\mathbf{r}'_i)w(\mathbf{r}_i)
\label{gkavCklatts}\end{aligned}$$
with $\mathbf{q}_0=0$. The integral here can be represented as the convolution $$f_k(\{\mathbf{q}\})=\int d^D p \prod_{i=1}^k |\tilde{w}(\mathbf{p}+\mathbf{q}_{i-1})|^2$$ of the Fourier transforms $$\tilde{w}(\mathbf{p})=(2\pi)^{-D/2}\int d^Dr w(\mathbf{r})
\exp(-i\mathbf{p}\mathbf{r})$$ of the Wannier function.
Consider a lattice with the size $L_l$ in the direction of the primitive vector $\mathbf{a}_l$, such that $0\leq n_l < L_l$, and use the equality $$\sum_{n_l=0}^{L_l-1} \exp(i \Delta\mathbf{q}_i \mathbf{a}_l n_l)=\frac{\sin (L_l \Delta\mathbf{q}_i \mathbf{a}_l/2)}
{\sin (\Delta\mathbf{q}_i \mathbf{a}_l/2)}
\exp(i \Delta\mathbf{q}_i \mathbf{a}_l \frac{L_l -1}{2}),
\label{sumexp}$$ were $\Delta\mathbf{q}_i=\mathbf{q}_{i-1}-\mathbf{q}_{({\mathcal{P}}i)-1}$. Equation [(\[sumexp\])]{} tends to $L_l$ in the limit of $\Delta\mathbf{q}_i\to 0$. The sum over ${\{j\}}$ in [Eq. (\[gkavCklatts\])]{} is the sum over all $j_i$ minus the sums over ${\{j\}}$ where two or more $j_i$ coincide. The sum over all $j_i$ is the product of the sums [(\[sumexp\])]{} over all $1\leq l\leq D$ and $1\leq i\leq k$. This sum has the maximal value of $\prod_{l=1}^D L_l^k=N^k$ when all $\Delta\mathbf{q}_i=0$. The sum where $k'$ of $j_i$ coincide will have the maximal value of $N^{k-k'}$. Thus the contributions of such sums can be neglected. The exponential factors in [Eq. (\[sumexp\])]{} will be canceled in all sums over ${\{j\}}$ since $
\sum_{i=1}^k\Delta\mathbf{q}_i=\sum_{i=1}^k \mathbf{q}_{i-1}- \sum_{i=1}^k\mathbf{q}_{({\mathcal{P}}i)-1}=0
$. As a result, we get $$\begin{aligned}
\langle g_k(\{\mathbf{q}\})\rangle_{C_k}&\approx&
\frac{(N-k)!}{N!g(C_k)}f_k(\{\mathbf{q}\})
\nonumber
\\
&&\times
\sum_{{\mathcal{P}}\in C_k}\prod_{i=1}^k \prod_{l=1}^D
\frac{\sin( L_l \mathbf{a}_l\Delta\mathbf{q}_i/2)}{\sin( \mathbf{a}_l\Delta\mathbf{q}_i/2)} ,
\phantom{qqq}
\label{gkavCklatt}
$$ oscillating as functions of each component of $\mathbf{q}_j$ with a period $\sim(L_la_l)^{-1}$, except for $C_k=\{\}$, when the arguments of sines in [Eq. (\[gkavCklatt\])]{} are equal to $0$.
If all $\mathbf{q}_{i}=0$, all exponents in [Eq. (\[gkavCklatts\])]{} will be equal to zero, and we get [Eq. ([11]{})]{} with no approximations.
Few-particle cases
==================
Consider the simplest non-trivial examples of calculation of correlation functions for few-particle cases. Using [Eq. (\[gclass\])]{} for numbers of permutations $g(C_k)$ in the conjugate classes $C_k$ of permutations of $k$ symbols, the universal factors [Eq. ([9]{})]{} for lowest-order correlations are expressed as $$\begin{aligned}
\tilde{\rho}^{[\lambda]}_2&=&1\pm \tilde{\chi}_\lambda (\{2\})
\nonumber
\\
\tilde{\rho}^{[\lambda]}_3&=&1\pm 3\tilde{\chi}_\lambda (\{2\})
+2\tilde{\chi}_\lambda (\{3\})
\\
\tilde{\rho}^{[\lambda]}_4&=&1\pm 6\tilde{\chi}_\lambda (\{2\})
+8\tilde{\chi}_\lambda (\{3\})
\nonumber
\\
&&\pm 6\tilde{\chi}_\lambda (\{4\})+3\tilde{\chi}_\lambda (\{2^2\}) .
\nonumber\end{aligned}$$ Here, the sign $+$ or $-$ is taken for bosons or fermions, respectively. The correlations depend on the Young diagrams and are independent of the particle spin whenever the diagram row number does not exceed the multiplicity $M=2s+1$. The particle spin $s$ below means only the minimal spin, allowing the considered Young diagrams, and all results are applicable to particles with higher spins.
For particles with spin $s=1$, the Fortran codes for the normalized characters and the universal factors are presented in the accompanying file `char_corr.f`. They are derived from the explicit expressions for the characters [@lassalle2008] and used in Fig. 2. In the simplest non-trivial case of $N=3$ the characters and universal factors are presented in Table \[ST\_M3N3\].
$\lambda $ $\tilde{\chi}_\lambda (\{2\})$ $\tilde{\chi}_\lambda (\{3\})$ $\tilde{\rho}^{[\lambda]}_2$ $\tilde{\rho}^{[\lambda]}_3$
------------ -------------------------------- -------------------------------- ------------------------------ ------------------------------
$[3]$ 1 1 $1 \pm 1$ $3 \pm 3$
$[2 1]$ 0 -1/2 $1$ $0$
$[1^3]$ -1 1 $1 \mp 1$ $3 \mp 3$
: The normalized characters and universal factors [Eq. ([9]{})]{} for $N=3$ particles of the spin $s=1$ in states associated with the Young diagrams $\lambda$. The upper and lower signs correspond to bosons and fermions, respectively. \[ST\_M3N3\]
Characters for few particles with higher spins are tabulated in [@kaplan]. The characters and universal factors for $N=4$ particles of the spin $s=\frac{3}{2}$ are presented in Table \[ST\_M4N4\]. As would be expected the correlations in Tables \[ST\_M3N3\] and \[ST\_M4N4\] agree with the correlation rules and the correlations for fermions are equal to the correlations for bosons with the conjugate Young diagram.
$\lambda $ $\tilde{\chi}_\lambda (\{2\})$ $\tilde{\chi}_\lambda (\{3\})$ $\tilde{\chi}_\lambda (\{4\})$ $\tilde{\chi}_\lambda (\{2^2\})$ $\tilde{\rho}^{[\lambda]}_2$ $\tilde{\rho}^{[\lambda]}_3$ $\tilde{\rho}^{[\lambda]}_4$
------------ -------------------------------- -------------------------------- -------------------------------- ---------------------------------- ------------------------------ ------------------------------ ------------------------------
$[4]$ 1 1 1 1 $1 \pm 1$ $3 \pm 3$ $12 \pm 12$
$[3 1]$ 1/3 0 -1/3 -1/3 $1\pm 1/3$ $1\pm 1$ $0$
$[2^2]$ 0 -1/2 0 1 $1$ 0 $0$
$[2 1^2]$ -1/3 0 1/3 -1/3 $1\mp 1/3$ $1\mp 1$ $0$
$[1^4]$ -1 1 -1 1 $1 \mp 1$ $3 \mp 3$ $12\mp 12$
As an example of multiple occupation of spatial orbitals consider $N=3$ non-interacting particles of the spin $s=1$ in two spatial orbitals, $\varphi_{1}(\mathbf{r})$ and $\varphi_{2}(\mathbf{r})$, with the occupations $\{\tilde{N_1}\}=2$ and $\{\tilde{N_2}\}=1$, respectively. For non-interacting particles only one term in the sum over $\{\tilde{N}\}$ remains in [Eq. (\[FkavCk\])]{}, and for a one-body observable it takes then the form $$\langle F_1\rangle_{C_N}=\frac{1}
{N g(C_N)\{\tilde{N}\}!}
\sum_{j=1}^N
\sum_{{\mathcal{R}}\in C_N}
\prod_{j'\neq j}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\langle \varphi_{m_{{\mathcal{R}}j}} |\hat{F}_1|\varphi_{m_{j}}\rangle,
$$ where $$\langle \varphi_{m'} |\hat{F}_1|\varphi_m\rangle = \int d^{D} r
\varphi^*_{m'}(\mathbf{r})\hat{F}_1
\varphi_{m}(\mathbf{r}) .$$ The product of Kronecker symbols above allows only permutations of equal quantum numbers. Then ${\mathcal{R}}$ can be either the identity permutation ${\mathcal{E}}$ or the transposition ${\mathcal{P}}_{12}$, which belong to the conjugate classes $\{\}$ or $\{2\}$, respectively. As a result [Eq. ([8]{})]{} contains two terms with $$\begin{aligned}
\langle F_1\rangle_{\{\}}&=&\frac{1}{6}(2 \langle \varphi_{1} |\hat{F}_1|\varphi_{1}\rangle + \langle \varphi_{2} |\hat{F}_1|\varphi_{2}\rangle) ,
\phantom{qqq}
\label{F1iden}
\\
\langle F_1\rangle_{\{2\}}&=&\frac{1}{3}\langle F_1\rangle_{\{\}}
\nonumber\end{aligned}$$ and the $\lambda$-multiplet-average is expressed as $$\bar{F}^{[\lambda]}_1=f_{\lambda}\langle F_1\rangle_{\{\}}
(1\pm \tilde{\chi}_\lambda (\{2\}))$$ with the sign $+$ or $-$ for bosons or fermions, respectively. Finally, using characters in Table \[ST\_M3N3\], we have $\bar{F}^{[3]}_1=\bar{F}^{[21]}_1=2\langle F_1\rangle_{\{\}}$, $\bar{F}^{[1^3]}_1=0$ for bosons and $\bar{F}^{[1^3]}_1=\bar{F}^{[21]}_1=2\langle F_1\rangle_{\{\}}$, $\bar{F}^{[3]}_1=0$ for fermions. The results can be applied, for example, to the one-body density matrix $\varrho(\mathbf{r},\mathbf{r}')$ with $\langle \varphi_{m} |\varrho(\mathbf{r},\mathbf{r}')|\varphi_m\rangle=\varphi^*_{m}(\mathbf{r})\varphi_{m}(\mathbf{r}')$ in [Eq. (\[F1iden\])]{}.
For non-interacting particles and a two-body observable $\hat{F}_2$, [Eq. (\[FkavCk\])]{} will be replaced by $$\begin{aligned}
\langle F_2\rangle_{C_N}=\frac{2}
{N (N-1) g(C_N)\{\tilde{N}\}!}
\sum_{j_1 < j_2}
\sum_{{\mathcal{R}}\in C_N}
\prod_{j_1 \neq j'\neq j_2}\delta_{m_{{\mathcal{R}}j'}m_{j'}}
\nonumber
\\
\times
\langle \varphi_{m_{{\mathcal{R}}j_1}} \varphi_{m_{{\mathcal{R}}j_2}}|\hat{F}_2|\varphi_{m_{j_1}}\varphi_{m_{j_2}}\rangle,
$$ where $$\begin{aligned}
&&\langle \varphi_{m''}\varphi_{m'''} |\hat{F}_2|\varphi_m\varphi_{m'}\rangle =
\\
&&\frac{1}{2}\int d^{D} r_1 d^{D} r_2
[\varphi^*_{m''}(\mathbf{r_1})\varphi^*_{m'''}(\mathbf{r_2})\hat{F}_2
\varphi_{m}(\mathbf{r_1}) \varphi_{m'}(\mathbf{r_1})
\\
&&+\varphi^*_{m'''}(\mathbf{r_1})\varphi^*_{m''}(\mathbf{r_2})\hat{F}_2
\varphi_{m'}(\mathbf{r_1}) \varphi_{m}(\mathbf{r_1})].
$$ For $N=3$, $\{\tilde{N_1}\}=2$ and $\{\tilde{N_2}\}=1$ the permutations ${\mathcal{R}}$, allowed by the Kronecker symbols, depend on $j'$, namely $$\begin{aligned}
j'=1 &:& {\mathcal{R}}\in \{{\mathcal{E}},{\mathcal{P}}_{12},{\mathcal{P}}_{23},{\mathcal{P}}_{123}\}
\\
j'=2 &:& {\mathcal{R}}\in \{{\mathcal{E}},{\mathcal{P}}_{12},{\mathcal{P}}_{13},{\mathcal{P}}_{132}\}
\\
j'=3 &:& {\mathcal{R}}\in \{{\mathcal{E}},{\mathcal{P}}_{12}\} ,\end{aligned}$$ where the cycles of length 3, ${\mathcal{P}}_{123}$ and ${\mathcal{P}}_{132}$, belong to the conjugate class $\{3\}$. Then [Eq. ([8]{})]{} contains three terms with $$\begin{aligned}
\langle F_2\rangle_{\{\}}&=&\frac{1}{6}(\langle \varphi_{1}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{1}\rangle+2\langle \varphi_{1}\varphi_{2} |\hat{F}_2|\varphi_1\varphi_{2}\rangle)
\\
\langle F_2\rangle_{\{2\}}&=&\frac{1}{18}(\langle \varphi_{1}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{1}\rangle+2\langle \varphi_{1}\varphi_{2} |\hat{F}_2|\varphi_1\varphi_{2}\rangle
\\
&&+2\langle \varphi_{2}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{2}\rangle)
\nonumber
\\
\langle F_2\rangle_{\{3\}}&=&\frac{1}{6}\langle \varphi_{2}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{2}\rangle
\nonumber\end{aligned}$$ and the $\lambda$-multiplet-average is expressed as $$\begin{aligned}
\bar{F}^{[\lambda]}_2&=&f_{\lambda}\Bigl[\Bigl(\frac{1}{6}\langle \varphi_{1}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{1}\rangle+\frac{1}{3}\langle \varphi_{1}\varphi_{2} |\hat{F}_2|\varphi_1\varphi_{2}\rangle\Bigr)
(1\pm \tilde{\chi}_\lambda (\{2\}))
\\
&&+\frac{1}{3}\langle \varphi_{2}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{2}\rangle(\tilde{\chi}_\lambda (\{3\})\pm \tilde{\chi}_\lambda (\{2\}))\Bigr] .\end{aligned}$$ Finally, using characters in Table \[ST\_M4N4\], we have for bosons $$\begin{aligned}
\bar{F}^{[3]}_2&=&\frac{1}{3}(\langle \varphi_{1}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{1}\rangle+2\langle \varphi_{1}\varphi_{2} |\hat{F}_2|\varphi_1\varphi_{2}\rangle+2\langle \varphi_{2}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{2}\rangle)
\\
\bar{F}^{[21]}_2&=&\frac{1}{3}(\langle \varphi_{1}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{1}\rangle+2\langle \varphi_{1}\varphi_{2} |\hat{F}_2|\varphi_1\varphi_{2}\rangle-\langle \varphi_{2}\varphi_{1} |\hat{F}_2|\varphi_1\varphi_{2}\rangle) .
\\
\bar{F}^{[1^3]}_2&=&0\end{aligned}$$ The averages for fermions are equal to the averages for bosons with the conjugate Young diagram. The averages of the local two-body correlations for bosons are expressed as $$\begin{aligned}
\bar{\rho}^{[3]}_2(0)=\frac{1}{3}\left( \int d^{D} r |\varphi_{1}(\mathbf{r})|^4
+4\int d^{D} r |\varphi_{1}(\mathbf{r})|^2|\varphi_{2}(\mathbf{r})|^2\right)
\\
\bar{\rho}^{[21]}_2(0)=\frac{1}{3}\left( \int d^{D} r |\varphi_{1}(\mathbf{r})|^4
+\int d^{D} r |\varphi_{1}(\mathbf{r})|^2|\varphi_{2}(\mathbf{r})|^2\right) .\end{aligned}$$
|
---
abstract: 'If $A\ot _{R, \sigma }V$ and $A\ot _{P, \nu }W$ are two Brzeziński crossed products and $Q:W\ot V\rightarrow V\ot W$ is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A\ot _{S, \theta }(V\ot W)$. This construction contains as a particular case the iterated twisted tensor product of algebras.'
author:
- |
Leonard Dăuş [^1]\
Department of Mathematical Sciences, UAE University\
PO-Box 15551, Al Ain, United Arab Emirates\
e-mail: leonard.daus@uaeu.ac.ae
- |
Florin Panaite[^2]\
Institute of Mathematics of the Romanian Academy\
PO-Box 1-764, RO-014700 Bucharest, Romania\
e-mail: Florin.Panaite@imar.ro
title: A new way to iterate Brzeziński crossed products
---
Introduction {#introduction .unnumbered}
============
${\;\;\;}$The [*twisted tensor product*]{} of the associative unital algebras $A$ and $B$ is a new associative unital algebra structure built on the linear space $A\ot B$ with the help of a linear map $R:B\ot A\rightarrow A\ot B$ called a [*twisting map*]{}. This construction, denoted by $A\ot _RB$, appeared in several contexts and has various applications ([@Cap], [@VanDaele]). Concrete examples come especially from Hopf algebra theory, like for instance the smash product.
It was proved in [@jlpvo] that twisted tensor products of algebras may be iterated. Namely, if $A\otimes _{R_1}B$, $B\otimes _{R_2}C$ and $A\otimes _{R_3}C$ are twisted tensor products and the twisting maps $R_1$, $R_2$, $R_3$ satisfy the braid relation $(id_A\otimes R_2)\circ
(R_3\otimes id_B)\circ (id_C\otimes R_1)=(R_1\otimes id_C)\circ (id_B\otimes R_3)\circ
(R_2\otimes id_A)$, then one can define certain twisted tensor products $A\otimes _{T_2}(B\otimes _{R_2}C)$ and $(A\otimes _{R_1}B)\otimes _{T_1}C$ that are equal as algebras (and this algebra is called the iterated twisted tensor product).
The Brzeziński crossed product, introduced in [@brz], is a common generalization of twisted tensor products of algebras and the Hopf crossed product (containing also as a particular case the quasi-Hopf smash product introduced in [@bpvo]). If $A$ is an associative unital algebra, $V$ is a linear space endowed with a distinguished element $1_V$ and $\sigma :V\ot V\rightarrow A\ot V$ and $R:V\ot A\rightarrow A\ot V$ are linear maps satisfying certain conditions, then the Brzeziński crossed product is a certain associative unital algebra structure on $A\ot V$, denoted by $A\ot _{R, \sigma }V$.
In [@iterated] was proved that Brzeziński crossed products may be iterated, in the following sense. One can define first a ”mirror version” of the Brzeziński crossed product, denoted by $W\overline{\otimes}_{P, \nu }D$ (where $D$ is an associative unital algebra, $W$ is a linear space and $P$, $\nu $ are certain linear maps). Examples are twisted tensor products of algebras and the quasi-Hopf smash product introduced in [@bpv]. Then it was proved that, if $W\overline{\otimes}_{P, \nu }D$ and $D\ot _{R, \sigma }V$ are two Brzeziński crossed products and $Q:V\ot W\rightarrow W\ot D\ot V$ is a linear map satisfying some conditions, then one can define certain Brzeziński crossed products $(W\overline{\otimes}_{P, \nu }D)\ot
_{\overline{R}, \overline{\sigma }}V$ and $W\overline{\otimes }_{\overline{P}, \overline{\nu }}
(D\ot _{R, \sigma }V)$ that are equal as algebras. Iterated twisted tensor products of algebras appear as a particular case of this construction, as well as the so-called quasi-Hopf two-sided smash product $A\#H\#B$ from [@bpvo].
The aim of this paper is to show that Brzeziński crossed products may be iterated in a different way, that will also contain as a particular case the iterated twisted tensor product of algebras. Namely, we prove that, if $A\ot _{R, \sigma }V$ and $A\ot _{P, \nu }W$ are two Brzeziński crossed products and $Q:W\ot V\rightarrow V\ot W$ is a linear map satisfying certain properties, then we can define two Brzeziński crossed products $A\ot _{S, \theta }(V\ot W)$ and $(A\ot _{R, \sigma }V)\ot _{T, \eta }W$ that are equal as algebras.
Our inspiration for looking at this new way of iterating Brzeziński crossed products came from the following result in graded ring theory: If $G$ is a group, $R$ is a $G$-graded ring, $A$ and $B$ are two finite left $G$-sets, then there exists a ring isomorphism between the smash products $R\#(A\times B)$ and $(R\# A)\# B$. This result was obtained in [@leo Corollary 3.2], and it is useful in the study of the von Neumann regularity of rings of the type $R\# A$, cf. [@leo]. The smash product $R\# A$ of the $G-$graded ring $R$ by a (finite) left $G-$set $A$ was introduced in the paper [@nrv] and it is a particular case of a more general construction. If $H$ is a Hopf algebra, $R$ an $H-$comodule algebra and $C$ an $H-$module coalgebra, then we may consider the category $_{R}^{C}\mathcal{M}(H)$ of Doi-Koppinen Hopf modules (i.e. left $R-$modules and left $C-$comodules which satisfy certain compatibility relations). Then, the smash product used in [@leo] is a particular smash product and it is the first example in the category $_{R}^{C}\mathcal{M}(H)$ (in the case when $H$ is the groupring $k[G]$, $R$ a $G-$graded ring and $C$ the grouplike coalgebra $k[A]$ on a $G-$set $A$).
Preliminaries
=============
[\[se:1\]]{} ${\;\;\;\;}$ We work over a commutative field $k$. All algebras, linear spaces etc. will be over $k$; unadorned $\ot $ means $\ot_k$. By ”algebra” we always mean an associative unital algebra. The multiplication of an algebra $A$ is denoted by $\mu _A$ or simply $\mu $ when there is no danger of confusion, and we usually denote $\mu _A(a\ot a')=aa'$ for all $a, a'\in A$. The unit of an algebra $A$ is denoted by $1_A$ or simply $1$ when there is no danger of confusion.
We recall from [@Cap], [@VanDaele] that, given two algebras $A$, $B$ and a $k$-linear map $R:B\ot A\rightarrow A\ot B$, with Sweedler-type notation $R(b\ot a)=a_R\ot b_R$, for $a\in A$, $b\in B$, satisfying the conditions $a_R\otimes 1_R=a\otimes 1$, $1_R\otimes b_R=1\otimes b$, $(aa')_R\otimes b_R=a_Ra'_r\otimes (b_R)_r$, $a_R\otimes (bb')_R=(a_R)_r\otimes b_rb'_R$, for all $a, a'\in A$ and $b, b'\in B$ (where $r$ and $R$ are two different indices), if we define on $A\ot B$ a new multiplication, by $(a\ot b)(a'\ot b')=aa'_R\ot b_Rb'$, then this multiplication is associative with unit $1\ot 1$. In this case, the map $R$ is called a [*twisting map*]{} between $A$ and $B$ and the new algebra structure on $A\ot B$ is denoted by $A\ot _RB$ and called the [*twisted tensor product*]{} of $A$ and $B$ afforded by the map $R$.
We recall from [@brz] the construction of Brzeziński’s crossed product:
([@brz]) \[defbrz\] Let $(A, \mu , 1_A)$ be an (associative unital) algebra and $V$ a vector space equipped with a distinguished element $1_V\in V$. Then the vector space $A\ot V$ is an associative algebra with unit $1_A\ot 1_V$ and whose multiplication has the property that $(a\ot 1_V)(b\ot v)=
ab\ot v$, for all $a, b\in A$ and $v\in V$, if and only if there exist linear maps $\sigma :V\ot V\rightarrow A\ot V$ and $R:V\ot A\rightarrow A\ot V$ satisfying the following conditions: $$\begin{aligned}
&&R(1_V\ot a)=a\ot 1_V, \;\;\;R(v\ot 1_A)=1_A\ot v, \;\;\;\forall
\;a\in A, \;v\in V, \label{brz1} \\
&&\sigma (1_V\ot v)=\sigma (v\ot 1_V)=1_A\ot v, \;\;\;\forall
\;v\in V, \label{brz2} \\
&&R\circ (id_V\ot \mu )=(\mu \ot id_V)\circ (id_A\ot R)\circ (R\ot id_A),
\label{brz3} \\
&&(\mu \ot id_V)\circ (id_A\ot \sigma )\circ (R\ot id_V)\circ
(id_V\ot \sigma ) \nonumber \\
&&\;\;\;\;\;\;\;\;\;\;
=(\mu \ot id_V)\circ (id_A\ot \sigma )\circ (\sigma \ot id_V), \label{brz4} \\
&&(\mu \ot id_V)\circ (id_A\ot \sigma )\circ (R\ot id_V)\circ
(id_V\ot R ) \nonumber \\
&&\;\;\;\;\;\;\;\;\;\;
=(\mu \ot id_V)\circ (id_A\ot R )\circ (\sigma \ot id_A). \label{brz5} \end{aligned}$$ If this is the case, the multiplication of $A\ot V$ is given explicitly by $$\begin{aligned}
&&\mu _{A\ot V}=(\mu _2\ot id_V)\circ (id_A\ot id_A\ot \sigma )\circ
(id_A\ot R\ot id_V),\end{aligned}$$ where $\mu _2=\mu \circ (id_A\ot \mu )=\mu \circ (\mu \ot id_A)$. We denote by $A\ot _{R, \sigma }V$ this algebra structure and call it the [*crossed product*]{} (or Brzeziński crossed product) afforded by the data $(A, V, R, \sigma )$.
If $A\ot _{R, \sigma }V$ is a crossed product, we introduce the following Sweedler-type notation: $$\begin{aligned}
&&R:V\ot A\rightarrow A\ot V, \;\;\;R(v\ot a)=a_R\ot v_R, \\
&&\sigma :V\ot V\rightarrow A\ot V, \;\;\;\sigma (v\ot v')=\sigma _1(v, v')
\ot \sigma _2(v, v'), \end{aligned}$$ for all $v, v'\in V$ and $a\in A$. With this notation, the multiplication of $A\ot _{R, \sigma }V$ reads $$\begin{aligned}
&&(a\ot v)(a'\ot v')=aa'_R\sigma _1(v_R, v')\ot \sigma _2(v_R, v'), \;\;\;
\forall \;a, a'\in A, \;v, v'\in V.\end{aligned}$$
A twisted tensor product is a particular case of a crossed product (cf. [@guccione]), namely, if $A\ot _RB$ is a twisted tensor product of algebras then $A\ot _RB=A\ot _{R, \sigma }B$, where $\sigma :B\ot B\rightarrow A\ot B$ is given by $\sigma (b\ot b')=1_A\ot bb'$, for all $b, b'\in B$.
The conditions (\[brz3\]), (\[brz4\]) and (\[brz5\]) for $R$, $\sigma $ may be written in Sweedler-type notation respectively as $$\begin{aligned}
&&(aa')_R\ot v_R=a_Ra'_r\ot (v_R)_r, \label{brz6} \\
&&\sigma _1(y, z)_R\sigma _1(x_R, \sigma _2(y, z))\ot
\sigma _2(x_R, \sigma _2(y, z)) \nonumber \\
&&\;\;\;\;\;\;\;\;\;\;=\sigma _1(x, y)\sigma _1(\sigma _2(x, y), z)\ot
\sigma _2(\sigma _2(x, y), z), \label{brz7}\\
&&(a_R)_r\sigma _1(v_r, v'_R)\ot \sigma _2(v_r, v'_R)
=\sigma _1(v, v')a_R\ot \sigma _2(v, v')_R, \label{brz8}\end{aligned}$$ for all $a, a'\in A$ and $x, y, z, v, v'\in V,$ where $r$ is another copy of $R$.
The main result and examples
============================
\[principala\] Let $A\ot _{R, \sigma }V$ and $A\ot _{P, \nu }W$ be two crossed products and $Q:W\ot V\rightarrow V\ot W$ a linear map, with notation $Q(w\ot v)=
v_Q\ot w_Q$, for all $v\in V$ and $w\in W$. Assume that the following conditions are satisfied:\
(i) $Q$ is unital, in the sense that $$\begin{aligned}
&&Q(1_W\ot v)=v\ot 1_W, \;\;\; Q(w\ot 1_V)=1_V\ot w, \;\;\; \forall \;
v\in V, \; w\in W. \label{unitQ}\end{aligned}$$ (ii) the braid relation for $R$, $P$, $Q$, i.e. $$\begin{aligned}
&&(id_A\ot Q)\circ (P\ot id_V)\circ (id_W\ot R)=
(R\ot id_W)\circ (id_V\ot P)\circ (Q\ot id_A), \end{aligned}$$ or, equivalently, $$\begin{aligned}
&&(a_R)_P\ot (v_R)_Q\ot (w_P)_Q=(a_P)_R\ot (v_Q)_R\ot (w_Q)_P,
\;\;\; \forall \;a\in A, \; v\in V, \; w\in W. \label{braidcoord}\end{aligned}$$ (iii) we have the following hexagonal relation between $\sigma $, $P$, $Q$: $$\begin{aligned}
&&(id_A\ot Q)\circ (P\ot id_V)\circ (id_W\ot \sigma )=
(\sigma \ot id_W)\circ (id_V\ot Q)\circ (Q\ot id_V), \end{aligned}$$ or, equivalently, $$\begin{aligned}
&&\sigma _1(v, v')_P\ot \sigma _2(v, v')_Q\ot (w_P)_Q=
\sigma _1(v_Q, v'_q)\ot \sigma _2(v_Q, v'_q)\ot (w_Q)_q, \label{PQsigma}\end{aligned}$$ for all $v, v'\in V$ and $w\in W$, where $q$ is another copy of $Q$.\
(iv) we have the following hexagonal relation between $\nu $, $R$, $Q$: $$\begin{aligned}
&&(id_A\ot Q)\circ (\nu \ot id_V)=(R\ot id_W)\circ (id_V\ot \nu )\circ
(Q\ot id_W)\circ (id_W\ot Q), \end{aligned}$$ or, equivalently, $$\begin{aligned}
&&\nu _1(w, w')\ot v_Q\ot \nu _2(w, w')_Q=\nu _1(w_q, w'_Q)_R\ot
((v_Q)_q)_R\ot \nu _2(w_q, w'_Q), \label{RQniu}\end{aligned}$$ for all $v\in V$ and $w, w'\in W$, where $q$ is another copy of $Q$.
Define the linear maps $$\begin{aligned}
&&S:(V\ot W)\ot A\rightarrow A\ot (V\ot W), \;\;\;
S:=(R\ot id_W)\circ (id_V\ot P), \\
&&\theta :(V\ot W)\ot (V\ot W)\rightarrow A\ot (V\ot W), \\
&&\theta :=(\mu _A\ot id_V\ot id_W)\circ (id_A\ot R\ot id_W)
\circ (\sigma \ot \nu )\circ (id_V\ot Q\ot id_W), \\
&&T:W\ot (A\ot V)\rightarrow (A\ot V)\ot W, \;\;\;
T:=(id_A\ot Q)\circ (P\ot id_V), \\
&&\eta :W\ot W\rightarrow (A\ot V)\ot W, \\
&&\eta (w\ot w')=(\nu _1(w, w')\ot 1_V)\ot \nu _2(w, w'), \;\;\; \forall \; w, w'\in W.\end{aligned}$$ Then we have a crossed product $A\ot _{S, \theta }(V\ot W)$ (with respect to $1_{V\ot W}:=1_V\ot 1_W$), we have a crossed product $(A\ot _{R, \sigma }V)\ot _{T, \eta }W$ and we have an algebra isomorphism $A\ot _{S, \theta }(V\ot W)\simeq (A\ot _{R, \sigma }V)\ot _{T, \eta }W$ given by the trivial identification.
We prove first that $A\ot _{S, \theta }(V\ot W)$ is a crossed product, i.e. we need to prove the relations (\[brz1\])-(\[brz5\]) with $R$ replaced by $S$, $\sigma $ replaced by $\theta $ etc. The relations (\[brz1\]) and (\[brz2\]) follow immediately by (\[unitQ\]) and the relations (\[brz1\]) and (\[brz2\]) for $R$, $\sigma $ and $P$, $\nu $. Note that the maps $S$ and $\theta $ are defined explicitely by $$\begin{aligned}
&&S(v\ot w\ot a)=(a_P)_R\ot v_R\ot w_P, \\
&&\theta (v\ot w\ot v'\ot w')=\sigma _1(v, v'_Q)\nu _1(w_Q, w')_R\ot
\sigma _2(v, v'_Q)_R\ot \nu _2(w_Q, w'), \end{aligned}$$ for all $v, v'\in V$, $w, w'\in W$ and $a\in A$. We will denote by $R=r={\mathcal R}=
\overline{R}$, $Q=q=\overline{Q}$ and $P=p$ some more copies of $R$, $Q$ and $P$.\
:\
${\;\;\;}$$S\circ (id_V\ot id_W\ot \mu _A)(v\ot w\ot a\ot a')$ $$\begin{aligned}
&=&S(v\ot w\ot aa')\\
&=&((aa')_P)_R\ot v_R\ot w_P\\
&\overset{(\ref{brz6})}{=}&(a_Pa'_p)_R\ot v_R\ot (w_P)_p\\
&\overset{(\ref{brz6})}{=}&(a_P)_R(a'_p)_r\ot (v_R)_r\ot (w_P)_p\\
&=&(\mu _A\ot id_V\ot id_W)((a_P)_R\ot (a'_p)_r\ot (v_R)_r\ot (w_P)_p)\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot S)
((a_P)_R\ot v_R\ot w_P\ot a')\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot S)\circ (S\ot id_A)(v\ot w\ot a\ot a'),
\;\;\;q.e.d.\end{aligned}$$ :\
$(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )\circ
(S\ot id_V\ot id_W)\circ (id_V\ot id_W\ot \theta )(v\ot w\ot v'\ot w'\ot v''\ot w'')$ $$\begin{aligned}
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )\circ
(S\ot id_V\ot id_W)(v\ot w\ot \sigma _1(v', v''_Q)\nu _1(w'_Q, w'')_R\\
&&\ot
\sigma _2(v', v''_Q)_R\ot \nu _2(w'_Q, w''))\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )
(([\sigma _1(v', v''_Q)\nu _1(w'_Q, w'')_R]_P)_r\ot v_r\ot w_P\\
&&\ot
\sigma _2(v', v''_Q)_R\ot \nu _2(w'_Q, w''))\\
&=&([\sigma _1(v', v''_Q)\nu _1(w'_Q, w'')_R]_P)_r
\sigma _1(v_r, (\sigma _2(v', v''_Q)_R)_q)
\nu _1((w_P)_q, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
\sigma _2(v_r, (\sigma _2(v', v''_Q)_R)_q)_{\mathcal R}\ot
\nu _2((w_P)_q, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz6})}{=}&(\sigma _1(v', v''_Q)_P(\nu _1(w'_Q, w'')_R)_p)_r
\sigma _1(v_r, (\sigma _2(v', v''_Q)_R)_q)
\nu _1(((w_P)_p)_q, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
\sigma _2(v_r, (\sigma _2(v', v''_Q)_R)_q)_{\mathcal R}\ot
\nu _2(((w_P)_p)_q, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz6})}{=}&(\sigma _1(v', v''_Q)_P)_{\overline{R}}
((\nu _1(w'_Q, w'')_R)_p)_r
\sigma _1((v_{\overline{R}})_r, (\sigma _2(v', v''_Q)_R)_q)
\nu _1(((w_P)_p)_q, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
\sigma _2((v_{\overline{R}})_r, (\sigma _2(v', v''_Q)_R)_q)_{\mathcal R}\ot
\nu _2(((w_P)_p)_q, \nu _2(w'_Q, w''))\\
&\overset{(\ref{braidcoord})}{=}&(\sigma _1(v', v''_Q)_P)_{\overline{R}}
((\nu _1(w'_Q, w'')_p)_R)_r
\sigma _1((v_{\overline{R}})_r, (\sigma _2(v', v''_Q)_q)_R)
\nu _1(((w_P)_q)_p, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
\sigma _2((v_{\overline{R}})_r, (\sigma _2(v', v''_Q)_q)_R)_{\mathcal R}\ot
\nu _2(((w_P)_q)_p, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz8})}{=}&(\sigma _1(v', v''_Q)_P)_{\overline{R}}
\sigma _1(v_{\overline{R}}, \sigma _2(v', v''_Q)_q)
(\nu _1(w'_Q, w'')_p)_R
\nu _1(((w_P)_q)_p, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
(\sigma _2(v_{\overline{R}}, \sigma _2(v', v''_Q)_q)_R)_{\mathcal R}\ot
\nu _2(((w_P)_q)_p, \nu _2(w'_Q, w''))\\
&\overset{(\ref{PQsigma})}{=}&
\sigma _1(v'_{\overline{Q}}, (v''_Q)_q)_{\overline{R}}
\sigma _1(v_{\overline{R}}, \sigma _2(v'_{\overline{Q}}, (v''_Q)_q))
(\nu _1(w'_Q, w'')_p)_R
\nu _1(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
(\sigma _2(v_{\overline{R}}, \sigma _2(v'_{\overline{Q}}, (v''_Q)_q))_R)
_{\mathcal R}\ot
\nu _2(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz7})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\sigma _1(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)
(\nu _1(w'_Q, w'')_p)_R
\nu _1(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))_{\mathcal R}\\
&&\ot
(\sigma _2(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)_R)
_{\mathcal R}\ot
\nu _2(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz6})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\sigma _1(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)
[\nu _1(w'_Q, w'')_p
\nu _1(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))]_R\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)_R
\ot
\nu _2(((w_{\overline{Q}})_q)_p, \nu _2(w'_Q, w''))\\
&\overset{(\ref{brz7})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\sigma _1(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)
[\nu _1((w_{\overline{Q}})_q, w'_Q)
\nu _1(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')]_R\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}}), (v''_Q)_q)_R
\ot \nu _2(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')\\
&\overset{(\ref{brz8})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\{[\nu _1((w_{\overline{Q}})_q, w'_Q)
\nu _1(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')]_R\}_r
\sigma _1(\sigma _2(v, v'_{\overline{Q}})_r, ((v''_Q)_q)_R)\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}})_r, ((v''_Q)_q)_R)
\ot \nu _2(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')\\
&\overset{(\ref{brz6})}{=}&
\sigma _1(v, v'_{\overline{Q}})
[\nu _1((w_{\overline{Q}})_q, w'_Q)_R
\nu _1(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')_{\mathcal R}]_r
\sigma _1(\sigma _2(v, v'_{\overline{Q}})_r, (((v''_Q)_q)_R)_{\mathcal R})\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}})_r, (((v''_Q)_q)_R)_{\mathcal R})
\ot \nu _2(\nu _2((w_{\overline{Q}})_q, w'_Q), w'')\\
&\overset{(\ref{RQniu})}{=}&
\sigma _1(v, v'_{\overline{Q}})
[\nu _1(w_{\overline{Q}}, w')
\nu _1(\nu _2(w_{\overline{Q}}, w')_Q, w'')_{\mathcal R}]_r
\sigma _1(\sigma _2(v, v'_{\overline{Q}})_r, (v''_Q)_{\mathcal R})\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}})_r, (v''_Q)_{\mathcal R})
\ot \nu _2(\nu _2(w_{\overline{Q}}, w')_Q, w'')\\
&\overset{(\ref{brz6})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\nu _1(w_{\overline{Q}}, w')_R
(\nu _1(\nu _2(w_{\overline{Q}}, w')_Q, w'')_{\mathcal R})_r
\sigma _1((\sigma _2(v, v'_{\overline{Q}})_R)_r, (v''_Q)_{\mathcal R})\\
&&\ot
\sigma _2((\sigma _2(v, v'_{\overline{Q}})_R)_r, (v''_Q)_{\mathcal R})
\ot \nu _2(\nu _2(w_{\overline{Q}}, w')_Q, w'')\\
&\overset{(\ref{brz8})}{=}&
\sigma _1(v, v'_{\overline{Q}})
\nu _1(w_{\overline{Q}}, w')_R
\sigma _1(\sigma _2(v, v'_{\overline{Q}})_R, v''_Q)
\nu _1(\nu _2(w_{\overline{Q}}, w')_Q, w'')_r\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}})_R, v''_Q)_r
\ot \nu _2(\nu _2(w_{\overline{Q}}, w')_Q, w'')\\
&=&(\mu _A\ot id_V\ot id_W)(\sigma _1(v, v'_{\overline{Q}})
\nu _1(w_{\overline{Q}}, w')_R\ot
\sigma _1(\sigma _2(v, v'_{\overline{Q}})_R, v''_Q)
\nu _1(\nu _2(w_{\overline{Q}}, w')_Q, w'')_r\\
&&\ot
\sigma _2(\sigma _2(v, v'_{\overline{Q}})_R, v''_Q)_r
\ot \nu _2(\nu _2(w_{\overline{Q}}, w')_Q, w''))\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )
(\sigma _1(v, v'_{\overline{Q}})
\nu _1(w_{\overline{Q}}, w')_R\\
&&\ot \sigma _2(v, v'_{\overline{Q}})_R
\ot \nu _2(w_{\overline{Q}}, w')\ot v''\ot w'')\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )\circ
(\theta \ot id_V\ot id_W)(v\ot w\ot v'\ot w'\ot v''\ot w''), \;\;\;q.e.d.\end{aligned}$$ :\
${\;\;\;}$ $(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )\circ (S\ot id_V\ot id_W)\circ (id_V\ot
id_W\ot S)(v\ot w\ot v'\ot w'\ot a)$ $$\begin{aligned}
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )\circ (S\ot id_V\ot id_W)
(v\ot w\ot (a_P)_R\ot v'_R\ot w'_P)\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot \theta )((((a_P)_R)_p)_r\ot v_r
\ot w_p\ot v'_R\ot w'_P)\\
&=&(((a_P)_R)_p)_r\sigma _1(v_r, (v'_R)_Q)\nu _1((w_p)_Q, w'_P)_{\mathcal R}
\ot \sigma _2(v_r, (v'_R)_Q)_{\mathcal R}\ot \nu _2((w_p)_Q, w'_P)\\
&\overset{(\ref{braidcoord})}{=}&(((a_P)_p)_R)_r\sigma _1(v_r, (v'_Q)_R)
\nu _1((w_Q)_p, w'_P)_{\mathcal R}
\ot \sigma _2(v_r, (v'_Q)_R)_{\mathcal R}\ot \nu _2((w_Q)_p, w'_P)\\
&\overset{(\ref{brz8})}{=}&\sigma _1(v, v'_Q)
((a_P)_p)_R
\nu _1((w_Q)_p, w'_P)_{\mathcal R}
\ot (\sigma _2(v, v'_Q)_R)_{\mathcal R}\ot \nu _2((w_Q)_p, w'_P)\\
&\overset{(\ref{brz6})}{=}&\sigma _1(v, v'_Q)
[(a_P)_p
\nu _1((w_Q)_p, w'_P)]_R
\ot \sigma _2(v, v'_Q)_R\ot \nu _2((w_Q)_p, w'_P)\\
&\overset{(\ref{brz8})}{=}&\sigma _1(v, v'_Q)
[\nu _1(w_Q, w')a_P]_R
\ot \sigma _2(v, v'_Q)_R\ot \nu _2(w_Q, w')_P\\
&\overset{(\ref{brz6})}{=}&\sigma _1(v, v'_Q)
\nu _1(w_Q, w')_R(a_P)_r
\ot (\sigma _2(v, v'_Q)_R)_r\ot \nu _2(w_Q, w')_P\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot S)(\sigma _1(v, v'_Q)
\nu _1(w_Q, w')_R\ot \sigma _2(v, v'_Q)_R\ot \nu _2(w_Q, w')\ot a)\\
&=&(\mu _A\ot id_V\ot id_W)\circ (id_A\ot S)\circ (\theta \ot id_A)
(v\ot w\ot v'\ot w'\ot a),\end{aligned}$$ so $A\ot _{S, \theta }(V\ot W)$ is indeed a crossed product. With a similar computation one can prove that $(A\ot _{R, \sigma }V)\ot _{T, \eta }W$ is a crossed product, so the only thing left to prove is that the multiplications of $A\ot _{S, \theta }(V\ot W)$ and $(A\ot _{R, \sigma }V)\ot _{T, \eta }W$ coincide. A straightforward computation shows that the multiplication of $A\ot _{S, \theta }(V\ot W)$ is given by the formula $$\begin{aligned}
&&(a\ot v\ot w)(a'\ot v'\ot w')=a(a'_P)_{\mathcal R}\sigma _1(v_{\mathcal R},
v'_Q)\nu _1((w_P)_Q, w')_r\ot
\sigma _2(v_{\mathcal R},
v'_Q)_r\ot \nu _2((w_P)_Q, w').\end{aligned}$$ We compute now the multiplication of $(A\ot _{R, \sigma }V)\ot _{T, \eta }W$:\
${\;\;\;}$$(a\ot v\ot w)(a'\ot v'\ot w')$ $$\begin{aligned}
&=&(a\ot v)(a'\ot v')_T\eta _1(w_T, w')\ot \eta _2(w_T, w')\\
&=&(a\ot v)(a'_P\ot v'_Q)\eta _1((w_P)_Q, w')\ot \eta _2((w_P)_Q, w')\\
&=&(a\ot v)(a'_P\ot v'_Q)(\nu _1((w_P)_Q, w')\ot 1_V)\ot \nu _2((w_P)_Q, w')\\
&=&(a\ot v)(a'_P\nu _1((w_P)_Q, w')_R\ot (v'_Q)_R)\ot \nu _2((w_P)_Q, w')\\
&=&a[a'_P\nu _1((w_P)_Q, w')_R]_r\sigma _1(v_r, (v'_Q)_R)\ot
\sigma _2(v_r, (v'_Q)_R)\ot \nu _2((w_P)_Q, w')\\
&\overset{(\ref{brz6})}{=}&a(a'_P)_{\mathcal R}(\nu _1((w_P)_Q, w')_R)_r
\sigma _1((v_{\mathcal R})_r, (v'_Q)_R)\ot
\sigma _2((v_{\mathcal R})_r, (v'_Q)_R)\ot \nu _2((w_P)_Q, w')\\
&\overset{(\ref{brz8})}{=}&a(a'_P)_{\mathcal R}\sigma _1(v_{\mathcal R},
v'_Q)\nu _1((w_P)_Q, w')_r\ot
\sigma _2(v_{\mathcal R},
v'_Q)_r\ot \nu _2((w_P)_Q, w'),\end{aligned}$$ and we can see that the two multiplications coincide.
*We recall from [@jlpvo] what was called there an iterated twisted tensor product of algebras. Let $A$, $B$, $C$ be associative unital algebras, $R_1:B\ot A\rightarrow A\ot B$, $R_2:C\ot B\rightarrow B\ot C$, $R_3:C\ot A\rightarrow A\ot C$ twisting maps satisfying the braid equation $$\begin{aligned}
&&(id_A\ot R_2)\circ (R_3\ot id_B)\circ (id_C\ot R_1)=
(R_1\ot id_C)\circ (id_B\ot R_3)\circ (R_2\ot id_A). \end{aligned}$$ Then we have an algebra structure on $A\ot B\ot C$ (called the iterated twisted tensor product) with unit $1_A\ot 1_B\ot 1_C$ and multiplication $$\begin{aligned}
&&(a\ot b\ot c)(a'\ot b'\ot c')=a(a'_{R_3})_{R_1}\ot b_{R_1}b'_{R_2}\ot
(c_{R_3})_{R_2}c'.\end{aligned}$$*
We define $V=B$, $W=C$, $R=R_1$, $P=R_3$, $Q=R_2$ and the linear maps $$\begin{aligned}
&&\sigma :V\ot V\rightarrow A\ot V, \;\;\; \sigma (b\ot b')=1_A\ot bb', \;\;\;
\forall \;b, b'\in V, \\
&&\nu :W\ot W\rightarrow A\ot W, \;\;\;\nu (c\ot c')=1_A\ot cc', \;\;\;
\forall \; c, c'\in W. \end{aligned}$$ Then, for the crossed products $A\ot _{R, \sigma }V=A\ot _{R_1}B$, $A\ot _{P, \nu }W=A\ot _{R_3}C$ and the map $Q$, one can check that the hypotheses of Theorem \[principala\] are satisfied and the crossed products $A\ot _{S, \theta }(V\ot W)\equiv (A\ot _{R, \sigma }V)\ot _{T, \eta }W$ (notation as in Theorem \[principala\]) coincide with the iterated twisted tensor product.
Let $A\ot _{R, \sigma }V$ be a crossed product and $W$ an (associative unital) algebra. Define the linear maps $$\begin{aligned}
&&P:W\ot A\rightarrow A\ot W, \;\;\; P(w\ot a)=a\ot w, \;\;\;
\forall \; a\in A, \; w\in W, \\
&&\nu :W\ot W\rightarrow A\ot W, \;\;\; \nu (w\ot w')=1_A\ot ww', \;\;\;
\forall \; w, w'\in W, \end{aligned}$$ so we have the crossed product $A\ot _{P, \nu }W$ which is just the ordinary tensor product of algebras $A\ot W$. Define the linear map $Q:W\ot V
\rightarrow V\ot W$, $Q(w\ot v)=v\ot w$, for all $v\in V$, $w\in W$. Then one can easily check that the hypotheses of Theorem \[principala\] are satisfied.
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[^1]: Research supported by the National Research Foundation - United Arab Emirates and by the United Arab Emirates University(NRF-UAEU) under Grant No. 31S076/2013.
[^2]: Work supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0635, contract nr. 253/5.10.2011.
|
---
date:
- 1 December 2012 Last Revised 22 August 2017
- '.'
---
=cmr8
.25in
[**$3$-List-Coloring Graphs of Girth at least Five on Surfaces**]{}
.4in
[**Luke Postle**]{} [^1]
Department of Combinatorics and Optimization
University of Waterloo
Waterloo, ON
Canada, N2L 3G1
**ABSTRACT**
=1.0truein 5.5truein
Grötzsch proved that every triangle-free planar graph is $3$-colorable. Thomassen proved that every planar graph of girth at least five is $3$-choosable. As for other surfaces, Thomassen proved that there are only finitely many $4$-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding $3$-colorablity for graphs of girth at least five on any fixed surface. Dvořák, Král’ and Thomas strengthened Thomassen’s result by proving that the number of vertices in a $4$-critical graph of girth at least five is linear in its genus. They used this result to prove Havel’s conjecture that a planar graph whose triangles are pairwise far enough apart is $3$-colorable. As for list-coloring, Dvořák proved that a planar graph whose cycles of size at most four are pairwise far enough part is $3$-choosable.
In this article, we generalize these results. First we prove a linear isoperimetric bound for $3$-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many $4$-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a $4$-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvořák’s result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is $3$-choosable. Finally, we generalize to surfaces Thomassen’s result that every planar graph of girth at least five has exponentially many distinct $3$-list-colorings. Specifically, we show that every graph of girth at least five that has a $3$-list-coloring has $2^{\Omega(n)-O(g)}$ distinct $3$-list-colorings.
Introduction
============
All graphs considered in this paper are simple and finite. Graph coloring is an important area of study in graph theory. Recall that a *coloring* of a graph $G$ is an assignment of colors to vertices such that no two adjacent vertices receive the same color. A *$k$-coloring* is a coloring that uses at most $k$ colors while a graph $G$ is *$k$-colorable* if there exists a $k$-coloring of $G$.
List coloring, also known as choosability, generalizes the concept of coloring and was introduced by Vizing [@Vizing] and independently by Erdős et al. [@Erdos].
A *list assignment* of $G$ is a function $L$ that assigns to each vertex $v\in V(G)$ a list $L(v)$ of colors. An *$L$-coloring* of $G$ is a function $\phi: V(G)\rightarrow \bigcup_{v\in V(G)} L(v)$ such that $\phi(v)\in L(v)$ for every $v\in V(G)$ and $\phi(u)\ne \phi(v)$ for every pair of adjacent vertices $u,v$ in $G$. We say $G$ is *$L$-colorable* if $G$ has an $L$-coloring.
A *$k$-list-assignment* is a list-assignment $L$ such that $|L(v)|\ge k$ for all $v\in V(G)$. A graph $G$ is *$k$-list-colorable* or *$k$-choosable* if $G$ is $L$-colorable for every $k$-list-assignment $L$.
A well-known result of Grötzsch [@Grotzsch] states that every triangle-free planar graph is $3$-colorable. This theorem does not extend to list-coloring as Voigt [@Voigt] constructed a triangle-free planar graph that is not 3-choosable. However, Thomassen [@ThomList] proved that every planar graph of girth at least $5$ is $3$-choosable where *girth* is the length of the smallest cycle.
A natural extension of such results is to graphs on surfaces. For terms related to graphs embedded in surfaces, we refer to [@MoharThom]. Since not every graph is $3$-colorable and coloring is a *monotone* property, that is, $\chi(H)\le \chi(G)$ for every $H \subseteq G$, it is natural to consider the minimal non-colorable graphs. Similarly, choosability is a monotone property. Hence the following definitions.
A graph $G$ is *$k$-critical* if $G$ is not $(k-1)$-colorable but every proper subgraph of $G$ is. A graph $G$ is *$k$-list-critical* if there exists a $k$-list-assignment $L$ for $V(G)$ such that $G$ is not $L$-colorable but every proper subgraph of $G$ is $L$-colorable.
Thomassen [@ThomRegular] proved that there are only finitely many $4$-critical graphs of girth at least five embeddable in a fixed surface. Dvořák, Král’ and Thomas [@DvoKraTho3] strengthened this result by proving that the number of vertices in a $4$-critical graph of girth at least five is linear in its genus. One of the main results of this paper is to generalize these results to list-coloring as follows.
\[LinearGenus\] There exists a constant $c$ such that if $G$ is a $4$-list-critical of girth at least five embedded in a surface of genus $g$, then $|V(G)|\le cg$.
As an immediate corollary of Theorem \[LinearGenus\], we have the following.
\[FinMany\] For every surface $S$, there exist only finitely many $4$-list-critical graphs of girth at least five embeddable in $S$.
Using a result of Eppstein [@Eppstein] that testing for a fixed subgraph on a fixed surface can be done in linear time, we obtain the following.
\[LinearAlg\] For a fixed surface $S$, testing if a graph $G$ of girth at least five embedded in $S$ is $3$-choosable can be done in linear time.
Moreover, Postle and Thomas [@PostleThomas] deduced from Theorem \[LinearGenus\] that, for a fixed surface $S$, testing if a graph of girth at least five embedded in $S$ can be colored from a given $3$-list-assignment can be done in linear time, which is a theorem of Dvořák and Kawarabayashi [@DvoKaw2].
Another rather immediate corollary of Theorem \[LinearGenus\] is that locally planar graphs of girth at least five are $3$-choosable. More precisely, recall that the *edge-width* of an embedded graph is the length of its shortest non-contractible cycle. The corollary then is as follows.
\[LocPlanar\] For every surface $S$, there exists a constant $c(S)$ such that if $G$ is a graph of girth at least five and the edge-width of $G$ is at least $c(S)$, then $G$ is $3$-choosable.
Corollary \[LocPlanar\] follows from Corollary \[FinMany\] by letting $c(S)$ be strictly larger than the maximum number of vertices in a $4$-list-critical graph embeddable in $S$. Hence it follows from Theorem \[LinearGenus\] that $c(S) = O(g)$ where $g$ is the genus of $S$. Postle and Thomas [@PostleThomas] improved this further by showing that $c(S) = O(\log g)$ which is best possible since there exists expander graphs with girth $\Omega(\log g)$ and high chromatic number.
For ordinary coloring, Thomassen [@ThomRegular] derived similar consequences about locally planar graphs and algorithms from his theorem that there are only finitely many $4$-critical graphs of girth at least five embeddable in a fixed surface. Indeed, A key approach developed by Thomassen to prove these kinds of results is to consider a subgraph $H$ of a graph $G$. We say a coloring of $H$ *extends* to a coloring of $G$ if the two colorings agree on all vertices of $H$. The key then is to prove that there exists a subgraph $G'$ of $G$, whose size depends only on the size of $H$, such that any coloring of $H$ extends to $G$ if and only if that coloring extends to $G'$.
In recent years, researchers have realized that proving a linear bound on such subgraphs, namely $|V(G')|=O(|V(H)|)$, is not only desirable in the sense that this is usually best possible but also has many striking consequences. Indeed, a linear bound for such precolored subgraphs for ordinary $3$-coloring is the key to the work of Dvořák, Král’ and Thomas [@DvoKraTho] alluded to above (see [@DvoKraTho2] and [@DvoKraTho3]).
Of central importance is the case when $G$ is a plane graph and $H$ is its outer cycle. Dvořák and Kawarabayashi [@DvoKaw] have proven a linear bound for $3$-list-coloring graphs of girth at least five when $G$ is planar and $H$ has one component. To be more precise, we need the following definition.
A graph $G$ is *$C$-critical* (with respect to some list assignment $L$) if for every proper subgraph $H\subset G$ such that $C\subseteq H$, there exists an $L$-coloring of $C$ that extends to an $L$-coloring of $H$, but not to an $L$-coloring of $G$.
Dvořák and Kawarabayashi [@DvoKaw] proved that if $G$ is a plane graph of girth at least five that is $C$-critical with respect to some $3$-list-assignment $L$ where $C$ is its outer cycle, then $|V(G)|\le \frac{37}{3}|V(C)|$. This linear bound implies that the family of $4$-list-critical graphs of girth at least five is “hyperbolic", a precise definition of which can be found in Section \[Exp\], but informally says that there exists a constant $K$ such that for every graph in the family, the number of vertices inside a disk is at most $K$ times the number of vertices on the boundary.
The theory of hyperbolic families, developed by Postle and Thomas [@PostleThomas], then implies a number of striking consequences from this fact, namely Corollary \[LocPlanar\] with $c(S)=O(\log g)$ and - following Dvořák and Kawarabayashi [@DvoKaw2] - a linear-time algorithm for $3$-list-coloring a graph of girth at least five on a fixed surface.
The main result of this paper is to prove that the family of $4$-list-critical graphs is in fact “strongly hyperbolic" (the precise definition of which can also be found in Section \[Exp\] but can be thought of as requiring a linear bound not only for disks but annuli). This family being strongly hyperbolic is implied by the following theorem (see [@PostleThomas] for the details of this implication), which an extension of Dvorak and Kawaravayashi’s result to two precolored cycles.
\[Cylinder\] Let $G$ be a plane graph of girth at least five, let $L$ be a $3$-list assignment of $G$ and let $C_1\ne C_2$ be facial cycles of $G$. If $G$ is $C_1\cup C_2$-critical with respect to $L$, then $|V(G)|\le 177(|V(C_1)|+|V(C_2)|)$.
Hence we obtain the following corollary.
\[StronglyHyper\] The family of $4$-list-critical graphs of girth at least five is strongly hyperbolic.
Postle and Thomas [@PostleThomas] proved that if ${{\mathcal}{F}}$ is a strongly hyperbolic family of graphs, then there exists $c_{{{\mathcal}{F}}}$ such that for every graph $G\in {{\mathcal}{F}}$, $|V(G)|\le c_{{{\mathcal}{F}}}g$ where $g$ is the genus of $G$. That theorem combined with Corollary \[StronglyHyper\] implies Theorem \[LinearGenus\]. Moreover the proof of Theorem \[Cylinder\] does not directly rely on the theorems of Thomassen, or Dvořák, Král’ and Thomas, or Dvořák and Kawarabayashi and hence provides independent proofs of these results (though in the latter case with a larger constant).
Thus the majority of this paper is devoted to proving Theorem \[Cylinder\]. However, the consequences of Corollary \[StronglyHyper\] go beyond even Theorem \[LinearGenus\] as explained in full detail in [@PostleThomas]. In the next few subsections, we highlight some of these further applications to related problems about $3$-list-coloring graphs of girth at least five on surfaces.
A Linear Bound for Precolored Subgraphs
---------------------------------------
Postle and Thomas [@PostleThomas] showed that Theorem \[Cylinder\] is equivalent to the strong hyperbolicity of a slightly more general family of graphs as follows. We say $(G,H)$ is a *graph with boundary* if $H$ is a subgraph of the graph $G$. We can then extend the notion of hyperbolicity and strong hyperbolicity to families of graph with boundaries by requiring that the disk (or annuli) not contain any edge or vertex of $H$ in its interior. With this terminology, Theorem \[Cylinder\] is equivalent to the following.
\[CylinderBoundary\] The family of graphs with boundary $(G,H)$ where $G$ is a graph of girth at least five such that $G$ is $H$-critical with respect to some $3$-list-assignment is strongly hyperbolic.
Postle and Thomas then showed that if ${{\mathcal}{F}}$ is a strongly hyperbolic family with boundary, then there exists $c_{{{\mathcal}{F}}}$ such that for every $(G,H)\in {{\mathcal}{F}}$, $|V(G)|=c_{{{\mathcal}{F}}}(g+|V(H)|)$ where $g$ is the genus of $G$. Combining that theorem with Theorem \[CylinderBoundary\] gives the following corollary.
\[LinearCriticalBound\] There exists a constant $c$ such that if $G$ is $H$-critical with respect to some $3$-list-assignment $L$ for some subgraph $H$ of $G$, then $|V(G)|\le c(g+|V(H)|)$ where $g$ is the genus of $G$.
Corollary \[LinearCriticalBound\] is a far-reaching generalization of Theorem \[Cylinder\] (where $g=0$ and $H$ has at most 2 components), though the constant is much larger than $177$ (around $5,000,000$ if one does the calculations in [@PostleThomas] using 177 and the constant of $37/3$ from Dvořák and Kawarabayashi for one cycle).
Precolored Cycles and Crossings Far Apart
-----------------------------------------
Using Theorem \[CylinderBoundary\], combined with a structure theorem for strongly hyperbolic families of graphs and the fact that every graph of girth at least five embeddable in the plane with at most one crossing is $3$-choosable (which can derived from Thomassen’s original proof of $3$-choosability of graphs of girth at least five), Postle and Thomas [@PostleThomas] proved the following.
\[Precolored2\] There exists $D>0$ such that the following holds: Let $G$ be a graph of girth at least five $2$-cell embedded in a surface $S$ of genus $g$ such that the edge-width of $G$ is $\Omega(g)$ and let $L$ be a $3$-list-assignment for $G$. If $X\subset V(G)$ such that $d(u,v)\ge D$ for all $u\ne v\in X$, then every $L$-coloring of $X$ extends to an $L$-coloring of $G$.
Indeed, Postle and Thomas proved a stronger version of Theorem \[Precolored2\] when there are precolored cycles of length at most four far enough apart.
\[PrecoloredCycles\] There exists $D>0$ such that the following holds: Let $G$ be a graph $2$-cell embedded in a surface $S$ of genus $g$ such that the edge-width of $G$ is $\Omega(g)$ and let $L$ be a $3$-list-assignment for $G$. Let ${{\mathcal}{C}}$ be the set of cycles of $G$ of length at most four. If $d(C_i,C_j)\ge D$ for all $C_i\ne C_j \in {{\mathcal}{C}}$ and each $C_i\in {{\mathcal}{C}}$ is homotopically trivial, then if $\phi$ is an $L$-coloring of the cycles in ${{\mathcal}{C}}$, then $\phi$ extends to an $L$-coloring of $G$.
When $S$ is the plane, this was proved by Dvořák [@Dvo] and hence Theorem \[PrecoloredCycles\] provides an independent proof of his result. His result is actually an analogue of Havel’s conjecture for list-coloring.
Havel’s conjecture [@Hav1; @Hav2] states that there exists $d>0$ such that if all the triangles in a planar graph $G$ are pairwise distance at least $d$ apart, then $G$ is $3$-colorable. Dvořák, Král’ and Thomas [@DvoKraTho5] proved Havel’s conjecture (see also [@DvoKraTho]). An essential ingredient of their proof is proving that the family of $4$-critical graphs of girth at least five is strongly hyperbolic, for which Theorem \[Cylinder\] provides an independent (and arguably shorter) proof. Dvořák’s result is a natural analogue of Havel’s conjecture for list-coloring as there exist triangle-free planar graphs which are not $3$-choosable.
Postle and Thomas [@PostleThomas] also deduced the following theorem from Theorem \[CylinderBoundary\].
\[CrossingSurface\] There exists $D>0$ such that the following holds: Let $G$ be a graph of girth at least five drawn in a surface $S$ of genus $g$ with a set of crossings $X$ and $L$ be a $3$-list-assignment for $G$. Let $G_X$ be the graph obtained by adding a vertex $v_x$ at every crossing $x\in X$. If the edge-width of $G_X$ is $\Omega(g)$ and $d(v_x,v_{x'})\ge D$ for all $v_x \ne v_{x'} \in V(G_X)\setminus V(G)$, then $G$ is $L$-colorable.
For ordinary $3$-coloring, the analogues of Theorems \[Precolored2\], \[PrecoloredCycles\], and \[CrossingSurface\] may be derived from Dvořák, Král’ and Thomas’ work [@DvoKraTho5].
Exponentially Many List Colorings
---------------------------------
For ordinary coloring, Thomassen gave a suprisingly short proof that every planar graph of girth at least five has at least $2^{\frac{|V(G)|}{9}}$ distinct $3$-colorings by using the edge-density of planar graphs of girth at last five and the fact that such a graph has at least one $3$-coloring by Grötzsch’s Theorem. Furthermore, Thomassen’s work easily implies that for every surface $S$, there exists $c_S$ such that if a graph $G$ of girth at least five embedded in $S$ has at least one $3$-coloring, then it has at least $c_S2^{\frac{|V(G)|}{9}}$ distinct $3$-colorings.
As for list-coloring, Thomassen [@ThomExp] in a deeper result proved that a planar graph $G$ of girth at least five has at least $2^{\frac{|V(G)|}{10000}}$ distinct $L$-colorings for any $3$-list-assignment $L$. Once again using Theorem \[CylinderBoundary\] and a structure theorem for strongly hyperbolic families of graphs, Postle and Thomas [@PostleThomas] extended this to all surfaces as follows.
\[ExpSurfacetheorem\] There exists $\epsilon > 0$ such that: For every surface $S$ there exists a constant $c_S>0$ such that following holds: Let $G$ be a graph of girth at least five embedded in $\Sigma$ and $L$ a $3$-list-assignment for $G$. If $G$ has an $L$-coloring, then $G$ has at least $c_S 2^{\epsilon|V(G)|}$ $L$-colorings of $G$.
Indeed, they prove a stronger version of Theorem \[ExpSurfacetheorem\] about extending a precoloring of a subset of the vertices as follows. Note if $G$ is a graph and $R\subseteq V(G)$, we let $G[R]$ denote the subgraph of $G$ induced by $R$.
\[ExpSurface2\] There exist constants $\epsilon,\alpha>0$ such that following holds: Let $G$ be a graph of girth at least five embedded in a surface $S$ of genus $g$, $R\subseteq V(G)$ and $L$ a $3$-list-assignment for $G$. If $\phi$ is an $L$-coloring of $G[R]$ such that $\phi$ extends to an $L$-coloring of $G$, then $\phi$ extends to at least $2^{\epsilon(|V(G)|-\alpha(g+|R|))}$ $L$-colorings of $G$.
To prove Theorem \[ExpSurface2\], they showed that it suffices to prove that the family of graphs of girth at least five with boundary which are ‘critical’ with respect to not having exponentially many extensions is strongly hyperbolic. In fact, they proved with some additional work that it suffices to prove such a family is hyperbolic. The proof of that fact however was intentionally omitted from their paper as it relies on the proof of Theorem \[Cylinder\]; we provide the proof of said hyperbolicity in Section \[Exp\], thereby completing the proof of Theorem \[ExpSurface2\].
Outline of Proof {#Outline}
----------------
The above theorems show the value in establishing that the family of $4$-list-critical graphs is strongly hyperbolic. Our main result - Theorem \[Cylinder\] - proves this fact. To prove Theorem \[Cylinder\], we will need Thomassen’s [@ThomShort] stronger inductive statement as follows.
\[Thom0\] Let $G$ be a plane graph of girth at least $5$ and $C$ the outer cycle of $G$. Let $P$ be a path in $G$ of length at most $5$, such that $V(P)\subseteq V(C)$. Let $L$ be an assignment of lists to the vertices of $G$ such that $|L(v)|$ = 3 for all $v \in V(G)\setminus V(C)$, $|L(v)|\ge 2$ for all $v\in V(C) \setminus V(P)$, and $|L(v)| = 1$ for all $v\in V (P)$. Further suppose that no vertex $v$ with $|L(v)| = 2$ is adjacent to a vertex $u$ such that $|L(u)|\le 2$. If there exists an $L$-coloring of the subgraph induced by $V(P)$, then there exists an $L$-coloring of $G$.
To prove Theorem \[Cylinder\], we then consider the structures which arise from Theorem \[Thom0\]. This is also the idea behind Dvořák and Kawarabayashi’s proof. However, they used a stronger version of Theorem \[Thom0\] (which they also proved) to yield a shorter list of structures and from these derived an inductive formula on the size of a $C$-critical graph which decreases if there are long faces (an idea also used by Dvořák, Král’ and Thomas in [@DvoKraTho2] and [@DvoKraTho3]).
For two cycles, we are not able to use Dvořák and Kawarabayashi’s stronger version of Theorem \[Thom0\]. Instead, we rely only on one additional result of Thomassen [@ThomRegular], which is the key to his proof that there are finitely many $4$-critical graphs of girth at least five embeddable in a fixed surface. That result (stated in this paper as Theorem \[CylinderStructure\]) says that two cycles that are far apart (distance at least three) and whose vertices of lists of size two form an independent set has a $3$-list-coloring (technically Thomassen’s proof is done in terms of ordinary coloring but it is easily adapted to work for list-coloring).
Thus our proof for two cycles must only use the structures arising from Theorem \[Thom0\]. Hence we also provide an independent proof of Dvořák and Karawabayashi’s result. To accomplish this, we also prove a general inductive formula on the size of a $C$-critical graph which decreases if there are long faces; crucially though, the formula also decreases if there are many edges from vertices in $C$ to vertices not in $C$. That subtlety is enough to allow us to use the weaker list of structures arising from Theorem \[Thom0\].
Outline of Paper
----------------
In Section \[Canvas\], we provide some necessary preliminaries and list the structures arising in $C$-critical graphs (see Lemma \[Structure3\]). In Section \[Linear\], we develop our general inductive formula (see Definition \[Parameters\] and Theorem \[StrongLinear\]) on the size of a $C$-critical graph and then show how it implies Theorem \[Cylinder\]. In Section \[Proof\], we prove said general formula. Finally, in Section \[Exp\], we provide as promised the proof that the family of exponentially critical graphs is hyperbolic thereby completing the proof of Theorem \[ExpSurface2\] (see Theorem \[ExpHyper\]).
Critical Canvases {#Canvas}
=================
In this section, we develop the necessary preliminaries and provide a key structural lemma (Lemma \[Structure3\]). Let us first define the graphs we will be working with as follows.
We say that $(G, S, L)$ is a *canvas* if $G$ is a plane graph of girth at least five, $S$ is a subgraph of $G$, $L$ is a list assignment for the vertices of $G$ such that $|L(v)|\ge 3$ for all $v\in V(G)\setminus V(S)$ and there exists an $L$-coloring of $S$. We call $S$ the *boundary* of the canvas. We say a canvas $(G,S,L)$ is *critical* if $G$ is $S$-critical with respect to the list assignment $L$.
We need the following lemma about subgraphs of critical graphs. The lemma is standard and can be found in [@PostleThomas2] but we include its proof for completeness. Note that if $G$ is a graph and $A,B$ are subgraphs of $G$, we let $A\cap B$ denote the graph where $V(A\cap B)=V(A)\cap V(B)$ and $E(A\cap B)=E(A)\cap E(B)$.
\[SComponent\] Let $S$ be a subgraph of a graph $G$ such that $G$ is $S$-critical graph with respect to a list assignment $L$. Let $A,B\subseteq G$ such that $A\cup B=G, S\subseteq A$ and $B\ne A\cap B$. Then $B$ is $A\cap B$-critical.
Since $G$ is $S$-critical, every isolated vertex of $G$ is in $S$, and thus every isolated vertex of $B$ is in $A\cap B$. Suppose for a contradiction that $B$ is not $A\cap B$-critical. Then, there exists an edge $e \in E(B) \setminus E(A\cap B)$ such that every $L$-coloring of $A\cap B$ that extends to $B \setminus e$ also extends to $B$.
Note that $e \not\in E(S)$. Since $G$ is $S$-critical, then there exists an $L$-coloring $\phi$ of $S$ that extends to an $L$-coloring $\phi$ of $G \setminus e$, but does not extend to an $L$-coloring of $G$. However, by the choice of $e$, the restriction of $\phi$ to $A\cap B$ extends to an $L$-coloring $\phi'$ of $B$. Let $\phi''$ be the coloring that matches $\phi'$ on $V(B)$ and $\phi$ on $V(G) \setminus V(B)$. Observe that $\phi''$ is an $L$-coloring of $G$ extending $\phi$, which is a contradiction.
Lemma \[SComponent\] has two useful corollaries. To state it, we need the following definitions.
Let $T=(G,S,L)$ be a canvas and $H\subseteq G$ such that $S\subseteq H$. We define the *subcanvas* of $T$ induced by $H$ to be $(H,S,L)$, which we denote $T|H$. Similarly, we define the *supercanvas* of $T$ induced by $H$ to be $(G,H,L)$, which we denote $T/H$.
First let us note a useful fact about canvases whose proof we omit.
\[CriticalSubgraph\] Let $T=(G,S,L)$ be a canvas. If there exists a proper $L$-coloring of $S$ that does not extend to $G$, then $T$ contains a critical subcanvas.
Here then is the first corollary of Lemma \[SComponent\].
\[Subcanvas\] Let $T=(G,S,L)$ be a critical canvas. If $H$ is a subgraph of $G$ such that $S$ is a proper subgraph of $H$ and every edge in $E(G)\setminus E(H)$ has no end in $V(H)\setminus V(S)$, then the subcanvas induced by $H$, that is $T|H$, is a critical canvas.
This follows from Lemma \[SComponent\] with $B=H$ and $A=G\setminus (V(H)\setminus V(S))$ since $A\cap B=S$ and $A\cup B=G$.
Finally here is the second corollary of Lemma \[SComponent\].
\[Supercanvas\] Let $T=(G,S,L)$ be a critical canvas. If $H$ is a proper subgraph of $G$ containing $S$, then the supercanvas induced by $H$, that is $T/H$, is critical.
This follows from Lemma \[SComponent\] with $B=G$ and $A=H$.
Critical Canvases with One Boundary Component
---------------------------------------------
We now prove a structure theorem for critical canvases. For that, we need the following structures.
Let $T=(G,S,L)$ be a canvas. Let $k\ge 1$ and let $P=p_1p_2\ldots p_{k+1}$ be a path in $G\setminus V(S)$. We say $P$ is a
- a *neighboring $k$-path* of $S$ if $p_i\in N(S)$ for all $i$ with $1\le i \le k+1$,
- a *semi-neighboring $3$-path* of $S$ if $k=3$ and $p_1,p_2,p_4\in N(S)$,
- a *semi-neighboring $5$-path* of $S$ if $k=5$ and $p_1,p_2,p_5,p_6\in N(S)$.
Finally let $v\in N(S)\setminus S$ and suppose that $v$ has three neighbors $u_1,u_2,u_3$ in $N(S)\setminus S$. Then we say that $G[\{v,u_1,u_2,u_3\}]$ is a *neighboring claw* of $S$.
We can first derive a set of simple structures from Theorem \[Thom0\] for critical canvases whose boundary has one component as follows.
\[Structure\] If $T=(G,S,L)$ is a critical canvas such that $S$ is connected, then there exists one of the following:
1. an edge not in $S$ with both ends in $V(S)$, or,
2. a vertex not in $V(S)$ with at least two neighbors in $V(S)$, or,
3. a neighboring $1$-path of $S$.
Suppose not. Let $\phi$ be an $L$-coloring of $S$ that does not extend to $G$. Let $L'(v)=L(v)\setminus \{\phi(u)| u\in N(v)\cap V(S)\}$. Since 1 does not hold, $V(G)\setminus V(S)\ne\emptyset$. Since 2 does not hold, $|L'(v)|\ge 2$ for all $v\in V(G)\setminus V(S)$. Since 3 does not hold, there does not exist an edge $uv \in E(G)$ with $u,v \in V(G)\setminus V(S)$ with $|L'(u)|=|L'(v)|=2$. But then by Theorem \[Thom0\] applied to $G\setminus V(S)$ and $L'$ (with $P=\emptyset$), there exists an $L'$-coloring $\phi'$ of $G$. But now $\phi''=\phi\cup\phi'$ is an $L$-coloring of $G$, a contradiction.
Lemma \[Structure\] is not useful for our proof however, since a neighboring $1$-path is unhelpful for reductions. Nevertheless, by coloring neighboring $1$-paths and using a second application of Theorem \[Thom0\] (or rather Theorem \[Structure\]), we can deduce a stronger outcome than Lemma \[Structure\](3) as follows.
\[StructureB\] If $T=(G,S,L)$ is a critical canvas such that $S$ is connected, then there exists one of the following:
1. an edge not in $S$ with both ends in $V(S)$, or,
2. a vertex not in $V(S)$ with at least two neighbors in $V(S)$, or,
3. a neighboring $2$-path of $S$, or
4. a semi-neighboring $3$-path of $S$, or,
5. a semi-neighboring $5$-path of $S$.
Suppose not. Since 3 does not hold, the components of $N(S)\setminus S$ have size at most two. Let $R$ be the union of all components of $N(S)\setminus S$ of size at most two. As 1 and 2 do not hold, it follows from Lemma \[Structure\] applied to $T$ that there exists a neighboring $1$-path of $S$ and hence $R\ne \emptyset$. Let $H=G[V(S)\cup V(R)]$. As the vertices in $R$ have degree at most two in $H$, it follows that $H$ is a proper subgraph of $G$. By Corollary \[Supercanvas\], $T/H$ is a critical canvas. Note that $H$ is connected and hence we can apply Lemma \[Structure\] to $T/H$.
Note that Lemma \[Structure\](1) does not hold for $T/H$ as $H$ is an induced subgraph of $G$. So let us suppose that Lemma \[Structure\](2) holds for $T/H$, that is there a vertex $v$ not in $V(H)$ with at least two neighbors in $V(H)$. Let $u,w$ be neighbors of $v$ in $V(H)$. As 2 does not hold for $T$, at least one of $u$ or $w$ must be in $R$. Suppose without loss of generality that $u$ is in $R$. Let $u'$ be the unique neighbor of $u$ in $R$. As $G$ has girth at least five, $u'\ne w$. As 3 does not hold for $T$, it follows that $w\in R$. But now $u'uvw$ is a semi-neighboring $3$-path of $S$ in $T$, that is 4 holds, a contradiction.
So we may assume that Lemma \[Structure\](3) holds for $T/H$, that is there is a neighboring $1$-path $P=p_1p_2$ of $H$ in $T/H$. Let $u_1$ be a neighbor of $p_1$ in $V(H)$ and let $u_2$ be a neighbor of $p_2$ in $V(H)$. Since $p_1,p_2\notin V(R)$, at least one of $u_1$ or $u_2$ is not in $S$. Suppose without loss of generality that $u_1$ is not in $S$. Hence $u_1\in R$. Let $u_1'$ be the unique neighbor of $u_1$ in $R$. If $u_2\in V(S)$, then $u_1'u_1p_1p_2$ is a semi-neighboring $3$-path of $S$ in $T$ and 4 holds, a contradiction. So we may assume $u_2\in R$. Let $u_2'$ be the unique neighbor of $u_2$ in $R$. Note that as $G$ has girth at least five, $u_1\ne u_2$ and $u_1$ is not adjacent to $u_2$. Hence $u_1,u_2,u_1',u_2'$ are all distinct. But then $u_1'u_1p_1p_2u_2u_2'$ is a semi-neighboring $5$-path of $S$ in $T$ and 5 holds, a contradiction.
Unfortunately, in our proof neighboring $2$-paths are also not strong enough for reductions. Yet neighboring $2$-paths just barely fail in this regard. To that end, we make the following definition.
Let $T=(G,S,L)$ be a canvas. Let $P=p_1p_2p_3$ be a neighboring $2$-path of $S$ such that for each $i\in\{1,2,3\}$, $p_i$ has a unique neighbor $u_i$ in $S$. Let $H=G\cup P \cup \{p_1u_1,p_2u_2,p_3u_3\}$. We say $T/H$ is obtained from $T$ by *relaxing* $P$ and that $T/H$ is a *relaxation* of $T$. We define $T$ to be a $0$-relaxation of itself. For $k\ge 1$, we say a supercanvas $T'$ of $T$ is a *$k$-relaxation* of $T$ if $T'$ is a relaxation of a $(k-1)$-relaxation of $T$.
Now by first coloring neighboring $2$-paths and applying Lemma \[StructureB\] a second time, we can upgrade the outcomes of Lemma \[StructureB\] (in particular outcome 3) at the cost of finding outcomes 4 or 5 in a $k$-relaxation for some $k\le 2$ as follows.
\[Structure2\] If $T=(G,S,L)$ is a critical canvas such that $S$ is connected, then there exists one of the following:
1. an edge not in $S$ with both ends in $S$, or,
2. a vertex not in $V(S)$ with at least two neighbors in $V(S)$, or,
3. a neighboring claw of $S$, or
4. a $k$-relaxation $T'=(G,S',L)$ of $T$ with $k\le 2$ and a semi-neighboring $3$-path of $S'$, or,
5. a $k$-relaxation $T'=(G,S',L)$ of $T$ with $k\le 2$ and a semi-neighboring $5$-path of $S'$.
Suppose not. Note that since $G$ has no semi-neighboring $3$-path of $S$ then $G$ has no neighboring $k$-path of $S$ for any $k\ge 3$. Let $R$ be the union of all components of $N(S)\setminus S$ of size exactly three. By Lemma \[StructureB\], we may assume that $R\ne\emptyset$ as otherwise one of 1, 2, 4 or 5 holds, a contradiction.
Let $H=G[V(C)\cup V(R)]$. As there is no neighboring claw or neighboring $3$-path of $S$, every vertex in $V(H)\setminus V(S)$ is in a unique neighboring $2$-path of $S$; let $P(u)$ denote said path for each $u\in V(H)\setminus V(S)$. Further note that if $Q$ is a neighboring $1$-path of $H$, then either $Q$ is a neighboring $1$-path of $S$ or the neighbors of $Q$ in $H$ are contained in a unique neighboring $2$-path of $C$, as otherwise there exists a semi-neighboring $3$-path or semi-neighboring $5$-path of $C$, that is 4 or 5 holds, a contradiction. In the latter case, let $P(Q)$ denote this unique neighboring $2$-path of $S$.
Note that $H$ is a proper subgraph of $G$ as there exist vertices in $V(H)\setminus V(S)$ which degree two in $H$ but degree at least three in $G$. By Lemma \[Supercanvas\], $T/H$ is critical. Apply Lemma \[StructureB\] to $T/H$. Note that Lemma \[StructureB\](1) does not hold for $T/H$ as $H$ is an induced subgraph of $G$ since 1 and 2 do not hold for $T$ and $R$ is a set of components of $N(S)\setminus S$.
So suppose Lemma \[StructureB\](1) holds, that is there exists a vertex $v\in V(G)\setminus V(H)$ with two neighbors $u_1,u_2$ in $V(H)$. Since 2 does not hold for $T$ and $R$ is a set of components of $N(S)\setminus S$, it follows that $u_1,u_2\not\in V(S)$ and hence $u_1,u_2\in V(R)$. As $G$ has girth at least five, $P(u_1)\ne P(u_2)$. But then there exists a semi-neighboring $3$-path of $S$ contained in $P(u_1)\cup P(u_2) \cup \{v\}$ and hence 4 holds, a contradiction.
Next suppose Lemma \[StructureB\](3) holds, that is there exists a neighboring $2$-path $P=p_1p_2p_3$ of $H$. Given that $V(P)\cap V(R)=\emptyset$, there exists $i\in \{1,2,3\}$ such that $p_i\in N(R)$. We may suppose without loss of generality that $i\in\{1,2\}$. But then the neighbors of $p_1p_2$ are contained in a unique neighboring $2$-path of $C$, $P(p_1p_2)$ as noted above. As $G$ has girth at least five, the neighbor of $p_1$ and the neighbor of $p_2$ must be the ends of $P(p_1p_2)$. Yet now the neighbors of $p_2p_3$ are contained in a unique neighboring $2$-path of $C$, $P(p_2p_3)$ and we find that $P(p_1p_2)=P(p_2p_3)$. As $G$ has girth at least five, the neighbor of $p_3$ in $P(p_1p_2)$, call it $x$, is distinct from the neighbor of $p_2$ in $P(p_1p_2)$. But then $xp_1p_2p_3$ is a $4$-cycle, contradicting that $G$ has girth at least five.
Next suppose Lemma \[StructureB\](4) holds, that is there exists a semi-neighboring $3$-path $P=p_1p_2p_3p_4$ of $H$, where $p_1,p_2,p_4\in N(V(H))\setminus V(H)$. As $p_1p_2$ is a neighboring $1$-path of $H$, the neighbors of $p_1p_2$ are either contained in a unique neighboring $2$-path of $S$, $P(p_1p_2)$, or $p_1p_2$ is a neighboring $1$-path of $S$. Let $y$ be the neighbor of $p_4$ in $V(H)$. Now either $y\in V(S)$ or $y\in V(R)$. In all cases, $P$ is a semi-neighboring $3$-path of a $\le 2$-relaxation $T'$ of $T$, where either $T'=T$, or $T'$ is obtained from $T$ by relaxing $P(p_1p_2)$, or by relaxing $P(y)$ or by relaxing both. Hence 4 holds, a contradiction.
Next suppose Lemma \[StructureB\](5) holds, that is there exists a semi-neighboring $5$-path $P=p_1p_2p_3p_4p_5p_6$ of $H$, where $p_1,p_2,p_5,p_6\in N(V(H))\setminus V(H)$. As $p_1p_2$ is a neighboring $1$-path of $H$, the neighbors of $p_1p_2$ are either contained in a unique neighboring $2$-path of $S$, $P(p_1p_2)$, or $p_1p_2$ is a neighboring $1$-path of $S$. Similarly, as $p_5p_6$ is a neighboring $1$-path of $H$, the neighbors of $p_5p_6$ are either contained in a unique neighboring $2$-path of $S$, $P(p_5p_6)$, or $p_5p_6$ is a neighboring $1$-path of $S$. In all cases, $P$ is a semi-neighboring $5$-path of a $\le 2$-relaxation $T'$ of $T$, where either $T'=T$, or $T'$ is obtained from $T$ by relaxing $P(p_1p_2)$, or by relaxing $P(p_5p_6)$ or by relaxing both. Hence 5 holds, a contradiction.
Critical Canvases with Two Boundary Components
----------------------------------------------
We will need a similar structural lemma for critical canvases whose boundary has two components. This can be done using the following theorem of Thomassen [@ThomRegular].
\[CylinderStructure\] Let $G$ be a connected plane graph of girth at least five, $C_1,C_2$ the boundaries of distinct faces of $G$, and $L$ a $3$-list-assignment for $G$ such that $|L(v)|\ge 2$ for all $v\in V(G)$ and $|L(v)|\ge 3$ for all $v\in V(G)\setminus (V(C_1)\cup V(C_2))$. Then there is an $L$-coloring of $G$ unless one of the following holds:
1. $G$ has a path $u_1u_2u_3$ each vertex of which has precisely two available colors, or
2. $G$ has a path $u_1u_2u_3u_4$ such that each of $u_1, u_2, u_4$ has precisely two available colors, or
3. $G$ has a path $u_1u_2u_3u_4u_5u_6$ such that each of $u_1, u_2, u_5, u_6$ has precisely two available colors, or
4. $G$ has a path $w_1w_2$ or $w_1xw_2$ such that $w_i$ is in $C_i$ and $|L(w_i)|=2$ for $i=1,2$, or
5. $G$ has a path $w_1w_2w_3w_4w_5$ such that $w_1, w_2$ are in one of $C_1,C_2$, $w_5$ is in the other, and $|L(w_1)|=|L(w_2)|=|L(w_5)|=2$, or
6. $G$ has a path $w_1w_2w_3w_4w_5w_6w_7$ such that $w_1, w_2$ are in one of $C_1,C_2$, $w_6,w_7$ are in the other, and $|L(w_1)|=|L(w_2)|=|L(w_6)|=|L(w_7)|=2$.
The proof of Theorem \[CylinderStructure\] is done for ordinary coloring but it can be easily modified to give the same result for list-coloring. More precisely, the proof at times uses the fact that list of vertices are the same so as to identify vertices or color them with the same color; when the lists are not the same, this is not always possible, but in such cases the proof can be modified to avoid these assumptions. Further note that condition $(viii)$ was erroneously omitted from the statement of Theorem 5.1 in [@ThomRegular] (and is necessary for the first inductive argument).
Using Theorem \[CylinderStructure\], we can now generalize Lemma \[StructureB\] to critical canvases whose boundary has two components as long as the distance between those components is at least $7$.
\[StructureC\] If $T=(G,S,L)$ is a critical canvas such that $S$ has at most two components and if two then they are distance at least $7$, then there exists one of the following:
1. an edge not in $S$ with both ends in $V(S)$, or,
2. a vertex not in $V(S)$ with at least two neighbors in $V(S)$, or,
3. a neighboring $2$-path of $S$, or
4. a semi-neighboring $3$-path of $S$, or,
5. a semi-neighboring $5$-path of $S$.
Suppose not and let $T$ be a counterexample such that $|V(G)|$ is minimized. By Lemma \[StructureB\], we may assume that $S$ has at least two components $S_1$ and $S_2$.
First suppose that $G$ is not connected. As $G$ is critical, by Theorem \[Thom0\], $G$ has two components $G_1,G_2$ such that $S_i\subseteq G_i$ for $i\in\{1,2\}$. Since $T$ is critical, for at least one $i\in\{1,2\}$, $S_i\ne G_i$. Let us assume without loss of generality that $S_1\ne G_1$. By Lemma \[SComponent\] applied to $G$ with $B=G_1$ and $A=S_1$, we find that $G_1$ is $S_1$-critical. Let $T_1=(G_1,S_1,L)$. Now $T_1$ is a critical canvas. As $S_1$ is connected, Lemma \[StructureB\] applied to $T_1$ implies that one of 1-5 holds, a contradiction. So we may assume that $G$ is connected.
Next suppose that $G\setminus V(S)$ is not connected. Let $H$ be a component of $G\setminus V(S)$. By Lemma \[Subcanvas\], $T'=T|(G[V(H)\cup V(S)])$ is a critical canvas and yet $|V(G[V(H)\cup V(S)])|$ is smaller than $|V(G)|$. Hence one of 1-5 holds for $T'$ and hence for $T$, a contradiction. So we may assume that $G\setminus V(S)$ is connected.
As 1 does not hold, $S$ is an induced subgraph of $G$. Let $\phi$ be an $L$-coloring of $S$ that does not extend to $G$. Let $L'(v)=L(v)\setminus \{\phi(u)| u\in N(v)\cap V(C)\}$. As $\phi$ does not extend to $G$, there does not exist an $L'$-coloring of $G\setminus V(S)$. As 2 does not hold, $|L'(v)|\ge 2$ for all $v\in V(G')$. Let $C_1,C_2$ be the boundaries of the two faces in $G'$ which contain vertices of $S$ in their interior. Note that $|L'(v)|\ge 3$ for all $v\in V(G')\setminus (V(C_1)\cup V(C_2))$.
Note that as 1 and 2 not hold, $N(V(S_1))\cap N(V(S_2)) = \emptyset$. Moreover $N(V(S_1))\setminus V(S_1)\ne \emptyset$ as $G$ is connected. Similarly $N(V(S_2))\setminus V(S_2) \ne \emptyset$. Let $G'$ be obtained from $G\setminus V(S)$ by adding a path of four new degree two vertices inside $C_1$ between consecutive (in the cyclic order) pairs of vertices in $N(V(S_1))\setminus V(S_1)$ and similarly in $C_2$ for pairs of vertices in $N(V(S_2))\setminus V(S_2)$ (if there is only one such vertex, then we add a path to itself). For each vertex $v$ in $V(G')\setminus (V(G)\setminus V(S))$, let $L'(v)$ be a list of three arbitrary colors.
Note that $G'$ is a connected plane graph of girth at least five and that $|L'(v)|\ge 2$ for all $v\in V(G')$. Moreover, there exists two faces $C_1',C_2'$ of $G'$ such that every vertex in $G'$ with $|L(v')|\ge 2$ is on the boundary of $C_1'$ or $C_2'$. Moreover, no vertex in $G'$ is on the boundary of both $C_1'$ and $C_2'$ by construction. Finally note that there does not exist an $L'$-coloring of $G'$ as otherwise there exists an $L'$-coloring of $G\setminus V(S)$, a contradiction.
By Theorem \[CylinderStructure\], as there is no $L'$-coloring of $G'$, one of Theorem \[CylinderStructure\](iii)-(viii) holds. As the distance between $S_1$ and $S_2$ is at least 7, none of (vi)-(viii) holds as otherwise there exists a path from a neighbor of a vertex in $S_1$ to a neighbor of a vertex in $S_2$ of length at most 4 and hence a path from $S_1$ to $S_2$ of length 6. If (iii) holds, then there exists a neighboring $2$-path of $S$ and 3 holds, a contradiction. If (iv) holds, then there exists a semi-neighboring $3$-path of $S$ and 4 holds, a contradiction. If (v) holds, then there exists a semi-neighboring $5$-path of $S$, a contradiction.
Finally we generalize Lemma \[Structure2\] to canvases whose boundary has two components as long as the distance between those components is at least $9$.
\[Structure3\] If $T=(G,S,L)$ is a critical canvas such that $S$ has at most two components and if two then the distance between the components is at least $9$, then there exists a $k$-relaxation $T'=(G,S',L)$ of $T$ with $k\le 2$ such that there exists one of the following:
1. an edge not in $S$ with both ends in $S$, or,
2. a vertex not in $S$ with at least two neighbors in $S$, or,
3. a neighboring claw of $S$, or
4. a semi-neighboring $3$-path of $S'$, or,
5. a semi-neighboring $5$-path of $S'$.
Suppose not. Note that since $G$ has no semi-neighboring $3$-path of $S$ then $G$ has no neighboring $k$-path of $S$ for any $k\ge 3$. Let $R$ be the union of all components of $N(S)\setminus S$ of size exactly three. By Lemma \[StructureC\], we may assume that $R\ne\emptyset$ as otherwise one of 1, 2, 4 or 5 holds, a contradiction.
Let $H=G[V(C)\cup V(R)]$. As there is no neighboring claw or neighboring $3$-path of $S$, every vertex in $V(H)\setminus V(S)$ is in a unique neighboring $2$-path of $S$; let $P(u)$ denote said path for each $u\in V(H)\setminus V(S)$. Note that $H$ is a proper subgraph of $G$ as there exist vertices in $V(H)\setminus V(S)$ which degree two in $H$ but degree at least three in $G$. By Lemma \[Supercanvas\], $T/H$ is critical. The proof now proceeds identically as Lemma \[Structure2\] except that we apply Lemma \[StructureC\] instead of Lemma \[StructureB\], which is permissible since if $H$ has two components then they are at distance at least 7 because if $S$ had two components then they were at distance at least 9 by assumption.
Linear Bound for Two Cycles {#Linear}
===========================
In this section, we develop the parameters necessary to state our general formula (Theorem \[StrongLinear\]), state said formula and derive Theorem \[Cylinder\] from it. As for the proof of Theorem \[StrongLinear\], it comprises the entirety of Section \[Proof\]. First a few definitions.
Deficiency
----------
First we need the following key parameter which essentially tracks how many edges the graph of a canvas is below the maximum imposed by Euler’s formula on a planar graph of girth five; hence the parameter is larger if the graph has many faces of length more than five.
Let $T=(G,S,L)$ be a canvas. We let $v(T):= |V(G)\setminus V(S)|$ and $e(T) := |E(G)\setminus E(S)|$. We define the *deficiency* of the canvas $T$ as
$${\rm def}(T):=3e(T) - 5v(T) + 10(c(S)-c(G)),$$
where $c(S)$ is the number of components of $S$ and $c(G)$ is the number of components of $G$.
We now prove the following very useful lemma, which says that the deficiency of a canvas equals the sum of the deficiencies of a subcanvas and its supercanvas.
\[defsum\] If $T=(G,C,L)$ is a canvas and $H$ is a subgraph of $G$ containing $S$, then
$${\rm def}(T) = {\rm def}(T|H) + {\rm def}(T/H).$$
Every edge of $E(G)\setminus E(S)$ is in exactly one of $E(H)\setminus E(S)$ or $E(G)\setminus E(H)$. Similarly, every vertex of $V(G)\setminus V(S)$ is in exactly one of $V(H)\setminus V(S)$ or $V(G)\setminus V(H)$. Lastly $c(S)-c(G) = c(S)-c(H) + c(H)-c(G)$. Combining these facts gives the desired formula.
More Complicated Parameters
---------------------------
Next we will need more complicated parameters. As alluded to in Section \[Outline\], we need to track the number of edges not in $S$ with an end in $S$. Also to show that the number of vertices is bounded, we will need to add a small additional weight to the vertices above and beyond what is already counted in deficiency. Thus we will need two small weights, $\alpha$ for tracking the edges out of $S$, and $\epsilon$ for the vertices.
We will prove our general formula for critical canvases (Theorem \[StrongLinear\]) assuming a number of inequalities on $\alpha,\epsilon$. Then to prove Theorem \[Cylinder\], we deduce the appropriate $\alpha$ and $\epsilon$ for the formula to hold. So for the benefit of the reader, we shall assume these are fixed but unspecified constants except in deriving Theorem \[Cylinder\] when it is needed to specify them. Strangely, while the formula holds for any small enough $\epsilon$, the value of $\alpha$ is more tightly controlled and needs to be slightly between $1/3$and $2/5$ (in fact any value in $(1/3,2/5)$ is acceptable if $\epsilon$ is made small enough).
\[Parameters\] Let $T=(G,S,L)$ be a canvas. Fix $\epsilon,\alpha > 0$. We define
$$q(T):= \sum_{v\in V(S)} {\rm deg}_{G-E(S)}(v),$$ $$s(T):=\epsilon v(T) + \alpha q(T),$$ $$d(T):={\rm def}(T)-s(T).$$
Thus $q(T)$ equals the number of edges not in $S$ with an end in $S$ (where an edge with both ends in $S$ is counted twice). We now prove that these new parameters satisfy natural relations for subcanvases and supercanvases as in Lemma \[defsum\].
\[surplussum\] Let $T=(G,S,L)$ be a canvas and $H$ a subgraph of $G$ containing $S$. The following statements hold:
- $v(T)=v(T|H)+v(T/H)$,
- $q(T)\le q(T|H)+q(T/H)$,
- $s(T)\le s(T|H)+s(T/H)$,
- $d(T)\ge d(T|H)+d(T/H)$.
The first statement follows as every vertex of $V(G)\setminus V(S)$ is in exactly one of $V(H)\setminus V(S)$ or $V(G)\setminus V(H)$. To prove the second statement, note that
$$q(T) = \sum_{v\in V(S)} {\rm deg}_{G-E(S)}(v) = \sum_{v\in V(S)} {\rm deg}_{G-E(H)}(v) + \sum_{v\in V(S)} {\rm deg}_{H-E(S)}(v).$$
But then as $V(S)\subseteq V(H)$,
$$q(T) \le \sum_{v\in V(H)} {\rm deg}_{G-E(H)}(v) + \sum_{v\in V(S)} {\rm deg}_{H-E(S)}(v) = q(T/H) + q(T|H),$$
which proves the second statement. The third statement follows from the first two. The fourth statement follows from the third and Lemma \[defsum\].
We can improve upon Proposition \[surplussum\] by noting an improved bound on $q(T)$. First, a definition.
If $T=(G,S,L)$ is a canvas and $H$ is a subgraph of $G$ containing $S$, then we let
$$q_T(H,S):=\sum_{v\in V(H)\setminus V(S)}{\rm deg}_{G-E(H)}(v)$$
and
$$d_T(T|H) := d(T|H) + \alpha q_T(H,S).$$
Note that if $T$ is critical, then the vertices in $V(H)\setminus V(S)$ have degree at least three. Hence, $q(H,S)$ is at least the number of vertices in $V(H)\setminus V(S)$ of degree two in $H$ plus twice the number of vertices of degree one in $H$.
\[surplussum2\] If $T=(G,S,L)$ is a canvas and $H$ is a subgraph of $G$ containing $S$, then
$$q(T) = q(T/H) + q(T|H) - q_T(H,S),$$
and hence
$$d(T) \ge d_T(T|H) + d(T/H).$$
As in the proof of Proposition \[surplussum\], we have that
$$q(T) = \sum_{v\in V(S)} {\rm deg}_{G-E(H)}(v) + \sum_{v\in V(S)} {\rm deg}_{H-E(S)}(v).$$
But then
$$q(T) = q(T/H)+q(T|H) - \sum_{v\in V(H)\setminus V(S)}{\rm deg}_{G-E(H)}(v) = q(T/H) + q(T|H) - q_T(H,S).$$
Small Canvases and a General Formula
------------------------------------
The next proposition determines $d$ for small canvases.
\[d0\] Let $T=(G,S,L)$ be a canvas.
1. If $G=S$, then $d(T)= 0$.
2. If $v(T)=0$, then $d(T)\ge (3-2\alpha)e(T)$.
3. If $v(T)=1$, $c(G)=c(S)$ and $v\in V(G\setminus S)$, then $d(T)\ge (3-2\alpha)e(T)-5+\deg(v)\alpha-\epsilon$.
If $G=S$, then $v(T)=q(T)=s(T)=0$. Moreover, $c(G)=c(S)$ and hence ${\rm def}(T) = 0$. So $d(T)= 0-0=0$ as desired. This proves (i).
If $v(T)=0$, then $q(T)=|E(G)\setminus E(S)|$. Thus $s(T)=2\alpha|E(G)\setminus E(S)|=2\alpha e(T)$. As $v(T)=0$, $c(G)\le c(S)$ and hence ${\rm def}(T)\ge 3e(T)$. So $d(T)\ge (3-2\alpha)e(T)$ as desired. This proves (ii).
Let $v\in V(G)\setminus V(S)$. As $v(T)=1$, $q(T)=2|E(G)\setminus E(S)|-\deg(v)=2e(T)-\deg(v)$. Thus $s(T)=\alpha(2e(T)-\deg(v))+\epsilon$. Yet as $c(G)=c(S)$, ${\rm def}(T)\ge 3e(T)-5v(T) = 3e(T)-5$. Combining, $d(T)\ge(3-2\alpha)e(T)-5+\deg(v)\alpha-\epsilon$ as desired. This proves (iii).
In particular, Proposition \[d0\] says that if $G$ consists of $S$ and one edge that is not in $S$ and yet has both ends in $S$, then $d(T)\ge 3-2\alpha$. Similarly if $G$ consists of $S$ and one vertex not in $S$ of degree three, then $d(T)\ge 4-3\alpha-\epsilon$. These two critical canvases are special for our proof in that they have the smallest value of $d$. To that end, we make the following definitions.
Let $T=(G,S,L)$ be a canvas. We say $T$ is a *chord* if $G$ consists of exactly $S$ and one edge not in $S$ with both ends in $S$. We say $T$ is a *tripod* if $G$ consists of exactly $S$ and one vertex not in $S$ of degree three. We say $T$ is *singular* if $T$ is a chord or a tripod and *non-singular* otherwise. We say $T$ is *normal* if no subcanvas of $T$ is singular.
We are now ready to state our generalization of the linear bound for two cycles. It asserts that the only exceptions are the two cases listed above.
\[StrongLinear\] Let $\epsilon, \alpha > 0$ satisfy the following where $\epsilon,\alpha$ are as in the definition of $s$ and $d$:
1. $9\epsilon \le \alpha$,
2. $2.5\alpha+5.5\epsilon\le 1$,
3. $11\epsilon + 1\le 3\alpha$,
If $T=(G,S,L)$ is a non-singular critical canvas with $c(S)\le 2$, then $d(T)\ge 3$.
The proof of Theorem \[StrongLinear\] is given in Section \[Proof\].
Deriving the Main Theorem
-------------------------
We proceed to derive Theorem \[Cylinder\] from Theorem \[StrongLinear\] as follows. First we determine the appropriate $\epsilon, \alpha$ and $\gamma$.
\[StrongLinear2\] If $G$ is a planar graph of girth at least five and $S$ is a subgraph of $G$ such that $G$ is $S$-critical for some $3$-list-assignment $L$ and $S$ has at most two components, then
$$\frac{1}{88}|V(G)\setminus V(S)| + \frac{3}{8}|E(S,G\setminus S)| \le 3|E(G)|-5|V(G)| + 5|V(S)|-3|E(S)| + 10(c(S)-c(G)).$$
Furthermore,
$$|V(G)| \le 177|V(S)|+528 \le 393|V(S)|.$$
Let $T=(G,S,L)$ and note that $T$ is a critical canvas. Let $\epsilon = 1/88$ and $\alpha=33\epsilon=3/8$. Note that inequality (1) of Theorem \[StrongLinear\] clearly holds. Moreover, $2.5 \alpha + 5.5 \epsilon = (82.5+5.5)\epsilon \le 1$ and hence inequality (2) holds.Moreover, $3\alpha= 99\epsilon \ge 11\epsilon+1$ and so inequality (3) holds. Thus $\epsilon$ and $\alpha$ satisfies the hypotheses of Theorem \[StrongLinear\]. The first formula now follow from Theorem \[StrongLinear\] and Proposition \[d0\] which give that $d(T)\ge 0$.
For the second formula, we note that by Euler’s formula since $G$ is planar and has girth at least five that $3|E(G)|\le 5|V(G)|-10$. Meanwhile, $c(G)\ge 1$ and $c(S)\le 2$, so $c(S)-c(G)\le 1$. Finally note that $|E(S)|\ge |V(S)|-2$ since $S$ has at most two components. Thus the right side of the first formula is at most $-10 + 2|V(S)|+6 + 10 = 2|V(S)|+6$. The left side is at least $\frac{1}{88}(|V(G)|-|V(S)|)$. The second formula now follows, where the last inequality holds as $|V(S)|\ge 1$ by Theorem \[Thom0\].
Theorem \[Cylinder\] would now be an immediate corollary of Theorem \[StrongLinear2\] with constant 393 in place of 99. However, we can do better. For example, in the case that $S$ is a facial cycle, we can derive a stronger bound as follows.
\[StrongLinear3\] If $G$ is a plane graph of girth at least five and $C$ is a cycle of $G$ such that $G$ is $C$-critical for some $3$-list-assignment $L$, then
$$\frac{1}{88}|V(G\setminus C)| + \frac{3}{8}|E(C,G\setminus C)| + \sum_{f\in {{\mathcal}{F}}(G)} (|f|-5) \le 2|V(C)|-10.$$
Furthermore, if $C$ is facial, then $|V(G)| \le 89|V(C)|$.
By Euler’s formula, $3|E(G)|-5|V(G)| = -10 - \sum_{f\in {{\mathcal}{F}}(G)} (|f|-5)$. Since $C$ is a cycle, $|V(C)|=|E(C)|$. Moreover, $c(G)\ge 1$ and $c(C)\le 1$. The first formula now follows from Theorem \[StrongLinear2\] with $C$ in place of $S$ as the right side of Theorem \[StrongLinear2\] is at most $2|V(C)|-10 - \sum_{f\in {{\mathcal}{F}}(G)} (|f|-5)$.
As for the second formula, if $C$ is facial, then as $G$ has girth at least five, we find that $\sum_{f\in F(G)}(|f|-5) \ge |C|-5$. Thus $\frac{1}{88}|V(G\setminus C)| \le |V(C)|-5$. Hence $|V(G)|\le 89|V(C)|$ as desired.
We are now ready to prove Theorem \[Cylinder\].
[**Proof of Theorem \[Cylinder\].**]{} By Euler’s formula, $3|E(G)|-5|V(G)| = -10 - \sum_{f\in {{\mathcal}{F}}(G)} (|f|-5)$. Since $C_1$ and $C_2$ are cycles, $|V(C_1)|=|E(C_1)|$ and $|V(C_2)|=|E(C_2)|$. Let $S=C_1\cup C_2$. Note that $c(S)\le 2$ while $c(G)\ge 1$. In addition, as $S=C_1\cup C_2$, $|V(S)|\le |E(S)|$ since if $S$ is connected, $S$ contains a cycle. Note that $G$ is $S$-critical by assumption. Since $C_1,C_2$ are facial cycles, $\sum_{f\in {{\mathcal}{F}}(G)} (|f|-5) \ge |V(C_1)|+|V(C_2)|-10$. Thus by Theorem \[StrongLinear2\], $\frac{1}{88}|V(G\setminus S)| \le |V(S)|+10(c(S)-c(G))$.
If $S$ is connected, then this is at most $|V(C_1)|+|V(C_2)|$ and we find that $|V(G)|\le 89 (|V(C_1)|+|V(C_2)|)$ as desired. So we may assume that $S$ is not connected, that is $C_1$ and $C_2$ are disjoint. As $G$ has girth at least five, $|V(C_1)|,|V(C_2)|\ge 5$. Thus $|V(S)|+10\le 2(|V(C_1)|+|V(C_2)|)$. Hence we find that $|V(G)|\le 177(|V(C_1)|+|V(C_2)|)$ as desired.
Proof of Theorem \[StrongLinear\] {#Proof}
=================================
We say a canvas $T_1=(G_1,S_1,L_1)$ is *smaller* than a canvas $T_2=(G_2,S_2,L_2)$ if either
- $v(T_1) < v(T_2)$, or
- $v(T_1)=v(T_2)$ and $e(T_1) < e(T_2)$, or
- $v(T_1)=v(T_2)$, $e(T_1)=e(T_2)$ and $\sum_{v\in V(G_1)\setminus V(S_1)} |L(v)| < \sum_{v\in V(G_2)\setminus V(S_2)} |L(v)|$.
Let $T_0=(G_0,S_0,L_0)$ be a counterexample to Theorem \[StrongLinear\] such that every canvas $T$ smaller than $T_0$ that satisfies the assumptions of Theorem \[StrongLinear\] have $d(T)\ge 4-\gamma$.
We say a canvas $T$ is *close* to $T_0$ if $T=T_0$ or $T$ is smaller than $T_0$ and $d(T_0)\ge d(T)-6\epsilon$.
Note that $3 > 4-3\alpha-\epsilon > 3-2\alpha$ by inequalities (1) and (2). Thus every critical canvas $T$ smaller than $T_0$ satisfies $d(T)\ge 3-2\alpha$. Finally it is useful to note one more inequality:
$$4. 2\alpha + 10\epsilon \le 1,$$
which follows from inequalities (1) and (2) since $1\ge 2.5\alpha + 5.5\epsilon \ge 2\alpha + 4.5\epsilon + 5.5\epsilon = 2\alpha + 10\epsilon$.
Properties of Close Canvases
----------------------------
For the remainder of the proof of Theorem \[StrongLinear\], let $T=(G,S,L)$ be a critical canvas close to $T_0$ such that $c(S)\le 2$. We proceed to establish many properties of such a $T$. In particular that $T$ has none of the following: an edge not in $S$ with both ends in $S$, a vertex not in $S$ with at least two neighbors in $S$, a neighboring claw of $S$, a semi-neighboring $3$-path of $S$ or a semi-neighboring $5$-path of $S$. Finally we will also show that a $\le 2$-relaxation of $T_0$ is close to $T_0$. Hence if we can apply Lemma \[Structure3\] to $T_0$, the proof will be complete; our next claim shows that and more.
\[Dist9\] If $S$ has two components $S_1$ and $S_2$, then $d(S_1,S_2)\le 9$.
Suppose not. Let $P$ be a shortest path from $S_1$ to $S_2$. By Corollary \[Supercanvas\], $T/(S\cup P)$ is critical. As $T/(S\cup P)$ is smaller than $T_0$, $T/(S\cup P)$ is not a counterexample to Theorem \[StrongLinear\]. Hence, $d(T/(S\cup P))\ge 2$. Furthermore if $|E(P)|\ge 3$, then $d(T/(S\cup P))\ge 3$ since in that case $|V(G) \setminus V(S\cup P)|\ge 2$.
Note that $e(T)-e(T/(S\cup P)) = |E(P)|$ and $v(T) - V(T/(S\cup P)) = |V(P)|-2$. Yet $S\cup P$ is connected where as $S$ is not. Finally, note that $q(T/(S\cup P))-q(T) \ge |V(P)|-4$ as the $|V(P)|-2$ internal vertices of $P$ have degree 3 in $G$ and so will count for at least one in $q(T/(S\cup P))$ while the first and last edges of $P$ which were counted in $q(T)$ will no longer count for $q(T/(S\cup P))$. Combining these observations we have
$$d(T) \ge 10 + d(T/(S\cup P)) +3|E(P)|-(5+\epsilon)(|V(P)|-2) + (|V(P)|-4)\alpha.$$
Since $|E(P)|=|V(P)|-1$,
$$d(T) \ge d(T/(S\cup P)) + 15 + \epsilon - 3\alpha -(2+\epsilon - \alpha)|E(P)|.$$
If $|E(P)|\le 2$, then as $d(T/(S\cup P))\ge 2$, we have that $d(T) \ge 13-\epsilon - \alpha $, but this at least $3+6\epsilon$ by inequality (2); hence $d(T_0)\ge 3$ as $T$ is close to $T_0$, a contradiction. So we may assume that $|E(P)|\ge 3$ and hence $d(T/(S\cup P))\ge 3$. Thus
$$d(T) \ge 18 + \epsilon - 3\alpha - (2+\epsilon-\alpha)|E(P)|.$$
Note that $2+\epsilon-\alpha$ is positive by inequality (2) and hence the right side is minimized when $|E(P)|$ is maximized. Since $|E(P)|\le 8$, it follows that
$$d(T) \ge 2-7\epsilon + 5\alpha.$$
Yet $5\alpha \ge 3\alpha + 2\alpha \ge (1+11\epsilon) + 18\epsilon = 1 = 29 \epsilon$ by inequalities (1) and (3). Thus we find that $d(T)\ge 3 + 29\epsilon$ and hence $d(T_0)\ge 3$, a contradiction.
So we may assume by Claim \[Dist9\] that either $c(S)=1$ or that if $S$ has two components $S_1,S_2$, then $d(S_1,S_2)\ge 9$. Hence by applying Lemma \[Structure3\] to $T_0$, there exists one of Lemma \[Structure3\](1)-(5). We shall proceed to show that the existence of each of these yields a contradiction as described above. However first we will need some further claims about subcanvases of $T$.
Chords and Neighbors of $S$
---------------------------
\[D1\] If $T|H=(H,S,L)$ is a proper subcanvas of $T$ such that $S$ is a proper subgraph of $H$, then $d_T(T|H) < 1+\epsilon$. Further, if $|V(G)\setminus V(H)|\ge 2$, then $d_T(T|H) < 6\epsilon$.
Suppose not. By Proposition \[surplussum2\], $d(T)\ge d_T(T|H) + d(T/H)$. By Corollary \[Supercanvas\], $T/H$ is critical. As $T/H$ is smaller than $T_0$, $d(T/H) \ge 3-2\alpha$. Thus if $d_T(T|H)\ge 1+\epsilon$, then $d(T)\ge 4-2\alpha \ge 3+6\epsilon$ by inequality (4); hence $d(T_0)\ge 3$, a contradiction. Further, if $|V(G)\setminus V(H)|\ge 2$, then $T/H$ is non-singular and hence $d(T/H) \ge 3$. So if $d_T(T|H)\ge 6\epsilon$, then $d(T)\ge 3+6\epsilon$ and hence $d(T_0)\ge 3$, a contradiction.
Claim \[D1\] has the following useful corollaries, namely that Lemma \[Structure3\](1) and Lemma \[Structure3\](2) do not exist in $T$.
\[Chord\] There does not exist an edge in $E(G)\setminus E(S)$ with both ends in $S$.
Suppose not. Let $H$ be the subgraph consisting of the union of $S$ and an edge in $E(G)\setminus E(S)$ with both ends in $S$. As $T$ is not a chord, $T|H$ is a proper subcanvas. Yet $d(T|H)\ge 3-2\alpha$ and hence $d(T|H)\ge 1+\epsilon$ by inequality (4). Since $d_T(T|H)\ge d(T|H)$, we find that $d_T(T|H)\ge 1+\epsilon$, contradicting Claim \[D1\].
\[2Neighbors\] There does not exist a vertex $v\in V(G)\setminus V(S)$ with at least two neighbors in $S$.
Suppose not. That is, there exists a vertex $v\in V(G)\setminus V(S)$ with two neighbors $u_1,u_2\in V(S)$. Let $H=S\cup \{u_1v, u_2v\}$. Note that $T|H$ is a proper subcanvas of $T$ as $v$ has degree two in $H$ but degree at least three in $G$. Yet $d(T|H)\ge 1-2\alpha-\epsilon > 6\epsilon$ by inequality (4). Thus by Claim \[D1\], $|V(G)\setminus V(H)|\le 1$. Hence $v(T)=2$. But then $d(T)\ge 3(5) - (5+\epsilon)(2) - 4\alpha = 5-2\epsilon-4\alpha$ which is at least $3+6\epsilon$ by inequality (4); hence $d(T_0)\ge 3$, a contradiction.
Hence by Corollary \[2Neighbors\], $|N(v)\cap V(S)|=1$ for all $v\in N(S)\setminus V(S)$. Moreover, by Claims \[Chord\] and \[2Neighbors\], $T$ is normal. Another useful corollary of Claim \[D1\] is the following claim which bounds the length of neighboring paths.
\[5Path\] There does not exist a neighboring $3$-path of $S$.
Suppose not. Let $P=p_1p_2p_3p_4$ be a neighboring $3$-path of $S$. Let $H$ be the subgraph of $G$ induced by $V(P)\cup V(S)$. Note that $v(T|H) = 4$, $q(T|H)=4$ and $e(T|H) = 7$ as each vertex of $P$ has a unique neighbor in $S$ by Claim \[2Neighbors\]. Thus $d(T|H)\ge 3(7)-(5+\epsilon)(4)-4\alpha = 1-4\epsilon-4\alpha$. Yet $q_T(H,S)=\sum_{v\in V(H)\setminus V(S)}({\rm deg}_{G-H}(v)\ge 2$ given that $\deg{H}(p_1)=\deg_{H}(p_4)= 2$. Thus $d_T(T|H) \ge 1-2\alpha-4\epsilon$. This is at least $6\epsilon$ by inequality (4).
Thus by Claim \[D1\], $|V(G)\setminus V(H)|\le 1$. As $p_1\not\sim p_4$ since $G$ has girth at least five, it follows that $|V(G)\setminus V(H)|=1$. Let $p_5\in V(G)\setminus V(H)$. It follows that $p_5$ is adjacent to $p_1$ and $p_4$ and exactly one vertex of $S$. It follows from Corollaries \[Chord\] and \[2Neighbors\] that $G$ consists of $S$ and a $5$-cycle $p_1p_2p_3p_4p_5$ of vertices of degree three. Thus $d(T)\ge 3(10)-(5+\epsilon)5 -5\alpha = 5-5\alpha-5\epsilon$, which is at least $3+6\epsilon$ by inequality (2), a contradiction.
The next claim shows that the components of $G[N(S)\setminus V(S)]$ have maximum degree two and hence are paths or cycles, that is Lemma \[Structure3\](3) does not exist in $T$.
\[NoTree\] There does not exists a neighboring claw of $S$.
Suppose not. Let $v\in N(S)\setminus V(S)$ and $v_1,v_2,v_3 \in N(v)\cap (N(S)\setminus V(S))$. Let $H=G[V(S)\cup \{v,v_1,v_2,v_3\}]$. It follows from Claim \[2Neighbors\] and the fact that $G$ has girth at least five that ${\rm deg}_{H}(v_i)=2$ for all $i\in\{1,2,3\}$. Hence $T|H$ is a proper subcanvas of $T$. Moreover since $G$ has girth at least five, $|V(G)\setminus V(H)|\ge 2$.
Note that $v(T|H) = 4$ and $e(T|H)=7$ since each of $\{v,v_1,v_2,v_3\}$ has a unique neighbor in $S$ by Corollary \[2Neighbors\]. Moreover, $q(T|H)=4$ and hence $d(T|H)=3(7)-(5+\epsilon)(4)-4\alpha = 1 -4\epsilon-4\alpha$. Yet $q_T(H,S) = \sum_{v\in V(H)\setminus V(S)}({\rm deg}_{G-H}(v)\ge 3$ since $\deg_{H}(v_i)=2$ for all $i\in\{1,2,3\}$. Thus $d_T(T|H) \ge 1-4\epsilon-\alpha$ which is at least $6\epsilon$ by inequality (4), contradicting Claim \[D1\].
Thus one of Lemma \[Structure3\](4) or Lemma \[Structure3\](5) exists in $T_0$. That is, there exists a $k$-relaxation $T_0'=(G_0,S_0',L_0)$ of $T_0$ with $k\le 2$ such that there exists a semi-neighboring $3$-path of $S_0'$, or a semi-neighboring $5$-path of $S_0'$. As mentioned before, we will prove that such a $T_0'$ is close to $T_0$. We shall also prove that $T$ does not have a semi-neighboring $3$-path of $S$ or a semi-neighboring $5$-path of $S$. Combined these facts will complete the proof, but before we can do that we will need improved bounds for subcanvases of canvases smaller than or equal to $T_0$, which the next subsection provides.
Proper Critical Subgraphs
-------------------------
Here is a very useful claim.
\[ProperCrit\] Suppose $T_1=(G_1,S_1,L_1)$ is a normal critical canvas such that $c(S_1)\le 2$ and either $T_1=T_0$ or $T_1$ is smaller than $T_0$. If $G_1$ contains a proper subgraph $H_1$ that is $S_1$-critical with respect to some $3$-list-assignment $L_1'$, then $d(T_1)\ge 6-\alpha$. Furthermore, if $|E(G_1)\setminus E(H_1)|\ge 2$, then $d(T_1)\ge 6$.
Suppose not. Let $T_1'= (H_1,S_1,L_1')$. By assumption, $T_1'$ is critical. Since $T_1$ is normal, $T_1'$ is non-singular. As $T_1'$ is smaller than $T$, $d(T_1')\ge 3$. Note that $d(T_1|H_1) = d(T_1')$ and hence $d(T_1|H_1)\ge 3$. As $H_1$ is proper, $T_1/H_1$ is critical by Corollary \[Supercanvas\]. By Proposition \[surplussum2\], $d(T_1)\ge d_{T_1}(T_1|H_1)+d(T_1/H_1)$.
If $T_1/H_1$ is non-singular, then since $T_1/H_1$ is smaller than $T_0$, we find that $d(T_1/H_1)\ge 3$ and hence $d(T_1)\ge 6$, a contradiction.
So we may suppose that $T_1/H_1$ is singular, that is $T_1/H_1$ is either a chord or tripod. First suppose $T_1/H_1$ is a tripod. Then $d(T_1/H_1)\ge 4-3\alpha-\epsilon$. Yet as there does not exist a vertex not in $S_1$ with at least three neighbors in $S_1$ as $T_1$ is normal, at least one of the edges in $E(G_1)\setminus E(H_1)$ does not have an end in $S_1$. Thus $q_{T_1}(H_1,S_1)\ge 1$. It follows that $d(T_1)\ge 7-2\alpha-\epsilon$, which is at least $6$ by inequality (2), a contradiction.
So we may suppose that $T_1/H_1$ is a chord. Hence $|E(G_1)\setminus E(H_1)|=1$ and so $d(T_1/H_1) = 3-2\alpha$. However, as $T_1$ is normal, $q_{T_1}(H_1,S_1)\ge 1$. Hence $d(T_1)\ge 3 + 3-2\alpha + \alpha = 6-\alpha$, a contradiction.
\[NoProperCrit\] There does not exist a proper subgraph of $G$ that is $S$-critical with respect to some list assignment $L'$.
Suppose not. Let $H$ be a proper subgraph of $G$ that is $S$-critical with respect to $L'$. By Corollaries \[Chord\] and \[2Neighbors\], $T$ is normal. By Claim \[ProperCrit\] with $T_1=T$, $G_1=G$, $H_1=H$ and $L_1=L'$, we find that $d(T)\ge 6-\alpha$ which is at least $3+6\epsilon$ by inequality (2). As $T$ is close to $T_0$, we have that $d(T_0)\ge 3$, a contradiction.
There exists a proper coloring $\phi$ of $S$ that does not extend to $G$ as $G$ is $S$-critical. Our next claim proves that $G$ is critical with respect to any such coloring. Actually we can prove more, but we need the following definition: we say a $3$-list-assignment $L'$ is *nice* for $T$ if $L'(v)\subseteq L(v)$ for all $v\in V(G)$, $L'(v)=L(v)$ for all $v\in V(S)$ and $|L'(v)|=3$ for all $v\in V(G)\setminus V(S)$.
\[Precolored\] Let $L'$ be a nice $3$-list-assignment for $T$. If $\phi$ is an $L$-coloring of $S$ that does not extend to an $L$-coloring of $G$, then $\phi$ extends to an $L'$-coloring of every proper subgraph of $G$ containing $S$.
Suppose not. That is, there exists a proper subgraph $H$ of $G$ such that $H$ contains $S$ and $\phi$ does not extend to an $L'$-coloring $H$. But then $H$ contains a subgraph $H'$ that is $S$-critical with respect to $L'$. As $H$ is a proper subgraph of $G$, $H'$ is also a proper subgraph of $G$, contradicting Corollary \[NoProperCrit\].
For the rest of the proof of Theorem \[StrongLinear\], we fix an $L$-coloring $\phi$ of $S$ which does not extend to $G$ and we fix a nice $3$-list-assignment $L'$ for $T$. Note that $\phi$ does not extend to an $L'$-coloring of $G$ (and hence an $L$-coloring of $G$) as $T$ is critical. For $v\not\in V(S)$, we let
$$A(v):=L'(v)\setminus \{\phi(u): u\in N(v)\cap V(S)\}.$$
Note that for every edge $e=uv$ with $u,v\not\in V(S)$, $L'(u)\cap L'(v)\ne \emptyset$ by Corollary \[Precolored\]. Similarly, for every vertex $v\not\in V(S)$, $|A(v)|=|L'(v)|-|N(v)\cap V(S)|=3-|N(v)\cap V(S)|$. This follows since by Corollary \[Precolored\], $\phi(u)\in L'(v)$ for all $u\in N(v)\cap V(S)$ and $\phi(u)\ne \phi(w)$ for all $u\neq w \in N(v)\cap V(S)$.
Here is the application of Claim \[ProperCrit\] that we repeatedly use for reductions.
\[Reduction\] Let $H$ be a subgraph of $G$ containing $S$ such that $T/H$ is normal. If there exists a proper subgraph $G'$ of $G$ such that $H\subseteq G'$ and there exists an $L'$-coloring of $H$ that does not extend to an $L'$-coloring of $G'$, then $d_T(T|H) < -3 + \alpha + 6\epsilon$. Further if $|E(G)\setminus E(G')|\ge 2$, then $d_T(T|H) < -3 + 6\epsilon$.
Suppose not. As there exists an $L'$-coloring of $H$ that does not extend to an $L'$-coloring of $G'$, there exists a subgraph $G''$ of $G'$ that is $H$-critical with respect to $L'$. Apply Claim \[ProperCrit\] with $T_1 = (G,H,L) = T/H$, $H_1=G''$ and $L_1'=L'$. Thus $d(T/H) \ge 6-\alpha$ and furthermore if $|E(G)\setminus E(G')|\ge 2$, then $d(T/H)\ge 6$.
By Proposition \[surplussum2\], $d(T)\ge d_T(T|H)+d(T/H)$. As $d(T) < 3+6\epsilon$, we find that $d_T(T|H) < 3 +6\epsilon - d(T/H)$. Hence $d(T/H) < 3 + 6\epsilon - (6-\alpha) = -3 + \alpha + 6\epsilon$. Furthermore if $|E(G)\setminus E(G')|\ge 2$, then $d_T(T|H) < 3 + 6\epsilon - (6) = -3 + 6\epsilon$, in either case a contradiction to the bounds above.
In what remains of the the proof of Theorem \[StrongLinear\], we will invoke Claim \[Reduction\] to appropriately chosen $G'$ and $H$ to show that $T$ does not have a semi-neighboring $3$-path or semi-neighboring $5$-path. Before we do that, we need a preliminary claim whose proof also relies on Claim \[Reduction\]. For the coloring arguments, it is useful to note the following claim.
\[ColorSubset\] If $v\in N(S)$ and $u\in N(v)\setminus V(S)$, then $A(v)\subseteq A(u)$.
Suppose not. Let $H=G[V(C)\cup v]$. Note that $d_T(T|H) \ge 3-(5+\epsilon) - \alpha + 2\alpha = -2-\epsilon+\alpha$ which is at least $-3+\alpha+6\epsilon$ as $7\epsilon \le 1$ by inequalities (1) and (2). Furthermore, it follows from Claims \[Chord\] and \[2Neighbors\] that there does not exists an edge in $E(G)\setminus E(H)$ with both ends in $V(H)$ or a vertex in $V(G)\setminus V(H)$ with three neighbors in $V(H)$ and hence $T/H$ is normal.
Let $\phi(v)\in A(v)\setminus A(u)$. Now $\phi$ is an $L'$-coloring of $H$. Let $G'=G\setminus \{vu\}$. Thus $G'$ is a proper subgraph of $G$ such that $H\subseteq G'$. Yet $\phi$ does not extend to an $L'$-coloring of $G'$ as otherwise $\phi$ extends to an $L'$-coloring of $G$ (and hence an $L$-coloring of $G$), contradicting that $T$ is critical. So we may assume that $\phi$ does not extend to an $L'$-coloring of $G'$. Thus by Claim \[Reduction\], $d_T(T|H) < -3+\alpha+6\epsilon$, a contradiction.
Neighboring Paths
-----------------
By Claim \[5Path\] only neighboring $1$-paths and neighboring $2$-paths of $S$ may exist. We cannot directly obtain a contradiction by their existence. Hence we also need more information on the degrees of vertices in neighboring $1$-paths which the following claim provides.
\[NotBothDeg3\] If $P=p_1p_2$ is a neighboring $1$-path of $S$ such that $P$ is a component of $N(S)\setminus S$ of size two, then either ${\rm deg}(p_1)\ge 4$ or ${\rm deg}(p_2)\ge 4$.
Suppose not. Hence ${\rm deg}(p_1)={\rm deg}(p_2)=3$. By Claim \[ColorSubset\], $A(p_1)=A(p_2)$. Let $u_1,u_2$ be such that $N(p_1)\setminus (V(S)\cup \{p_2\})=\{u_1\}$ and $N(p_2)\setminus (V(S)\cup \{p_1\})=\{u_2\}$. Note that $u_1,u_2\not\in V(S)$ by Claim \[2Neighbors\]. Furthermore, $u_1\ne u_2$ as $G$ has girth at least five. As $L'$ is nice, $|L'(u_1)|=|L'(u_2)|=3$. Yet, $u_1, u_2\not\in N(S)$ by assumption and hence $|A(u_1)|=|A(u_2)|=3$. We may assume without loss of generality that ${\rm deg(u_1)} \ge {\rm deg}(u_2)$.
Let $H=G[V(C)\cup \{u_1,p_1,p_2\}]$. Note that $e(T|H) = e(T)-4$, $v(T|H)=v(T)-3$ and $q(T|H)=2$. Yet $q_T(H,S) = {\rm deg}(u_1)$ as ${\rm deg}_{G-H}(u_1) = {\rm deg}(u_1)-1$ and ${\rm deg}_{G-H}(p_2)=1$. Combining we find that $$d_T(T|H) = 3(4)-(5+\epsilon)(3) - 2\alpha + {\rm deg}(u_1)\alpha = -3 + ({\rm deg}(u_1)-2)\alpha-3\epsilon,$$ which is at least $-3+\alpha - 3\epsilon$ as $\deg(u_1)\ge 3$. By inequality (1), $-3+\alpha-3\epsilon \ge -3 + 6\epsilon$. Moreover, this is at least $-3+\alpha+6\epsilon$ if ${\rm deg}(u_1)\ge 4$ as $\alpha\ge 9\epsilon$ by inequality (1). Furthermore, it follows from Corollaries \[Chord\] and \[2Neighbors\] and the fact that $G$ has girth at least five that there does not exist an edge in $E(G)\setminus E(H)$ with both ends in $V(H)$ or a vertex in $V(G)\setminus V(H)$ with three neighbors in $V(H)$. Hence $T/H$ is normal.
Let $\phi(u_1)\in A(u_1)\setminus A(p_1)$. Then let $\phi(p_2)\in A(p_2)$ and $\phi(p_1)\in A(p_1)\setminus \{\phi(p_2)\}$.
First suppose ${\rm deg}(u_1)\ge 4$. Let $G'=G\setminus \{p_2u_2\}$. Now if $\phi$ extends to an $L'$-coloring of $G'$, then there exists an $L'$-coloring of $G$ by recoloring $p_2$ different from $u_2$ and then recoloring $p_1$ different from $p_2$, contradicting that $T$ is critical. Thus by Claim \[Reduction\], $d_T(T|H) < -3+\alpha+6\epsilon$, a contradiction to the bound given above.
So we may suppose that ${\rm deg}(u_1)=3$. Thus ${\rm deg}(u_2)=3$ since we assumed that $\deg(u_1)\ge \deg(u_2)$. Let $G'=G\setminus \{u_2\}$. Now if $\phi$ extends to an $L'$-coloring of $G'$, then there exists an $L'$-coloring of $G$ by coloring $u_2$ and then recoloring $p_2,p_1$ in that order, contradicting that $T$ is critical. So we may assume that $\phi$ does not extend to an $L'$-coloring of $G'$. Yet $|E(G)\setminus E(G'')|\ge 2$. Thus by Claim \[Reduction\], $d_T(T|H) < -3+6\epsilon$, a contradiction to the bound given above.
We also need more information about the degrees of vertices in neighboring $2$-paths as follows.
\[NotAllDeg3\] If $P=p_1p_2p_3$ is a neighboring $2$-path of $S$, then ${\rm deg}(p_1)+{\rm deg}(p_2)+\deg(p_3)\ge 10$.
Suppose not. As $G$ is $S$-critical, ${\rm deg}(p_i)\ge 3$ for all $i\in\{1,2,3\}$. It now follows that ${\rm deg}(p_1)={\rm deg}(p_2) = {\rm deg}(p_3) = 3$. Let $u_1$ be the unique neighbor of $p_1$ in $V(G)\setminus (V(S)\cup \{p_2\})$. Similarly let $u_3$ be the unique neighbor of $p_3$ in $V(G)\setminus (V(S)\cup \{p_2\})$. Note that $u_1,u_3\not\in V(S)$ by Claim \[2Neighbors\]. Furthermore, $u_1\ne u_3$ as $G$ has girth at least five. As $L'$ is nice, $|L'(u_1)|=|L'(u_3)|=3$. Yet, $u_1, u_3\not\in N(S)$ by Claim \[5Path\] and hence $|A(u_1)|=|A(u_3)|=3$.
Let $H=G[V(C)\cup \{u_1,p_1,p_2,p_3\}]$. Note that $e(T|H) = e(T)-6$, $v(T|H)=v(T)-4$ and $q(T|H)=3$. Yet $q_T(H,S) = {\rm deg}(u_1)$ as ${\rm deg}_{G-H}(u_1) = {\rm deg}(u_1)-1$ and ${\rm deg}_{G-H}(p_3)=1$. Combining we find that $$d_T(T|H) = 3(6)-(5+\epsilon)(4) - 3\alpha + {\rm deg}(u_1)\alpha = -2 + ({\rm deg}(u_1)-3)\alpha-4\epsilon,$$ which is at least $-2 - 4\epsilon$ as $\deg(u_1)\ge 3$. By inequality (4), $-2-4\epsilon \ge -3 \alpha + 6\epsilon$. Moreover, it follows from Corollaries \[Chord\] and \[2Neighbors\] and the fact that $G$ has girth at least five that there does not exist an edge in $E(G)\setminus E(H)$ with both ends in $V(H)$ or a vertex in $V(G)\setminus V(H)$ with three neighbors in $V(H)$. Hence $T/H$ is normal.
Let $\phi(u_1)\in A(u_1)\setminus A(p_1)$. Then let $\phi(p_3)\in A(p_3)$, $\phi(p_2)\in A(p_2)\setminus \{\phi(p_3)\}$ and $\phi(p_1)\in A(p_1)\setminus \{\phi(p_2)\}$.
Let $G'=G\setminus \{p_3u_3\}$. Now if $\phi$ extends to an $L'$-coloring of $G'$, then there exists an $L'$-coloring of $G$ by recoloring $p_3$ different from $u_3$ and then recoloring $p_2,p_1$ in that order if necessary, contradicting that $T$ is critical. So we may assume that $\phi$ does not extend to an $L'$-coloring of $G'$. Thus by Claim \[Reduction\], $d_T(T|H) < -3+\alpha+6\epsilon$, a contradiction to the bound given above.
Claim \[NotAllDeg3\] is enough to show that a relaxation of $T$ has a value of $d$ close to that of $T$ as follows.
\[Relax1\] If $T'$ is a relaxation of $T$, then $d(T)\ge d(T')-3\epsilon$.
By the definition of relaxation, there exists a neighboring $2$-path $P=p_1p_2p_3$ of $S$ such that for each $i\in\{1,2,3\}$ there is a unique neighbor of $p_i$ in $S$, call it $u_i$, such that if $H=G\cup P \cup \{p_1u_1,p_2u_2,p_3u_3\}$, then $T'=T/H$.
Note that $v(T|H)=3$, $e(T|H)=5$ and $q(T|H)=3$. Hence $d(T|H) = 3(5)-(5+\epsilon)3 - 3\alpha = -3\epsilon - 3\alpha$. By Claim \[NotAllDeg3\], ${\rm deg}_G(p_1)+{\rm deg}_G(p_2)+\deg_G(p_3)\ge 10$. Yet ${\rm deg}_H(p_1)+{\rm deg}_H(p_2)+\deg_H(p_3)=7$. Thus $q_T(H,S) \ge 3$. So $d_T(T|H) = d(T|H) +\alpha q_T(H,S) \ge -3\epsilon$. By Proposition \[surplussum2\], $$d(T)\ge d(T/H) + d_T(T|H) \ge d(T')-3\epsilon,$$ as desired.
From Corollary \[Relax1\], we have that a $\le 2$-relaxation of $T_0$ is close to $T_0$ as follows.
\[Relax2\] If $T_1$ is a $k$-relaxation of $T_0$ with $k\le 2$, then $T_1$ is close to $T_0$.
We proceed by induction on $k$. If $k=0$, then $T_1=T_0$ and hence is close to $T_0$ as desired. If $k=1$, then as $T_0$ is close to $T_0$, it follows from Corollary \[Relax1\] that $d(T_0)\ge d(T_1)-3\epsilon$ and hence $T_1$ is close to $T_0$ as desired. So we may assume that $k=2$. But then $T_1$ is a $1$-relaxation $T_1'$ of $T_0$. By induction, $T_1'$ is close to $T_0$. So by Corollary \[Relax1\], $d(T_1')\ge d(T_1)-3\epsilon$. Yet $d(T_0)\ge d(T_1')-3\epsilon$ by Corollary \[Relax1\] and hence $d(T_0)\ge d(T_1)-6\epsilon$ and so $T_1$ is close to $T_0$ as desired.
Semi-Neighboring Paths
----------------------
We now proceed to show that there does not exist a semi-neighboring $3$-path or semi neighoring $5$-path of $S$.
\[Semi3\] There does not exist a semi-neighboring $3$-path of $S$.
Suppose not. Let $P=p_1\ldots p_4$ be a semi-neighboring $3$-path of $C_1$. By Claim \[2Neighbors\], $|N(p_i)\cap V(C)| = 1$ for $i\in \{1,2,4\}$ and hence $|A(p_i)|=2$ for $i\in\{1,2,4\}$. As there does not exist a neighboring $3$-path of $C$ by Claim \[5Path\], we find that $p_3\not\in N(C)$ and hence $|A(p_3)|=3$. By Claim \[ColorSubset\], we find that $A(p_1)=A(p_2)\subset A(p_3)$.
Let $R= (N(p_1)\cup N(p_2))\cap (N(S) \setminus \{p_1,p_2\})$. By Claims \[5Path\] and \[NoTree\], $|R|\le 1$. Let $H=G[V(S)\cup \{p_1,p_2,p_3\} \cup R]$.
\[DFirst\] $d_T(T|H) \ge -3+\alpha+6\epsilon$.
First suppose $|R|=1$ and let $x\in R$. Hence $e(T|H)=e(T)-6$, $v(T|H)=v(T)-4$ and $q(T|H)=3$. Note that ${\rm deg}_{G-H}(p_3)\ge 2$. Yet ${\rm deg}_G(x) + {\rm deg}_G(p_1) + {\rm deg}_G(p_2) \ge 10$ by Claim \[NotAllDeg3\] while ${\rm deg}_H(x) + {\rm deg}_H(p_1) + {\rm deg}_H(p_2) = 8$. Hence ${\rm deg}_{G-H}(x) + {\rm deg}_{G-H}(p_1) + {\rm deg}_{G-H}(p_2) \ge 2$. Thus $q_T(H,S)\ge 4$. Combining, we find that
$$d_T(T|H)\ge 3(6)-(5+\epsilon)(4) - 3\alpha + 4\alpha = -2-3\epsilon+\alpha,$$
which is at least $-3+\alpha+6\epsilon$ as claimed since $9\epsilon\le 1$ by inequalities (1) and (2).
So we may assume that $|R|=0$. Hence $e(T|H)=e(T)-4$, $v(T|H)=v(T)-3$ and $q(T|H)=2$. Note that ${\rm deg}_{G-H}(p_3)\ge 2$. Yet ${\rm deg}_G(p_1) + {\rm deg}_G(p_2) \ge 7$ by Claim \[NotBothDeg3\] while ${\rm deg}_H(p_1) + {\rm deg}_H(p_2) = 5$. Hence ${\rm deg}_{G-H}(p_1) + {\rm deg}_{G-H}(p_2) \ge 2$. Thus $q_T(H,S)\ge 4$. Combining, we find that
$$d_T(T|H) \ge 3(4)-(5+\epsilon)(3)-2\alpha + 4\alpha= -3 + 2\alpha-3\epsilon,$$
which is at least $-3+\alpha+6\epsilon$ as claimed since $9\epsilon \le \alpha$ by inequality (1).
\[No3First\] $T/H$ is normal.
Suppose not. There does not exist an edge with both endpoints in $V(H)$ by Claims \[Chord\], \[2Neighbors\] and \[NoTree\] and the fact that $G$ has girth at least five. So we may assume there there exists a vertex $v\in V(G)\setminus V(H)$ with at least three neighbors in $V(H)$. By Claim \[2Neighbors\], $v$ would have to have at least two neighbors in $V(H)\setminus V(S)$. As $G$ has girth at least five, it follows that these two neighbors have to be $p_3$ and a vertex $x$ in $R$. But then $v$ has another neighbor in $V(H)$ and hence $v\in N(S)$. Moreover since $G$ has girth at least five, $x$ is adjacent to $p_1$. Thus $p_2p_1xp_4$ is a neighboring $3$-path of $S$, contradicting Claim \[5Path\].
Let $\phi(p_3)\in A(p_3)\setminus A(p_4)$. Then extend $\phi$ to an $L'$-coloring of $H$ by coloring $p_2,p_1$ and $R$ (if it exists) in that order. Let $G'=G\setminus \{p_3p_4\}$. Now if $\phi$ extends to an $L'$-coloring of $G'$, then there exists an $L'$-coloring of $G$, contradicting that $T$ is critical. So we may assume that $\phi$ does not extend to an $L'$-coloring of $G'$. Given Subclaim \[No3First\], it now follows from Claim \[Reduction\] that $d_T(T|H) < -3+\alpha+6\epsilon$, contradicting Subclaim \[DFirst\].
\[Semi5\] There does not exist a semi-neighboring $5$-path of $S$.
Suppose not. Then there exists a semi-neighboring $5$-path $P=p_1\ldots p_6$ where $p_1,p_2,p_5,p_6\in N(C)$. We may assume without loss of generality that $\deg(p_3)\ge \deg(p_4)$. By Claim \[ColorSubset\], $A(p_1)=A(p_2)\subset A(p_3)$ and $A(p_6)=A(p_5)\subset A(p_4)$.
Let $U_1=(N(p_1)\cup N(p_2))\cap (N(S)\setminus (V(S)\cup\{p_1,p_2\}))$ and similarly let $U_2=(N(p_5)\cup N(p_6))\cap (N(C)\setminus (V(S)\cup \{p_5,p_6\}))$. If $U_1\ne \emptyset$, then it follows from Claims \[5Path\] and \[NoTree\] that $|U_1|=1$. Similarly if $U_2\ne \emptyset$, then $|U_2|=1$.
Let $S_1=\{p_1,p_2\}\cup U_1$ and let $S_2=\{p_5,p_6\}\cup U_2$. By Claim \[5Path\], $S_1\cap S_2 = \emptyset$ and there does not exist an edge with one end in $S_1$ and the other end in $S_2$. By Claim \[Semi3\], there does not exist a vertex in $V(G)\setminus V(S)$ with a neighbor in both $S_1$ and $S_2$. Similarly by Claim \[Semi3\], $p_3\not\in N(S_2)$ and $p_4\not\in N(S_1)$. Hence as $G$ has girth at least five, the graph induced by $G$ on $S_1\cup S_2\cup\{p_3,p_4\}$ has precisely $|S_1|+|S_2|+1 = 5 + |U_1|+|U_2|$ edges.
Let $H = V(S) \cup S_1\cup S_2 \cup \{p_3\}$.
\[D\] $d_T(T|H) \ge -4 -5\epsilon + \deg(p_3)\alpha$.
First note that $e(T|H) = 7 + 2|U_1|+2|U_2|$ and $v(T|H)=5+|U_1|+|U_2|$. Hence ${\rm def}(T|H)= 3e(T|H) - 5v(T|H) = 3(7+2|U_1|+2|U_2|)-5(5+|U_1|+|U_2|) = -4 + |U_1|+|U_2|$. Moreover, $q(T|H)=|S_1|+|S_2| = 4+|U_1|+|U_2|$ and hence $s(T|H)=(4+|U_1|+|U_2|)\alpha + (5+|U_1|+|U_2|)\epsilon$. Thus
$$d(T|H) = -4 - 4\alpha - 5\epsilon + (|U_1|+|U_2|)(1-\alpha -\epsilon).$$
Let $B_1 = \sum_{v\in S_1} \deg_{G-H}(v)$. Note that $B_1\ge 1$. Furthermore, if $|U_1|=0$, then $\deg_G(p_1)+\deg_G(p_2)\ge 7$ by Claim \[NotBothDeg3\]. Hence if $|U_1|=0$, then $B_1\ge 2$. Thus $B_1\ge 2 - |U_1|$. Similarly let $B_2=\sum_{v\in S_2} \deg_{G-H}(v)$. Note that $B_2\ge 2$. Furthermore, if $|U_2|=0$, then $\deg(p_5)+\deg(p_6)\ge 7$ by Claim \[NotBothDeg3\]. Hence if $|U_2|=0$, then $B_2=3$. Thus $B_2\ge 3-|U_2|$. Also note that $\deg_{G-H}(p_3)=\deg_G(p_3)-1$. Hence $q_T(H,S) \ge B_1+B_2+\deg(p_3)-1$. So $d_T(T|H) \ge d(T|H) + (B_1+B_2+\deg(p_3)-1)\alpha$.
Using the bounds above, we find that $$\begin{aligned}
d_T(T|H) &\ge -4 - 4\alpha - 5\epsilon + (|U_1|+|U_2|)(1-\alpha-\epsilon) + (4+\deg(p_3) - |U_1|-|U_2|)\alpha \\
&= -4 - 5\epsilon +\deg(p_3)\alpha + (|U_1|+|U_2|)(1-2\alpha-\epsilon).\end{aligned}$$
Since $2\alpha+\epsilon \le 1$ by inequality (2), $d_T(T|H) \ge -4-5\epsilon +\deg(p_3)\alpha$ as claimed.
\[No3\] $T/H$ is normal.
Suppose not. Since $H$ is an induced subgraph of $G$ there does not exist an edge of $E(G)\setminus E(H)$ with both ends in $V(H)$. So we may assume there exists a vertex $v\in V(G)\setminus V(H)$ with at least three neighbors in $V(H)$. If $v\in N(S)$, then by Claim \[2Neighbors\], $v$ would have to have at least two neighbors in $V(H)\setminus V(S)$. Hence $v$ has at least one neighbor in either $S_1$ or $S_2$, contradicting either Claim \[5Path\] or Claim \[NoTree\]. So we may assume that $v\not\in N(S)$. But then as $G$ has girth at least five, $v$ has a neighbor in both $S_1$ and $S_2$, contradicting Claim \[Semi3\].
If $\deg(p_4)\ge 4$, let $G'=G\setminus \{p_3p_4\}$ and if $\deg(p_4)=3$, let $G'=G\setminus p_4$. If $A(p_3)\cap A(p_5)\ne\emptyset$, let $\phi(p_3)= \phi(p_5)\in A(p_3)\cap A(p_5)$. If $A(p_3)\cap A(p_5)=\emptyset$, then $A(p_3)\setminus A(p_4)\ne \emptyset$ and in that case let $\phi(p_3)\in A(p_3)\setminus A(p_4)$ and $\phi(p_5)\in A(p_5)$. In either case, then extend $\phi$ to the rest of $V(H)$ by coloring $p_2,p_1,u_1$ and $p_6,u_2$ in that order. Now if $\phi$ extends to an $L'$-coloring of $G'$, then $\phi$ extends to an $L'$-coloring of $G$, contradicting that $T$ is critical. So we may assume that $\phi$ does not extend to an $L'$-coloring of $G'$.
Given Subclaim \[No3\], it follows from Claim \[Reduction\] that $d_T(T|H) < -3+\alpha+6\epsilon$, and further if $\deg(p_4)=3$, then $d_T(T|H) < -3+6\epsilon$. If $\deg(p_4)\ge 4$, then $d_T(T|H) \ge -4 -5\epsilon + 4\alpha$ by Subclaim \[D\]. Since $3\alpha \ge 1 + 11\epsilon$ by inequality (3), $d_T(T|H) \ge - 3 + \alpha+6\epsilon$, a contradiction. So we may assume that $\deg(p_4)=3$ and hence $d_T(T|H)\ge -4 - 5\epsilon +3\alpha$ by Subclaim \[D\]. Since $3\alpha \ge 1 + 11\epsilon$ by inequality (3), $d_T(T|H)\ge -3+6\epsilon$, a contradiction.
We now finish the proof by applying Lemma \[Structure3\] to $T_0$. If Lemma \[Structure3\](1) holds, then as $T_0$ is close to itself, this contradicts Claim \[Chord\]. If Lemma \[Structure3\](2) holds, then as $T_0$ is close to itself, this contradicts Claim \[2Neighbors\]. If Lemma \[Structure3\](3) holds, then as $T_0$ is close to itself, this contradicts Claim \[NoTree\]. If Lemma \[Structure3\](4) holds, then there exists a $\le 2$-relaxation $T_0'=(G_0,S_0',L_0)$ of $T_0$ such that there exists a semi-neighboring $3$-path of $S_0'$. But then $T_0'$ is close to $T_0$ by Corollary \[Relax2\] and so by Claim \[Semi3\], there does not exist a semi-neighboring $3$-path of $S_0'$, a contradiction.
So we may assume that Lemma \[Structure3\](5) holds. That is, there exists a $\le 2$-relaxation $T_0'=(G_0,S_0',L_0)$ of $T_0$ such that there exists a semi-neighboring $5$-path of $S_0'$. But then $T_0'$ is close to $T_0$ by Corollary \[Relax2\] and so by Claim \[Semi5\], there does not exist a semi-neighboring $5$-path of $S_0'$, a contradiction. This concludes the proof of Theorem \[StrongLinear\].
Exponentially Many Extensions of a Precoloring of a Cycle {#Exp}
=========================================================
In this section, we provide as promised the proof that the family of exponentially critical graphs is hyperbolic thereby completing the proof of Theorem \[ExpSurface2\] (see Theorem \[ExpHyper\]). Before we do that, we need to recall a number of defintions from [@PostleThomas].
A *ring* is a cycle or a complete graph on one or two vertices. A *graph with rings* is a pair $(G,{{\mathcal}{R}})$, where $G$ is a graph and ${{\mathcal}{R}}$ is a set of vertex-disjoint rings in $G$.
We say that a graph $G$ with rings ${{\mathcal}{R}}$ is *embedded in a surface $\Sigma$* if the underlying graph $G$ is embedded in $\Sigma$ in such a way that for every ring $R \in {{\mathcal}{R}}$ there exists a component $\Gamma$ of the boundary of $\Sigma$ such that $R$ is embedded in $\Gamma$, no other vertex or edge of $G$ is embedded in $\Gamma$, and every component of the boundary of $\Sigma$ includes some ring of $G$.
Now let us state the formal definition of a hyperbolic family.
Let ${{\mathcal}{F}}$ be a family of non-null embedded graphs with rings. We say that ${{\mathcal}{F}}$ is *hyperbolic* if there exists a constant $c>0$ such that if $G\in{{\mathcal}{F}}$ is a graph with rings that is embedded in a surface $\Sigma$, then for every closed curve $\gamma: \mathbb{S}^1 \rightarrow \Sigma$ that bounds an open disk $\Delta$ and intersects $G$ only in vertices, if $\Delta$ includes a vertex of $G$, then the number of vertices of $G$ in $\Delta$ is at most $c(|\{x\in \mathbb{S}^1: \gamma(x)\in V(G)\}|-1)$. We say that $c$ is a *Cheeger constant* of ${{\mathcal}{F}}$.
Let us also state the definition of strongly hyperbolic for those who are interested though we do not need it.
Let ${{\mathcal}{F}}$ be a hyperbolic family of embedded graphs with rings, let $c$ be a Cheeger constant for ${{\mathcal}{F}}$, and let $d := \lceil 3(2c + 1) \log_2(8c + 4)\rceil$. We say that ${{\mathcal}{F}}$ is *strongly hyperbolic* if there exists a constant $c_2$ such that for every $G \in {{\mathcal}{F}}$ embedded in a surface $\Sigma$ with rings and for every two disjoint cycles $C_1,C_2$ of length at most $2d$ in $G$, if there exists a cylinder $\Lambda \subseteq \Sigma$ with boundary components $C_1$ and $C_2$, then $\Lambda$ includes at most $c_2$ vertices of $G$. We say that $c_2$ is a *strong hyperbolic constant* for ${{\mathcal}{F}}$.
Thomassen [@ThomExp] proved the following in [@ThomExp].
\[ThomExp\] If $G$ is planar graph of girth at least five and $L$ is a $3$-list-assignment of $V(G)$, then $G$ has at least $2^{|V(G)|/10000}$ distinct $L$-colorings.
In fact, Thomassen proved a stronger result as follows.
\[ThomExp2\]\[cf. Theorem 4.3 in [@ThomExp]\] Let $T=(G,S,L)$ be a canvas such that $S$ is path on at most two vertices. If $\phi$ is an $L$-coloring of $S$, then $\phi$ extends to at least $2^{|V(G)|/10000}$ distinct $L$-colorings of $G$.
We prove the following generalization of Theorem \[ThomExp\] about the number of extensions of a coloring of a cycle, which we define as follows.
If $T=(G,S,L)$ is a canvas and $\phi$ is an $L$-coloring of $S$, then we let $E_T(\phi)$ denote the number of *extensions* of $\phi$ to $G$, that is the number of distinct $L$-colorings of $G$ whose restriction on $S$ is equal to $\phi$.
\[ExpManyDisc\] If $T=(G,S,L)$ is a canvas such that $S$ is connected and $\phi$ is an $L$-coloring of $S$ that extends to an $L$-coloring of $G$, then
$$\log E_T(\phi) \ge (v(T) + 265(3|E(S)|-5|V(S)|))/10000.$$
We proceed by induction on $v(T)+e(T)$. We may assume that $S$ is induced as otherwise the lemma follows by induction applied to $T/G[S]$. We may assume that $G$ is connected, as otherwise we may apply induction to the component of $G$ containing $S$ and Theorem \[ThomExp\] to the components of $G$ not containing $S$.
We may also assume that $v(T) > 0$ as otherwise the lemma follows as $\phi$ extends to at least one $L$-coloring of $G$.
Let $v\in N(S)\setminus S$. Suppose that $|N(v)\cap V(S)|\ge 2$. Let $H=G[V(S) \cup v]$. Extend $\phi$ to $H$ such that the coloring also extends to $G$. Note that $v(T/H) < v(T)$. Hence by induction, $$10000 \log E_{T/H}(\phi) \ge v(T/H)+265(3|E(H)|-5|V(H)|).$$ Yet $E_T(\phi)\ge E_{T/H}(\phi)$. As $v(T/H)=v(T)-1$, $|E(H)|\ge|E(S)|+2$ and $|V(H)|=|V(S)|+1$, we find that $$\begin{aligned}
v(T/H)+148(3|E(H)|-5|V(H)|)&\ge v(T)-1 + 265(3(|E(S)|+2)-5(|V(S)|+1))\\
&= v(T) + 265 (3|E(S)|-5|V(S)|) + 264,\end{aligned}$$ and the result follows.
So we may suppose that $|N(v)\cap V(S)|=1$. Let $H=G[V(S)\cup v]$. Let $S(v)= L(v)\setminus \{\phi(u)| u\in N(v)\cap V(S)\}$. As $|N(v)\cap V(S)|=1$, $|S(v)|\ge 2$. Let $c_1,c_2\in S(v)$. For $i\in\{1,2\}$, let $\phi_i(v)=c_i$ and $\phi_i(u)=\phi(u)$ for all $u\in V(S)$. If both $\phi_1$ and $\phi_2$ extend to $L$-colorings of $G$ (and hence $H$), it follows by induction applied to $T/H$ that $$10000 \log E_{T/H}(\phi_i)\ge (v(T/H) + 265(3|E(H)|-5|V(H)|).$$ As $v(T/H)=v(T)-1$, $|E(H)|=|E(S)|+1$ and $|V(H)|=|V(S)|+1$ we find that $$\begin{aligned}
10000 \log E_{T/H} (\phi_i)&\ge v(T)-1 + 265 (3(|E(S)|+1) - 5 (|V(S)|+1)) \\
&= v(T)+265(3|E(S)|-5|V(S)|) - 529.\end{aligned}$$ Yet $E_T(\phi)\ge E_{T/H}(\phi_1)+E_{T/H}(\phi_2)$ and hence $$10000 \log E_T(\phi)\ge v(T)+265(3|E(S)|-5|V(S)|),$$ and the lemma follows.
So we may suppose without loss of generality that $\phi_1$ does not extend to an $L$-coloring of $G$. Hence there exists a critical subcanvas $T'=(G', H, L)$ of $T/H$. By Theorem \[StrongLinear\], $v(T') \le 88{\rm def}(T')$. By Theorem \[StrongLinear\], it also follows that ${\rm def}(T')\ge 3$. Thus $v(T|G') = v(T')+1$. Moreover, ${\rm def}(T')\le {\rm def}(T|G')+2$ and hence ${\rm def}(T|G')\ge 1$. Hence $$v(T|G')\le 1+88{\rm def}(T') \le 1+88({\rm def}(T|G')+2) \le 265{\rm deg}(T|G').$$ Also recall that ${\rm def}(T|G')= 3e(T|G')-5v(T|G')$.
As $\phi$ extends to an $L$-coloring of $G$, $\phi$ extends to an $L$-coloring of $H$. By induction, $$10000 \log E_{T/G'}(\phi) \ge v(T/G')+265(3|V(G')|-5|V(G')|).$$ Note that $|V(G')|=v(T|G')+|V(S)|$ and $|E(G')|=e(T|G')+|E(S)|$. Thus $$\begin{aligned}
10000 \log E_T(\phi) &\ge v(T)-v(T|G') + 265(3(e(T|G')+|E(S)|)-5(v(T|G')+|V(S)|)) \\
&= v(T)+265(3|E(S)|-5|V(S)|) + 265{\rm deg}(T|G') - v(T|G').\end{aligned}$$ Since from above $265{\rm def}(T|G')\ge v(T|G')$, the lemma follows.
Let us now define a notion of criticality for having exponentially many extensions as follows.
Let $\epsilon,\alpha>0$. Let $G$ be a graph with rings ${{\mathcal}{R}}$ embedded in a surface $\Sigma$ of Euler genus $g$, let $R$ be the total number of ring vertices and let $L$ be a list-assignment of $G$. We say that $G$ is *$(\epsilon,\alpha)$-exponentially-critical with respect to $L$* if $G\ne \bigcup {{\mathcal}{R}}$ and for every proper subgraph $G'\subseteq G$ that includes all the rings there exists an $L$-coloring $\phi$ of $\bigcup {{\mathcal}{R}}$ such that exist $2^{\epsilon (|V(G')\setminus V(H)|-\alpha(g+R))}$ distinct $L$-colorings of $G'$ extending $\phi$ but there does not exist $2^{\epsilon(|V(G)\setminus V(H)|-\alpha(g+R))}$ distinct $L$-colorings of $G$ extending $\phi$.
We are now ready to prove that the family of graphs of girth at least five with rings which are $(\epsilon,\alpha)$-exponentially-critical with respect to some $3$-list assignment are hyperbolic as long $\epsilon \in (0, 1/20000)$ and $\alpha\ge 0$. In fact, we can prove the stronger result where we relax the girth condition to the condition that every cycle of four or less is equal to a ring.
\[ExpHyper\] Let $\epsilon>0$ and $\alpha\ge0$, and let ${{\mathcal}{F}}$ be the family of embedded graphs $G$ with rings such that every cycle of length four or less is equal to a ring and $G$ is $(\epsilon,\alpha)$-exponentially-critical with respect to some $3$-list assignment. If $\epsilon<1/20000$, then the family ${{\mathcal}{F}}$ is hyperbolic with Cheeger constant $6908$ (independent of $\epsilon$ and $\alpha$).
Let $G$ be a graph with rings ${{\mathcal}{R}}$ embedded in a surface $\Sigma$ of Euler genus $g$ such that every cycle of length four or less is not null-homotopic and such that $G$ is $(\epsilon,\alpha)$-exponentially-critical with respect to a $3$-list assignment $L$, let $R$ be the total number of ring vertices, and let $\gamma:{\mathbb S}^1\to\Sigma$ be a closed curve that bounds an open disk $\Delta$ and intersects $G$ only in vertices. To avoid notational complications we will assume that $\gamma$ is a simple curve; otherwise we split vertices that $\gamma$ visits more than once to reduce to this case. We may assume that $\Delta$ includes at least one vertex of $G$, for otherwise there is nothing to show. Let $X$ be the set of vertices of $G$ intersected by $\gamma$. Then $|X|\ge 2$ by Theorem \[ThomExp2\] and further if $|X|=2$, then $X$ is an independent set.
Let $G_0$ be the subgraph of $G$ consisting of all vertices and edges drawn in the closure of $\Delta$. Let $G_1$ be obtained from $G_0$ as follows. For every pair of vertices $u,v\in X$ that are consecutive on the boundary of $\Delta$ we do the following. We delete the edge $uv$ if it exists and then we introduce a path of two new degree two vertices joining $u$ and $v$, embedding the new edges and vertices in a segment of $\gamma$.
Thus $G_1$ has a cycle $C_1$ embedded in the image of $\gamma$, and hence $G_1$ may be regarded as a plane graph with outer cycle $C_1$. For $v\in V(G_0)$ let $L_1(v):=L(v)$, and for $v\in V(G_1)-V(G_0)$ let $L_1(v)$ be an arbitrary set of size three.
Let $T=(G_1,C_1,L_1)$. Note that $G_1$ has girth at least five and so $T$ is a canvas. Also note that $|C_1|\le 3|X|$.
We may assume for a contradiction that $$|V(G_0)-X|> 6908(|X|-1)\ge 3454|X|,$$ where the last inequality follows since $|X|\ge 2$.
Let $G_2$ be the smallest subgraph of $G_1$ such that $G_2$ includes $C_1$ as a subgraph and every $L_1$-coloring of $C_1$ that extends to an $L_1$-coloring of $G_2$ also extends to an $L_1$-coloring of $G_1$. Then $G_2$ is $C_1$-critical with respect to $L_1$. Hence $T|G_2$ is a critical canvas or $T|G_2=T|C_1$. By Corollary \[StrongLinear3\], $$|V(G_2)-V(C_1)|\le 88 |V(C_1)|\le 274|X|.$$
Let $G_0'=G\setminus (V(G_1)- V(G_2))$. Thus $G_0'$ is a proper subgraph of $G$, and, in fact, $$\begin{aligned}
|V(G)-V(G_0')|&= |V(G_0)-V(G_0')|\\
&=|V(G_0)-X|-|V(G_2)-V(C_1)|\\
&\ge 3454|X|-274|X| = 3180|X|.\end{aligned}$$
As $G$ is $(\epsilon,\alpha)$-exponentially-critical with respect to $L$, there exists an $L$-coloring $\phi$ of $\bigcup{{\mathcal}{R}}$ such that $\phi$ extends to at least $2^{\epsilon (|V(G_0')|-\alpha(g+R))}$ distinct $L$-colorings of $G_0'$, but does not extend to at least $2^{\epsilon (|V(G)|-\alpha(g+R))}$ distinct $L$-colorings of $G$.
\[Extend\] If $\phi'$ is an $L$-coloring of $G_0'$ that extends $\phi$, then $\phi'$ extends to at least $2^{( |V(G)-V(G_0')|)/20000}$ distinct $L$-colorings of $G$.
Note that $\phi'$ extends to an $L_1$-coloring $\phi''$ of $G_2$ by colorings the paths of degree two vertices in $G_1-V(G_0)$. By the definition of $G_2$, every $L_1$-coloring of $G_2$ extends to an $L_1$-coloring of $G_1$. Hence $\phi''$ extends to an $L_1$-coloring of $G_1$. Thus by Lemma \[ExpManyDisc\]
$$10000 \log E_{T/G_2}(\phi'') \ge v(T/G_2) + 265(3|E(G_2)|-5|V(G_2)|).$$
However, every such extension induces a different extension of $\phi'$ to $G$. Note that $v(T/G_2)=|V(G_0)|-|V(G_2)|=|V(G)|-|V(G_0')|$. Moreover, $3|E(G_2)|-5|V(G_2)| = {\rm def}(T|G_2) + (3|E(C_1)|-5|V(C_1)|)$. As $T|G_2$ is either critical or equal to $T|C_1$, it follows from Theorem \[StrongLinear\] that ${\rm def}(T|G_2)\ge 0$. Since $|E(C_1)|=|V(C_1)|$ and $|V(C_1)|\le 3|X|$, we find that $$3|E(G_2)|-5|V(G_2)|\ge - 6|X|.$$ Hence, $$10000 \log E_{T/G_2}(\phi'') \ge |V(G)|-|V(G_0')| - 1590|X| \ge (|V(G)|-|V(G_0')|)/2,$$
where the last inequality follows since $|V(G)|-|V(G_0')|\ge 3180|X|$.
The coloring $\phi$ extends to at least $2^{\epsilon (|V(G_0')|-\alpha(g+R))}$ distinct $L$-colorings of $G_0'$. By Claim \[Extend\], each such extension $\phi'$ extends to at least $2^{|V(G)-V(G_0')|/20000}$ distinct $L$-colorings of $G$. But then as $\epsilon\le 1/20000$, there exist at least $2^{\epsilon (|V(G)|-\alpha(g+R))}$ distinct $L$-colorings of $G$ extending $\phi$, a contradiction.
[99]{} Z. Dvořák, $3$-choosability of planar graphs with ($\le 4$)-cycles far apart. arXiv:1101.4275 \[math.CO\] Z. Dvořák and K. Kawarabayashi, Choosability of planar graphs of girth $5$. arXiv:1109.2976 \[math.CO\] Z. Dvořák and K. Kawarabayashi, List-coloring embedded graphs. arXiv:1210.7605 \[cs.DS\] Z. Dvořák, D. Král’ and R. Thomas, Three-coloring triangle-free graphs on surfaces, [*Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)*]{}, New York, NY (2009), 120–129. See arXiv:1010.2472 \[cs.DM\] Z. Dvořák, D. Král’ and R. Thomas, Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk, arXiv:1302.2158. Z. Dvořák, D. Král’ and R. Thomas, Three-coloring triangle-free graphs on surfaces III. Graphs of girth five, arXiv:1402.4710. Z. Dvořák, D. Král’ and R. Thomas, Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies, arXiv:0911.0885. D. Eppstein, Subgraph isomorphism in planar graphs and related problems, [*J. Algor. and Appl.*]{} [**3**]{} (1999), 1–27. P. Erdős, A. Rubin, H. Taylor, Choosability in graphs, Proc. West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, Congressus Numerantium, 26 (1979), 125–157. H. Grötzsch, Ein Dreifarbensatz fur dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 8 (1959), 109–120. I. Havel, On a conjecture of B. Grunbaum, [*J*. Combinatorial Theory]{} [**7**]{} (1969), 184–186.
I. Havel, O zbarvitelnosti rovinnych grafu tremi barvami, Mathematics (Geometry and Graph Theory), Univ. Karlova, Prague (1970), 89–91. B. Mohar, C. Thomassen. Graphs on Surfaces. John Hopkins University Press, 2001. L. Postle, R. Thomas, $5$-List-Coloring Graphs on Surfaces II. A Linear Bound For Critical Graphs in a Disk, J. Comb. Theory Ser. B 119 (2016), 42–65. L. Postle, R. Thomas, Hyperbolic families and coloring graphs on surfaces, manuscript. C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Comb. Theory Ser. B 64 (1995), 101–107. C. Thomassen, The chromatic number of a graph of girth 5 on a fixed surface, J. Combin. Theory Ser. B 87 (2003), 38–71. C. Thomassen, A short list color proof of Grotzsch’s theorem, J. Combinatorial Theory B 88 (2003) 189–192. C. Thomassen, Many 3-colorings of triangle-free planar graphs, J. Combin. Theory Ser. B 97 (3) (2007) 334–349. V. G. Vizing, Vertex colorings with given colors (in Russian), Metody Diskret. Analiz, Novosibirsk 29 (1976), 3–10. M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995), 325–328.
[^1]: `Partially supported by NSERC under Discovery Grant No. 2014-06162, the Ontario Early Researcher Awards program and the Canada Research Chairs program. Email: lpostle@uwaterloo.ca`
|
---
author:
- 'W. S. Dias'
- 'B. S. Alessi'
- 'A. Moitinho'
- 'J. R. D. Lépine'
date: 'Received <date>/ Accepted <date>'
title: 'New catalogue of optically visible open clusters and candidates [^1] '
---
Introduction
============
In this work, we introduce a new catalogue of the open clusters of our Galaxy. Open clusters have long been recognized as important tools to investigate the kinematics of star formation regions, aspects of Galactic structure such as the location of spiral arms, Galactic dynamics, or even the chemical abundance gradients in the disk.
With the publication of the Hipparcos Catalogue [@ESA1997] and its sub-products, the Tycho [@ESA1997] and Tycho2 [@Hog2000] catalogues , and with individual works using CCDs for photometry and/or astrometry, we have seen a large growth of the available data on open clusters in a short time.
Among the recent results, we note the discovery of new open clusters by different authors: @Platais1998 discovered 12 new objects using Hipparcos data, @Chereul1999 discovered 3 new probable loose open clusters, and @Dutra2001 discovered 42 objects at infra-red wavelengths using the 2MASS survey. Important contributions were given by @Baumgardt2000 and @Dias2001 [@Dias2002] who determined the mean proper motions of more than a hundred clusters, using the Hipparcos and Tycho2 catalogues, respectively. @Dias2001 [@Dias2002] also computed the membership probabilities of the stars in the cluster fields. Other recent results are the determination of the fundamental parameters of 423 clusters by @Loktin2000 and the discussion of the problem of the differences in the distances obtained with parallaxes and by photometric main sequence fitting . The latest publications on open clusters are divulged in the SCYON electronic newsletter hosted by the University of Heidelberg [^2] in parallel with the WEBDA database[^3].
Most of the basic data, as well as other results, are included in the WEBDA database [@Mermilliod1995], which is the most complete open cluster database presently available. The WEBDA database includes not only the data contained in the @Lynga1987 catalogue, which is also a basic reference much used in the literature, but also provides a huge amount of additional information. Most of this information is, however, presented in separate files, available individually for each cluster. Also, the database is not updated in what concerns recently discovered clusters and new designations proposed in the literature (as discussed in next section). Therefore, the main reasons that prompted us to prepare a new catalogue, instead of simply adding newly discovered objects, were the need to have the relevant information in a single file, for easiness of use, and more important, the fact that the previous catalogues do not provide the open clusters’ proper motions and radial velocities in a systematic way.
In this work, we inserted the available information on open clusters’ fundamental parameters, kinematics and metalicity in a single file. We believe that this list will be an important tool for all types of research on open clusters. Sect. \[sec:cat\] describes the contents of the catalogue and the main reference sources. In Sect. \[sec:comm\] we comment on the new data included.
The catalogue \[sec:cat\]
=========================
In this new open cluster catalogue, we used the previous ones like the WEBDA, ESO Catalogue [@Lauberts1982] and @Lynga1987 as a starting point. The basic data contained in these catalogues are coordinates, age, apparent diameter, colour excess, and distance. We inserted new objects, and when available, kinematical and metalicity data. We made extensive use of the Simbad database and of the literature to find data on the clusters or on individual stars of the clusters, to obtain radial velocities and proper motions averaged over a number of stars. We do not claim, however, that the catalogue is the result of a complete survey of all the bibliography on open clusters.
Our catalogue (Tab. 1a) consists of a single list of fundamental parameters and kinematical data, with bibliographic notes. The file is self-explanatory and fully documented internally. The present version of the catalogue includes information for 1537 open clusters. For each cluster we list its equatorial coordinates in J2000.0 and the following parameters, when available: angular apparent diameter; distance; colour excess; age; mean proper motions and errors; number of stars used in the proper motion computation and references; mean radial velocity and error; number of stars used in the radial velocity determination and references; mean metalicity and errors; number of stars used in the metalicity determination. An identical list (Tab. 1b) is also provided with positions and proper motions in galactic coordinates. The full bibliographic references are given in a separate file (Tab. 2).
In total, 94.7$\%$ of the objects have estimates of their apparent diameters, and 37$\%$ have distance, $E(B-V)$ and age determinations. Concerning the data on kinematics, 18$\%$ have their mean proper motions listed, 12$\%$ their mean radial velocities, and 9$\%$ have both information simultaneously.
Many objects in the list were visually checked in the Digitized Sky Survey [^4] (DSS) plates, and in several cases the central coordinates of the clusters were corrected. This is the case of clusters like , , and , just to mention a few, that present great differences in position.
Throughout our visual inspection of the DSS plates, there were also many cases in which no cluster could be found (eg. several Ruprecht, Collinder and Loden clusters), even in large fields around their catalogued coordinates. They were nevertheless kept in the catalogue, but a comment was added. We shall refer to these “objects” as “non-identified clusters”. Among these, are the NGC objects flagged as “non-existent” in the *The Revised New General Catalogue of Non stellar Astronomical Objects* (RNGC)[@Sulentic1973]. On the other hand, some clusters noted as “non-existent” in the RNGC seem to be actual clusters (eg. , , , , , ). These have been marked as “recovered” in our catalogue.
A complementary table (Tab. 3) of the clusters with available photometric data was also built. Tab. 3 consists of four columns: cluster name; bands observed with CCD; bands observed with photomultipiers; bands observed with photographic plates. For each cluster, only bands with more than ten observed stars are listed. At the present, the table only lists the UBVRI bands, but it will be extended to other commonly used photometric systems (eg. uvby$\beta$, Geneva, Washington, Vilnius, etc.). The data table was assembled using data collected from WEBDA and from searches in the literature.
Comments on new information and new data included \[sec:comm\]
==============================================================
In this section we comment on some important information given in the catalogue.
[ *designations*]{} - An additional remark on the nature of the open clusters is provided. Among others, we flag the [*POCR*]{} [Possible Open Cluster Remnant, @Bica2001]. There are 34 objects located at relatively high galactic latitudes ($b\geq
15^{\degr}$) which appear to be late stages of star cluster dynamical evolution. The categories also include possible moving groups like the objects catalogued by Latysev and non-identified clusters.
[ *kinematics*]{} - Recently, many open clusters were investigated and their mean proper motions [@Dias2001; @Dias2002; @Baumgardt2000] could be determined. New mean proper motions for 280 objects, and radial velocities for 182 were inserted in the list.
[ *Fundamental parameters*]{} - The main source of the fundamental parameters (reddening, distance and age) was the WEBDA which uses the information compiled by @Lynga1987, @Loktin2000, @Dambis1998 and @Malysheva1997.
All the clusters investigated by @Baumgardt2000 had their distances estimated from the mean Hipparcos parallaxes of the stars considered as members. Recently we investigated 4 open clusters and determined the mean Hipparcos parallax of stars with membership probability provided by Tycho2 proper motions [@Dias2001]. The catalogue includes distances derived from mean parallaxes for Ruprecht 147, Stock 10, vdB-Hagen 23, vdB-Hagen 34, all within 1 kpc. Also, a number of parameters from isolated studies were added.
[ *Newly discovered open clusters* ]{} - The list includes 191 clusters not present in the previous catalogues. To mention some cases: @Platais1998 - 12 open clusters were discovered using Hipparcos data. They are nearby and extended objects; ESO-SC - these objects (more than 100) were published as probable new open clusters in the ESO catalogue @Lauberts1982; Loiano 1- A photometric study of the surrounding stellar field [@Bernabei2001] revealed that this object lies inside the sky area of a previously undetected open cluster of intermediate age. Alessi 1 to 12 are non catalogued objects in the solar vicinity. Their fundamental parameters were recently determined showing that they are located at $d \leq 1 kpc$ [@Alessi2002].
Summary and conclusions
=======================
We have presented a new list of open clusters containing revised data compiled from old catalogues and from isolated papers recently published. This catalogue (Tab. 1a) has been developed mainly to be an efficient tool for open cluster studies since it presents all the available basic data (fundamental parameters and kinematics) in a single easy-to-use list. The catalogue is regularly updated, and the latest version is available at *http://www.iagusp.usp.br/\~wilton/*. An alternative list (Tab. 1b) with positions and proper motions in galactic coordinates is also made available. Since it is expected that the catalogue will be used in the selection of observational targets, an additional table of open clusters with available photometric data (Tab. 3) is also provided. Finally, Tab. 2 includes the references to the data used in Tabs. 1a and 1b.
In this edition, 1537 objects are given, of which 356 are not given in the catalogue compiled by @Lynga1987. The new objects include 191 open cluster published in the literature, and 11 recently discovered open cluster with fundamental parameters determined by our group and yet unpublished.
Nearly all the clusters (94.7$\%$) have estimates of their apparent diameters. Distances, $E(B-V)$ and ages are listed for 37$\%$ . Concerning the data on kinematics, 18$\%$ have mean proper motions determinations, 12$\%$ mean radial velocities, and 9$\%$ have both information simultaneously. These results point out to the observers that a large effort is still needed to improve the data on kinematics. Our group is presently working in this direction.
We use data from Digitized Sky Survey which were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. Extensive use has been made of the Simbad and WEBDA databases. This project was supported by FAPESP (grant number 99/11781-4).
[19]{} natexlab\#1[\#1]{}
Alessi, B. S., Dias, W. S., & Lépine, J. R. D. 2002, in preparation
Baumgardt, H., Dettbarn, C., & Wielen, R. 2000, A&AS, 146, 251
, S. & [Polcaro]{}, V. F. 2001, A&A, 371, 123
, E., [Santiago]{}, B. X., [Dutra]{}, C. M., [Dottori]{}, H., [de Oliveira]{}, M. R., & [Pavani]{}, D. 2001, A&A, 366, 827
, E., [Cr[' e]{}z[' e]{}]{}, M., & [Bienaym[' e]{}]{}, O. 1999, A&AS, 135, 5
Dambis, A. K. 1998, AstL, 25, 10
Dias, W. S., Lépine, J. R. D., & Alessi, B. S. 2001, A&A, 376, 441
—. 2002, submitted to A&A
, C. M. & [Bica]{}, E. 2001, A&A, 376, 434
. 1997, [The Hipparcos and Tycho Catalogues]{} (European Space Agency (ESA))
, E., [Fabricius]{}, C., [Makarov]{}, V. V., [Urban]{}, S., [Corbin]{}, T., [Wycoff]{}, G., [Bastian]{}, U., [Schwekendiek]{}, P., & [Wicenec]{}, A. 2000, A&A, 355, L27
, A. 1982, [ESO/Uppsala survey of the ESO(B) atlas]{} (Garching: European Southern Observatory (ESO))
Loktin, A. V., Gerasimenko, T. P., & Malisheva, T. 2000, astron. Astrophys. Trans. in press
Lyng[å]{}, G. 1987, Computer Based Catalogue of Open Cluster Data, 5th ed. (Strasbourg: CDS)
, L. K. 1997, Astronomy Letters, 23, 585
Mermilliod, J. C. 1995, in Information and On-Line Data in Astronomy, ed. D. Egret & M. A. Albrecht (Dordrecht: Kluwer), 127
, M. H., [Stauffer]{}, J., [Soderblom]{}, D. R., [King]{}, J. R., & [Hanson]{}, R. B. 1998, ApJ, 504, 170+
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Sulentic, J. W. & Tifft, W. G. 1973, The Revised New General Catalogue of Nonstellar Astronomical Objects (Tucson: U. of Arizona Press)
[^1]: Tables 1a, 1b, 2 and 3 are only available in electronic form at http://www.iagusp.usp.br/\~wilton/, or at the CDS via anonymous ftp to ...
[^2]: http://www.rzuser.uni-heidelberg.de/\~s17/scyon/current.html
[^3]: http://obswww.unige.ch/webda/
[^4]: http://archive.stsci.edu/dss/
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author:
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[NP]{}Stanisław Jastrzębski[^1] staszek.jastrzebski@gmail.com\
Jagiellonian University -0.09in [NP]{}Quentin de Laroussilhe underflow@google.com\
-0.33in [NP]{}Mingxing Tan tanmingxing@google.com\
-0.33in [NP]{}Xiao Ma xima@google.com\
-0.33in [NP]{}Neil Houlsby neilhoulsby@google.com\
-0.35in [NP]{}Andrea Gesmundo agesmundo@google.com\
title: Neural Architecture Search Over a Graph Search Space
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[^1]: Work done during an internship at Google
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author:
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F. P. Gavriil,$^{1,2,\ast}$ M. E. Gonzalez,$^{3}$ E. V. Gotthelf,$^{4}$ V. M. Kaspi,$^{3}$\
M. A. Livingstone,$^{3}$ and P. M. Woods$^{5,6}$\
\
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title: 'Magnetar-like Emission from the Young Pulsar in Kes 75'
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We report detection of magnetar-like X-ray bursts from the young pulsar [PSR J1846$-$0258]{}, at the center of the supernova remnant Kes 75. This pulsar, long thought to be rotation-powered, has an inferred surface dipolar magnetic field of 4.9$\mathbf{\times}$10$\mathbf{^{13}}$ G, higher than those of the vast majority of rotation-powered pulsars, but lower than those of the $\mathbf{\sim}$12 previously identified magnetars. The bursts were accompanied by a sudden flux increase and an unprecedented change in timing behavior. These phenomena lower the magnetic and rotational thresholds associated with magnetar-like behavior, and suggest that in neutron stars there exists a continuum of magnetic activity that increases with inferred magnetic field strength.
Magnetars are young, isolated neutron stars having ultra-high magnetic fields[@td95; @td96a]. Observational manifestations of these exotic objects include the Soft Gamma Repeaters (SGRs) and the Anomalous X-ray Pulsars (AXPs). Magnetars exhibit a variety of forms of radiative variability unique to their source class; these include short ($<$1 s) X-ray and gamma-ray bursts, and sudden flux enhancements that decay on time scales of weeks to months, both of which are too bright to be powered by rotational energy loss[@wt06]. A major puzzle in neutron star physics has been what distinguishes magnetars from neutron stars that have comparably high fields, yet no apparent magnetar-like emission[@km05].
The 326-ms [PSR J1846$-$0258]{} is the central isolated neutron star associated with the young shell-type supernova remnant (SNR) Kes 75 (SNR G29.6$+$0.1; see ref for details). Assuming standard magnetic dipole braking, this pulsar has among the largest dipolar magnetic fields of the known young rotation-powered pulsars and the sixth largest overall, $B\equiv
3.2$$\times$$10^{19}~\mathrm{G}~\sqrt{P\dot{P}} =
4.9$$\times$$10^{13}$ G, where $P$ is in seconds. In addition, its spin-down age of $\tau \equiv P/(n-1)\dot P = 884$ yr is the smallest of all known pulsars[@gvbt00; @lkgk06]. The observed X-ray luminosity of [PSR J1846$-$0258]{} is $L = 4.1$$\times$$10^{34}\left(d/
6~\mathrm{kpc} \right)^2$ erg s$^{-1}$ in the 3$-$10 keV band, assuming a distance of $d\sim6$ kpc, the mean distance found from HI and $^{13}$CO spectral measurements[@lt08]. The pulsar has all the hallmarks of being rotation-powered – a radiative output well under its spin-down luminosity ($\dot E \equiv
3.9$$\times$$10^{46}\dot{P}/P^3~\mathrm{erg~s}^{-1} =
8.1$$\times$$10^{36}$ erg s$^{-1}$), an otherwise unremarkable braking index ($n =2.65$)[@lkgk06], and a bright pulsar wind nebula (see Fig. 1). This pulsar is one of only $\sim$3 young rotation-powered pulsars for which no radio emission is detected, although this may be due to beaming.
Observations in the direction of Kes 75 obtained with the *Rossi X-ray Timing Explorer* ([[*RXTE*]{}]{}) have revealed several short bursts of cosmic origin lasting $\sim$0.1 s (see Fig. 2). We discovered four bursts in a 3.4 ks observation made on 2006 May 31 and a 5th in a 3.5 ks observation made on 2006 July 27.
These data were obtained with the Proportional Counter Array (PCA) onboard [[*RXTE*]{}]{} which provides $\sim$$\mu$s time resolution and 256 spectral channels over the $\sim$2$-$60 keV bandpass, and consists of 5 independent sub-units (PCUs). The bursts are plotted in Fig. 2 and their properties are listed in Table 1. We quantified the burst properties as we have for those seen in bursting AXPs (see supporting online text ). All five bursts were highly significant, and were recorded in all operational PCUs simultaneously. We found no additional bursts in the 21.4 Msec of available data of this field collected by [[*RXTE*]{}]{} over the past 7 years.
Because of the PCA’s large ($1^{\circ}$$\times$$1^{\circ}$) field-of-view, the origin of the bursts was not immediately apparent. However, we could unambiguously identify [PSR J1846$-$0258]{} as their origin because the bursts coincided with a dramatic rise in its pulsed flux, which lasted $\sim$2 months (see Fig. 2) and was remarkably similar to those observed from AXPs[@kgw+03; @ims+04; @gk04]. The pulsed flux was extracted according to the method detailed in ref and corrected for collimator response and exposure for each PCU. We model the recovery from the pulsed flux enhancement as an exponential decay (with $1/e$ time constant 55.5$\pm$5.7 day) and estimate a total 2$-$60 keV energy release of 3.8$-$4.8$\times$$10^{41}\left(d/ 6~\mathrm{kpc} \right)^2$ erg, assuming isotropic emission. If we assume a power-law model for the flux decay, commonly used for the magnetars, we obtain an index of $-$0.63$\pm$0.06. However, this model is rejected with $\chi^2_{\nu}(51 \ \rm{DoF}) =1.31$ compared with $\chi^2_{\nu}(51
\ \rm{DoF})=0.95$ for the exponential model. At the onset of the outburst, the timing noise of the source changed dramatically from that typical of a young rotation-powered pulsar to that typical of AXPs. [PSR J1846$-$0258]{} was spinning down smoothly with a braking index of $n=$2.65$\pm$0.01[@lkgk06] until phase coherence was lost on MJD 53886, the same observation in which the first four bursts were observed. This loss of phase coherence could signal a spin-up glitch as has been seen to accompany other AXP radiative events[@kgw+03; @dkg07; @tgd+07]. The dramatic sudden timing noise makes the determination of accurate glitch parameters via phase-coherent timing difficult. In the most recent data, the timing noise appears to have settled somewhat, though has not relaxed to its pre-burst behavior. We also examined archival high-resolution CCD images of Kes 75 obtained with the [*Chandra X-ray Observatory*]{} ([[*CXO*]{}]{}) both before (2000 Oct) and very fortuitously during (2006 June) the event. This allows us to identify the dramatic change in the flux of the pulsar relative to its bright, but relatively constant, pulsar wind nebula (see Fig. 1 and supporting online text).
The [[*CXO*]{}]{}-measured spectrum at the outburst epoch softened significantly relative to quiescence. A fit to a power-law model in 2006 produced a larger value for the photon index, with $\Gamma$$=$1.89$^{+0.04}_{-0.06}$ and 1.17$^{+0.15}_{-0.12}$ for epochs 2006 and 2000, respectively (3-$\sigma$ errors). Interestingly, the larger value of the photon index is now closer to those seen in magnetars ($\Gamma$$\sim$2$-$4). Due to this softening, the 0.5–2 keV flux showed the largest increase, a factor of $17^{+11}_{-6}$, while the 2$-$10 keV flux increased by a factor of $5.5^{+4.5}_{-2.7}$ (3-$\sigma$ errors, see Fig. 1). Though the 2006 spectrum is softer, the large absorption precludes the identification of any significant thermal components. Note that the [[*CXO*]{}]{} spectral analysis was non-trivial due to the brightness of the source and associated CCD pile-up; see online supporting text for details.
The coincidence of the bursts with the flux enhancement (see Fig. 2), the distinct changes in the pulsar spectral properties (see Fig. 1), and the timing anomaly and sudden change in timing noise properties all firmly establish [PSR J1846$-$0258]{} as the origin of the bursts.
This is the first detection of X-ray bursts from an apparent rotation-powered pulsar. It is instructive to compare the burst properties with those of SGRs and AXPs. SGRs are characterized by their frequent, hyper-Eddington ($\sim$$10^{41}$ erg s$^{-1}$), and short ($\sim$0.1 s) repeat X-ray bursts. AXPs also emit such bursts, albeit less frequently[@gkw02]. The bursts from [PSR J1846$-$0258]{} were short ($<$0.1 s), showed no emission lines in their spectra, and occurred preferentially at pulse maximum. The peak luminosities ($L_p$) of all bursts were greater than the Eddington luminosity ($L_E$) for a 1.4 $M_{\odot}$ neutron star, assuming isotropic emission and a distance of $d=6$ kpc[@lt08] (burst 2 had $L_p> 10 L_E$). Considering the distribution of SGR and AXP burst temporal, energetic and spectral properties[@gkw+01; @gkw04], the Kes 75 bursts are indistinguishable from many of the bursts seen in AXPs and SGRs.
[PSR J1846$-$0258]{}’s pulsed flux flare is also a magnetar hallmark. A twisted magnetosphere and associated magnetospheric currents induce enhanced surface thermal X-ray emission, and resonant upscattering thereof[@tlk02; @bt07]. Flux enhancements and their subsequent decay in AXPs have been interpreted as sudden releases of energy (either above or below the crust) followed by thermal afterglow, in which case there is an abrupt rise with a gradual decay. A power-law fit was an excellent characterization of AXP [1E 2259$+$586]{}’s flux decay after its 2002 outburst. For [PSR J1846$-$0258]{}, such a model did not fit the data as well as an exponential (see Fig. 2). Spectral changes are also expected with these enhancements. The softening of the source’s spectrum suggests that it underwent a transition from a purely magnetospheric-type spectrum, typical of energetic rotation-powered pulsars, to one consistent with the persistent emission from magnetars. For this reason, it is difficult to directly compare the spectral characteristics of this flux enhancement to those of other magnetars in outburst. The total 2$-$10 keV energy released during the flux enhancement ($3.3-3.8$$\times$$10^{41}\left(d/ 6~\mathrm{kpc} \right)^2$ erg, assuming isotropic emission) is comparable that released in the 2007 flux enhancement [@tgd+07] of AXP [1E 1048.1$-$5937]{} ($\sim$5$\times$$10^{42}\left(d/ 9~\mathrm{kpc} \right)^2$ erg), the most most energetic enhancement yet seen from this AXP. It is also comparable to the energy released during the rapid ($\sim$3$\times$$10^{39}\left(d/ 3~\mathrm{kpc} \right)^2$ erg) and gradual ($\sim$2$\times$$10^{41}\left(d/ 3~\mathrm{kpc}
\right)^2$ erg) decay components of the 2002 outburst of AXP [1E 2259$+$586]{}[@wkt+04]. Similar to AXP [1E 2259$+$586]{}’s 2002 outburst [@wkt+04], the energy released by [PSR J1846$-$0258]{} during the observed short bursts represents only a small ($\sim$0.03%) fraction of the total outburst energy.
Prior to showing magnetar-like emission, [PSR J1846$-$0258]{} exhibited timing noise and a glitch in 2001 [@lkgk06] that were both similar to what has been seen observed in other comparably aged (i.e. $\tau$$\simeq$1 kyr) rotation-powered pulsars. By contrast, in 2007, [PSR J1846$-$0258]{} exhibited much larger timing noise, such that the root mean square phase residual after subtracting a model including the spin frequency, and its first and second derivative is a factor of $\sim$33 larger than before, for the same duration of observations. Such a dramatic, sudden change in timing noise characteristics has never been seen before in a rotation-powered pulsar. The coincidence of the enhanced timing noise with the flux flare is also reminiscent of behavior exhibited by AXP [1E 1048.1$-$5937]{}[@gk04].
Our discovery of distinctly magnetar-like behavior from what previously seemed like a *bona fide* rotation-powered pulsar may shed new light on the magnetic evolution of these objects, and whether their extreme fields originate from a dynamo operating in a rapidly rotating progenitor[@td93a], magnetic flux conservation[@fw06], or a strongly magnetized core, initially with crustal shielding currents[@bs07]. In the first two scenarios, magnetars are born with high magnetic fields which subsequently decay. In the third recently proposed scenario, the very large magnetic fields of magnetars slowly emerge as the shielding currents decay[@bs07]. This source has a well measured braking index ($n
=2.65$$\pm$0.01)[@lkgk06], at least before outburst, which is significantly less than 3, suggesting that its spin properties, and hence magnetic field are headed towards the magnetar regime[@lyn04]. In this case, the timescale for magnetic field decay, given by the magnetic field divided by its decay rate will be $B/(\partial B/ \partial t$)$\sim$8 kyr, at which point [PSR J1846$-$0258]{} will have $P\sim$1.3 s. However, other mechanisms, such as the interaction between a strong relativistic pulsar wind nebula (PWN) and the magnetosphere[@hck99], can also yield the value of $n$ measured for [PSR J1846$-$0258]{}. In this case, the magnetar-like behavior could be a result of the moderately high $B$, with no $B$ evolution occurring.
There have been suggestions of magnetar-related emission from other high-magnetic-field radio pulsars, e.g. PSR J1119$-$6127[@gkc+05], but, until now, nothing that could not also be explained within the constraints of rotation-powered pulsar physics. It has been suggested (see ref ) that the high-$B$ field pulsars are related to transient AXPs, magnetars generally in quiescence whose X-ray emission can grow by factors of $\sim$hundreds in outburst. Interestingly, the first two reports of radio pulsations from a magnetar were from transient AXPs after outburst[@crh+06; @crh+07]. Despite a lack of radio emission, the behavior of [PSR J1846$-$0258]{} reinforces the connection between transient AXPs and high-$B$ rotation-powered pulsars, and suggests that careful monitoring of other high-$B$ rotation-powered pulsars[@km05] is warranted.
The addition of [PSR J1846$-$0258]{} to the list of sources which emit magnetar-like events provides insight into the origin of this activity. Extreme magnetic activity is prevalent in the SGRs which exhibit giant flares with energy releases upwards of 10$^{44}$ ergs (see ref for an example) and are also prolific busters, emitting bursts fairly frequently, typically multiple times per year, with larger outbursts occurring every few years. AXPs can be considered milder versions of SGRs, with several showing sporadic short SGR-like events, though more rarely than in SGRs, with even modest outbursts occurring only once or twice per decade. Now, Kes 75, weakly magnetized by magnetar standards, shows properties of both rotation powered pulsars and AXPs, and seems to produce an outburst only roughly every decade. The detection of magnetar-like emission from Kes 75 suggests that there is a continuum of “magnetar-like” activity throughout all neutron stars which depends on spin-inferred magnetic field strength.
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We thank Tod Strohmayer for assistance and Alice Harding and Demosthenes Kazanas for discussion. MAL is a Natural Sciences and Engineering Research Council (NSERC) PGS-D fellow. Support for this work was also provided by an NSERC Discovery Grant Rgpin 228738-03, an R. Howard Webster Fellowship of the Canadian Institute for Advanced Research, Les Fonds de la Recherche sur la Nature et les Technologies, a Canada Research Chair and the Lorne Trottier Chair in Astrophysics and Cosmology to VMK, and [[*RXTE*]{}]{} grants NNG05GM87G and N5-RXTE05-34 to EVG . This research has made use of data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center.
{width="\columnwidth"}
[**Fig. 1.**]{} High resolution [[*Chandra*]{}]{} X-ray images (0.5$-$10 keV) of [PSR J1846$-$0258]{} in SNR Kes 75 centered on the pulsar and its surrounding PWN, obtained before and during the 2006 outburst. Following the bursts, the pulsar became brighter as well as softer. These images were made using archival ACIS-S3 observations obtained on 2000 Oct 15-16 ([*left*]{}) and very fortuitously 2006 June 5, 7-8, 9, 12-13 ([*right*]{}) and are background-subtracted, exposure-corrected, smoothed with a constant Gaussian with width $\sigma$=0.5$''$ and finally displayed using the same brightness scale.
{width="0.8\columnwidth"}
[**Fig. 2.**]{} Top: Pulsed flux history of [PSR J1846$-$0258]{} showing the prominent outburst of June 2006 as recorded in the 2$-$60 keV band by [[*RXTE*]{}]{}. The horizontal dotted line represents the persistent flux level. Epochs corresponding to [[*CXO*]{}]{} observations are indicated with arrows. Middle: The light curve around the outburst. The vertical dashed lines indicate the epochs of the observations containing the bursts, 2006 May 31 (4 bursts) and 2006 July 27 (1 burst). The leftmost vertical dashed line also coincides with the time when phase coherence was first lost. Bottom: The 2$-$60 keV [[*RXTE*]{}]{} X-ray lightcurves corresponding to five bursts detected from [PSR J1846$-$0258]{}, sampled with 5 ms bins. The bursts lasted for $\sim$0.1 s and were detected with high significance from two data sets obtained on 2006 May 31 and 2006 July 27. Notice that in 7 years of [[*RXTE*]{}]{} observations the only bursts found either occur at the onset of the $\sim$2 month X-ray outburst (4 bursts) or at the end of the decay (1 burst).
[lccccc]{} Table 1 &\
& & & & &\
\
Burst day (MJD) & 53886 & 53886 & 53886 & 53886 & 53943\
Burst start time & 0.92113966(5) & 0.93247134(1) & 0.93908845(2) & 0.94248467(5) & 0.45543551(1)\
(fraction of day) & & & & &\
Rise time, $t_r$ (ms) & 4.2$^{+3.5}_{-2.0}$ & 1.1$^{+0.9}_{-0.5}$ & 1.90$^{+1.7}_{-0.9}$ & 4.1$^{+3.1}_{-1.9}$ & 0.9$^{+2.2}_{-0.7}$\
$T_{90}$ (ms) & 71.8$^{+38.0}_{-5.5}$ & 42.9$^{+0.3}_{-0.2}$ & 137.0$^{+11.4}_{-36.2}$ & 33.4$^{+29.1}_{-23.1}$ & 65.3$^{+0.7}_{-0.5}$\
Phase (cycles) & -0.49(1) & -0.04(1) & -0.20(1) & -0.05(1) & -0.08(1)\
\
$T_{90}$ Fluence & 8.9$\pm$0.7 & 712.8$\pm$2.5 & 18.3$\pm$0.7 & 18.4$\pm$0.7 & 18.4$\pm$1.1\
(counts/PCU) &&&&&\
$T_{90}$ Fluence & 4.1$\pm$2.4 & 289.9$\pm$13.1 & 6.6$\pm$2.5 & 5.8$\pm$1.7 & 5.3$\pm$2.0\
(10$^{-10}$ erg/cm$^{2}$) & & & & &\
Flux for 64 ms & 57$\pm$36 & 4533$\pm$227 & 99$\pm$41 & 97$\pm$31 & 79$\pm$32\
(10$^{-10}$ erg/s/cm$^{2}$) &&&&&\
Flux for $t_r$ & 678$\pm$427 & 5783$\pm$885 & 810$\pm$385 & 828$\pm$284 & 2698$\pm$1193\
(10$^{-10}$ erg/s/cm$^{2}$) &&&&&\
\
Power-law index & 0.89$\pm$0.58 & 1.05$\pm$.04 & 1.14$\pm$0.34 & 1.36$\pm$0.25 & 1.41$\pm$0.31\
$\chi^2/$DoF (DoF) & 0.42 (1) & 1.16 (55) & 0.97 (3) & 0.35 (2) & 1.18 (2)\
[**Table 1.**]{} All the quoted errors represent 1-$\sigma$ uncertainties unless otherwise indicated. All times are given in units of UTC corrected to the Solar System barycenter using the source position R.A.=$18^{\mathrm{h}}46^{\mathrm{m}}24$94, decl=$-02^{\circ}58^\prime30$1 and the JPL DE200 ephemeris[@hcg03].
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abstract: |
We investigate electromagnetic corrections to the rare $B$-meson leptonic decay $B_{s,d} \to \mu^+\mu^-$ from scales below the bottom-quark mass $m_b$. Contrary to QCD effects, which are entirely contained in the $B$-meson decay constant, we find that virtual photon exchange can probe the $B$-meson structure, resulting in a “non-local annihilation” effect. We find that this effect gives rise to a dynamical enhancement by a power of $m_b/\Lambda_{\rm QCD}$ and by large logarithms. The impact of this novel effect on the branching ratio of $B_{s,d}\to\mu^+\mu^-$ is about $1\%$, of the order of the previously estimated non-parametric theoretical uncertainty, and four times the size of previous estimates of next-to-leading order QED effects due to residual scale dependence. We update the Standard Model prediction to $\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm SM}
= (3.57 \pm 0.17) \cdot 10^{-9}$.
author:
- Martin Beneke
- Christoph Bobeth
- Robert Szafron
date: 'August 25, 2017'
title: ' Enhanced electromagnetic correction to the rare $B$-meson decay $B_{s,d} \to \mu^+ \mu^-$ '
---
Rare leptonic decays $B_q\to \ell^+\ell^-$ of neutral $B$ mesons ($q = d, s$ and $\ell = e, \mu, \tau$) provide important probes of flavour-changing neutral currents, since the decay rate in the Standard Model (SM) is predicted to be helicity- and loop-suppressed. Both suppressions can be lifted, for example, in models with extended Higgs sectors, in which case the leptonic decays constrain the scalar masses far above current direct search limits.
Only the muonic decay $B_s\to \mu^+\mu^-$ has been observed to date [@Aaij:2013aka; @Chatrchyan:2013bka]. The most recent measurement of the LHCb experiment for the untagged time-integrated branching ratio finds $\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm LHCb} =
(3.0^{+0.7}_{-0.6}) \cdot 10^{-9}$ [@Aaij:2017vad], compatible with the SM prediction [@Bobeth:2013uxa] $$\label{eq:SM:Br}
\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm SM}
= (3.65 \pm 0.23) \cdot 10^{-9}\,.$$ With higher experimental statistics and improvement in the knowledge of SM parameters, the accuracy of both results is expected to increase in the future, eventually providing one of the most important precision tests in flavour physics.
The neutral $B$-meson leptonic decays are indeed well suited for precision physics, because long-distance strong-interaction (QCD) effects, which cannot be computed with perturbative methods, are under exceptionally good control. This follows from the purely leptonic final state and the fact that the decay is caused by the effective local interaction $$Q_{10} = \frac{\alpha_{\rm em}}{4\pi} \,\big(\bar{q} \gamma^\mu P_L b\big)
\big(\bar{\ell} \gamma_\mu \gamma_5 \ell\big) \,, \quad
P_L \equiv \frac{1 - \gamma_5}{2}\,.$$ The strong interaction effects are therefore confined to the matrix element $$\langle 0|\bar{q} \gamma^\mu \gamma_5 b|\bar{B}_q(p)\rangle
= i f_{B_q} p^\mu\,,
\label{fB}$$ which defines the $B$-meson decay constant. $f_{B_q}$ can be computed non-perturbatively with few percent accuracy within the framework of lattice QCD [@Aoki:2016frl].
In this Letter, we report on an investigation of electromagnetic (QED) quantum corrections to the leptonic decay which even at the one-loop order reveals a surprisingly complex pattern. As a consequence, the suppression of the correction due to the small electromagnetic coupling is partially compensated by a power-like enhancement in the ratio of the $B$-meson mass $m_B\approx 5~$GeV and the strong interaction scale $\Lambda_{\rm QCD}
\approx 200~$MeV. While logarithmic enhancements due to collinear and soft radiation are well-known in QED and also appear in the process under consideration, the power-like enhancement arises due to a dynamical mechanism that to our knowledge has not been observed before. A virtual photon exchanged between the final-state leptons and the light spectator antiquark $\bar q$ in the $\bar B_q$ meson effectively acts as a weak probe of the QCD structure of the $B$ meson. The scattering “smears out” the spectator–$b$-quark annihilation over the distance $1/\sqrt{m_B\Lambda_{\rm QCD}}$ inside the $B$ meson, as opposed to the local annihilation through the axial-vector current in Eq. (\[fB\]). This provides power-enhancement and also shows that at first order in electromagnetic interactions, the strong interaction effects can no longer be parameterized by $f_{B_q}$ alone. Our calculation below shows that the effect is of the same order as the non-parametric theoretical uncertainty previously assumed to obtain Eq. (\[eq:SM:Br\]).
Before discussing the main result, we briefly review the computations and theoretical uncertainties entering Eq. (\[eq:SM:Br\]), referring to Ref. [@Bobeth:2013uxa] for further details. The general framework employs the effective weak interaction Lagrangian, which generalizes the Fermi theory to the full SM, includes all short-distance quantum effects systematically by matching, and sums large logarithms between the scale $m_W$ of the $W$-boson mass and $\mu_b \sim m_b$ of the order of the bottom-quark mass, $m_b$. The SM prediction includes next-to-leading order (NLO) electroweak (EW) [@Bobeth:2013tba] and next-to-next-to-leading order QCD [@Hermann:2013kca] corrections and the resummation of large logarithms $\ln(\mu_W/\mu_b)$ due to QCD and QED radiative corrections by means of the renormalization-group (RG) evolution [@Bobeth:2003at; @Huber:2005ig] down to $\mu_b$ at the same accuracy. Relevant to this work is the observation that unlike QCD effects, which are contained in $f_{B_q}$ to any order, QED corrections below the bottom mass scale $\mu_b$ have not been fully considered even at NLO.
The largest uncertainties in the SM prediction are of parametric origin: $4\%$ from the $B_s$ meson decay constant $f_{B_s}$, $4.3$% from the quark-mixing element $V_{cb}$ [^1], and $1.6$% from the top-quark mass. These uncertainties will reduce as lattice QCD calculations and measurements of SM parameters improve. Non-parametric uncertainties are due to the omission of higher-order corrections $\alpha_s^3,
\alpha_{\rm em}^2, \alpha_s \alpha_{\rm em}$ in the QCD and QED couplings $\alpha_s$ and $\alpha_{\rm em}$, respectively, and also $m_b^2/m_W^2$ from higher-dimension operators in the weak effective Lagrangian. Altogether, the non-parametric uncertainties have been estimated to be about $1.5$% [@Bobeth:2013uxa]. Among these, the renormalization scale dependence of $\overline{{\cal B}}(B_q \to \ell^+\ell^-)$ due to higher-order QED corrections accounts for only $0.3\%$. In view of such extraordinary precision, it is necessary to exclude the existence of unaccounted theoretical effects at the level of 1%.
Although NLO electromagnetic effects above the $b$-quark mass scale $\mu_b$ are completely included in Eq. (\[eq:SM:Br\]), this is not the case for photons with energy or virtuality below this scale. Since the decay involves electrically charged particles in the final state, only a suitably defined decay rate $\Gamma(B_q\to \ell^+\ell^-) +
\Gamma(B_q\to \ell^+\ell^- + n \, \gamma )_{\rm cut}$ including photon radiation and virtual photon corrections is infrared finite and well-defined. Energetic photons are usually vetoed in the experiment and accordingly neglected on the theory side. Soft-photon emission from the final-state leptons is accounted for by experiments [@Aaij:2013aka; @Chatrchyan:2013bka; @Aaij:2017vad]. Initial-state soft radiation has been estimated to be very small based on heavy-hadron chiral perturbation theory [@Aditya:2012im]. The quoted measured branching fraction is corrected for soft emission and actually refers to the non-radiative branching ratio [@Buras:2012ru], as does Eq. . For the purpose of the SM prediction [@Bobeth:2013uxa] it was assumed that other NLO QED corrections below $\mu_b$ can not exceed the natural size of $\alpha_{\rm em}/\pi \sim 0.3$%. However, as we discuss now, the true size of so far neglected QED effects is substantially larger and in fact of the same order as the non-parametric theoretical uncertainty of 1.5%.
The primary challenge of NLO QED computations below $\mu_b$ consists in the reliable computation of non-local matrix elements. For example, a virtual photon connecting the spectator quark with one of the final-state leptons involves the QCD matrix element $$\label{eq:NLO:QED:ME}
\langle 0|\int d^4 x\,
T\{j_{\rm QED}(x), \mathcal{L}_{\Delta B=1}(0) \}
|\bar{B}_q \rangle ,$$ where $j_{\rm QED} = Q_q \bar q\gamma^\mu q$ is the electromagnetic quark current and $\mathcal{L}_{\Delta B=1}$ denotes the (QCD part of the) weak effective Lagrangian for $\Delta B = 1$ transitions. This matrix element bears close resemblance to the hadronic tensor that contains the strong-interaction physics of $B^+\to \ell^+\nu_\ell\gamma$ decay, which is known to be highly non-trivial (for example, Ref. [@Beneke:2011nf]) despite its apparently purely non-hadronic final state.
In the following we focus on the muonic final state $\mu^+\mu^-$. We have analyzed the complete NLO electromagnetic corrections below the bottom mass scale $\mu_b$, counting the muon mass $m_\mu$ and spectator quark mass $m_q$ as $m_\mu\sim m_q \sim
\Lambda_{\rm QCD} \ll m_b$ to organize the result in an expansion in $\Lambda_{\rm QCD}/m_b$. We then find that the electromagnetic correction to the decay amplitude is enhanced by one power of $m_B/\Lambda_{\rm QCD}$ compared to the pure-QCD amplitude. In the following we discuss only this formally dominant power-enhanced contribution, leaving the analysis of the complete QED correction to a separate publication. Note that the standard collinear and soft electromagnetic logarithms belong to these further, non power-enhanced terms, and are therefore not discussed here.
We then find that the leading-order $\bar{B}_q\to \ell^+\ell^-$ decay amplitude plus the electromagnetic correction can be represented as
-0.5cm $$\begin{aligned}
i \mathcal{A} &=& m_\ell f_{B_q} {\cal N}\,C_{10} \,\bar{\ell} \gamma_5 \ell
\nonumber\\
&& +\,\frac{\alpha_{\rm em}}{4\pi} Q_\ell Q_q\,
m_\ell m_B f_{B_q} {\cal N}\,\bar{\ell} (1+\gamma_5) \ell
\times\Bigg\{
\int_0^1 du \,(1-u)\,C_9^{\rm eff}(u m_b^2)\,
\int_0^\infty\frac{d\omega}{\omega}\,\phi_{B+}(\omega)
\left[\ln\frac{m_b\omega}{m_\ell^2}+\ln\frac{u}{1-u}\right]
\nonumber\\
&& -\,Q_\ell C_7^{\rm eff}
\int_0^\infty\frac{d\omega}{\omega}\,\phi_{B+}(\omega)
\left[
\ln^2\frac{m_b\omega}{m_\ell^2}
-2\ln\frac{m_b\omega}{m_\ell^2}+\frac{2\pi^2}{3}
\right]
\Bigg\}+\ldots\,,
\label{eq:mainresult}\end{aligned}$$
where the overall factor $$\mathcal{N} = V_{tb} V^*_{tq} \frac{4 G_F}{\sqrt{2}}
\frac{\alpha_{\rm em}}{4 \pi}$$ contains CKM quark-mixing elements, the Fermi constant $G_F$, and $Q_\ell=-1$, $Q_q=-1/3$ denote the lepton and quark electric charge, respectively. We use the short-hands $\bar{\ell}=
\bar{u}(p_{\ell^-})$, $\ell=v(p_{\ell^+})$ for the external lepton spinors. Omitted terms are power-suppressed. The two terms in the electromagnetic correction in the above equation arise from the four-fermion operator $Q_9 = \frac{\alpha_{\rm em}}{4 \pi}
(\bar{q} \gamma^\mu P_L b)(\bar{\ell} \gamma_\mu\ell)$ and the electric dipole operator $Q_7$ in the effective weak interaction Lagrangian $$\label{eq:eff:Lagr}
{\cal L}_{\Delta B=1}
= \frac{4 G_F}{\sqrt{2}}\,\sum_{i=1}^{10} C_i Q_i + \mbox{h.c.} \,,$$ with the effective operators $Q_i$ as defined in Ref. [@Chetyrkin:1996vx]. The effective short-distance coefficients [@Bobeth:1999mk; @Beneke:2001at] $$\begin{aligned}
&&C_7^{\rm eff} = C_7-\frac{C_3}{3}-\frac{4 C_4}{9}-\frac{20 C_5}{3}
-\frac{80 C_6}{9}
\\[0.1cm]
&&C_9^{\rm eff}(q^2) = C_9+Y(q^2)\end{aligned}$$ account for the quark-loop induced contributions. The relevant Feynman diagrams are shown in Fig. \[fig:diagrams\].
-4.5cm ![Feynman diagrams that contain the power-enhanced electromagnetic correction. Symmetric diagrams with order of vertices on the leptonic line interchanged are not displayed.[]{data-label="fig:diagrams"}](D1 "fig:"){width="18.00000%"}
3.7cm ![Feynman diagrams that contain the power-enhanced electromagnetic correction. Symmetric diagrams with order of vertices on the leptonic line interchanged are not displayed.[]{data-label="fig:diagrams"}](D2 "fig:"){width="25.00000%"}\
![Feynman diagrams that contain the power-enhanced electromagnetic correction. Symmetric diagrams with order of vertices on the leptonic line interchanged are not displayed.[]{data-label="fig:diagrams"}](D3 "fig:"){width="25.00000%"}
An important observation on Eq. (\[eq:mainresult\]) is that the non-perturbative strong-interaction physics is no longer contained in the $B$-meson decay constant $f_{B_q}$ alone. Rather, the exchange of an energetic photon between the lepton pair and the spectator antiquark $\bar q$ probes correlations between the constituents in the $B$ meson separated at large but light-like distances. The corresponding strong-interaction physics is parameterized by the inverse moment of the $B$-meson light-cone distribution amplitude (LCDA) $\lambda_B$, introduced in Ref. [@Beneke:1999br], $$\begin{aligned}
&&\frac{1}{\lambda_B(\mu)}
\equiv \int_0^\infty \frac{d\omega}{\omega} \,\phi_{B+}(\omega, \mu),
\\[0.2cm]
&& \frac{\sigma_n(\mu)}{\lambda_B(\mu)}
\equiv \int_0^\infty \frac{d\omega}{\omega}
\ln^n\frac{\mu_0}{\omega} \,\phi_{B+}(\omega, \mu)\end{aligned}$$ and the first two inverse-logarithmic moments, which we define as in Ref. [@Beneke:2011nf] with fixed $\mu_0 = 1$ GeV. These parameters have frequently appeared in other exclusive $B$-meson decays. In the numerical analysis below we shall adopt [@Beneke:2011nf] $\lambda_B(1~\mbox{GeV}) = (275\pm 75)~\mbox{MeV}$, $\sigma_1(1~\mbox{GeV}) = 1.5\pm 1$, and $\sigma_2(1~\mbox{GeV}) = 3\pm 2$. The non-locality of $\bar{q} b$ annihilation due to the photon interaction removes a suppression factor of the local annihilation process. The enhancement of the electromagnetic correction by a factor $m_B/\Lambda_{\rm QCD}$ in Eq. (\[eq:mainresult\]) arises from $$m_B \int_0^\infty\frac{d\omega}{\omega}\,\phi_{B+}(\omega)
\,\ln^k\omega
\sim \frac{m_B}{\lambda_B}\times \sigma_k\,.$$ There is a further single-logarithmic enhancement of order $\ln m_b\Lambda_{\rm QCD}/m_\mu^2\sim 5$ for the $C_9^{\rm eff}$ term, and even a double-logarithmic enhancement of the $C_7^{\rm eff}$ term.
We obtained Eq. (\[eq:mainresult\]) in two different ways. First, from a standard computation of QED corrections to the four-point amplitude with two external lepton lines, one heavy-quark and one light-quark line, and second, from a method-of-region computation [@Beneke:1997zp] in the framework of soft-collinear effective theory (SCET) [@Bauer:2000yr; @Beneke:2002ph]. The second method is instructive as it reveals the origin of the enhancement from the hard-collinear virtuality $\mathcal{O}(m_b\Lambda_{\rm QCD})$ of the spectator-quark propagator. A further single-logarithmic enhancement arises from the contribution of both hard-collinear and collinear (virtuality $\Lambda_{\rm QCD}^2\sim m_\ell^2$) photon and lepton virtuality. The double logarithm in the $C_7^{\rm eff}$ term is caused by an endpoint-singularity as $u\to 0$ in the hard-collinear and collinear convolution integral for the box diagrams, whereby the hard photon from the electromagnetic dipole operator becomes hard-collinear. The singularity is cancelled by a soft contribution, where the leptons in the final state interact with each other through the exchange of a soft lepton. The relevance of soft-fermion exchange is interesting by itself since it is beyond the standard analysis of logarithmically enhanced terms in QED. We shall therefore return to a full analysis within SCET in a detailed separate paper.
We now proceed to the numerical evaluation of the power-enhanced QED correction. Let us denote $m_B$ times the curly bracket in Eq. (\[eq:mainresult\]) by $\Delta_{\rm QED}$. Since the scalar $\bar \ell\ell$ term in the amplitude $\mathcal{A}$ does not interfere with the pseudoscalar tree-level amplitude, the QED correction can be included in the expression for the tree-level $B_s \to \ell^+\ell^-$ branching fraction [^2], $$\label{eq:SMtree}
\frac{\tau_{B_q}m_{B_q}^3 f_{B_q}^2}{8\pi}\,|\mathcal{N}|^2\,
\frac{m_\ell^2}{m_{B_q}^2}
\sqrt{1-\frac{4m_\ell^2}{m_{B_q}^2}}\,|C_{10}|^2\,,$$ by the substitution $$C_{10} \to C_{10} +
\frac{\alpha_{\rm em}}{4\pi} Q_\ell Q_q \Delta_{\rm QED}\,.$$ We calculate the Wilson coefficients $C_i(\mu_b)$ entering $\Delta_{\rm QED}$ at the scale $\mu_b=5\,$GeV at next-to-next-to-leading logarithmic accuracy in the renormalization-group evolution from the electroweak scale, evaluate the convolution integrals in Eq. (\[eq:mainresult\]) with $m_b=4.8\,$GeV, and express them in terms of $\lambda_B(1\,\mbox{GeV})$, $\sigma_1(1\,\mbox{GeV})$, $\sigma_2(1\,\mbox{GeV})$ specified above. We then find $$\Delta_{\rm QED} = (33 - 119) + i\,(9-23)\qquad (\ell=\mu)\,,$$ where the large range is entirely due to the independent variation of the poorly known parameters of the $B$-meson LCDA. In this result the total effect is reduced by a factor of three by a cancellation between the $C_9^{\rm eff}(q^2)$ and $C_7^{\rm eff}$ term. With $C_{10} = -4.198$, this results in a $(0.3-1.1)\%$ reduction of the muonic $B_s \to \ell^+\ell^-$ branching fraction. We update the SM prediction to $$\begin{aligned}
\label{eq:SMBrnew}
\overline{{\cal B}}(B_s \to \mu^+\mu^-)_{\rm SM}
&=& (3.57 \pm 0.17) \cdot 10^{-9}\,,
\quad\end{aligned}$$ which supersedes the one from Eq. (\[eq:SM:Br\]). To obtain this result we proceeded as in Ref. [@Bobeth:2013uxa] and used the same numerical input except for updated values of the strong coupling $\alpha_s^{(5)}(m_Z) = 0.1181(11)$ and $1/\Gamma_H^{s} = 1.609(10)$ ps [@Patrignani:2016xqp], $f_{B_s} = 228.4(3.7)$ MeV ($N_f=2+1$) [@Aoki:2016frl], $|V_{tb}^* V_{ts}/V_{cb}| = 0.982(1)$ [@Bona:2016dys] and the inclusive determination of $|V_{cb}|=0.04200(64)$ [@Gambino:2016jkc]. The parametric ($\pm 0.167$) and non-parametric non-QED ($\pm 0.043$) uncertainty and the uncertainty from the QED correction (${}^{+0.022}_{-0.030}$) have been added in quadrature. Quite surprisingly, the QED uncertainty (which itself is almost exclusively parametric, from the $B$-meson LCDA) is now almost as large as the non-parametric non-QED uncertainty.
The generation of a scalar $\bar{\ell}\ell$ amplitude in Eq. (\[eq:mainresult\]) leads to further interesting effects. The time-dependent rate asymmetry for $B_s$ decay into a muon pair $\mu^+_\lambda\mu^-_\lambda$ in the $\lambda=L,R$ helicity configuration is given by $$\begin{aligned}
&& \frac{\Gamma(B_s(t)\to \mu^+_\lambda\mu^-_\lambda)-
\Gamma(\bar B_s(t)\to \mu^+_\lambda\mu^-_\lambda)}
{\Gamma(B_s(t)\to \mu^+_\lambda\mu^-_\lambda)+
\Gamma(\bar B_s(t)\to \mu^+_\lambda\mu^-_\lambda)}
\nonumber\\
&&=\,\frac{C_\lambda \cos(\Delta M_{B_s}t)+S_\lambda \sin(\Delta M_{B_s}t)}
{\cosh(y_s t/\tau_{B_s})+\mathcal{A}^\lambda_{\Delta \Gamma}
\sinh(y_s t/\tau_{B_s}) }\,,\end{aligned}$$ where all quantities are defined in Ref. [@DeBruyn:2012wk]. For example, the mass-eigenstate rate asymmetry $A_{\Delta\Gamma}^\lambda$ equals exactly $+1$, if only a pseudo-scalar amplitude exists, and is therefore assumed to be very sensitive to new flavour-changing interactions, with essentially no uncertainty from SM background. We now see that the SM itself generates a small “contamination” of the observable, given by $$\begin{aligned}
\mathcal{A}^\lambda_{\Delta\Gamma} &=&
1-r^2 |\Delta_{\rm QED}|^2 \approx 1- 1.0\cdot 10^{-5}\,,
\\[0.2cm]
C_\lambda&=&-\eta_\lambda\,2r \,\mbox{Re}(\Delta_{\rm QED})
\approx \eta_\lambda\,0.6\%\,,
\\[0.2cm]
S_\lambda&=&2r \,\mbox{Im}(\Delta_{\rm QED})
\approx -0.1\%\,,\end{aligned}$$ where $r\equiv \frac{\alpha_{\rm em}}{4\pi} \frac{Q_\ell Q_q}{C_{10}}$ and $\eta_{L/R} = \pm 1$. Present measurements [@Aaij:2017vad] set only very weak constraints on the deviations of $A_{\Delta\Gamma}^\lambda$ from unity, and $C_\lambda$, $S_\lambda$ have not yet been measured, but the uncertainty in the $B$-meson LCDA is in principle a limiting factor for the precision with which New Physics can be constrained from these observables.
The power-enhanced QED correction reported here may appear also relevant to the leptonic charged $B$-meson decay $B^+\to\ell^+\nu_\ell$, but cancels due to the V–A nature of the charged current. While we discussed only the case $\ell=\mu$ above, the other leptonic final states $\ell=e,\tau$ are also of interest. However, whereas the muon mass is numerically of the order of the strong interaction scale, the much larger mass of the tau lepton, and the much smaller electron mass imply that the results are not exactly the same. We therefore conclude that the systematic study of hitherto neglected electromagnetic corrections to exclusive $B$ decays reveals an unexpectedly complex structure. Its further phenomenological and theoretical implications are currently under investigation.\
We thank H. Patel for helpful communication on Package-X [@Patel:2016fam]. This work is supported by the DFG Sonderforschungsbereich/Transregio 110 “Symmetries and the Emergence of Structure in QCD”.
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[^1]: The determination of $V_{cb}$ from inclusive $b\to c\ell\bar\nu_\ell$ has been used.
[^2]: The given expression refers to the “instantaneous” branching fraction at $t=0$, which differs from the untagged time-integrated branching fraction (\[eq:SM:Br\]) by the factor $(1-y_s^2)/
(1+y_s\mathcal{A}_{\Delta \Gamma})$ [@DeBruyn:2012wk], where $y_s$ is related to the lifetime difference of the two $B_s$ mass eigenstates.
|
---
abstract: 'We present new model of $D=4$ relativistic massive particle with spin and we describe its quantization. The model is obtained by an extension of standard relativistic phase space description of massive spinless particle by adding new topological Souriau-Wess-Zumino term which depends on spin fourvector variable. We describe equivalently our model as given by the free two-twistor action with suitable constraints. An important tool in our derivation is the spin-dependent twistor shift, which modifies standard Penrose incidence relations. The quantization of the model provides the wave function with correct mass and spin eigenvalues.'
---
[**Massive twistor particle with spin** ]{}
[**generated by Souriau-Wess-Zumino term** ]{}
[**and its quantization**]{}
[**Sergey Fedoruk**]{},${}^{\dagger}$[^1] ${}^{\ast}$
${}^{\dagger}$[*Bogoliubov Laboratory of Theoretical Physics, JINR,*]{}\
[*Joliot-Curie 6, 141980 Dubna, Moscow region, Russia*]{}\
[fedoruk@theor.jinr.ru]{}\
${}^{\ast}$[*Institute for Theoretical Physics, University of Wroc[ł]{}aw,*]{}\
[*pl. Maxa Borna 9, 50-204 Wroc[ł]{}aw, Poland*]{}\
[lukier@ift.uni.wroc.pl]{}\
PACS: 11.10.Ef, 11.25.Mj
Keywords: twistors, massive particle, spin
Introduction
============
In order to introduce in geometric way the spin degrees of freedom one has to enlarge the space-time description of relativistic point particles. Well-known introduction of spin degrees of freedom is provided by superspace extension of space-time, with anticommuting Grassmann algebra attached to each space-time point. Other way of introducing the geometric spin degrees of freedom is to consider twistorial particle models, with primary spinorial coordinates. The single twistor space has the degrees of freedom describing massless particles with arbitrary helicity [@PenrM]–[@HugTod]. In order to describe in twistor space the massive particles with arbitrary spin one should consider particle models in two-twistor space [@Hugh]–[@Bette].
The Penrose twistor approach [@Penr1; @PenrM; @Hugh; @Penr2] has been shown to be a powerful tool for the analysis of different point–like and extended objects. In last time there is some renaissance of the twistor methods connected with successful application of twistors in description of amplitudes in (super)Yang-Mills and (super)gravity theories (see, for example, [@Wit]–[@Mas]). It should be added that the twistor approach has been considered mainly for massless (super)particles (see e.g. [@Sor] for approximately complete list of more references on this subject), but its application to massive particles, especially with non–zero spin, were investigated in rather limited number of papers [@Perj; @Bette], [@SV]–[@MezRTown]. The description of particles with a nonconformal mass parameter, and nonvanishing spin requires additional degrees of freedom which has been studied in space-time as well as in the twistorial approach. In space-time formalism one introduces an additional Pauli-Lubanski spin fourvector $w_\mu$ which satisfies the subsidiary conditions [@Tod; @Hugh; @Bette] $$\label{sab-cond}
w_\mu p^\mu=0\,,\qquad w^2\equiv w_\mu w^\mu=-m^2j^2\,,$$ with with relativistic spin-shell described by $j^2$ and fourmomenta satisfying the mass-shell condition $p^2=m^2$. Alternatively, in twistor approach the two-twistor space is required to describe the phase space of massive particle with arbitrary spin, and one construct from two twistors the composite spin fourvector $w_\mu$ satisfying the constraints (\[sab-cond\]).
In our presentation we shall generalize from $D\,{=}\,3$ to $D\,{=}\,4$ the arguments of Mezincescu, Routh and Townsend [@MezRTown], who demonstrated that for $D=3$ massive particle the nonvanishing spin is generated in phase space $(x^\mu,p_\nu)$ by adding the term in the action described by the pullback to the world-line of the following symplectic $D=3$ two-form $$\label{Om-3}
\Omega_2^{(D=3)}=\frac{s}{2(p^2)^{3/2}}\,\epsilon^{\mu\nu\rho}p_\mu dp_\nu\wedge dp_\rho\,,$$ satisfying $d\Omega_2^{(D=3)}=$. It appears that such a term describes in $D\,{=}\,3$ action the Lorentz-Wess-Zumino (LWZ) term $\Omega_1^{(D=3)}$ which is the solution of the equation $\Omega_2^{(D=3)}=d\Omega_1^{(D=3)}$ [@Schon; @MezTown]. Calculating in two-twistor formulation the LWZ term one can see that it generates the twistor shift which modifies standard Penrose incidence relations as follows $$\label{inc-3}
\omega_\alpha^i=x_{\alpha\beta}\lambda^{\beta i}+\frac{s}{m}\,\lambda_{\alpha}^{i}\,.$$ Using modified incidence relations (\[inc-3\]) one can obtain in the twistorial action of $D\,{=}\,3$ massive particle with spin the kinetic term for twistors, which implies standard twistor Poisson brackets (PB). Moreover as it was shown in [@MezRTown], the eigenvalues of the Casimir operators of $D=3$ Poincare algebra corresponds to massive states with the $D\,{=}\,3$ counterpart of spin $s$. Note that twistorial shift in twistorial models is more important for massive particles, because for $D\,{=}\,3,4$ massless particles it does not produce any change of the particle helicities [@SSTV].
In our paper we provide an analogous scheme by introducing in place of (\[Om-3\]) for $D\,{=}\,4$ the symplectic two-form introduced by Souriau [@Sour; @Tod; @Ku] $$\label{Sor-form-in}
\Omega_2^{(D=4)}=\frac{1}{2m^2}\,\epsilon_{\mu\nu\rho\sigma}w^\rho p^\sigma\left(
\frac{1}{m^2}\,dp^\mu \wedge dp^\nu + \frac{1}{w^2}\,dw^\mu \wedge dw^\nu\right)\,,$$ where $w^\mu$ is the Pauli-Lubanski vector satisfying the relations (\[sab-cond\]). In Sect.2 we consider firstly the $D\,{=}\,4$ spinless massive particle and we recall that such a model can be formulated in three equivalent ways (see e.g. [@FFLM])
– by using relativistic phase space description ($x^\mu, p_\mu$)
– by employing mixed space-time/spinor description (Shirafuji formulation [@Shir])
– by using two-twistor framework.\
We obtain that in $D=4$ two-twistor space our model is described by free action with added six constraints: two related with mass-shell condition, three describing vanishing spin and sixth introducing vanishing $U(1)$ charge.
In Sect.3 we add in $D=4$ space-time formulation the Souriau-Wess-Zumino (SWZ) topological term which depends on the spin four-vector $w_\mu$ (see (\[sab-cond\])). After passing to the spinorial description one can calculate the SWZ term by the pullback to the world-line of the Souriau symplectic two-form (\[Sor-form-in\]). Subsequently, using spin dependent twistor shift we obtain the model depending on two-twistor coordinates and auxiliary spin three-vector which span the coordinates of two-sphere. We shall recall how to derive from topological action such semi-dynamical spin variables, which satisfy $SU(2)$ PB bracket relations. In two-twistor description the model is described by free bilinear action with four first class and two second class constraints imposed by Lagrange multiplier method, what leaves eight unconstrained physical degrees of freedom. Further, in Sect.4, using the two-twistor formulation of our particle model, we obtain the relativistic wave functions with mass and properly quantized spin values. In final Sect.5 we summarize main results and point out some possible generalizations of presented scheme.
Massive spinless particle
=========================
The three equivalent descriptions of massive spinless particle are known but we present them here in order to prepare the ground for the generalization in Sect.3 to the case of the massive particle with spin.
Relativistic phase space formulation of massive spinless particle is defined by well-known action $$\label{act-st-0}
\tilde S_1=\int \,d\tau
\Big[p_{\mu}\,\dot x^{\mu}+
e\left(p_{\mu}p^{\mu}-m^2 \right)\Big]\,.$$ Here, $x^{\mu}(\tau)$, $\mu=0,1,2,3$ are the coordinates of position, $\dot x^{\mu}=dx^{\mu}/d\tau$ and $p_{\mu}$ is fourvector of momenta. We use the metric with plus time signature, $\eta_{\mu\nu}={\rm diag}(+---)$.
In order to pass to mixed space-time/spinorial Shirafuji formulation we should use the Cartan-Penrose formula expressing the relativistic fourmomenta by a pair of Weyl commuting spinors ($k=1,2$) [^2] $$\label{P-spin-expr}
p_{\alpha\dot\alpha} = \lambda^k_{\alpha}\bar\lambda_{\dot\alpha k}\,,$$ where $$\label{P-spin-expr1}
\lambda^k_{\alpha}=\left(\lambda^1_{\alpha},\lambda^2_{\alpha} \right),\qquad
\bar\lambda_{\dot\alpha k}=(\overline{\lambda^k_{\alpha}})=\left(\bar\lambda_{\dot\alpha 1},\bar\lambda_{\dot\alpha 2} \right).$$ Massive spinless particle dynamics is described by the extension of Shirafuji approach [@Shir] $$\label{act-shir-0}
\tilde S_2=\int \,d\tau
\Big[\lambda^k_{\alpha}\bar\lambda_{\dot\beta k}\,\dot x^{\dot\beta\alpha}+
g\left(\lambda^{\alpha k}\lambda_{\alpha k}-2M \right)+
\bar g\left(\bar\lambda^{\dot\alpha k}\bar\lambda_{\dot\alpha k}-2\bar M \right)
\Big]\,.$$ where $x^{\dot\beta\alpha}=\frac{1}{\sqrt{2}}\,\tilde\sigma_{\mu}^{\dot\beta\alpha}x^{\mu}$ and $M$ is a complexified mass parameter. In action (\[act-shir-0\]) there are incorporated the mass-shell constraints [^3] $$\label{constr-lambda}
\lambda^{\alpha i}\lambda_{\alpha k}=M \,\delta^i_k \,,\qquad
\bar\lambda^{\dot\alpha i}\bar\lambda_{\dot\alpha k}=\bar M \,\delta^i_k$$ or equivalently $$\label{constr-lambda1}
\lambda^{\alpha k}\lambda_{\beta k}=M \,\delta^\alpha_\beta \,,\qquad
\bar\lambda^{\dot\alpha k}\bar\lambda_{\dot\beta k}=\bar M \,\delta^{\dot\alpha}_{\dot\beta}\,.$$ Due to the constrains (\[constr-lambda\]) we have the following real mass-shell condition ($p^{\dot\beta\alpha}=\epsilon^{\alpha\gamma}\epsilon^{\dot\beta\dot\delta}p_{\gamma\dot\delta}$) $$\label{P-mass}
p^{\dot\beta\alpha}p_{\alpha\dot\beta} = 2\,|M|^2$$ and comparing with (\[act-st-0\]) we get $$\label{m-M}
m = \sqrt{2}\,|M|\,.$$
The pair of spinors $\lambda^k_{\alpha}$, $\bar\lambda_{\dot\alpha k}$ describe one-half of two-twistor components. Remaining twistorial components are defined by the Penrose incidence relations (see e.g. [@Penr1; @PenrM; @Penr2]) $$\label{mu-spin0}
\mu^{\dot\alpha k}=x^{\dot\alpha\beta}\lambda^k_{\beta}\,,\qquad
\bar\mu^{\alpha}_{k}=\bar\lambda_{\dot\beta k}x^{\dot\beta\alpha}\,.$$ The relations (\[P-spin-expr\]) and (\[mu-spin0\]) link the Poisson brackets (PB) of space-time and twistor space approaches. Namely, when the relations (\[mu-spin0\]) are satisfied then $$\label{sim-str-0}
p_\mu\,\dot x^{\mu}=
\lambda^k_{\alpha}\bar\lambda_{\dot\beta k}\,\dot x^{\dot\beta\alpha}=
\lambda^k_{\alpha}\dot{\bar\mu}^{\alpha}_{k}+\bar\lambda_{\dot\alpha k}\dot\mu^{\dot\alpha k} + \mbox{(total derivative)}$$ and we get the kinematic terms which lead to canonical PB in relativistic phase space as well as in two-twistor space.
If space-time coordinates are real twistor incident relations (\[mu-spin0\]) lead to the following conditions $$\label{cond-0}
\lambda^i_{\alpha}\bar\mu^{\alpha}_{k}-
\bar\lambda_{\dot\alpha k}\mu^{\dot\alpha i}=0\,,$$ which should be imposed as the constraints. Thus, after using (\[sim-str-0\]) the pure twistor formulation is described by the action (see also [@MezRTown]) $$\label{act-tw-0}
\tilde S_3=\int \,d\tau
\Big[\lambda^k_{\alpha}\dot{\bar\mu}^{\alpha}_{k}+\bar\lambda_{\dot\alpha k}\dot\mu^{\dot\alpha k}+
gT+\bar g\bar T +\Lambda^r S^r +\Lambda S
\Big]$$ incorporating the mass constraints (see also (\[constr-lambda\])) $$\label{mass-constr}
T\equiv\lambda^{\alpha k}\lambda_{\alpha k}-2M\approx0\,,\qquad
\bar T\equiv\bar\lambda^{\dot\alpha k}\bar\lambda_{\dot\alpha k}-2\bar M\approx0$$ and the $U(2)$ constraints $$\begin{aligned}
\label{Sr-0}
S^r&\equiv&-{\textstyle\frac{i}{2}}\left(\lambda^i_{\alpha}\bar\mu^{\alpha}_{k}-
\bar\lambda_{\dot\alpha k}\mu^{\dot\alpha i}\right)(\tau^r)_i{}^k\approx0\,,\qquad r=1,2,3\\[6pt]
S&\equiv&i\left(\lambda^i_{\alpha}\bar\mu^{\alpha}_{i}- \bar\lambda_{\dot\alpha i}\mu^{\dot\alpha i}\right)\approx0\,,\label{S-0}\end{aligned}$$ which are the traceless and trace parts of the conditions (\[cond-0\]) (in (\[Sr-0\]) the $2\times 2$ matrices $(\tau^r)_i{}^k$, $i,k=1,2$, $r=1,2,3$ are the usual Pauli matrices).
The action (\[act-tw-0\]) yields canonical twistor Poisson brackets $$\label{PB-tw-0}
\{ \bar\mu^{\alpha}_{k}, \lambda^j_{\beta} \}_{{}_P}=\delta^{\alpha}_{\beta}\delta_k^j\,,\qquad
\{ \mu^{\dot\alpha i} ,\bar\lambda_{\dot\beta k}\}_{{}_P}=\delta^{\dot\alpha}_{\dot\beta}\delta^k_j,.$$ Then, nonvanishing Poisson brackets of the constraints (\[mass-constr\]), (\[Sr-0\]), (\[S-0\]) are $$\label{PB-constr-0}
\{ S^p, S^r \}_{{}_P}=\epsilon^{prs}S^s\,,$$ $$\label{PB-constr-0a}
\{ S ,T\}_{{}_P}=2iT+4iM\,,\quad \{ S ,\bar T\}_{{}_P}=-2i\bar T-4i\bar M\,,$$ where the constraints ($S^p$, $S$) describe $U(2)$ PB algebra. One can check easily that we can choose four real constraints $S^r$, $(\bar MT+M\bar T)$ as first class constraints whereas two real constraints $S$ and $i(\bar MT-M\bar T)$ are second class. We get therefore six unconstrained degrees of freedom what coincides with number of degrees of freedom in standard space-time formulation (\[act-st-0\]) of massive particle.
In twistor formulation the Poincare generators $p_\mu$ and $m_{\mu\nu}=x_\mu p_\nu-x_\nu p_\mu$ are represented by the expressions (\[P-spin-expr\]) and $$\label{M-tw}
m_{\mu\nu}=-\sigma_{\mu\nu}^{\alpha\beta}m_{\alpha\beta}+\tilde\sigma_{\mu\nu}^{\dot\alpha\dot\beta}\bar m_{\dot\alpha\dot\beta}\,,\qquad\qquad
m_{\alpha\beta}=\lambda^k_{(\alpha}\bar\mu_{\beta)k}\,,\qquad \bar m_{\dot\alpha\dot\beta}=\bar\lambda_{(\dot\alpha k}\mu_{\dot\beta)}^{ k}\,.$$ Then, Pauli-Lubanski vector $w_\mu=\frac12\,\epsilon_{\mu\nu\lambda\rho}p^\nu m^{\lambda\rho}$ has the following twistor representation $$\label{W-tw}
w_{\alpha\dot\alpha}=S^r u^r_{\alpha\dot\alpha}\,,$$ where $S^r$ are defined by (\[Sr-0\]) and (see e.g. [@FedZ2]) $$\label{u-expr}
u^r_{\alpha\dot\alpha} = \lambda^i_{\alpha}(\tau^r)_i{}^k\bar\lambda_{\dot\alpha k}\,.$$ Due to equation (\[m-M\]) and the constraints (\[mass-constr\]) the vectors (\[u-expr\]) satisfy $$\label{u-norm}
u^r_{\mu}u^s{}^{\mu} = - m^2\delta^{rs}\,.$$ Therefore due to the constraints (\[Sr-0\]) and formulae (\[W-tw\])-(\[u-norm\]) in consistency with (\[sab-cond\]) we have $$\label{W2-0}
p^{\mu}w_{\mu} =0\,,\qquad w^{\mu}w_{\mu} = - m^2S^{r}S^{r}\,,$$ $$\label{S2-0}
S^{r}S^{r}=j^2\,.$$ In conclusion the spin of the massive particle described by the twistor action (\[act-tw-0\]) vanishes, i.e. we should put $j=0$.
Massive particle with spin and twistor shift
============================================
We define $D{=}4$ massive spin particle in space-time formulation with help of the action $$\label{act-st-s}
S_1=\tilde S_1 + \int \Omega_1^{(D=4)}+ \int d\tau\left[l_1(p^{\mu}w_{\mu})+l_2(w^{\mu}w_{\mu} + m^2j^2) \right]\,,$$ where first term $\tilde S_1$ is given by (\[act-st-0\]), one-form $\Omega_1^{(D=4)}$ is defined by Souriau symplectic two-form (\[Sor-form-in\]) as follows $$\label{Sor-form}
\Omega_2^{(D=4)}=d\Omega_1^{(D=4)}\,,$$ and the constraints on Pauli-Lubanski four-vector $w^\mu$ are imposed by Lagrange multipliers.
Using the expressions (\[P-spin-expr\]), (\[W-tw\]) and the property that $M$, $\bar M$ are constants we obtain the following twistorial expression for Souriau two-form $$\label{Sor-tw}
\Omega_2^{(D=4)}=-{\textstyle\frac{i}{2M\bar M}}\,S^r (\tau^r)_i{}^k \Big(
\bar M\,d\lambda^{\alpha i} \wedge d\lambda_{\alpha k} + M\,d\bar\lambda^{\dot\alpha i} \wedge d\bar\lambda_{\dot\alpha k}\Big)\,,$$ where the three-vector $S^r$ satisfies the constraint (\[S2-0\]).
We recall here that in the theory of massive relativistic free fields with spin the Pauli-Lubanski four-vector satisfies the relations (\[W2-0\]) with $s^r$ described by the nondynamical matrix realization of $SU(2)$ algebra. Further, in order to obtain that $\Omega_2^{(D=4)}$ in relation (\[Sor-tw\]) satisfies the condition $d\Omega_2^{(D=4)}=0$ we shall postulate that $$\label{S-dot}
\dot S^r=0\qquad \rightarrow\qquad S^r=s^r\,,\quad s^rs^r=j^2$$ with the variables $s^r\in \mathbb{S}^2$ as classical counterparts of quantum spin components endowed with $SU(2)$ PB relation $$\label{S-br-s}
\{ s^p, s^r \}_{{}_P}=\epsilon^{prq}s^q\,.$$ Using (\[S-dot\]) one sees easily that Liouville one-form $\Omega_1^{(D=4)}$ satisfying (\[Sor-form\]) takes the form $$\label{Lio-tw}
\Omega_1^{(D=4)}=-{\textstyle\frac{i}{2M\bar M}}\,s^r (\tau^r)_i{}^k \Big(
\bar M\,\lambda^{\alpha i} d\lambda_{\alpha k} + M\,\bar\lambda^{\dot\alpha i} d\bar\lambda_{\dot\alpha k}\Big)$$ and the action (\[act-st-s\]) becomes the following Shirafuji-like action $$\label{act-shir-s}
\begin{array}{rcl}
S_2&=&{\displaystyle \int \,d\tau
\Big[\lambda^k_{\alpha}\bar\lambda_{\dot\alpha k}\,\dot x^{\dot\alpha\alpha}+
g\left(\lambda^{\alpha k}\lambda_{\alpha k}-2M \right)+
\bar g\left(\bar\lambda^{\dot\alpha k}\bar\lambda_{\dot\alpha k}-2\bar M \right)
\Big]}\\[7pt]
&& {\textstyle -\frac{i}{2M\bar M}} {\displaystyle\int} d\tau\,
s^r (\tau^r)_i{}^k \Big(
\bar M\,\lambda^{\alpha i} \dot\lambda_{\alpha k} + M\,\bar\lambda^{\dot\alpha i} \dot{\bar\lambda}_{\dot\alpha k}\Big)\,.
\end{array}$$ It appears that due to relation (\[W-tw\]) the constraint $p^{\mu}w_{\mu}=0$ is valid as identity, thus the action (\[act-shir-s\]) becomes the sum of the action (\[act-shir-0\]) and the twistorial Souriau-Wess-Zumino topological term, represented by second integral in (\[act-shir-s\]).
It should be stressed that the postulated PB relations (\[S-br-s\]) can be derived from the dynamical formulation if we supplement the action (\[act-shir-s\]) with the following topological (Chern-Simons) coupling term (see e.g. [@HoTo]–[@FedIvLech]) $$\label{de-act-shir-s}
\triangle \,S_2=\int \,d\tau
\Big[\mathcal{A}^r(s)\dot s^r+
l\left(s^r s^r-j^2 \right)
\Big]\,,$$ where three-potential $\mathcal{A}^r(S)$ is such that $$\label{strengts}
{\mathcal F}^{rq}=\partial^r\mathcal{A}^q-\partial^q\mathcal{A}^r=-j\epsilon^{rqt}s^t/|s|^3\,.$$ In order to derive the conditions $\dot s^r=0$ one should then pass to twistor formulation and fix the local $SU(2)$ gauge which are generated by first class constraints defined below (see (\[Sr-s\])).
Let us eliminate the space-time variables $x^\mu$ and pass to pure twistorial formulation in two-twistor space. This requires to define second twistorial spinors. As first attempt one can use the relations (\[mu-spin0\]) as defining the second pair of Weyl twistors $\mu^{\dot\alpha k}$, but if we use the spinor variables $\lambda^k_{\beta}$, $\mu^{\dot\alpha k}$ the terms with derivatives in the action (\[act-shir-s\]) $$\label{sim-str-mixs}
\lambda^k_{\alpha}\bar\lambda_{\dot\beta k}\,\dot x^{\dot\beta\alpha}-{\textstyle\frac{i}{2M\bar M}} \,
s^r (\tau^r)_i{}^k \Big(
\bar M\,\lambda^{\alpha i} \dot\lambda_{\alpha k} + M\,\bar\lambda^{\dot\alpha i} \dot{\bar\lambda}_{\dot\alpha k}\Big)$$ take the form $$\label{sim-str-s1}
\lambda^k_{\alpha}\dot{\bar\mu}^{\alpha}_{k}+\bar\lambda_{\dot\alpha k}\dot\mu^{\dot\alpha k}
-{\textstyle\frac{i}{2M\bar M}} \, s^r (\tau^r)_i{}^k \Big(
\bar M\,\lambda^{\alpha i} \dot\lambda_{\alpha k} + M\,\bar\lambda^{\dot\alpha i} \dot{\bar\lambda}_{\dot\alpha k}\Big)
+ \mbox{(total derivative)}\,.$$ The kinetic terms given by (\[sim-str-s1\]) show that the variables $\lambda^k_{\beta}$, $\mu^{\dot\alpha k}$ and their complex conjugated do not satisfy the canonical twistorial Poisson brackets.
In order to obtain the canonical twistorial PB we should redefine the half of twistor variables by the following modified incidence relations $$\label{mu-spin-s}
\begin{array}{rcl}
\omega^{\dot\alpha k}&=&\mu^{\dot\alpha k}+{\textstyle\frac{i}{2\bar M}} \,s^r (\tau^r)_j{}^k \bar\lambda^{\dot\alpha j}=
x^{\dot\alpha\beta}\lambda^k_{\beta}+{\textstyle\frac{i}{2\bar M}} \,s^r (\tau^r)_j{}^k \bar\lambda^{\dot\alpha j}\,,\\[6pt]
\bar\omega^{\alpha}_{k}&=&\bar\mu^{\alpha}_{k}+{\textstyle\frac{i}{2M}} \,s^r (\tau^r)_k{}^j \lambda^{\alpha}_{j}=
\bar\lambda_{\dot\beta k}x^{\dot\beta\alpha}+{\textstyle\frac{i}{2M}} \,s^r (\tau^r)_k{}^j \lambda^{\alpha}_{j}\,.
\end{array}$$ The formulae (\[mu-spin-s\]) describe the spin-dependent twistor shift from Weyl spinors $\lambda^k_{\alpha}$, $\mu^{\dot\alpha k}$ to $\lambda^k_{\alpha}$, $\omega^{\dot\alpha k}$. It appears that subsequently the kinetic terms (\[sim-str-mixs\]) take (even without (\[S-dot\])) the standard form $$\label{sim-str-s}
\lambda^k_{\alpha}\dot{\bar\omega}^{\alpha}_{k}+\bar\lambda_{\dot\alpha k}\dot\omega^{\dot\alpha k}
+ \mbox{(total derivative)}\,.$$ We see that the variables ($\lambda^k_{\beta}$, $\omega^{\dot\alpha k}$) and ($\bar\lambda_{\dot\beta k}$, $\bar\omega^{\alpha}_{k}$) are the canonical twistor variables for particle with spin and they are obtained by the twistor shift applied to standard Penrose incidence relations for spinless particle (compare (\[mu-spin-s\]) with (\[mu-spin0\])).
If the space-time coordinates are real, the twistor incidence relations (\[mu-spin-s\]) lead to the following conditions $$\label{cond-s}
\lambda^i_{\alpha}\bar\omega^{\alpha}_{k}-
\bar\lambda_{\dot\alpha k}\omega^{\dot\alpha i}=-is^r(\tau^r)_k{}^i\,,$$ which generalize the constraints (\[Sr-0\]) in the presence of nonvanishing spin variables $s^r$. Thus, in two-twistor formulation we have the constraints (\[mass-constr\]) and the modified constraints (\[Sr-0\])-(\[S-0\]) [^4] $$\begin{aligned}
\label{Sr-s}
\mathcal{V}^r\equiv V^r+s^r&\equiv&-{\textstyle\frac{i}{2}}\left(\lambda^i_{\alpha}\bar\omega^{\alpha}_{k}-
\bar\lambda_{\dot\alpha k}\omega^{\dot\alpha i}\right)(\tau^r)_i{}^k+s^r\approx0\,,\qquad r=1,2,3\\[6pt]
V&\equiv&i\left(\lambda^i_{\alpha}\bar\omega^{\alpha}_{i}- \bar\lambda_{\dot\alpha i}\omega^{\dot\alpha i}\right)\approx0\,,\label{S-s}\end{aligned}$$ which traceless and trace parts of the conditions (\[cond-s\]) supplemented by the condition (\[S2-0\]). Thus, pure twistor formulation is described by the action $$\label{act-tw-s}
S_3=\int \,d\tau
\Big[\lambda^k_{\alpha}\dot{\bar\omega}^{\alpha}_{k}+\bar\lambda_{\dot\alpha k}\dot\omega^{\dot\alpha k}+
gT+\bar g\bar T +\Lambda^r (V^r+s^r) +\Lambda V
\Big]\,.$$ Semi-dynamical variables $s^r$ satisfy the PB (\[S-br-s\]) and can be described by the action (\[de-act-shir-s\]); all the constraints in the model are introduced by using Lagrange multipliers.
In the formulation (\[act-tw-s\]) of our model Poincare generators are given by the expressions (\[P-spin-expr\]) and Lorentz generators are $$\label{M-tw-s}
M_{\alpha\beta}=\lambda^k_{(\alpha}\bar\omega_{\beta)k}\,,\qquad \bar M_{\dot\alpha\dot\beta}=\bar\lambda_{(\dot\alpha k}\omega_{\dot\beta)}^{ k}\,.$$ The Pauli-Lubanski vector $W_{\alpha\dot\alpha}=ip_{\alpha}^{\dot\beta} \bar M_{\dot\alpha\dot\beta}-
ip_{\dot\alpha}^{\beta} M_{\alpha\beta}$ has the following twistor representation $$\label{W-tw-s}
W_{\alpha\dot\alpha}=V^r u^r_{\alpha\dot\alpha}\,,$$ where $V^r$ are defined in (\[Sr-s\]) and $u^r_{\alpha\dot\alpha}$ by (\[u-expr\]). Further due to the constraints (\[Sr-s\]) and relation (\[S2-0\]) we get $$\label{W2-s}
W^{\mu}W_{\mu} = - m^2(V^{r}V^{r})=- m^2(s^{r}s^{r})=- m^2s^{2}\,.$$
The action (\[act-tw-s\]) yields the canonical twistor Poisson brackets $$\label{PB-tw-s}
\{ \bar\omega^{\alpha}_{k}, \lambda^j_{\beta} \}_{{}_P}=\delta^{\alpha}_{\beta}\delta_k^j\,,\qquad
\{ \omega^{\dot\alpha i} ,\bar\lambda_{\dot\beta k}\}_{{}_P}=\delta^{\dot\alpha}_{\dot\beta}\delta^k_j\,.$$ The twistorial PB of the quantities $V^r$ are the same as these for $s^r$ in (\[S-br-s\]) $$\label{PB-V-s}
\{ V^p, V^r \}_{{}_P}=\epsilon^{prq}V^q$$ what will provide the relations (\[Sr-s\]) as first class constraints. Because twistor coordinates and variables $s^r$ are kinematically independent, nonvanishing Poisson brackets of all constraints (\[mass-constr\]), (\[Sr-s\]), (\[S-s\]) are the following $$\label{PB-cV-s}
\{ \mathcal{V}^p,\mathcal{V}^r \}_{{}_P}=\epsilon^{prq}\mathcal{V}^q\,,$$ $$\label{PB-constr-s}
\{ V ,T\}_{{}_P}=2iT+4iM\,,\quad \{ V ,\bar T\}_{{}_P}=-2i\bar T-4i\bar M\,.$$ We see that in present model four constraints are first class: three constraints $\mathcal{V}^r$ and the constraint $(\bar MT+M\bar T)$. Other two constraints $V$ and $i(\bar MT-M\bar T)$ are second class. In comparison with spinless case, we have additional two degrees of freedom in $s^r$, describing spin degrees of freedom and the number of unconstrained degrees is $18-10=8$.
Quantization and field twistor transform
========================================
We obtained the system, which is described in phase space by the variables $\lambda^j_{\alpha}$, $\bar\lambda_{\dot\alpha k}$, $\bar\omega^{\alpha}_{k}$, $\omega^{\dot\alpha i}$, $s^r$, with canonical brackets (\[PB-tw-s\]), (\[S-br-s\]) and the constraints $T$, $\bar T$ (see (\[mass-constr\])), $\mathcal{V}^r$ (see (\[Sr-s\])) and $V$ (see (\[S-s\])). The constraints $V$ and $i(\bar MT-M\bar T)$ are second class. We shall introduce the gauge fixing condition $$\label{gauge}
G=\lambda^i_{\alpha}\bar\omega^{\alpha}_{i}+ \bar\lambda_{\dot\alpha i}\omega^{\dot\alpha i}\approx0$$ for the local gauge transformations generated by the constraint $\bar MT+M\bar T$, i.e. we get second pair of second class constraints. After introducing Dirac bracket for the second class constraints ($V$, $i(\bar MT-M\bar T)$), ($\bar MT+M\bar T$, $G$) will should only impose three first class constraints $\mathcal{V}^r$.
Nonvanishing PBs of the constraint (\[gauge\]) are $$\label{PB-G-s1}
\{ G ,T\}_{{}_P}=2T+4M\,,\quad \{ G ,\bar T\}_{{}_P}=2\bar T+4\bar M\,.$$ Then, the Dirac brackets (DB) for second class constraints $V$, $G$ and $$\label{F-s1}
F_1= \bar MT+M\bar T\,,\quad F_2=i(\bar MT-M\bar T)$$ are given by formula $$\label{DB}
\!\!\!\!\!\!\begin{array}{c}
\{ A,B \}_{{}_D}=\{ A,B \}_{{}_P}+ \\[5pt]
{\textstyle\frac{1}{8M\bar M}}\Big[
\{ A,G \}_{{}_P}\{ F_1,B \}_{{}_P}-\{ A,F_1 \}_{{}_P}\{ G,B \}_{{}_P}
-\{ A,V \}_{{}_P}\{ F_2,B \}_{{}_P}+\{ A,F_2 \}_{{}_P}\{ V,B \}_{{}_P}
\Big]\,.
\end{array}$$ The DBs for twistor spinor components take the form $$\label{DB-tw-l}
\{ \lambda^k_{\alpha}, \lambda^j_{\beta} \}_{{}_D}=
\{ \bar\lambda_{\dot\alpha k} ,\bar\lambda_{\dot\beta j}\}_{{}_D}=
\{ \lambda^k_{\alpha} ,\bar\lambda_{\dot\beta j}\}_{{}_D}=0\,,$$ $$\label{DB-tw-2}
\{ \bar\omega^{\alpha}_{k}, \lambda^j_{\beta} \}_{{}_D}=\delta^{\alpha}_{\beta}\delta_k^j+
{\textstyle\frac{1}{2M}}\lambda^{\alpha}_{k}\lambda^j_{\beta}\,,\qquad
\{ \omega^{\dot\alpha k} ,\bar\lambda_{\dot\beta j}\}_{{}_D}=\delta^{\dot\alpha}_{\dot\beta}\delta^k_j-
{\textstyle\frac{1}{2\bar M}}\bar\lambda^{\dot\alpha k} \bar\lambda_{\dot\beta j}\,,$$ $$\label{DB-tw-3}
\{ \omega^{\dot\alpha k}, \lambda^j_{\beta} \}_{{}_D}=0\,,\qquad
\{ \bar\omega^{\alpha}_{k},\bar\lambda_{\dot\beta j}\}_{{}_D}=0\,,$$ $$\label{DB-tw-4}
\{ \bar\omega^{\alpha}_{k}, \bar\omega^{\beta}_{j} \}_{{}_D}=-{\textstyle\frac{1}{M}}
\left(\lambda^{\alpha}_{k}\bar\omega^{\beta}_{j}-\lambda_j^{\beta} \bar\omega^{\alpha}_{k}\right)\,,\qquad
\{ \omega^{\dot\alpha k} ,\omega^{\dot\beta j}\}_{{}_D}=
{\textstyle\frac{1}{\bar M}}\left(\bar\lambda^{\dot\alpha k}\omega^{\dot\beta j}- \bar\lambda^{\dot\beta j}\omega^{\dot\alpha k}\right),$$ $$\label{DB-tw-5}
\{ \bar\omega^{\alpha}_{k}, \omega^{\dot\beta j}\}_{{}_D}=0\,.$$
Further we consider $(\lambda,\bar\lambda)$-coordinate representation. In such spinorial Schrödinger representation for the commutator algebra obtained by quantization of DB (\[DB-tw-l\])-(\[DB-tw-5\]) the spinorial momentum operators under suitable ordering ($\lambda$’s on the left, $\omega$’s on the right) are realized in the following way $$\label{op-om}
\hat{\bar\omega}{}^{\alpha}_{k}=i\frac{\partial}{\partial\lambda_{\alpha}^{k}}+
\frac{i}{2M}\,\lambda_{\alpha}^{k}\,\lambda_{\beta}^{j} \frac{\partial}{\partial\lambda_{\beta}^{j}}\,,\qquad
\hat{\omega}{}^{\dot\alpha k}=i\frac{\partial}{\partial\bar\lambda_{\dot\alpha k}}-
\frac{i}{2\bar M}\,\bar\lambda_{\dot\alpha k}\,\bar\lambda_{\dot\beta j} \frac{\partial}{\partial\bar\lambda_{\dot\beta j}}\,.$$ It is important that second terms in the operators (\[op-om\]) do not contribute to the realization of quantum counterpart $\hat V^r$ of the quantities $V^r$ (see (\[Sr-s\])): $$\label{S-qu}
D^r\equiv \hat V^r={\frac{1}{2}}\left(\lambda^i_{\alpha}\frac{\partial}{\partial\lambda_{\alpha}^{k}}-
\bar\lambda_{\dot\alpha k}\frac{\partial}{\partial\bar\lambda_{\dot\alpha i}}\right)(\tau^r)_i{}^k\,.$$ After quantization $s^r \to \hat s^r$ of the classical PB algebra (\[S-br-s\]) we get the $SU(2)$ algebra $$\label{S-br-qu}
[ \hat s^p, \hat s^r ]=i\epsilon^{prq}\hat s^q\,,$$ with classical constraint (\[S2-0\]) becoming an operator identity $$\label{ss-s2-qu-cl}
\hat s^r\hat s^r=j^2\,.$$ Because the quantum constraint (\[ss-s2-qu-cl\]) describe the eigenvalue condition of $SU(2)$ Casimir operator, for the unitary finite-dimensional representations of spin algebra (\[S-br-qu\]) the value of $j^2$ are quantized in known way $$\label{ss-s2-qu}
j^2=J(J+1)\,,$$ where $J$ is a non-negative half-integer number, i.e. $2J\in \mathbb{N}$.
For fixed $J$ the operators $\hat s^r$ are realized as $(2J+1)\times (2J+1)$ matrices.[^5] Therefore, twistor wave function of massive particle of spin $J$ has $(2J+1)$ components which are functions of $\lambda^i_{\alpha}$, $\bar\lambda_{\dot\alpha i}$, constrained by strong conditions (\[constr-lambda\]). Because $\hat s^r\hat s^r$ commutes with $\hat s^3$, the wave function for fixed spin $J$ still depends on eigenvalues $\mathcal{J}=(-J,-J+1,\ldots,J-1,J)$ of the spin projection $\hat s^3$. The wave function ($\lambda\equiv \lambda_\alpha^i$, $\bar\lambda=\bar\lambda_{\dot \alpha i}$) $$\label{wf-qu}
\Psi^{(J)}_{\mathcal{J}}=\Psi^{(J)}_{\mathcal{J}}(\lambda,\bar\lambda)\,,$$ satisfies the matrix equations $$\label{Sr-qu}
(D^r+\hat S^r)\Psi^{(J)}=0\,,$$ which is the quantum counterpart of the first class constraints (\[Sr-s\]). We get the equations $$\label{Sr-qu-v}
D^rD^r\,\Psi^{(J)}_{\mathcal{J}}=J(J+1)\,\Psi^{(J)}_{\mathcal{J}}\,,\qquad D^3\,\Psi^{(J)}_{\mathcal{J}}=-\mathcal{J}\,\Psi^{(J)}_{\mathcal{J}}\,.$$
From (\[S-qu\]) follows that $D^r$ are the $SU(2)$ generators acting on indices $i,k$ of twistor spinors $\lambda^i_{\alpha}$, $\bar\lambda_{\dot\alpha k}$ and $\hat s^r$ are the $SU(2)$ $(2J+1)\times (2J+1)$ matrix representation acting on index $\mathcal{J}$ of twistor wave function $\Psi^{(J)}_{\mathcal{J}}$. The formula (\[Sr-qu\]) links the parameters of both transformations and provide the following transformations of the twistor wave function under $SU(2)$ local transformations ($\lambda^\prime{}^i_{\alpha}=h^i_k\lambda^k_{\alpha}$; $h\in SU(2)$): $$\label{su2-tr}
\Psi^{\prime(J)}_{\,\,\mathcal{J}}(\lambda^\prime)=\mathbf{D}^{(J)}_{\mathcal{J}\mathcal{K}}(h)\,\Psi^{(J)}_{\mathcal{K}}(\lambda)\,,$$ where $\mathbf{D}^{(J)}_{\mathcal{J}\mathcal{K}}(h)$ is the matrix of irreducible $SU(2)$ representation of weight $J$. We can represent equivalently the index ${\mathcal{J}}=-J,-J+1,\ldots,J$ as obtained by symmetrized $2J$ two-component spinor indices $i,j,k,...$ describing fundamental representation of the $SU(2)$ algebra (\[S-br-qu\]) and we get the twistor wave function as symmetric multispinor wave function $\Psi^{(J)}_{\mathcal{J}}(\lambda)=\Psi^{(J)}_{(i_1...i_{2J})}(\lambda)$.
The space-time fields are obtained from twistor fields (\[wf-qu\]) by integral transform containing massive generalization of field twistor transform [@PenrM; @Penr2; @GaHoTo; @FedZ2; @Fed]. Such transform is obtained if we construct $SU(2)$ invariant quantities by contraction of the twistor fields (\[wf-qu\]) with symmetrized multispinor indices $(i_1...i_{2J})$ with $\lambda^i_{\alpha}$, $\bar\lambda_{\dot\alpha i}$ and performing integral with $SU(2)$-invariant measure with build-in mass-shell condition $$\label{meas}
d\mu_6(\lambda,\bar\lambda)=d^4 \lambda d^4\, \bar\lambda\, \delta(\lambda^{\alpha k}\lambda_{\alpha k}-2M)\,
\delta(\bar\lambda^{\dot\alpha k}\bar\lambda_{\dot\alpha k}-2\bar M)$$ We use the Fourier transform with exponent $e^{ix^\mu p_\mu}$ containing the four-momentum which is expressed by bilinear twistor formula (\[P-spin-expr\]). The twistorial field with $2J$ $SU(2)$ indices produces by the suitable integration with measure (\[meas\]) the collection of $2J{+}1$ multispinor space-time fields with Lorentz multispinor indices $$\label{st-fields}
\begin{array}{rcl}
\Phi^{(2J,0)}_{\alpha_1\ldots\alpha_{2J}}(x)&=& {\displaystyle\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\lambda^{i_1}_{\alpha_1}\ldots\lambda^{i_{2J}}_{\alpha_{2J}}\Psi^{(J)}_{i_1\ldots i_{2J}}(\lambda,\bar\lambda)}\,,\\[6pt]
\Phi^{(2J-1,1)}_{\alpha_1\ldots\alpha_{2J-1}}{}^{\dot\beta_1}(x)&=&{\displaystyle
\int d\mu_6(\lambda,\bar\lambda) \, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\lambda^{i_1}_{\alpha_1}\ldots\lambda^{i_{2J}}_{\alpha_{2J-1}}\bar\lambda^{\dot\beta_1 i_{2J}}\Psi^{(J)}_{i_1\ldots i_{2J}}(\lambda,\bar\lambda)}\,,\\[6pt]
&& \hspace{2cm}..............\\[6pt]
\Phi^{(0,2J)}{}^{\dot\beta_1\ldots\dot\beta_{2J-1}}(x)&=&{\displaystyle
\int d\mu_6(\lambda,\bar\lambda) \, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\bar\lambda^{\dot\beta_1 i_{1}}\ldots\bar\lambda^{\dot\beta_{2J} i_{2J}}\Psi^{(J)}_{i_1\ldots i_{2J}}(\lambda,\bar\lambda)}\,.
\end{array}$$ In general case the wave functions (\[st-fields\]) contain $n$ undotted symmetrized indices and $(2J{-}n)$ dotted symmetrized ones ($n=0,1,\ldots,2J$). These space-time fields satisfy massive Dirac-like equations which reproduce in two-spinor notations the Bargmann-Wigner fields.
Let us illustrate below the cases with lowest spins $J=0,\frac12,1$.
[**Spin 0:**]{} In this case twistor wave function $\Psi(\lambda,\bar\lambda)$ is a scalar field. Integral transform (\[st-fields\]) gives us the scalar space-time field $$\label{sc-tr}
\Phi^{(0,0)}(x)= {\displaystyle\int d\mu_6(\lambda,\bar\lambda) \, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\Psi^{(0)}(\lambda,\bar\lambda)}\,,$$ which due to (\[constr-lambda\])-(\[m-M\]) satisfies the Klein-Gordon equation $$\label{KG-eq}
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(0,0)}(x)= 0\,,$$ i.e. describes in space-time the relativistic particle with mass $m$ and spin zero.
[**Spin 1/2:**]{} In this case due to integral transformations (\[st-fields\]) we obtain two Weyl spinor fields $$\label{Ws12-tr}
\begin{array}{rcl}
\Phi^{(1,0)}_{\alpha}(x)&=& {\displaystyle\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\lambda^{i}_{\alpha}\Psi^{(1/2)}_{i}(\lambda,\bar\lambda)}\,,\\[6pt]
\Phi^{(0,1)}{}^{\dot\beta}(x)&=&{\displaystyle
\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\bar\lambda^{\dot\beta i}\Psi^{(1/2)}_{i}(\lambda,\bar\lambda)}\,.
\end{array}$$ These space-time fields due to algebraic properties of Weyl spinors satisfy the following generalized Dirac equations with complex mass $M$ $$\label{Dir-eq}
i\partial^{\dot\beta\alpha}\Phi^{(1,0)}_{\alpha}+M\Phi^{(0,1)}{}^{\dot\beta}= 0\,,\qquad
i\partial_{\alpha\dot\beta}\Phi^{(0,1)}{}^{\dot\beta}+\bar M\Phi^{(1,0)}_{\alpha}= 0\,.$$ We note however that phase $e^{i\varphi}$ of $M=|M|e^{i\varphi}$ can be absorbed into space-time spinor fields by the redefinition $(\Phi^{(1,0)}_{\alpha},\Phi^{(0,1)}{}^{\dot\beta})\rightarrow (e^{i\varphi/2}\Phi^{(1,0)}_{\alpha},e^{-i\varphi/2}\Phi^{(0,1)}{}^{\dot\beta})$. Thus, the fields (\[Ws12-tr\]) provide four-component Dirac spinor $(\Phi^{(1,0)}_{\alpha},\Phi^{(0,1)}{}^{\dot\beta})$ providing standard Dirac equation with real mass $m$ and describe spin $1/2$ massive particle. Finally it can be shown that even for complex mass $M$ the equations (\[Dir-eq\]) imply Klein-Gordon equations $$\label{KG-Dir-eq}
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(1,0)}_{\alpha}= 0\,,\qquad
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(0,1)}{}^{\dot\beta}= 0\,.$$
[**Spin 1:**]{} As the result of twistor transform (\[st-fields\]) we obtain the following three space-time fields $$\label{Ws-1}
\begin{array}{rcl}
\Phi^{(2,0)}_{\alpha_1\alpha_{2}}(x)&=& {\displaystyle\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\lambda^{i_1}_{\alpha_1}\lambda^{i_{2}}_{\alpha_{2}}\Psi^{(1)}_{i_1 i_{2}}(\lambda,\bar\lambda)}\,,\\[6pt]
\Phi^{(1,1)}{}_{\alpha}^{\dot\beta}(x)&=&{\displaystyle
\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\lambda^{i_1}_{\alpha}\bar\lambda^{\dot\beta i_{2}}\Psi^{(1)}_{i_1 i_{2}}(\lambda,\bar\lambda)}\,,\\[6pt]
\Phi^{(0,2)}{}^{\dot\beta_1\dot\beta_{2}}(x)&=&{\displaystyle
\int d\mu_6(\lambda,\bar\lambda)\, e^{ix^{\dot\gamma\gamma} \lambda^k_{\gamma}\bar\lambda_{\dot\gamma k}}
\bar\lambda^{\dot\beta_1 i_{1}}\bar\lambda^{\dot\beta_{2} i_{2}}\Psi^{(1)}_{i_1 i_{2}}(\lambda,\bar\lambda)}\,.
\end{array}$$ From these definition it follows that these fields satisfy Dirac-like equations $$\label{Dir-eq1a}
i\partial^{\dot\beta\gamma}\Phi^{(2,0)}_{\gamma\alpha}+M\Phi^{(1,1)}{}_{\alpha}^{\dot\beta}= 0\,,\qquad
i\partial_{\alpha\dot\gamma}\Phi^{(0,2)}{}^{\dot\gamma\dot\beta}+\bar M\Phi^{(1,1)}{}_{\alpha}^{\dot\beta}= 0\,,$$ $$\label{Dir-eq1b}
i\partial^{\dot\alpha\gamma}\Phi^{(1,1)}{}_{\gamma}^{\dot\beta}+M\Phi^{(0,2)}{}^{\dot\alpha\dot\beta}= 0\,,\qquad
i\partial_{\alpha\dot\gamma}\Phi^{(1,1)}{}_{\beta}^{\dot\gamma}+\bar M\Phi^{(2,0)}_{\alpha\beta}= 0\,.$$ Further, the formulae (\[Dir-eq1a\]), (\[Dir-eq1b\]) even for complex $M$ lead to the Klein-Gordon equations for all fields (\[Ws-1\]) $$\label{KG-Dir-eq1}
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(2,0)}_{\alpha\beta}= 0\,,\qquad
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(1,1)}_{\alpha\dot\beta}= 0\,,\qquad
(\partial^\mu\partial_\mu+m^2)\,\Phi^{(0,2)}_{\dot\alpha\dot\beta}= 0\,.$$ The equations (\[Dir-eq1b\]) imply transversality of four-vector field $\Phi^{(1,1)}_{\alpha\dot\beta}={\textstyle\frac{1}{\sqrt{2}}}\,\sigma^\mu_{\alpha\dot\beta}A_\mu$ $$\label{trans-1}
\partial^{\dot\alpha\beta}\Phi^{(1,1)}_{\beta\dot\alpha}= 0\qquad\leftrightarrow\qquad\partial^\mu A_\mu= 0\,.$$ We can consider vector field $\Phi^{(1,1)}_{\alpha\dot\beta}$ as primary field with spin $1$ and remaining two fields $\Phi^{(2,0)}_{\alpha\beta}$, $\Phi^{(0,2)}{}^{\dot\alpha\dot\beta}$ as derivable from $\Phi^{(1,1)}_{\alpha\dot\beta}$ by the formulae (\[Dir-eq1b\]) defining selfdual and anti-selfdual $J=1$ field strengths. The masses in the equations (\[Dir-eq1a\]), (\[Dir-eq1b\]) can be made real after the redefinition $(\Phi^{(2,0)}_{\alpha\beta},\Phi^{(0,2)}_{\dot\alpha\dot\beta},\Phi^{(1,1)}_{\alpha\dot\beta})\rightarrow
(e^{i\varphi}\Phi^{(2,0)}_{\alpha\beta},e^{-i\varphi}\Phi^{(0,2)}{}^{\dot\alpha\dot\beta},\Phi^{(1,1)}_{\alpha\dot\beta})$, where $e^{i\varphi}$ is the phase of complex mass $M$. If we define the $J=1$ field strength (see also [@FFLM]) $$\label{Proca-fields}
F_{\mu\nu}={\textstyle\frac{im}{\sqrt{2}}}\left(\sigma_{\mu\nu}^{\alpha\beta}\Phi^{(2,0)}_{\alpha\beta}+
\tilde\sigma_{\mu\nu}^{\dot\alpha\dot\beta}\Phi^{(0,2)}_{\dot\alpha\dot\beta}\right)\,,$$ due to the equations (\[Dir-eq1a\]), (\[Dir-eq1b\]), (\[trans-1\]) the fields (\[Proca-fields\]) satisfy the Proca equations $$\label{Proca-eq}
\partial^\mu A_\mu= 0\,,\qquad \partial_\mu A_\nu-\partial_\nu A_\mu=F_{\mu\nu}\,,\qquad
\partial^\mu F_{\mu\nu}+m^2 A_\nu=0$$ and the Bianchi identity $\partial_{[\mu} F_{\nu\rho]}=0$.
For arbitrary $J$ one can derive in analogous way the general form of the Bargmann-Wigner equations for massive fields with arbitrary spin $J$.
Outlook
=======
Twistor theory aims at providing a new geometric framework for the description of classical and quantum dynamical models, and one of its basic aims is to formulate the twistor theory of free and interacting particles. The theory in single $D=4$ twistor space describes conformal space-time geometry and provides six-dimensional phase space of massless particles with remaining two degrees of freedom describing $U(1)$ gauge and discrete set of helicities. After quantization the twistor theory via so-called twistor transform provides new method for solving the field equations for massless fields with arbitrary helicity (see e.g. [@EPW]). These techniques were further extended to curved twistor theory and provided new way of solving Einstein and Yang-Mills equations for selfdual and anti-selfdual cases (see e.g. [@Pen76; @Ward]).
The subject studied in this paper is the twistor description of free massive particles with arbitrary spin. In order to introduce in twistor theory time-like fourmomentum vector it is necessary to consider the two-twistor geometry, with sixteen real degrees of freedom. Relativistic spin is described by the Pauli-Lubanski fourvector which carries for definite mass and spin two new degrees of freedom. These new degrees we describe as parametrizing two-dimensional fuzzy sphere $\mathbb{S}^2$ with nonAbelian $SU(2)$ Poisson brackets. In this paper we did show that
- in space-time framework the particle dynamics with nonvanishing spin is obtained adding Souriau-Wess-Zumino term;
- in order to get pure twistorial formulation of massive particles with spin we should modify the standard Penrose incidence relations, which can be obtained by suitable shift of the twistor components;
- in two-twistor space the model is described by free two-twistor Lagrangian with suitable chosen six constraints bilinear in twistor variables;
- the degrees of freedom described by the three-vector $s^r$ due to the constraints (\[Sr-s\]) can be treated as specifying the choice of conformal-invariant scalar products of the pair of twistors, i.e. in such a way in physical phase space the variables $s^r$ are determined as well by the twistor components;
- in order to obtain Pauli-Lubanski spin fourvector one should multiply (see (\[W-tw\]) and (\[S-dot\])) the three-vector $s^r$ with internal three-vector indices by three fourvectors $u_\mu^r$ describing the soldering between internal and space-time descriptions of spin degrees of freedom.
The methods presented in this paper can be extended in several ways, in particular to particle models which generalize the presented here $D\,{=}\,4$ case. In particular
- one can consider the theory of supersymmetric particles and study the supertwistor description [@Ferb] of free massive superparticles with nonvanishing superspin. The superspin should be described by supersymmetric extension of Pauli-Lubanski fourvector [@Sok]. The formalism after using the first quantization rules will provide various known $D\,{=}\,4$ free massive superfields;
- it should be recalled that infinite higher spin multiplets have been obtained by spinorial and twistorial formulations of the free particle models in extended space-time with tensorial coordinates generated by tensorial central charges (for $D\,{=}\,4$ the extended tensorial space-time is ten-dimensional [@BaLu]–[@Vas]). These models used only the set of single twistorial variables and were describing massless higher spin fields. It is interesting to consider the massive two-twistor models linked with tensorially extended space-time which can be obtained by dimensional reduction of higher-dimensional massless spinorial theory in extended tensorial space-time. This idea has been already outlined in our previous paper [@FedLuk] (see also [@SV]), with the description of two-twistor $D\,{=}\,3$ massive spinorial model as obtained by the dimensional reduction from $D=4$ massless spinorial model.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge a support from the grant of the Bogoliubov-Infeld Programme and RFBR grants 12-02-00517, 13-02-91330 (S.F.), as well as Polish National Center of Science (NCN) research projects No. 2011/01/ST2/03354 and No. 2013/09/B/ST2/02205 (J.L.). S.F. thanks the members of the Institute of Theoretical Physics at Wroc[ł]{}aw University for their warm hospitality.
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[^1]: On leave of absence from V.N.Karazin Kharkov National University, Ukraine
[^2]: We shall use $D=4$ two-spinor notation, i.e. $p_{\alpha\dot\beta}=\frac{1}{\sqrt{2}}\,\sigma^{\mu}_{\alpha\dot\beta}p_{\mu}$, $p_{\mu}=\frac{1}{\sqrt{2}}\,\tilde\sigma_{\mu}^{\dot\beta\alpha}p_{\alpha\dot\beta}$, where $(\sigma^\mu)_{\alpha\dot\alpha}=(1_2,\vec{\sigma})_{\alpha\dot\alpha}$, $(\tilde\sigma^\mu)^{\dot\alpha\alpha}=\epsilon^{\alpha\beta}\epsilon^{\dot\alpha\dot\beta}(\sigma^\mu)_{\beta\dot\beta}=
(1_2,-\vec{\sigma})^{\dot\alpha\alpha}$, $\sigma^{\mu\nu}=\frac{1}{2}\,\sigma^{[\mu}\tilde\sigma^{\nu]}$, $\tilde\sigma^{\mu\nu}=\frac{1}{2}\,\tilde\sigma^{[\mu}\sigma^{\nu]}$ $\sigma^{\mu\nu}_{\alpha\beta}=\epsilon_{\beta\gamma}(\sigma^{\mu\nu})_{\alpha}{}^{\gamma}$, $\tilde\sigma^{\mu\nu}_{\dot\alpha\dot\beta}=\epsilon_{\dot\beta\dot\gamma}(\tilde\sigma^{\mu\nu})^{\dot\gamma}{}_{\dot\alpha}$. So, we have $p^{\dot\beta\alpha}p_{\alpha\dot\beta} = p^{\mu}p_{\mu}$. We use weight coefficient in (anti)symmetrization, i.e. $A_{(\alpha}B_{\beta)}=\frac12\,(A_{\alpha}B_{\beta}+A_{\beta}B_{\alpha})$, $A_{[\alpha}B_{\beta]}=\frac12\,(A_{\alpha}B_{\beta}-A_{\beta}B_{\alpha})$.
[^3]: We go up and down the indices $\alpha,\beta,\gamma,...$ and $i,j,k,...$ by antisymmetric tensors $\epsilon_{\alpha\beta}$, $\epsilon_{ij}$, $\epsilon^{\alpha\beta}$, $\epsilon^{ij}$: $A_\alpha=\epsilon_{\alpha\beta}A^\beta$, $A^\alpha=\epsilon^{\alpha\beta}A_\beta$, $A_i=\epsilon_{ij}A^j$, $A^i=\epsilon^{ij}A_j$. We take $\epsilon_{12}=\epsilon^{21}=1$.
[^4]: We denote by $V^r$, $V$ the expressions (\[Sr-0\])-(\[S-0\]) for $S^r$, $S$ with the replacement of $\mu^{\dot\alpha k}$ by $\omega^{\dot\alpha k}$ (see (\[mu-spin-s\])). The constraints (\[Sr-0\]) are additionally modified by inhomogeneous terms proportional to $s^r$.
[^5]: The constraints (\[Sr-s\]) were already proposed in [@FedZ2], however with the Schwinger realization of the algebra (\[S-br-qu\]) in terms of supplementary oscillators.
|
---
author:
- Marius Crainic and Ieke Moerdijk
title: '$\check{C}$ech-De Rham theory for leaf spaces of foliations[^1]'
---
Introduction {#introduction .unnumbered}
============
This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold $M$ equipped with a foliation ${\ensuremath{\mathcal{F}}}$ it is well-known that the coarse (naive) leaf space $M/{\ensuremath{\mathcal{F}}}$, obtained from $M$ by identifying each leaf to a point, contains very little information. In the literature, various models for a finer leaf space $M/{\ensuremath{\mathcal{F}}}$ are used for defining its cohomology. For example, one considers the cohomology of the classifying space of the foliation [@Bott; @Dupon; @Haefl; @LaPa], the sheaf cohomology of its holonomy groupoid [@CrMo; @difcoh; @conj], or the cyclic cohomology of its convolution algebra [@Co3; @Cra]. Each of these methods has considerable drawbacks. E.g. they all involve non-Hausdorff spaces in an essential way. More specifically, the classifying space, which is probably the most common model for the “fine” leaf space, is a space which in general is infinite dimensional and non-Hausdorff, it is not a CW-complex, and it has lost all the smooth structure of the original foliation. In particular, it is not suitable for constructing cohomology theories with compact support. For this reason, the construction of characteristic classes in the cohomology of the classifying space of the foliation proceeds in a very indirect way, and many of the standard geometrical constructions have to be replaced by or supplied with abstract non-trivial arguments. The same applies to the construction of “universal” characteristic classes in the cohomology of the classifying space of the Haefliger groupoid $\Gamma^q$. It is possible to construct interesting classes of (foliated or transversal) bundles over foliations by explicit geometrical methods [@Bott; @KT], but these classes are constructed in the cohomology of the manifold $M$ rather than that of the leaf space $M/{\ensuremath{\mathcal{F}}}$.\
The purpose of this paper is to present a “$\check{C}$ech-De Rham” model for the cohomology of leaf spaces (Section \[CDRcomplex\]), which circumvents the problems mentioned above. This $\check{C}$ech-De Rham model lends itself to the construction of (known) characteristic classes, now by explicit geometrical constructions which are immediate extensions of the standard constructions for manifolds (Section \[classes\]). As a consequence, for any transversal principal bundle over a foliated manifold $(M, {\ensuremath{\mathcal{F}}})$, we are able to lift the characteristic classes constructed in $H^*(M)$ by the methods of [@KaTo], to the $\check{C}$ech-De Rham cohomology $H^*(M/{\ensuremath{\mathcal{F}}})$, and establish all the relations, such as the Bott vanishing theorem, at the level of $H^*(M/{\ensuremath{\mathcal{F}}})$ (see Theorem \[theorem2\] below).\
We want to emphasize that the construction of the $\check{C}$ech-De Rham model and of the characteristic classes makes no reference to (holonomy) groupoids or classifying spaces. In particular, there are no non-Hausdorffness problems, and these constructions can be understood by anyone having some background in differential geometry, including familiarity with the very basic definitions concerning foliations.\
To prove that our $\check{C}$ech-De Rham model gives in fact the same cohomology as the other models (Theorem \[theorem1\]), we use étale groupoids (Section \[etale\]). In fact, our model and the associated method for constructing characteristic classes applies to any étale groupoid, not just to holonomy groupoids (see Theorem \[theoreml\], and \[CWetalgr\]). In particular, when used in the context of the Haefliger groupoid $\Gamma^q$, it provides an explicit geometric construction of the universal geometrical characteristic classes (as a map from Gelfand-Fuchs cohomology into the cohomology of $B\Gamma^q$ [@Bott]). In this way we rediscover (and explain) the Thurston formula and the Bott formulas for cocycles on diffeomorphism groups [@Boform] (for these explicit formulas, see Section \[explicit\]). Other groupoids of interest, different from holonomy groupoids, are the monodromy groupoids of foliations. Our methods also show that the characteristic classes of foliated bundles [@KaTo] actually live in the cohomology of the monodromy groupoid of the foliation, rather in the cohomology of $M$ itself.\
Our $\check{C}$ech-De Rham cohomology also has a natural version with compact supports, which is related to the one with arbitrary supports by an obvious duality. When passing to the cohomology of holonomy groupoids, this duality becomes the Poincaré duality of [@CrMo] (Proposition \[pdutr\]). This new proof of Poincaré duality for leaf spaces appears as a straightforward extension of the standard arguments [@BoTu] from manifolds to leaf spaces. Moreover, this duality extends the known one for basic cohomology of Riemannian foliations [@Ser].\
There are several other cohomology theories associated to foliations which are easier to describe and are perhaps more familiar, such as basic cohomology (see e.g. [@minimal; @Ser]) and foliated cohomology (see e.g. [@Alv; @Hei; @KT; @MoSo]). In the last two sections of our paper, we use our $\check{C}$ech-De Rham model to explicitly describe the relations between the cohomology of leaf spaces and the basic and foliated cohomology.
Transverse structures on foliations {#transverse}
===================================
In this section we recall some basic notions concerning the transverse structures of foliations, which formalize the idea of structures over the leaf space. Throughout, we will work in the smooth context.
[**Holonomy**]{} *Let $M$ be a manifold of dimension $n$, equipped with a foliation ${\ensuremath{\mathcal{F}}}$ of codimension $q$. A [*transversal section*]{} of ${\ensuremath{\mathcal{F}}}$ is an embedded $q$-dimensional submanifold $U\subset M$ which is everywhere transverse to the leaves. Recall that if $\alpha$ is a path between two points $x$ and $y$ on the same leaf, and if $U$ and $V$ are transversal sections through $x$ and $y$, then $\alpha$ defines a transport along the leaves from a neighborhood of $x$ in $U$ to a neighborhood of $y$ in $V$, hence a germ of a diffeomorphism $hol(\alpha): (U, x){\longrightarrow}(V, y)$, called the [*holonomy*]{} of the path $\alpha$. Two homotopic paths always define the same holonomy. The familiar [*holonomy groupoid*]{} [@CoOp; @Haefl; @Wi] is the groupoid $Hol(M, {\ensuremath{\mathcal{F}}})$ over $M$ where arrows $x{\longrightarrow}y$ are such germs $hol(\alpha)$. If the above transport “along $\alpha$” is defined in all of $U$ and embeds $U$ into $V$, this embedding $h: U\hookrightarrow V$ is sometimes also denoted by $hol(\alpha): U\hookrightarrow V$. Embeddings of this form will be called [*holonomy embeddings*]{}. Note that composition of paths also induces an operation of composition on those holonomy embeddings. (In section \[etale\] below we will present a more general definition of the so-called “embedding category”).*
[**Transversal basis**]{}\[rmks2.1\] *Transversal sections $U$ through $x$ as above should be thought of as neighborhoods of the leaf through $x$ in the leaf space. This motivates the definition of a [*transversal basis*]{} for $(M, {\ensuremath{\mathcal{F}}})$ as a family ${\ensuremath{\mathcal{U}}}$ of transversal sections $U\subset M$ with the property that, if $V$ is any transversal section through a given point $y\in M$, there exists a holonomy embedding $h: U\hookrightarrow V$ with $U\in {\ensuremath{\mathcal{U}}}$ and $y\in h(U)$.\
Typically, a transversal section is a $q$-disk given by a chart for the foliation. Accordingly, we can construct a transversal basis ${\ensuremath{\mathcal{U}}}$ out of a basis $\tilde{{\ensuremath{\mathcal{U}}}}$ of $M$ by domains of foliation charts $\phi_{U}: \tilde{U}\tilde{{\longrightarrow}} \mathbb{R}^{n-q}\times U$, $\tilde{U}\in \tilde{{\ensuremath{\mathcal{U}}}}$, with $U=\mathbb{R}^q$. Note that each inclusion $\tilde{U}\hookrightarrow \tilde{V}$ between opens of $\tilde{{\ensuremath{\mathcal{U}}}}$ induces a holonomy embedding $h_{U, V}: U{\longrightarrow}V$ defined by the condition that the plaque in $\tilde{U}$ through $x$ is contained in the plaque in $\tilde{V}$ through $h_{U, V}(x)$.*
\[trbd\][**Transversal bundles**]{} *Let $G$ be a Lie group and let $\pi: P{\longrightarrow}M$ be a principal $G$-bundle over $M$. Recall [@KaTo] that $P$ is said to be [*foliated*]{} if $P$ is equipped with a $G$-equivariant foliation $\tilde{{\ensuremath{\mathcal{F}}}}$, of the same dimension as ${\ensuremath{\mathcal{F}}}$, whose leaves are transversal to the fibers of $\pi$ and mapped by $\pi$ to those of ${\ensuremath{\mathcal{F}}}$. The vectors tangent to $\tilde{{\ensuremath{\mathcal{F}}}}$ define a flat partial connection on $P$. In particular, any path $\alpha$ in a leaf $L$ from $x$ to $y$ defines a parallel transport $P_x{\longrightarrow}P_y$ which depends only on the homotopy class of $\alpha$. We call $P$ a [*transversal*]{} principal bundle if the transport depends just on the holonomy of $\alpha$. A vector bundle $E$ on $M$ is said to be foliated (transversal) if the associated principal $GL_r$-bundle is foliated (transversal). By the usual relation between Cartan-Ehresmann connections and Koszul connections, we see that a foliated vector bundle is a vector bundle $E$ over $M$ endowed with a “flat ${\ensuremath{\mathcal{F}}}$-connection”, i.e. an operator $$\nabla: \Gamma({\ensuremath{\mathcal{F}}})\times \Gamma(E){\longrightarrow}\Gamma(E)$$ satisfying the usual relations $\nabla_{fX}(s)= f\nabla_{X}(s)$, $\nabla_{X}(fs)= f\nabla_{X}(s)+ X(f)s$, as well as the flatness relation $\nabla_{[X, Y]}= [\nabla_{X}, \nabla_{Y}]$, for all $X, Y\in \Gamma({\ensuremath{\mathcal{F}}})$, $f\in C^{\infty}(M)$, $s\in \Gamma(s)$.\
Notice that if $P$ is a transversal (principal or vector) bundle, any holonomy embedding $h: U\hookrightarrow V$ induces a well-defined map $h_*: P|_{U} {\longrightarrow}P|_{V}$, which is functorial in $h$. We will usually just write $h: P|_{U} {\longrightarrow}P|_{V}$ again for this map.\
The basic example of a transversal vector bundle is the normal bundle of the foliation, $\nu= TM/{\ensuremath{\mathcal{F}}}$. The associated Koszul connection is precisely the Bott connection [@Bott], $\nabla_{X}(\overline{Y})= \overline{[X, Y]}$. It is a transversal bundle by the very definition of (linear) holonomy.*
\[trsh\][**Transversal sheaves**]{} *Analogous definitions apply to sheaves. A sheaf ${\ensuremath{\mathcal{A}}}$ on $M$ is called [*foliated*]{} if its restriction to each leaf is locally constant. Thus, (the homotopy class of) a path $\alpha$ from $x$ to $y$ in a leaf $L$ induces an isomorphism between stalks $\alpha_*: {\ensuremath{\mathcal{A}}}_x{\longrightarrow}{\ensuremath{\mathcal{A}}}_y$. The sheaf is transversal if this isomorphism only depends on the holonomy of $\alpha$. A global section $s\in \Gamma(M, {\ensuremath{\mathcal{A}}})$ is called [*invariant*]{} if $s$ is invariant under transport along leaves, i.e. $\alpha_*s(x)= s(y)$ in the notations above.\
If ${\ensuremath{\mathcal{A}}}$ is a transversal sheaf, any holonomy embedding $h: U\hookrightarrow V$ gives a well-defined restriction $h^*: \Gamma(V, {\ensuremath{\mathcal{A}}}){\longrightarrow}\Gamma(U, {\ensuremath{\mathcal{A}}})$. The global section $s$ is invariant if and only if $h^*(s|_{V})= s|_{U}$ for each such $h$.\
An example of a transversal sheaf is the sheaf $\Omega_{bas}^{0}$ of smooth functions which are locally constant along the leaves. One similarly has the transversal sheaves $\Omega^{k}_{bas}$ of germs of basic differential $k$-forms. More generally, any foliated vector bundle $E$ induces a foliated sheaf $\Gamma_{\nabla}(E)$ defined as follows. We denote by $\Gamma_{\nabla}(M; E)$ the space of sections of $E$ which are $\nabla$-constant. Over $M$, $\Gamma_{\nabla}(E)$ is the sheaf whose space of sections over an open $U$ is $\Gamma_{\nabla}(E|_{\, U})$. Using the parallel transport with respect to $\nabla$ we see that this sheaf is locally constant when restricted to leaves, hence it is foliated. Clearly $\Gamma_{\nabla}(E)$ is transversal if $E$ is. For instance, if $E= \Lambda^{k}\nu^*$, then $\Gamma_{\nabla}(\Lambda^{k}\nu^*)= \Omega^{k}_{bas}$.\
Another important example is the (real) transversal orientation sheaf of the foliation, which we denote by ${\ensuremath{\mathcal{O}}}$. When restricted to a transversal open $U$, $\Gamma(U; {\ensuremath{\mathcal{O}}})= H^{q}_{c}(U)^{\vee}$. The foliation is transversally orientable if and only if ${\ensuremath{\mathcal{O}}}$ is constant.*
The transversal $\check{C}$ech-De Rham complex {#CDRcomplex}
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Let $(M, {\ensuremath{\mathcal{F}}})$ be a foliated manifold and let ${\ensuremath{\mathcal{U}}}$ be a transversal basis. Consider the double complex which in bi-degree $k, l$ is the vector space $$C^{k, l}= \check{C}^{k}({\ensuremath{\mathcal{U}}}, \Omega^{l})= \prod_{U_{0}\stackrel{h_1}{{\longrightarrow}} \ldots \stackrel{h_k}{{\longrightarrow}} U_k} \Omega^l(U_0) .$$ Here the product ranges over all $k$-tuples of holonomy embeddings between transversal sections from the given basis ${\ensuremath{\mathcal{U}}}$, and $\Omega^k(U_0)$ is the space of differential $k$-forms on $U_0$. For elements $\omega\in C^{k, l}$, we denote its components by $\omega(h_1, \ldots, h_k)\in \Omega^k({\ensuremath{\mathcal{U}}}_0)$. The vertical differential $C^{k, l}{\longrightarrow}C^{k, l+1}$ is $(-1)^kd$ where $d$ is the usual De Rham differential. The horizontal differential $C^{k, l}{\longrightarrow}C^{k+1, l}$ is $\delta= \sum(-1)^{i}\delta_{i}$ where $$\label{deltas} \delta_{i}(h_1, \ldots , h_{k+1})= \left\{ \begin{array}{lll}
h_{1}^{*}\omega(h_2, \ldots , h_{k+1}) \ \ \mbox{if $i=0$}\\
\omega(h_1, \ldots, h_{i+1}h_{i}, \ldots, h_{k+1}) \ \ \mbox{if $0<i< k+1$}\\
\omega(h_1, \ldots, h_k) \ \ \mbox{if $i= k+1$}
\end{array}
\right.$$ This double complex is actually a bigraded differential algebra, with the usual product $$(\omega\cdot\eta)(h_1, \ldots , h_{k+k\,'})= (-1)^{kk\,'}\omega(h_1, \ldots , h_{k}) h_{1}^{*} \ldots h_{k}^{*} \eta(h_{k+1}, \ldots h_{k+k\,'})$$ for $\omega\in C^{k, l}$ and $\eta\in C^{k\,', l\,'}$. We will also write $\check{C}({\ensuremath{\mathcal{U}}}, \Omega)$ for the associated total complex, and refer to it as the [*$\check{C}$ech-De Rham complex*]{} of the foliation. The associated cohomology is denoted $$\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}) ,$$ and referred to as the $\check{C}$ech-De Rham cohomology of the leaf space $M/{\ensuremath{\mathcal{F}}}$, w.r.t. the cover ${\ensuremath{\mathcal{U}}}$.\
Note that, when ${\ensuremath{\mathcal{F}}}$ is the codimension $n$ foliation by points, then ${\ensuremath{\mathcal{U}}}$ is a basis for the topology of $M$, and $C^{k, l}$ is the usual $\check{C}$ech-De Rham complex [@BoTu]. Thus in this case $\check{H}_{{\ensuremath{\mathcal{U}}}}^*(M/{\ensuremath{\mathcal{F}}})= H^*(M)$ is the usual De Rham cohomology of $M$.\
In general, choosing a transversal basis ${\ensuremath{\mathcal{U}}}$ and a basis $\tilde{{\ensuremath{\mathcal{U}}}}$ of $M$ as in \[rmks2.1\], there is an obvious map of double complexes $C^{k, l}({\ensuremath{\mathcal{U}}}) {\longrightarrow}C^{k, l}(\tilde{{\ensuremath{\mathcal{U}}}})$ into the $\check{C}$ech-De Rham complex for the manifold $M$. Hence a canonical map $$\label{zero}\label{map2.1}
\pi^{*}: \check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*}(M; \mathbb{R}) \ ,$$ which should be thought of as the pull-back along the “quotient map” $\pi: M{\longrightarrow}M/{\ensuremath{\mathcal{F}}}$.\
The standard way [@Co3; @minimal] to model the leaf space of a foliation $(M, {\ensuremath{\mathcal{F}}})$ is by the classifying space $BHol(M, {\ensuremath{\mathcal{F}}})$ of the holonomy groupoid. Thus, the following theorem can be interpreted as a $\check{C}$ech-De Rham theorem for leaf spaces.
\[theorem1\]There is a natural isomorphism $$\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}) \cong H^{*}(B Hol(M, {\ensuremath{\mathcal{F}}}); \mathbb{R})\ ,$$ between the $\check{C}$ech-De Rham cohomology and the cohomology of the classifying space. In particular, the left hand side is independent of the choice of a transversal basis ${\ensuremath{\mathcal{U}}}$.
For the proof of this theorem, we choose a complete transversal section $T$ which contains every $U\in {\ensuremath{\mathcal{U}}}$, and we consider the “reduced holonomy groupoid” $Hol_{T}(M, {\ensuremath{\mathcal{F}}})$, defined as the restriction of $Hol(M, {\ensuremath{\mathcal{F}}})$ to $T$. We may assume that ${\ensuremath{\mathcal{U}}}$ is a basis for the topology of $T$. By a well known Morita-invariance argument, the classifying spaces $B Hol(M, {\ensuremath{\mathcal{F}}})$ and $B Hol_{T}(M, {\ensuremath{\mathcal{F}}})$ are weakly homotopyc equivalent. The advantage of passing to a complete transversal is that $Hol_{T}(M, {\ensuremath{\mathcal{F}}})$ becomes an étale groupoid (see section \[etale\] for the precise definitions). For such groupoids ${\ensuremath{\mathcal{G}}}$ there is a standard cohomology $H^{*}({\ensuremath{\mathcal{G}}}; -)$ with coefficients, which was also defined by Haefliger [@difcoh] in terms of bar-complexes, and which is known [@conj] to be isomorphic to the cohomology of the classifying space. In section \[etale\] we will recall all the basic definitions. The theorem will then follow from the following proposition, which is a particular case of the Theorem \[theoreml\] below.
\[lema1\] For any complete transversal $T$ and any basis ${\ensuremath{\mathcal{U}}}$ of $T$, there is a natural isomorphism $$\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}) \cong H^{*}(Hol_{T}(M, {\ensuremath{\mathcal{F}}}); \mathbb{R}) \ .$$
We mention here that there are several variations of Theorem \[theorem1\]. For instance, for any transversal sheaf ${\ensuremath{\mathcal{A}}}$ there is a $\check{C}$ech complex $\check{C}({\ensuremath{\mathcal{U}}}, {\ensuremath{\mathcal{A}}})$. In degree $k$, $$\check{C}^{k}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})= \prod_{U_{0}\stackrel{h_1}{{\longrightarrow}} \ldots \stackrel{h_k}{{\longrightarrow}} U_k} \Gamma(U_0; {\ensuremath{\mathcal{A}}}) \ ,$$ with the boundary $\delta= \sum(-1)^{i}\delta_{i}$ given by the formulas (\[deltas\]). A consequence of the more general Theorem \[theoreml\] says that, if ${\ensuremath{\mathcal{A}}}|_{U}$ is acyclic for all $U\in {\ensuremath{\mathcal{U}}}$, then $\check{C}({\ensuremath{\mathcal{U}}}, {\ensuremath{\mathcal{A}}})$ computes the cohomology of the classifying space (of the reduced holonomy groupoid) with coefficients in a sheaf $\tilde{{\ensuremath{\mathcal{A}}}}$ naturally associated to ${\ensuremath{\mathcal{A}}}$.\
Another variation applies to the cohomology with compact supports (see section \[etale\]). Note that all these are actually extensions of the usual “$\check{C}$ech-De Rham isomorphisms” [@BoTu] from manifolds to leaf space. Accordingly, an immediate consequence will be the Poincaré duality for leaf spaces (see Section \[secbasic\]), which is one of the main results of [@CrMo]. With Theorem \[theorem1\] and its analogue for compact supports available, the new proof of Poincaré duality is this time a rather straightforward extension of the classical proof [@BoTu] from manifolds to leaf spaces.
The transversal Chern-Weil map {#classes}
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To illustrate the usefulness of the transversal $\check{C}$ech-De Rham complex we will adapt the standard geometric construction of characteristic classes of principal bundles to transversal bundles, so as to obtain explicit classes in this complex. We will use the Weil-complex formulation, which we recall first (for an extensive exposition, see [@KaTo; @Duff]).
\[clCW\][**Classical Chern-Weil:** ]{}[@Car] Recall that the Weil algebra of the Lie algebra $\mathfrak{g}$ (of a Lie group $G$) is the algebra $$W(\mathfrak{g})= S(\mathfrak{g}^*)\otimes \Lambda(\mathfrak{g}^*) .$$ It is a graded commutative dga (graded as $W(\mathfrak{g})^n= \oplus_{2p+q= n} S^p(\mathfrak{g}^*)\otimes \Lambda^q(\mathfrak{g}^*))$, equipped with operations $i_X$ and $L_X$ (linear in $X\in \mathfrak{g}$) which satisfy the usual Cartan identities. In the language of [@KaTo], this means that $W(\mathfrak{g})$ is a $\mathfrak{g}$-dga. If $P$ is a principal $G$-bundle over a manifold $M$, the algebra $\Omega^*(P)$ of differential forms on $P$ with its usual operations $i_X$ and $L_X$ is another example of a $\mathfrak{g}$-dga. A connection $\nabla$ on $P$ is uniquely determined by its connection form $\omega\in \Omega^1(P)\otimes\mathfrak{g}$. This can be viewed as a map $\omega: W(\mathfrak{g})^1= \mathfrak{g}^*{\longrightarrow}\Omega^1(P)$, which extends uniquely to a map of $\mathfrak{g}$-dga’s, $$\label{charact}
\tilde{k}(\nabla): W(\mathfrak{g}) {\longrightarrow}\Omega(P) \ .$$ (On $\mathfrak{g}^*= S^1(\mathfrak{g}^*)\subset W(\mathfrak{g})^2$, it restricts to the curvature $\Omega= d\omega+ \frac{1}{2} [\omega, \omega]$.) The restriction of this map (\[charact\]) to basic elements (elements annihilated by $i_X$ and $G$-invariant) gives a map of dga’s $$S(\mathfrak{g}^*)^{G} {\longrightarrow}\Omega^*(M)$$ (zero differential on $S(\mathfrak{g}^*)^{G}$, the usual De Rham differential on $\Omega(M)$), hence a map $$\label{classicalCW}
k(\nabla): S(\mathfrak{g}^*)^{G} {\longrightarrow}H^{*}(M) ,$$ known as the Chern-Weil map for the principal $G$-bundle $P$. Because of the $2p$ in the grading of the Weil algebra, $k(\nabla)$ maps invariant polynomials of degree $p$ to degree $2p$ cohomology classes. Moreover, $k(\nabla)$ does not depend on $\nabla$. This follows from the Chern-Simons construction (see below). A more refined characteristic map is obtained if one uses a maximal compact subgroup $K$ of $G$. Since $P/K{\longrightarrow}M$ has contractible fibers, the map induced in De Rham cohomology is an isomorphism. Hence, to get down to the base manifold, it suffices to consider the $K$-basic elements of (\[charact\]). Denoting by $W(\mathfrak{g}, K)$ the subcomplex of $W(\mathfrak{g})$ of $K$-basic elements, one obtains a characteristic map $H^*(W(\mathfrak{g}, K)){\longrightarrow}H^{*}(M)$.
\[Simons\][**Chern-Simons:** ]{}Given $k$ connections $\nabla_0, \ldots , \nabla_k$ on $P$, we consider the convex combination $$\label{doi}
\nabla= t_0\nabla_0+ \ldots + t_k\nabla_k$$ which defines a connection on the principal bundle ${\bf \Delta}^{k}\times P$ over ${\bf \Delta}^{k}\times M$, where ${\bf \Delta}^{k}= \{ (t_0, \ldots, t_k): t_i\geq 0, \sum t_{i}= 1\}$ is the standard $k$-simplex. We define $$\label{patruu}
\tilde{k}(\nabla_0, \ldots, \nabla_k)= (-1)^{k}\int_{{\bf \Delta}^{k}}\tilde{k}(\nabla): W(\mathfrak{g}) {\longrightarrow}\Omega^{*-k}(P)\ ,$$ where $\int_{{\bf \Delta}^{k}}: \Omega^*({\bf \Delta}^{k}\times P){\longrightarrow}\Omega^{*-k}(P)$ is the integration along the fibers ${\bf \Delta}^{k}$. Let us summarize the main properties of this construction:
(i) the map (\[patruu\]) commutes with the action of $G$, and with the operators $i_X$, $L_X$, and it vanishes on all elements $\alpha\otimes \beta\in W(\mathfrak{g})$ with $\alpha$ a polynomial of degree $> dim(M)$.
(ii) $$\label{Stokes}
[\tilde{k}(\nabla_0, \ldots, \nabla_k), d]= \sum_{i=0}^{k} (-1)^{i} \tilde{k}(\nabla_0, \ldots , \widehat{\nabla_i}, \ldots , \nabla_k)\ ,$$
(iii) (\[patruu\]) is natural w.r.t. isomorphisms of principal $G$-bundles.
[*Proof:*]{} (ii) is just a version of Stokes’ formula (see also [@Bott]), while (iii) is obvious. We prove the vanishing result of (i). Denote by $d$ the degree of the polynomial $\alpha$ and by $q$ the dimension of $M$. We prove that when $d< k$ or $2d> q+k$, our expression $$\theta= \tilde{k}(\nabla_0, \ldots, \nabla_k)(\alpha\otimes\beta)$$ vanishes (note that if $d> q$, then at least one of these two equalities holds). First assume that $d<k$. We have $\tilde{k}(\nabla)(\alpha\otimes \beta)= \alpha(\Omega)\wedge \beta(\omega)$, where $\nabla$ is the affine combination (\[doi\]), $\omega$ is the associated $1$-form, and $\Omega$ is its curvature. Let us say that a homogeneous form $f dt_{i_1}\ldots dt_{i_r}dx_{j_1}\ldots dx_{j_s}$ on ${\bf \Delta}^{k}\times P$ has bi-degree $(r, s)$. Since $\omega$ has bi-degree $(0, 1)$, $\Omega$ is a sum of forms of bi-degree $(1, 1)$ and $(0, 2)$, so $\int_{{\bf \Delta}^{k}} \alpha(\Omega)\wedge \beta(\omega)= 0$ because no bi-degree $(r, s)$ with $r= k$ will be involved.\
We now turn to the case $2d> q+k$. Let $l$ be the degree of $\beta$. Because of the similar property for $\beta$, we have $i_{X_1}\ldots i_{X_{l+1}}\theta = 0$ for any vertical vector fields $X_i$. On the other hand, $i_{Y_1}\ldots i_{Y_{q+1}}\theta = 0$ for any horizontal vector fields $Y_i$. Since $deg(\theta)= 2d+l-k> l+ q$, it follows that $\theta= 0$. ${\raisebox{.8ex}{\framebox}}$
\[trChW\][**Construction of the transversal Chern-Weil map:** ]{}Now let $P$ be a transversal principal $G$-bundle on a foliated manifold $(M, {\ensuremath{\mathcal{F}}})$. Consider the $\check{C}$ech-De Rham complex $$\check{C}^{k}({\ensuremath{\mathcal{U}}}, \Omega^{l}(P))= \prod_{U_{0}\stackrel{h_1}{{\longrightarrow}} \ldots \stackrel{h_k}{{\longrightarrow}} U_k} \Omega^l(P|_{U_{0}})\ ,$$ defined exactly as in section \[CDRcomplex\] (except that $\Omega^l(U_0)$ is replaced by $\Omega^l(P|_{U_{0}})$, and hence the horizontal differential $\delta$ involves the maps $h_{1}: P|_{U_{0}}{\longrightarrow}P|_{U_{1}}$ discussed in section \[transverse\]). Choose a system $\nabla= \{ \nabla_{U}\}$ of connections, one connection $\nabla_{U}$ on $P|_{U}$ for each $U$ in a transversal basis ${\ensuremath{\mathcal{U}}}$. In general we cannot assume this choice to be respected by holonomy embeddings $h: U{\longrightarrow}V$, i.e. $\nabla_U$ is in general different from the connection on $P|_{U}$ induced by $\nabla_{V}$ via the isomorphism $h: P|_{U} {\longrightarrow}P|_{h(U)}$. Denote this last connection by $\nabla_{h}$. For a string $U_{0}\stackrel{h_1}{{\longrightarrow}} \ldots \stackrel{h_k}{{\longrightarrow}} U_k$ of holonomy embeddings, we consider the map (see \[clCW\] above) $$\label{patru}
\tilde{k}(\nabla_{U_0},
\nabla_{h_1}, \nabla_{h_2h_1}, \ldots ,
\nabla_{h_k \ldots h_2h_1}): W(\mathfrak{g}) {\longrightarrow}\Omega^{*-k}(P|_{U_{0}}) .$$ Doing this for all such strings, we obtain a map into the total complex $$\label{cinci}
\tilde{k}(\omega): W(\mathfrak{g}) {\longrightarrow}\prod_{U_{0}\stackrel{h_1}{{\longrightarrow}} \ldots \stackrel{h_k}{{\longrightarrow}} U_k} \Omega^{*-k}(P|_{U_{0}})\ .$$ This map respects the total degree, and it is obviously compatible with the operations $i_X$ and the $G$-action. So, by restricting to basic elements it yields a map into the transversal $\check{C}$ech-De Rham complex $$\label{sase}
\tilde{k}(\omega): S(\mathfrak{g}^{*})^{G} {\longrightarrow}\check{C}^{*}({\ensuremath{\mathcal{U}}}, \Omega^*)$$ (mapping degree $p$ polynomials into elements of total degree $2p$).
\[theorem2\] The Chern-Weil map of a transversal principal $G$-bundle $P$ over $(M, {\ensuremath{\mathcal{F}}})$ has the following properties:
(i) The maps [(\[cinci\])]{} and [(\[sase\])]{} respect the differential, hence they induce a map $$\label{sapte}
k_{P}:= k(\nabla) : S(\mathfrak{g}^{*})^{G} {\longrightarrow}\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M/{\ensuremath{\mathcal{F}}})\ ,$$
(ii) This map [(\[sapte\])]{} does not depend on the choice of the connections $\{\nabla_{U}\}$, and respects the products.
(iii) Composed with the pull-back map $\pi^{*}: \check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*}(M)$, see [(\[map2.1\])]{}, it gives the usual Chern-Weil map [(\[classicalCW\])]{} of $P$.
(iv) [(]{}“Bott vanishing theorem”[)]{} The image of the map [(\[sapte\])]{} is zero in degrees $> 2q$, where $q$ is the codimension of ${\ensuremath{\mathcal{F}}}$.
The classical Bott vanishing theorem [@Bott] (for the normal bundle of the foliation) and its extensions to foliated bundles [@KaTo] are at the level of $H^*(M)$. The point of Theorem \[theorem2\] is that, using [*classical geometrical arguments*]{}, one can prove these vanishing results and construct the resulting characteristic classes at the level of the leaf space, i.e. in the cohomology of the classifying space (cf. Theorem \[theorem1\]).\
[*Proof of Theorem \[theorem2\]:*]{} (i) and (iv) clearly follow from the main properties of the Chern-Simons construction \[Simons\]. Also (iii) will follow from the independence of the connections. Indeed, it suffices to check that, if ${\ensuremath{\mathcal{F}}}$ is the foliation by points, then the resulting map $k_{\nabla}: S(\mathfrak{g}^{*})^{G}{\longrightarrow}\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M)$ composed with $\check{C}$ech-De Rham isomorphism $\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M)\cong H^{*}(M)$ (induced by the inclusion $\Omega^*(M)\subset C^*({\ensuremath{\mathcal{U}}}, \Omega^*)$ [@BoTu]) gives the usual Chern-Weil map. But this is clear even at the chain level, provided we choose $\nabla_{U}= \nabla |_{U}$ for some globally defined connection $\nabla$.\
(ii) For two different choices $\nabla= \{\nabla_{U}\}$ and $\nabla\,'= \{\nabla_{U}\,'\}$ of connections, the map $H: W(\mathfrak{g}){\longrightarrow}C^*({\ensuremath{\mathcal{U}}},\Omega^*)$ defined by $$H^{*}(w)(h_1, \ .\ .\ .\ , h_k)=
\sum_{i=0}^{k}(-1)^i k(\nabla_{h_i \ldots h_2h_1}, \ldots , \nabla_{h_k \ldots h_2h_1}, \nabla^{\,'}_{U_0}, \nabla^{\,'}_{h_1}, \ldots ,
\nabla^{\, '}_{h_i \ldots h_2h_1})(w) \ .$$ provides an explicit chain homotopy. To prove the compatibility with the products, one can either proceed as in [@KaTo] using the simplicial Weil complex (see [@Crath] for details), or, since the characteristic map is constructed through the double complex $\check{C}^{p}({\ensuremath{\mathcal{U}}}, \Omega^{p+q}({\bf \Delta}^{q} \times P))$ by integration over the simplices, one can use the simplicial De Rham complex and Theorem 2.14 of [@Dupon]. ${\raisebox{.8ex}{\framebox}}$
\[exotic\][**Exotic characteristic classes:** ]{}The vanishing result of Theorem \[theorem2\] shows that the construction of the “exotic” classes also lifts to the $\check{C}$ech-De Rham complex. To describe all the relevant characeristic classes, we consider the complex $W(\mathfrak{g}, K)$ of $K$-basic elements described in \[clCW\], together with its $q$-th truncation $W_q(\mathfrak{g}, K)$ defined as the quotient by the ideal generated by the elements of polynomial degree $> q$. By the vanishing result (more precisely from the proof above), the map (\[cinci\]) induces a chain map $W_q(\mathfrak{g}, K) {\longrightarrow}\check{C}^{*}({\ensuremath{\mathcal{U}}}, \Omega^*(P/K)$. Using the contractibility of $G/K$ as in \[clCW\], we obtain the following refinement of the characteristic map of Theorem \[theorem2\].
\[corex\] The Chern-Weil construction of [\[trChW\]]{} gives a well-defined algebra map $$\label{exoticone}
k^{ex}_{P}:= k^{ex}(\nabla): H^*(W_q(\mathfrak{g}, K)) {\longrightarrow}\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}) \ ,$$ again independent of the choice of connections. Moreover, composed with the pull-back map $\pi^{*}: \check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*}(M)$ (see [(\[map2.1\])]{}), it gives the exotic characteristic map of the foliated bundle $P$ [[@KaTo]]{}.
The $\check{C}$ech-De Rham complex of an étale groupoid {#etale}
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In this section we prove Theorem \[theorem1\], as well as some generalizations and variants, in the context of étale groupoids. Our general goal is to describe the (hyper-) homology and cohomology of étale groupoids in terms of the (hyper-) homology and cohomology of small categories. We begin by introducing some standard terminology.
[**Smooth étale groupoids:** ]{}A [*smooth groupoid*]{} is a groupoid ${\ensuremath{\mathcal{G}}}$ for which the sets ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$ and ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}1{{\raise 1pt\hbox{\tiny )}}}}}}$ of objects and arrows have the structure of a smooth manifold, all the structure maps are smooth, and the source and the target maps are moreover submersions. The holonomy groupoid $Hol(M, {\ensuremath{\mathcal{F}}})$ of a foliation is an example of a smooth groupoid. Such a smooth groupoid is said to be étale if the source and the target maps are local diffeomorphisms. In this case the manifolds ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$ and ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}1{{\raise 1pt\hbox{\tiny )}}}}}}$ have the same dimension, to which we refer as [*the dimension of*]{} ${\ensuremath{\mathcal{G}}}$. An example of an étale groupoid of dimension $q$ is the universal Haefliger groupoid $\Gamma^q$ for codimension $q$ foliations [@Haefl]. There is an important notion of [*Morita equivalence*]{} between smooth groupoids, see e.g. [@Co3; @Haefl; @Fourier; @Mrcun]. For any foliation, the holonomy groupoid $Hol(M, {\ensuremath{\mathcal{F}}})$ is Morita equivalent to an étale groupoid, namely to its restriction to any complete transversal $T$, denoted $Hol_{T}(M, {\ensuremath{\mathcal{F}}})$. A Morita equivalence between smooth groupoids induces a weak homotopy equivalence between their classifying spaces.
\[bar\][**Sheaves and cohomology:** ]{}For a smooth étale groupoid ${\ensuremath{\mathcal{G}}}$, a ${\ensuremath{\mathcal{G}}}$-sheaf is a sheaf ${\ensuremath{\mathcal{A}}}$ over the space ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$, equipped with a continuous ${\ensuremath{\mathcal{G}}}$-action. For any such sheaf there are natural cohomology groups $H^n({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ whose definition we recall. Denote by ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}k{{\raise 1pt\hbox{\tiny )}}}}}}$ the space of composable arrows $$\label{string}
x_0\stackrel{g_1}{{\longrightarrow}} \ldots \stackrel{g_k}{{\longrightarrow}} x_k$$ of ${\ensuremath{\mathcal{G}}}$, and by $\epsilon_{k}: {\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}k{{\raise 1pt\hbox{\tiny )}}}}}}{\longrightarrow}{\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$ the map which sends (\[string\]) to $x_0$. The bar complex of ${\ensuremath{\mathcal{A}}}$ is defined by $B^k({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= \Gamma({\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}k{{\raise 1pt\hbox{\tiny )}}}}}}; \epsilon_{k}^{*}{\ensuremath{\mathcal{A}}})$, hence consists on continuous functions $c$ which associate to a string of arrows (\[string\]) an element $c(g_1, \ldots, g_k)\in {\ensuremath{\mathcal{A}}}_{x_0}$. The boundary is $\delta= \sum (-1)^{i}\delta_i$ with the same formulas as in (\[deltas\]). If ${\ensuremath{\mathcal{A}}}$ is “good” in the sense that ${\ensuremath{\mathcal{A}}}$ and its pull-backs $\epsilon_{k}^{*}{\ensuremath{\mathcal{A}}}$ are injective sheaves, then $H^n({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ is computed by the bar complex $B({\ensuremath{\mathcal{G}}}, {\ensuremath{\mathcal{A}}})$. In general, one chooses a resolution ${\ensuremath{\mathcal{S}}}^{*}$ of ${\ensuremath{\mathcal{A}}}$ by “good” ${\ensuremath{\mathcal{G}}}$-sheaves, and $H^n({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ is computed by the double complex $B^k({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{S}}}^l)$. In general, these cohomology groups coincide with the cohomology groups of the classifying space $B{\ensuremath{\mathcal{G}}}$ [@conj].\
Similarly, using compact supports and direct sums in the definition of the bar complex, one defines the homology groups $H_{*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ [@CrMo] (sometimes denoted $H_{c}^{*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= H_{-*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$), which should be thought of as a good model for the compactly supported cohomology of the classifying space.
[**$\check{C}$ech complexes:** ]{}Let ${\ensuremath{\mathcal{G}}}$ be an étale groupoid and let ${\ensuremath{\mathcal{U}}}$ be a basis of opens in ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$. A [*${\ensuremath{\mathcal{G}}}$-embedding*]{} $\sigma: U{\longrightarrow}V$ is a smooth family $\sigma(x)$, $x\in U$, where each $\sigma(x): x{\longrightarrow}y$ is an arrow in ${\ensuremath{\mathcal{G}}}$ from $x$ to some point $y\in V$; moreover, the map $x{\longrightarrow}$target$(\sigma(x))$ should define an embedding of $U$ into $V$. As in the first section, we can now define the $\check{C}$ech complex $\check{C}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$ for any ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$, $$\check{C}^{k}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})= \prod_{U_0 {\longrightarrow}\ldots {\longrightarrow}U_k} \Gamma(U_0, {\ensuremath{\mathcal{A}}})\ ,$$ where the product is over all strings of ${\ensuremath{\mathcal{G}}}$-embeddings between opens $U\in {\ensuremath{\mathcal{U}}}$, and the boundary $\delta= \sum(-1)^{i} \delta_i$ is given by the same formulas as in (\[deltas\]).\
We say that ${\ensuremath{\mathcal{A}}}$ is ${\ensuremath{\mathcal{U}}}$-acyclic if $H^{i}(U; {\ensuremath{\mathcal{A}}})= 0$ for each $i>0$ and each $U\in {\ensuremath{\mathcal{U}}}$. In this case define $\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ as the cohomology of $\check{C}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$. In general, we define $\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ as the cohomology of the double complex $\check{C}^{k}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{S}}}^{l})$, where $0{\longrightarrow}{\ensuremath{\mathcal{A}}}{\longrightarrow}{\ensuremath{\mathcal{S}}}^0{\longrightarrow}\ldots {\longrightarrow}{\ensuremath{\mathcal{S}}}^d{\longrightarrow}0$ is a bounded resolution by ${\ensuremath{\mathcal{U}}}$-acyclic sheaves, $d= dim({\ensuremath{\mathcal{G}}})$. By the usual arguments, such resolutions always exist, and the definition does not depend on the choice of the resolution.
\[CDRetale\][**Examples:** ]{}The ${\ensuremath{\mathcal{G}}}$-sheaf $\Omega_{{\ensuremath{\mathcal{G}}}}^{l}$ of $l$-differential forms with its natural ${\ensuremath{\mathcal{G}}}$-action is always ${\ensuremath{\mathcal{U}}}$-acyclic, as is any soft ${\ensuremath{\mathcal{G}}}$-sheaf. We obtain the $\check{C}$ech-De Rham (double) complex of ${\ensuremath{\mathcal{G}}}$, $\check{C}({\ensuremath{\mathcal{U}}}; \Omega)$, computing $\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; \mathbb{R})$. If the basis ${\ensuremath{\mathcal{U}}}$ consists of contractible opens (balls), then any locally constant ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$ is ${\ensuremath{\mathcal{U}}}$-acyclic, hence $\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ is computed by $\check{C}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$.
Similarly one defines the $\check{C}$ech complex with compact supports $\check{C}_{c}^{*}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$ using $$\bigoplus_{U_0 {\longrightarrow}\ldots {\longrightarrow}U_k} \Gamma_{c}(U_0, {\ensuremath{\mathcal{A}}})\ .$$ In order to get a cochain complex, we associate the degree $-k$ to the direct sums over strings of $k$ ${\ensuremath{\mathcal{G}}}$-embeddings. If ${\ensuremath{\mathcal{A}}}$ is $c$-soft, then $\check{H}_{c,\, {\ensuremath{\mathcal{U}}}}^{*}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ is defined by $\check{C}_{c}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$. In general, one uses a resolution $0{\longrightarrow}{\ensuremath{\mathcal{A}}}{\longrightarrow}{\ensuremath{\mathcal{S}}}^0{\longrightarrow}\ldots {\longrightarrow}{\ensuremath{\mathcal{S}}}^d{\longrightarrow}0$ by $c$-soft ${\ensuremath{\mathcal{G}}}$-sheaves, and the double complex $\check{C}^{k}_{c}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{S}}}^{l})$. The resulting cohomology is denoted $\check{H}^{*}_{c,\, {\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$.
[**The embedding category:** ]{}\[embdcat\]The notion of ${\ensuremath{\mathcal{G}}}$-embedding originates in [@embd], where the second author has introduced a small category $Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$ for each basis ${\ensuremath{\mathcal{U}}}$ of open sets. The objects of $Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$ are the members $U$ of ${\ensuremath{\mathcal{U}}}$, and the arrows are the ${\ensuremath{\mathcal{G}}}$-embeddings between the opens of ${\ensuremath{\mathcal{U}}}$. The main result of [@embd] was that the classifying space $B{\ensuremath{\mathcal{G}}}$ is weakly homotopy equivalent to the CW-complex $BEmb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$, provided each of the basic opens in ${\ensuremath{\mathcal{U}}}$ is contractible.\
Now any ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$ defines an obvious contravariant functor $\Gamma({\ensuremath{\mathcal{A}}})$ on $Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$ sending $U$ to $\Gamma(U; {\ensuremath{\mathcal{A}}})$, and $\check{C}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$ is just the usual (bar) complex computing the cohomology $H^*(Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}); \Gamma({\ensuremath{\mathcal{A}}}))$ of the discrete category $Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$ with coefficients. Hence [@embd] proves that $H^*({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= \check{H}^{*}_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})$ provided all the opens $U\in {\ensuremath{\mathcal{U}}}$ are contractible and ${\ensuremath{\mathcal{A}}}$ is (locally) constant. We now prove a stronger “$\check{C}$ech-De Rham isomorphism” which applies to more general coefficients, and also to compact supports.
\[theoreml\] Let ${\ensuremath{\mathcal{G}}}$ be an étale groupoid, and let ${\ensuremath{\mathcal{U}}}$ be a basis for ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$ as above. Then for any ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$, there are natural isomorphisms $$H^n({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= \check{H}^{n}_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) ,\ \ H^{n}_{c}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= \check{H}^{n}_{c,\, {\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) \ .$$
[*Proof:*]{} The proofs of the isomorphisms in the statement are similar, and we only prove the first one (an explicit proof of the second one also occurs in [@Crath]). By comparing resolutions of the ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$, it suffices to find a suitable complex $C({\ensuremath{\mathcal{A}}})$ and explicit quasi-isomorphisms $$B({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) {\longleftarrow}C({\ensuremath{\mathcal{A}}}) {\longrightarrow}\check{C}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}})$$ natural in ${\ensuremath{\mathcal{A}}}$, for the case where ${\ensuremath{\mathcal{A}}}$ is “good” in the sense of \[bar\]. For this we consider the bisimplicial space $S_{p, q}$, whose $p, q$-simplices are of the form $$\label{doublestring}
x_0\stackrel{g_1}{{\longrightarrow}} \ldots \stackrel{g_q}{{\longrightarrow}} x_q\stackrel{g}{{\longrightarrow}} U_0 \stackrel{\sigma_1}{{\longrightarrow}} \ldots \stackrel{\sigma_p}{{\longrightarrow}} U_p \ ,$$ where $\sigma_1, \ldots , \sigma_p$ are ${\ensuremath{\mathcal{G}}}$-embeddings, and $g_1, \ldots , g_q, g$ are arrows in the groupoid ${\ensuremath{\mathcal{G}}}$, the notation $x_q\stackrel{g}{{\longrightarrow}} U_0$ indicating that the target of $g$ is in $U_0$. The topology on $S_{p, q}$ is the topology induced from the topology on ${\ensuremath{\mathcal{G}}}$, $$S_{p, q}= \coprod_{U_0{\longrightarrow}\ldots {\longrightarrow}U_p} {\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}q{{\raise 1pt\hbox{\tiny )}}}}}} \ .$$ The ${\ensuremath{\mathcal{G}}}$-sheaf ${\ensuremath{\mathcal{A}}}$ induces a sheaf ${\ensuremath{\mathcal{A}}}_{p, q}$ on $S_{p, q}$ by pull-back along the projection which maps (\[doublestring\]) to $x_q$. Consider the double complex $C= C({\ensuremath{\mathcal{A}}})$, $$C^{p, q}= \Gamma(S_{p, q}, {\ensuremath{\mathcal{A}}}_{p, q}) \ .$$ For a fixed $p$, the complex $C^{p, *}$ is a product of complexes, namely, for each string $U_0{\longrightarrow}\ldots{\longrightarrow}U_p$, the bar complex (see \[bar\]) of the (étale) comma groupoid ${\ensuremath{\mathcal{G}}}/U_0$ with coefficients in the pull-back of the sheaf ${\ensuremath{\mathcal{A}}}$. Since the groupoid ${\ensuremath{\mathcal{G}}}/U_0$ is Morita equivalent to the space $U_0$, this cohomology is $H^{*}(U_0; {\ensuremath{\mathcal{A}}})$. Since ${\ensuremath{\mathcal{A}}}$ is assumed to be good, $H^*(U_0;{\ensuremath{\mathcal{A}}})$ vanishes in positive degrees, and we conclude that the canonical map $$\check{C}^{p}({\ensuremath{\mathcal{U}}}; {\ensuremath{\mathcal{A}}}) {\longrightarrow}C^{p, *}$$ is a quasi-isomorphism for each fixed $p$. Write $\pi_{p, q}: S_{p, q}{\longrightarrow}{\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}q{{\raise 1pt\hbox{\tiny )}}}}}}$ for the projection of (\[doublestring\]) to the string $x_0 {\longrightarrow}\ldots {\longrightarrow}x_q$. Then $C^{p, q}= \Gamma((\pi_{p, q})_*({\ensuremath{\mathcal{A}}}_{p, q}))$. The stalk of $(\pi_{p, q})_*({\ensuremath{\mathcal{A}}}_{p, q})$ at $x_0 {\longrightarrow}\ldots {\longrightarrow}x_q$ is $$\label{colimit}
\lim_{\overrightarrow{x_q\in U}}( \prod_{U{\longrightarrow}U_0{\longrightarrow}\ldots {\longrightarrow}U_p} \Gamma(U; {\ensuremath{\mathcal{A}}}) )\ ,$$ where the colimit is taken over all basic open neighborhoods $U$ of $x_q$. For a fixed $U$, the complex inside the $\lim$ in (\[colimit\]) computes the cohomology of the (discrete) comma category $U/Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$ with coefficients in the constant group $\Gamma(U, {\ensuremath{\mathcal{A}}})$. Since the comma category is contractible, so is this complex. Taking the colimit, we see that for each $q$ the map ${\ensuremath{\mathcal{A}}}{\longrightarrow}(\pi_{p, q})_*({\ensuremath{\mathcal{A}}}_{p, q})$ is a quasi-isomorphism of (complexes of) injective sheaves on ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}q{{\raise 1pt\hbox{\tiny )}}}}}}$. Thus the natural map $$B^{q}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}})= \Gamma({\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}q{{\raise 1pt\hbox{\tiny )}}}}}}; \epsilon_{q}^{*}{\ensuremath{\mathcal{A}}}) {\longrightarrow}\Gamma((\pi_{*, q})_*({\ensuremath{\mathcal{A}}}_{*, q}))= C^{*, q}$$ is a quasi-isomorphism, and the proof is complete. ${\raisebox{.8ex}{\framebox}}$\
Regarding the relation with the embedding category \[embdcat\] and its cohomology, let us point out the following immediate consequence, which is an improvement of the result of [@embd].\
If ${\ensuremath{\mathcal{G}}}$ is an étale groupoid, $\tilde{{\ensuremath{\mathcal{U}}}}$ is a basis of opens of ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$, and ${\ensuremath{\mathcal{A}}}$ is a ${\ensuremath{\mathcal{G}}}$-sheaf with the property that $H^k(U, {\ensuremath{\mathcal{A}}}|_{\,U})= 0$ for all $U\in {\ensuremath{\mathcal{U}}}$, $k\geq 1$, then $$H^*({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) \cong H^{*}(Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}); \Gamma({\ensuremath{\mathcal{A}}}))\ .$$ Similarly, if $H^{k}_{c}(U, {\ensuremath{\mathcal{A}}}|_{\,U})= 0$ for all $U\in {\ensuremath{\mathcal{U}}}$, $k\geq 1$, then $$H^{*}_{c}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) \cong H_{-*}(Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}); \Gamma_{c}({\ensuremath{\mathcal{A}}}))\ .$$ Also, if each $U\in {\ensuremath{\mathcal{U}}}$ is contractible, and ${\ensuremath{\mathcal{A}}}$ is locally constant as a sheaf on ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$, then $$H^*({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) \cong H^{*}(Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}); \Gamma({\ensuremath{\mathcal{A}}})),\ \ H^{*}_{c}({\ensuremath{\mathcal{G}}}; {\ensuremath{\mathcal{A}}}) \cong H_{d-*}(Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}}); H^{d}_{c}(-; {\ensuremath{\mathcal{A}}}))$$ (where $d$ is the dimension of the base space ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$).
\[CWetalgr\][**Chern-Weil for étale groupoids:** ]{}Clearly all the constructions of Section \[classes\] apply to any étale groupoid ${\ensuremath{\mathcal{G}}}$, provided we use the $\check{C}$ech-De Rham complexes mentioned in \[CDRetale\]. Hence, for any principal $G$-bundle $P$ endowed with a smooth action of ${\ensuremath{\mathcal{G}}}$, one has an associated Chern-Weil map $$S(\mathfrak{g}^{*})^G {\longrightarrow}H^{*}({\ensuremath{\mathcal{G}}}; \mathbb{R})$$ whose image vanishes in degrees $> 2d$, where $d= dim({\ensuremath{\mathcal{G}}})$. The refined characteristic map, $$H^{*}(W_{d}(\mathfrak{g}, K)) {\longrightarrow}H^{*}({\ensuremath{\mathcal{G}}}; \mathbb{R})$$ defines the exotic characteristic classes. Of particular interest is the (frame bundle of the) tangent space of ${\ensuremath{{\ensuremath{\mathcal{G}}}^{{{\raise 1pt\hbox{\tiny (}}}0{{\raise 1pt\hbox{\tiny )}}}}}}$, which is naturally endowed with an action of ${\ensuremath{\mathcal{G}}}$, and which induces the exotic characteristic map of ${\ensuremath{\mathcal{G}}}$, $$\label{exetale}
k_{{\ensuremath{\mathcal{G}}}}: H^{*}(WO_{d}) {\longrightarrow}\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}({\ensuremath{\mathcal{G}}})\cong H^{*}(B{\ensuremath{\mathcal{G}}}; \mathbb{R}) \ .$$
When ${\ensuremath{\mathcal{G}}}= Hol_{T}(M, {\ensuremath{\mathcal{F}}})$ this is the map discussed in section \[classes\]. But this is not the only interesting example. For instance, if one works with foliated bundles which are not necessarily transversal (as e.g. in [@KaTo]), then one has to replace the holonomy groupoid $Hol_{T}(M, {\ensuremath{\mathcal{F}}})$ by the monodromy groupoid $Mon_{T}(M, {\ensuremath{\mathcal{F}}})$. The new versions of Theorem \[theorem2\] and Corollary \[corex\] for foliated bundles then yield characteristic classes in $H^*(BMon_{T}(M, {\ensuremath{\mathcal{F}}}))$. These classes are refinements of the characteristic classes in $H^*(M)$, already constructed in [@KaTo].\
Another interesting example is when ${\ensuremath{\mathcal{G}}}$ is Haefliger’s $\Gamma^q$. The importance of this example lies into the fact that $\Gamma^q$ plays a classifying role for codimension $q$ foliations, hence its cohomology consists on “universal” classes. We will elaborate this in \[univfor\] of the next section.\
Explicit formulas {#explicit}
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In this section we illustrate our constructions in the case of normal bundles. In particular we deduce Bott’s formulas for cocycles associated to group actions [@Boform], as well as Thurston’s formula.
[**Explicit formulas for the normal bundle:** ]{}We now apply the construction of the exotic characteristic map of Section \[classes\] to the normal bundle $\nu$. Corollary \[corex\] applied to the (principal $GL_q$-bundle associated to) $\nu$ provides us with a characteristic map $$k_{{\ensuremath{\mathcal{F}}}}: H^*(WO_q) {\longrightarrow}\check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}}) \ ,$$ which, when composed with the pull-back $\pi^{*}: \check{H}_{{\ensuremath{\mathcal{U}}}}^*(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^*(M)$, gives the familiar exotic characteristic classes [@Bott] of ${\ensuremath{\mathcal{F}}}$. Here $WO_{q}$ is the standard [@Bott] simplification of the truncated relative Weil complex $W_q(\mathfrak{gl}_{q}, O(q))$ that we now recall. The idea is that the relative Weil complex $W(\mathfrak{gl}_{q}, O(q))$ (see \[clCW\]) is quasi-isomorphic to a smaller subcomplex, namely the dg algebra $S[c_1,\, .\, .\, .\, , c_q] \otimes E(h_1, h_3, \, .\, .\, .\, , h_{2[\frac{q+1}{2}]-1})$ generated by elements $c_i$ of degree $2i$ (namely the polynomials $c_i(A)= Tr(A^i)$), elements $h_{2i+1}$ of degree $4i+1$ (any elements which transgress $c_{2i+1}$), with the boundary $$d(c_i)= 0, \ \ d(h_{2i+1})= c_{2i+1} .$$ Truncating by polynomials of degree $> q$, the resulting inclusion $$WO_q:= S_q[c_1,\, .\, .\, .\, , c_q] \otimes E(h_1, h_3, \, .\, .\, .\, , h_{2[\frac{q+1}{2}]-1}){\longrightarrow}W_q(\mathfrak{gl}_{q}, O(q))$$ induces isomorphism in cohomology. With this simplification, the desired cohomology can be computed explicitly. Apart from the classical Chern elements $c_i$ (non-trivial only for $i< q/2$ even) there are new exotic classes. Referring to [@Godb] for the complete description of $H^*(WO_q)$, we recall here that the simplest such class is the Godbillon-Vey class $gv= [h_{1}c_{1}^{q}]\in H^{2q+1}(WO_q)$. We denote by $gv_{{\ensuremath{\mathcal{F}}}}\in \check{H}_{{\ensuremath{\mathcal{U}}}}^{*}(M/{\ensuremath{\mathcal{F}}})$ the resulting cohomology class $k_{{\ensuremath{\mathcal{F}}}}(gv)$. Its pull-back to $H^{*}(M)$ is the usual Godbillon-Vey class of ${\ensuremath{\mathcal{F}}}$. More generally, the Bott-Godbillon-Vey classes $gv^{\alpha}= [u_1c_{\alpha_1} \ldots c_{\alpha_t}]$ (and their images $gv^{\alpha}_{{\ensuremath{\mathcal{F}}}}$) are defined for any partition $\alpha= (\alpha_1, \ldots , \alpha_t)$ of $q$ (i.e. $q= \sum \alpha_i$).\
For explicit formulas, let us choose a basis ${\ensuremath{\mathcal{U}}}$ so that $\tilde{{\ensuremath{\mathcal{U}}}}$ are also domains of trivialization charts for $\nu$ (as in \[rmks2.1\]). Let $J_{h}: U{\longrightarrow}GL_q$ denote the Jacobian of $h: U{\longrightarrow}V$ (any holonomy embedding). Then the $J_{h}$’s are the associated transition functions of the transversal bundle $\nu$. Locally, we choose the trivial connection $\nabla_U$ over $U$. The corresponding $\nabla(h)$ are then given by the connection $1$-forms: $$\omega_{h}:= J_{h}^{-1} dJ_{h} \in \Omega^1(U; \mathfrak{gl}_q),$$ for $h: U{\longrightarrow}V$. We see that the Chern character $Ch_{\nu}\in \check{C}^2({\ensuremath{\mathcal{U}}},\Omega^*)$ is given by: $$(h_1, \ldots , h_p)\mapsto (-1)^p\int_{t_0+ t_1+ \ldots + t_p\leq 1} exp(\ (t_1\omega_{h_1}+ t_2\omega_{h_2h_1}+ \ldots + t_p\omega_{h_p
\ldots h_2h_1})^2\ ) dt_0dt_1 . . . dt_p\ .$$ For instance, the first class $C_1= ch_1(\nu)\in \check{C}^*({\ensuremath{\mathcal{U}}},\Omega^*)$ has the components $$C_1^{{{\raise 1pt\hbox{\tiny (}}}1, 1{{\raise 1pt\hbox{\tiny )}}}}(h)= Tr( J_{h}^{-1}dJ_{h}), \ C_1^{{{\raise 1pt\hbox{\tiny (}}}0, 2{{\raise 1pt\hbox{\tiny )}}}}= C_1^{{{\raise 1pt\hbox{\tiny (}}}2, 0{{\raise 1pt\hbox{\tiny )}}}}= 0\ .$$ As we know, this class is cohomologically trivial. This can be seen directly, since $U_1\in \check{C}^1({\ensuremath{\mathcal{U}}},\Omega^*)$, $$U_{1}^{{{\raise 1pt\hbox{\tiny (}}}0, 1{{\raise 1pt\hbox{\tiny )}}}}= 0, \ U_{1}^{{{\raise 1pt\hbox{\tiny (}}}1, 0{{\raise 1pt\hbox{\tiny )}}}}(h)= log(\mid det(J_{h})\mid)$$ transgresses $C_1$. Computing the resulting closed cocycle $U_1C_{1}^{q}$ we see that
\[godvey\] The Godbillon-Vey class $gv_{{\ensuremath{\mathcal{F}}}}\in\check{H}^{2q+1}(M/{\ensuremath{\mathcal{F}}})$ is represented in the $\check{C}ech$- De Rham complex by the cocycle $gv_{{\ensuremath{\mathcal{F}}}}$ living in bi-degree $(q+1, q)$: $$\label{forgv}
gv_{{\ensuremath{\mathcal{F}}}}( h_1, \ldots , h_{q+1})= log(\mid
det(J_{h_1})\mid) h_{1}^* Tr(\omega_{h_2})
h_{1}^*h_{2}^*Tr(\omega_{h_3}) \ldots
h_{1}^* . . . h_{q}^* Tr(\omega_{h_{q+1}}).$$
Similarly, computing $U_1C_{\alpha_1}\ldots C_{\alpha_t}$ for a partition $\alpha= (\alpha_1, \ldots , \alpha_t)$ of $q$, we obtain the following formula, which explains Bott’s definition of the cocycles associated to group actions [@Boform].
The Bott-Godbillon-Vey class $gv^{\alpha}_{{\ensuremath{\mathcal{F}}}}\in\check{H}^{2q+1}(M/{\ensuremath{\mathcal{F}}})$ is represented in the $\check{C}ech$- DeRham complex by the closed cocycle $gv^{\alpha}_{{\ensuremath{\mathcal{F}}}}$ living in bi-degree $(q+1, q)$: $$gv^{\alpha}_{{\ensuremath{\mathcal{F}}}}( h_1, \ldots , h_{q+1})= log(\mid
det(J_{h_1})\mid)\cdot h_{1}^*\{
Tr[\ \omega_{h_2}\cdot h_{2}^*(\omega_{h_3})\cdot \ldots (h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}-1{{\raise 1pt\hbox{\tiny )}}}}
\ldots h_2)^*(\omega_{h_{\alpha_1}})]\ \} \cdot$$ $$(h_{\alpha_1}\ldots h_2 h_1)^*\{ Tr[ \
\omega_{h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}+1{{\raise 1pt\hbox{\tiny )}}}}}\cdot
h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}+1{{\raise 1pt\hbox{\tiny )}}}}^*(h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}+2{{\raise 1pt\hbox{\tiny )}}}})\cdot \cdots
(h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}+\alpha_{2}- 1{{\raise 1pt\hbox{\tiny )}}}} \cdots h_{{{\raise 1pt\hbox{\tiny (}}}\alpha_{1}+1{{\raise 1pt\hbox{\tiny )}}}})^*
(\omega_{h_{\alpha_2}})]\ \}\cdot \cdots$$
\[univfor\][**Universal formulas.**]{} As pointed out in the previous section, the constructions that we described for foliations apply to any étale groupoid. Due to its classifying properties, the case of the Haefliger groupoid $\Gamma^q$ is of particular interest. We wish to explain how the $\check{C}$ech-De Rham model for $\Gamma^q$ can be used to derive, in an explicit and straightforward way, the known formulas and properties of universal characteristic classes for codimension $q$ foliations. We emphasize that all these properties are now part of the folklore on characteristic classes for foliations, but they are usually derived by non-trivial abstract arguments at the level of classifying spaces.\
First of all we make a slight simplification of the $\check{C}$ech-De Rham complex of $\Gamma^q$. Choosing the basis ${\ensuremath{\mathcal{U}}}$ of $\mathbb{R}^q$ by discs, since any such disc is diffeomorphic to $\mathbb{R}^q$, we see that the category $Emb_{{\ensuremath{\mathcal{U}}}}(\Gamma^q)$ is equivalent to the category which has only one object, and all the embeddings $\mathbb{R}^q{\longrightarrow}\mathbb{R}^q$ as arrows. Accordingly, we define $\check{C}(\Gamma^q; \Omega)$ as in the previous sections, except that we take products only over strings $$\mathbb{R}^q \stackrel{\sigma_1}{{\longrightarrow}} \ldots \stackrel{\sigma_k}{{\longrightarrow}} \mathbb{R}^q$$ of embeddings $\mathbb{R}^q{\longrightarrow}\mathbb{R}^q$. The main theorem of this section implies
The $\check{C}$ech-De Rham complex $\check{C}(\Gamma^q; \Omega)$ computes $H^*(B\Gamma^q; \mathbb{R})$.
Now we can describe the main (cohomological) universal properties of $\Gamma^q$ in an explicit (and obvious) fashion. First of all, the universal property of $\Gamma^q$ can be seen easily in cohomology: given any codimension $q$ foliation, choosing a basis $\tilde{{\ensuremath{\mathcal{U}}}}$ for $M$ and a transversal basis ${\ensuremath{\mathcal{U}}}$ as in \[rmks2.1\], there is an obvious map $\check{H}(\Gamma^q) {\longrightarrow}\check{H}_{\tilde{{\ensuremath{\mathcal{U}}}}}(M)$, to be seen as the map induced in cohomology by the classifying map $M{\longrightarrow}B\Gamma^q$ of ${\ensuremath{\mathcal{F}}}$ (well defined up to homotopy). This map is the composition of the pull-back (\[zero\]) with another obvious map $$\label{clasmap}
\check{H}^*(\Gamma^q) {\longrightarrow}\check{H}^{*}_{{\ensuremath{\mathcal{U}}}}(M/{\ensuremath{\mathcal{F}}})$$ (compare to [@Bott]). Now, all the characteristic maps for codimension $q$ foliations are just the composition of the (\[clasmap\])’s with a universal map $$\label{univchar}
k_{q}: H^*(WO_q) {\longrightarrow}\check{H}^*(\Gamma^q)\ .$$ Again, with the $\check{C}$ech-De Rham complexes at hand this is obvious, and $k_{q}$ is not at all abstract: it is just the characteristic map (\[exetale\]) applied to ${\ensuremath{\mathcal{G}}}= \Gamma^q$ and can be described in terms of the trivial connection on $\mathbb{R}^q$ (compare to [@Bott]). In particular, all the formulas of section \[classes\] come from similar universal formulas in $\check{C}(\Gamma^q; \Omega)$.\
At the price of more complicated formulas, we can further simplify the complex $\check{C}(\Gamma^q; \Omega)$. Indeed, since the cohomology of $\Omega^*(\mathbb{R}^q)$ is $\mathbb{R}$ concentrated in degree zero (Poincaré lemma), we see that $\check{H}(\Gamma^q)$ is also computed by the $\check{C}$ech (subcomplex) with constant coefficients $$\check{C}(\Gamma^q):\ \ \ 0{\longrightarrow}\mathbb{R} {\longrightarrow}\prod_{\mathbb{R}^q\stackrel{\sigma_1}{{\longrightarrow}}\mathbb{R}^{q}} \mathbb{R} {\longrightarrow}\prod_{\mathbb{R}^q\stackrel{\sigma_1}{{\longrightarrow}}\mathbb{R}^{q}\stackrel{\sigma_2}{{\longrightarrow}}\mathbb{R}^{q}} \mathbb{R} {\longrightarrow}\ldots$$ To pass from $\check{C}(\Gamma^q; \Omega)$ to $\check{C}(\Gamma^q)$ one has to repeatedly apply the Poincaré lemma. After a lengthy but straightforward computation (for the details see Lemma 3.3.8 in [@Crath]) we obtain:
\[uuv\] An $n$-cocycle in the $C$ech-De Rham complex: $$u= u_0+ u_1+ \ldots + u_n, \ \ u_k\in \check{C}^k(\Gamma^q,\Omega^{n-k})$$ represents the same class in $\check{H}^{n}(\Gamma^q)$ as the $n$-cocycle $\tilde{u}$ in the $\check{C}$ech complex $\check{C}^*(\Gamma^q)$, given by: $$\tilde{u}(\sigma_1, \ldots , \sigma_n)= \sum_{s=0}^{n} (-1)^{ n{{\raise 1pt\hbox{\tiny (}}}s-1{{\raise 1pt\hbox{\tiny )}}}+ \frac{s{{\raise 1pt\hbox{\tiny (}}}s-1{{\raise 1pt\hbox{\tiny )}}}}{2}}\int_{I_{\sigma_1, \ldots , \sigma_s}} u_{n-s}(\sigma_{s+1}, \ldots , \sigma_n) .$$ Here, $I_{\sigma_1, \ldots , \sigma_s}$ is the $s$-cube: $$I_{\sigma_1, \ldots , \sigma_s}(t_1, \ldots , t_s)= \sigma_s(\sigma_{s-1}( \ldots \sigma_3(\sigma_2(\sigma_1(0)t_1)t_2) \ldots ) t_{s-1}) t_s .$$
If we apply this to the Godbillon-Vey cocycle (i.e. to the formula (\[forgv\]) in the $\check{C}$ech-De Rham complex $\check{C}(\Gamma^q; \Omega)$), we obtain the well-known Thurston’s formula:
The universal Godbillon-Vey class $GV\in H^3(B\Gamma^1)\cong \check{H}^{3}(\Gamma^1)$ is represented in $\check{C}(\Gamma^1)$ by the cocycle: $$\tilde{gv}_1(\sigma_1, \sigma_2, \sigma_3)= \int_{0}^{\sigma_1(0)}
log(\mid \sigma_{2}'(t)\mid)
\frac{\sigma_{3}''(\sigma_2(t))}{\sigma_{3}'(\sigma_2(t))}
\sigma_{2}'(t) dt .$$
Relations to basic cohomology {#secbasic}
=============================
In the previous sections we have seen various models for the cohomology of the leaf space, all canonically isomorphic. Let us put $$\label{cohls}
H^{*}(M/{\ensuremath{\mathcal{F}}})= H^{*}(Hol_{T}(M, {\ensuremath{\mathcal{F}}})),\ \ H^{*}_{c}(M/{\ensuremath{\mathcal{F}}})= H^{*}_{c}(Hol_{T}(M, {\ensuremath{\mathcal{F}}}))\ .$$ The reader may choose one of the many models: Haefliger’s model (as indicated by the above notations) i.e. \[bar\] applied to the holonomy groupoid reduced to any complete transversal $T$, the $\check{C}$ech-De Rham model that we have described in section \[CDRcomplex\] (cf. Proposition \[lema1\]), or the classifying-space model (cf. Theorem \[theorem1\]). We emphasize however that the last model only works for the cohomology without restriction on the supports!\
Here and in the next section we explain why these cohomology theories are suitable theories for the leaf space. We first compare them to the more familiar [*basic cohomology*]{} (see e.g. [@minimal; @Ser]), which is a different cohomology theory for leaf spaces.\
[**Basic cohomology.**]{} Choosing a basis ${\ensuremath{\mathcal{U}}}$ of opens of a complete transversal $T$ (or any transversal basis for ${\ensuremath{\mathcal{F}}}$), one defines $\Omega^{k}_{bas}(T/{\ensuremath{\mathcal{F}}})$ as the cohomology of $\check{C}^{*}({\ensuremath{\mathcal{U}}}, \Omega^{k})$ in degree $*= 0$. This complex consists on $k$-forms on $T$ which are invariant under holonomy, hence it does not depend on the choice of $T$ (up to canonical isomorphisms, of course). The resulting cohomology is denoted $H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}})$. There is an obvious map (induced by an inclusion of complexes) $$\label{jmap}
j: H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*}(M/{\ensuremath{\mathcal{F}}})\ .$$ Similarly one defines the basic cohomology with compact supports $H^{*}_{c, bas}(M/{\ensuremath{\mathcal{F}}})$ [@minimal]. The corresponding complex $\Omega^{k}_{c, bas}(T/{\ensuremath{\mathcal{F}}})$ is the homology of $\check{C}^{*}_{c}({\ensuremath{\mathcal{U}}}, \Omega^{k})$ in degree $*= 0$, i.e., as in [@minimal], the quotient of $\oplus_{U\in {\ensuremath{\mathcal{U}}}} \Omega^{k}_{c}(U)$ by the span of elements of type $\omega- h^{*}\omega$ ($h: U{\longrightarrow}V$ is a holonomy embedding, and $\omega\in \Omega_{c}^{k}(V)$). Again, there is an obvious map $$\label{jcmap}
j_{c}: H^{*}_{c}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*}_{c, bas}(M/{\ensuremath{\mathcal{F}}})\ .$$ In general, the maps (\[jmap\]) and (\[jcmap\]) are not isomorphisms. The basic cohomologies are much smaller then $H^{*}(M/{\ensuremath{\mathcal{F}}})$; for instance $H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}})= 0$ in degrees $*>q$, and they are finite dimensional if ${\ensuremath{\mathcal{F}}}$ is riemannian and $M$ is compact. The price to pay is the failure of most of the familiar properties from algebraic topology (e.g., as discussed below, Poincare duality and characteristic classes). However we point out that (\[jmap\]) and (\[jcmap\]) are isomorphisms when the naive leaf space is an orbifold. This was explained in 4.9 of [@CrMo], but the reader should think about the similar statement for actions of finite groups on manifolds, and the fact that the cohomology (over $\mathbb{R}$) of finite groups is trivial. In particular (see also [@Mol]), we have\
\[cptl\] If $(M, {\ensuremath{\mathcal{F}}})$ is a riemannian foliation with compact leaves, then [(\[jmap\])]{} and [(\[jcmap\])]{} are isomorphisms.
Another fundamental property of our cohomologies (\[cohls\]) is
\[pdutr\](Poincaré duality) For any codimension $q$ foliation $(M, {\ensuremath{\mathcal{F}}})$, $$\label{pppo}
H^{*}(M/{\ensuremath{\mathcal{F}}}; {\ensuremath{\mathcal{O}}}) \cong H^{q-*}_{c}(M/{\ensuremath{\mathcal{F}}})^{\vee} \ .$$
This (and the more general Verdier duality) has been proved in [@CrMo]. Note however that, with the $\check{C}$ech model in hand, the theorem becomes obvious. This new proof of Poincaré duality can be viewed as a rather straightforward extension of the classical proof for manifolds [@BoTu] (and can also be interpreted as the obvious duality between the homology and the cohomology of the discrete category $Emb_{{\ensuremath{\mathcal{U}}}}({\ensuremath{\mathcal{G}}})$, cf. \[embdcat\]). In contrast, the basic cohomologies $H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}})$ and $H^{*}_{c, bas}(M/{\ensuremath{\mathcal{F}}})$ satisfy Poincaré duality only in the riemannian case [@Ser]. In this case these dualities are compatible via (\[jmap\]) and (\[jcmap\]), and they coincide if the leaves are compact (see Proposition \[cptl\]).
[**Characteristic classes.**]{} As we have seen, one of the main features of $H^*(M/{\ensuremath{\mathcal{F}}})$ is that it contains the characteristic classes of the bundles over the leaf space (i.e. transversal bundles), and the Bott vanishing theorem and the construction of the exotic classes hold at this level. Regarding the groups $H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}})$, again, they are too small to contain these characteristic classes. But, as before, this is not seen in the case of riemannian foliations. The reason is that, if ${\ensuremath{\mathcal{F}}}$ is riemannian, then the transversal metric induces a [*transversal connection*]{}, i.e. a connection which is invariant under holonomy. Using this type of connections in the construction the characteristic maps $k_{\nu}$ of the normal bundle $\nu$, we see that $k_{\nu}: S(gl_{q}^{*})^{inv}{\longrightarrow}H^*(M/{\ensuremath{\mathcal{F}}})$ vanishes in degrees $> q$. This stronger vanishing result (at the level of $H^*(M)$), together with the construction of the refined exotic characteristic map, appears in [@LaPa]. Moreover, using the explicit constructions of section \[classes\], we see that $k_{\nu}$ (and its exotic versions) factors through the basic cohomology groups. This obviously applies to general transversal bundles. In conclusion,
If $P$ is a transversal principal $G$-bundles over $(M, {\ensuremath{\mathcal{F}}})$ which admits a transversal connection then the characteristic map $k_{P}: S(\mathfrak{g}^{*})^{G}{\longrightarrow}H^*(M/{\ensuremath{\mathcal{F}}})$ of $P$ (cf. Theorem [\[theorem2\]]{}) vanishes in degrees $> q$. Moreover the map $k_{P}$ (and its exotic version, cf. Corollary [\[corex\]]{}) factors through the basic cohomology groups:$$\xymatrix{
S(\mathfrak{g})^{G} \ar@{.}[rrr] \ar[rrrd]^-{k_{P}}
& & & H^{*}_{bas}(M/{\ensuremath{\mathcal{F}}}) \ar[d]^-{j} \\
& & & H^{*}(M/{\ensuremath{\mathcal{F}}})
}$$
\[intf\][**Integration along the leaves.**]{} Haefliger’s original motivation [@minimal] for introducing $H^{*}_{c, bas}(M/{\ensuremath{\mathcal{F}}})$ is the existence of an integration over the leaves map $\int_{{\ensuremath{\mathcal{F}}}}^{\,'}: H^{*}_{c}(M){\longrightarrow}H^{*-p}_{c, bas}(M/{\ensuremath{\mathcal{F}}})$ when the bundle of vectors tangent to the leaves is oriented. We want to point out the existence of a refined integration, $$\label{integr2}
\int_{{\ensuremath{\mathcal{F}}}}: H^{*}_{c}(M){\longrightarrow}H^{*-p}_{c}(M/{\ensuremath{\mathcal{F}}})\ ,$$ which, composed with the canonical map (\[jcmap\]), gives precisely Haefliger’s integration. Using the $\check{C}$ech model this map becomes obvious: choosing ${\ensuremath{\mathcal{U}}}$, $\tilde{{\ensuremath{\mathcal{U}}}}$ as in \[rmks2.1\], the integration over the plaques (with the induced orientation), $\int: \Omega^{*}_{c}(\tilde{U}){\longrightarrow}\Omega^{*-p}_{c}(U)$, induces a map at the level of the $\check{C}$ech-De Rham complexes associated to ${\ensuremath{\mathcal{U}}}$ and $\tilde{{\ensuremath{\mathcal{U}}}}$.\
An alternative abstract definition of $\int_{{\ensuremath{\mathcal{F}}}}$ follows e.g. from the spectral sequences of [@CrMo] by standard methods of algebraic topology (“integration over the fiber” as an edge map). The Hochschild-Serre spectral sequence (i.e. Theorem 4.4 of [@CrMo] applied to $\pi: M{\longrightarrow}M/{\ensuremath{\mathcal{F}}}$) takes the form $H^{s}_{c}(M/{\ensuremath{\mathcal{F}}}; {\ensuremath{\mathcal{L}}}^{t})\Longrightarrow H^{s+t}_{c}(M)$, where ${\ensuremath{\mathcal{L}}}^{t}$ is a transversal sheaf whose stalk above a leaf $L$ is $H^{t}_{c}(\tilde{L})$. This second description provides us with qualitative information. E.g., if the holonomy covers of the leaves are $k$-connected, we find that $\int_{{\ensuremath{\mathcal{F}}}}$ is isomorphism in degrees $n-k\leq * \leq n$. Using Poincaré duality, it follows that the pull-back map $H^{*}(M/{\ensuremath{\mathcal{F}}}){\longrightarrow}H^*(M)$ is isomorphism in degrees $0\leq *\leq k$.
Relations to foliated cohomology
================================
Another standard cohomology theory in foliation theory is the [*foliated cohomology*]{} of foliations (see e.g. [@Alv; @Hei; @KT; @MoSo]). In contrast to the other cohomologies that we have seen so far (transversal cohomologies), the foliated cohomology contains a great deal of longitudinal information. In this section we describe its relation to our $\check{C}$ech-De Rham cohomology.
[**Foliated cohomologies.**]{} The foliated cohomology $H^*({\ensuremath{\mathcal{F}}})$ is defined in analogy with the De Rham cohomology of $M$, which we recover if ${\ensuremath{\mathcal{F}}}$ has only one leaf. The defining complex is $\Omega^*(M, {\ensuremath{\mathcal{F}}})= \Gamma(\Lambda^*{\ensuremath{\mathcal{F}}})$, with the boundary defined by the usual Koszul-formula $$\begin{aligned}
\label{differential}
d(\omega)(X_1, \ldots , X_{p+1}) & = & \sum_{i<j}
(-1)^{i+j-1}\omega([X_i, X_j], X_1, \ldots , \hat{X_i}, \ldots ,
\hat{X_j}, \ldots X_{p+1})) \nonumber \\
& + & \sum_{i=1}^{p+1}(-1)^{i}
L_{X_i}(\omega(X_1, \ldots, \hat{X_i}, \ldots , X_{p+1})) .\end{aligned}$$ Here $L_{X}(f)= X(f)$. For later reference, we note the existence of an obvious (restriction to ${\ensuremath{\mathcal{F}}}$) $$\label{restr}
r: H^*(M){\longrightarrow}H^*({\ensuremath{\mathcal{F}}})$$ There is also a version with compact supports, as well as versions $H^*({\ensuremath{\mathcal{F}}}; E)$ with coefficients in any transversal (or foliated) vector bundle $E$: one uses $E$-valued forms on ${\ensuremath{\mathcal{F}}}$, and one replaces the $L_{X_i}$ in the previous formula, by the derivatives $\nabla_{X_i}$ w.r.t. the Koszul connection of $E$ (see \[trbd\]).
[**Remarks.**]{} In [@MoSo], the cohomology $H^*({\ensuremath{\mathcal{F}}})$ is called “tangential cohomology”, and is denoted $H^{*}_{\tau}(M)$. The groups $H^*({\ensuremath{\mathcal{F}}}; \nu)$ with coefficients in the normal bundle (see \[trbd\]) first appeared in [@Hei] in the study of deformations of foliations, while those with coefficients in the exterior powers $\Lambda\nu$ show up e.g. in the spectral sequence relating the foliated cohomology with De Rham cohomology [@Alv; @KT]. The groups $H^*({\ensuremath{\mathcal{F}}}; E)$ with general coefficients can also be viewed as an instance of algebroid cohomology [@McK]. Regarding the characteristic classes, since the Bott connection (see \[trbd\]) is flat, it follows that the characteristic classes of the normal bundle are annihilated by $r$. This new vanishing result at the level of foliated cohomology produces new (“secondary”) classes, $u_{4k-1}({\ensuremath{\mathcal{F}}})\in H^{4k-1}({\ensuremath{\mathcal{F}}})$. These appear in [@Gold] and have been described in great detail in [@Crave] in the more general context of algebroids. In particular, these new classes come from the cohomology groups $H^*(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas})$ (via the map (\[Phi\]) below). Still related to [@Crave], let us mention that if ${\ensuremath{\mathcal{F}}}$ is the foliation induced by a regular Poisson structure on $M$, then one has an induced foliated bundle $K$ (the kernel of the anchor map), and $H^2({\ensuremath{\mathcal{F}}}; K)$ contains obstructions to the integrability of the Poisson structure.
As explained in [@MoSo] in the case of trivial coefficients, and in [@Hei] in the case of the normal bundle as coefficients, the foliated cohomology can be expressed as the cohomology of certain sheaves on $M$. For general coefficients $E$ we consider the sheaf $\Gamma_{\nabla}(E)$ described in \[trsh\]. A version of Poincaré’s lemma with parameters shows that $H^k({\ensuremath{\mathcal{F}}}; E)= 0$ in degrees $k>0$ if ${\ensuremath{\mathcal{F}}}$ is the standard $p$-dimensional foliation of $M= \mathbb{R}^p\times \mathbb{R}^q$. Since always $\Gamma_{\nabla}(M; E)= H^{0}({\ensuremath{\mathcal{F}}}; E)$, we deduce that $U\mapsto \Omega^*({\ensuremath{\mathcal{F}}}|_{\, U}; E|_{\, U})$ is a flabby resolution of $\Gamma_{\nabla}(E)$, hence
\[folsheaf\] For any foliated vector bundle $E$ over $(M, {\ensuremath{\mathcal{F}}})$, $H^*({\ensuremath{\mathcal{F}}}; E)$ is isomorphic to $H^{*}(M; \Gamma_{\nabla}(E))$, the cohomology of $M$ with coefficients in the sheaf of $\nabla$-constant sections of $E$. In particular, $H^{*}({\ensuremath{\mathcal{F}}})\cong H^{*}(M; \Omega_{bas}^{0})$. The same holds for compact supports.
[**Comparison.**]{} We now note the existence of a canonical map $$\label{Phi}
\Phi: H^{*}(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas}) {\longrightarrow}H^*({\ensuremath{\mathcal{F}}})\ .$$ Recall that $\Omega^{0}_{bas}$, as a sheaf on $M$, is the sheaf of smooth function which are constant on the leaves. This map has various interpretations. First of all, it can be viewed as a version with coefficients of the pull-back map (\[map2.1\]) (cf. also Proposition \[folsheaf\]). Accordingly, the simplest description is in terms of the $\check{C}$ech-De Rham model. Choosing ${\ensuremath{\mathcal{U}}}$ and $\tilde{{\ensuremath{\mathcal{U}}}}$ as in \[rmks2.1\], the left hand side of (\[Phi\]) is computed by the cochain complex $\check{C}^*({\ensuremath{\mathcal{U}}}; C^{\infty}(U))$, which is obviously a subcomplex of the $t= 0$ column of $\check{C}^s(\tilde{{\ensuremath{\mathcal{U}}}}; \Omega^t(\tilde{U},{\ensuremath{\mathcal{F}}})$. Now (\[Phi\]) is the map induced in cohomology. Alternatively, at least when the holonomy groupoid is Hausdorff, $H^{*}(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas})$ coincide with the differentiable cohomology [@difcoh] of the holonomy groupoid of ${\ensuremath{\mathcal{F}}}$, and $\Phi$ is precisely the associated Van Est map described in [@WeXu]. It then follows from one of the main results of [@Crave] (applied to the holonomy groupoid) that $\Phi$ is an isomorphism in degrees $\leq k$ provided the leaves (or their holonomy covers) are $k$-connected. As in the previous section (see \[intf\]), the same result follows e.g. from the spectral sequences of [@CrMo].
\[intfle\][**Integration along the leaves.**]{} If ${\ensuremath{\mathcal{F}}}$ is oriented, then we have an integration map $$\label{integr3}
\int_{{\ensuremath{\mathcal{F}}}}: H^{*}_{c}({\ensuremath{\mathcal{F}}}){\longrightarrow}H^{*-p}_{c}(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas})\ .$$ This map is dual to the Van Est map (\[Phi\]) and can be viewed as a version of the integration map (\[integr2\]) with coefficients in the normal bundle (accordingly, there are similar maps for any transversal vector bundle $E$ over $M$, cf. also \[trsh\] and Proposition \[folsheaf\]). Again, as in the previous section (see \[intf\]), this map (\[integr3\]) becomes obvious if one uses the $\check{C}$ech-De Rham model.\
We want to point out here that the integration over the fibers that we have described clarifies the construction of the Ruelle-Sullivan current of a measured foliation (cf. e.g. Section 3 of [@CoOp], or [@MoSo] p 126), and also gives new qualitative information about it. Fix a transversal basis ${\ensuremath{\mathcal{U}}}$ for ${\ensuremath{\mathcal{F}}}$. A smooth transverse measure $\mu$ is just a measure on each $U\in {\ensuremath{\mathcal{U}}}$, which is invariant w.r.t. holonomy embeddings. Hence the integration against $\mu$ is simply a linear map $$\int_{\mu}: \Omega^{0}_{c, bas}(M/{\ensuremath{\mathcal{F}}})= H^{0}_{c}(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas}){\longrightarrow}\mathbb{R}\ .$$ Combining this with the integrations along the leaves (\[integr2\]), (\[integr3\]), we can arrange our maps into a diagram (see also (\[restr\])) $$\xymatrix{
H^{p}_{c}(M) \ar[r]^-{\int_{{\ensuremath{\mathcal{F}}}}} \ar[d]_-{r} & H_{c}^{0}(M/{\ensuremath{\mathcal{F}}}) \ar[d] & \\
H^{p}_{c}({\ensuremath{\mathcal{F}}})\ar[r]^-{\int_{{\ensuremath{\mathcal{F}}}}} & H_{c}^{0}(M/{\ensuremath{\mathcal{F}}}; \Omega^{0}_{bas}) \ar[r]^-{\int_{\mu}} & \mathbb{R}
}$$ The resulting map $\int_{C}: H^{p}_{c}(M) {\longrightarrow}\mathbb{R}$ is precisely the integration of [@CoOp] against the Ruelle-Sullivan current $C= C_{\mu}$ (and this defines $C$ as a degree $p$ element in the closed homology of $M$). As pointed out in [@MoSo], $C$ actually comes from the closed homology of ${\ensuremath{\mathcal{F}}}$. In terms of our diagram this simply means that $\int_{C}$ factors through $H^{p}_{c}({\ensuremath{\mathcal{F}}})$.
[**Spectral sequences.**]{} Almost all of the maps that we have described in the last two sections figure in certain the spectral sequences. First of all, the filtration on $\Omega^*(M)$ induced by ${\ensuremath{\mathcal{F}}}$ (cf. e.g. [@Alv; @KT]) induces a spectral sequence $$E^{s, t}_{1}= H^{s}({\ensuremath{\mathcal{F}}}; \Lambda^t\nu) \Longrightarrow H^{s+t}(M)\ .$$ Similarly, the filtration of the $\check{C}$ech-De Rham double complex induces a spectral sequence $$\bar{E}^{s, t}_{1}= H^{s}(M/{\ensuremath{\mathcal{F}}}; \Omega^{t}_{bas}) \Longrightarrow H^{s+t}(M/{\ensuremath{\mathcal{F}}})\ .$$ Note that $E^{0, t}_{2}= H^{t}_{bas}(M/{\ensuremath{\mathcal{F}}})$. These two spectral sequences are related by the pull-back map (\[map2.1\]), and by the Van Est map (\[Phi\]) with coefficients, $\Phi: H^{s}(M/{\ensuremath{\mathcal{F}}}; \Omega^t){\longrightarrow}H^{s}({\ensuremath{\mathcal{F}}}; \Lambda^t\nu)$. With the same arguments as above, these maps are isomorphisms in degrees $0\leq s\leq k$, if the holonomy covers of the leaves are $k$-connected. The version with compact supports of this discussion involves (\[integr2\]) and the integrations $\int_{{\ensuremath{\mathcal{F}}}}: H^{s}_{c}({\ensuremath{\mathcal{F}}}; \Lambda^t\nu){\longrightarrow}H^{s-p}_{c}(M/{\ensuremath{\mathcal{F}}}; \Omega^{t}_{bas})$ (cf. \[intfle\] above).
[xxxx]{}
[^1]: Research supported by NWO
|
---
abstract: |
The ultraluminous broad absorption line quasar APM08279+5255 is one of the most luminous systems known. Here, we present an analysis of its nuclear CO(1-0) emission. Its extended distribution suggests that the gravitational lens in this system is highly elliptical, probably a highly inclined disk. The quasar core, however, lies in the vicinity of naked cusp, indicating that APM08279+5255 is truly the only odd-image gravitational lens. This source is the second system for which the gravitational lens can be used to study structure on sub-kpc scales in the molecular gas associated with the AGN host galaxy. The observations and lens model require CO distributed on a scale of $\sim
400$ pc. Using this scale, we find that the molecular gas mass makes a significant, and perhaps dominant, contribution to the total mass within a couple hundred parsecs of the nucleus of APM08279+5255.
author:
- |
Geraint F. Lewis$^{1}$, Chris Carilli$^{2}$, Padeli Papadopoulos$^{3,4}$ & R. J. Ivison$^{5}$\
$^{1}$ Anglo-Australian Observatory, P.O. Box 296, Epping, NSW 1710, Australia: Email\
$^{2}$ National Radio Astronomy Observatories , PO Box 0, Socorro, New Mexico 87801-0387, USA: Email\
$^{3}$ Astrophysics Division, Space Science Dept. of ESA, ESTEC, Postbus 299, NL-2200 AG, Noordwijk, The Netherlands\
$^{4}$ Sterrewacht Leiden, P.O. Box 9513, 2300 RA Leiden, The Netherlands: Email\
$^{5}$ Astronomy Technology Centre, Royal Observatory, Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK: Email\
title: |
Resolved nuclear CO(1-0) emission in APM08279+5255:\
Gravitational lensing by a naked cusp?
---
Ultraluminous Galaxies: Quasars; individual: APM08279+5255
Introduction
============
Identified serendipitously in a search for high latitude carbon stars, the $z=3.9$ broad absorption line quasar [APM08279+5255]{} is coincident with an IRAS source with a flux of 0.95Jy at 100$\mu$m (Irwin et al. 1998). Observations with SCUBA reveal that [APM08279+5255]{} possesses a significant submillimetre flux of 75mJy at 850$\mu$m (Lewis et al. 1998), implying a bolometric luminosity of $\sim5\times10^{15}L_\odot$. Imaging reveals that [APM08279+5255]{} is not point-like, but rather is extended over a fraction of an arcsecond with a structure indicative of gravitational lensing (Irwin et al. 1998). The composite nature of [APM08279+5255]{} was confirmed in adaptive optics (AO) images obtained by Ledoux et al. (1998), with the system appearing as a pair of point-like images separated by 0.4 arcsec. Observations with NICMOS on the Hubble Space Telescope (Ibata et al. 1999) and AO images taken with the Keck telescope (Egami et al. 2000) also uncovered a fainter third component between the brighter two. Gravitational lens models derived from these observations suggest that the quasar continuum source has been magnified by $\sim90$.
Using IRAM, Downes et al. (1999) detected emission in CO(4-3) and CO(9-8), revealing the presence of warm circumnuclear gas in [APM08279+5255]{}. Papadopoulos et al. (2001) were able to search for CO(1-0) and CO(2-1) in this system using the Very Large Array (VLA). Both were clearly detected associated with the quasar nucleus, as well as a more extended component located several arcsecs from the quasar images. Using locally established values of the CO-to-H$_2$ ratio, this lone cloud represents $\sim10^{11}M_\odot$ of cold and/or subthermally excited gas.
In this paper, we present an analysis of nuclear CO(1-0) emission in [APM08279+5255]{} using VLA [^1] at high spatial resolution (0.3 arcsec). The CO appears as a partial ring of $\sim$0.6 arcsec diameter. These data suggest a total revision in the gravitational lens model for this source, with the new model involving a ‘naked cusp’, which naturally accounts for the observed odd-number of images. They also imply that the nuclear CO must be spatially extended on a scale of at least 400 pc, making this the second source in which gravitational lensing can be used as a ‘telescope’ to explore sub-kpc scale structure of molecular gas in the AGN host galaxy.
Observations and Results {#observations}
========================
Observations of the CO(1-0) emission from [APM08279+5255]{} were made in March and April 2001 using the VLA in the B (10 km) configuration. A total of 20 hours were spent on the source. Due to limitations with the VLA correlator, we chose to observe in continuum mode using two 50 MHz bandwidth IFs, each with two polarizations. One IF was centered on the redshifted CO(1-0) line, corresponding to a frequency of 23.465GHz, while the second IF was tuned away from the emission line to a frequency of 23.365GHz. This observing set-up maximizes sensitivity since the effective bandwidth (45 MHz, 575 km/s) closely matches the obseved CO linewidth (Downes et al. 1999), but sacrifices velocity information (Carilli, Menten, & Yun 1999). The source 3C286 was used for absolute gain calibration. The rms noise on the final image is 35$\mu$Jy beam$^{-1}$ with an effective spatial resolution of FWHM=$0.39''\times 0.28''$ with a major axis position angle of $-70^o$. Fast switching phase calibration was employed with a calibration duty cycle of 130 seconds (Carilli & Holdaway 1999). The phase stability for all the observations was excellent, such that the array was easily phase coherent over the calibration cycle time. This was demonstrated by imaging the phase calibrator (0824+558) with a similar calibration cycle time, and by imaging the radio continuum emission from [APM08279+5255]{} itself.
The images of the line and continuum emission from [APM08279+5255]{} are shown in Figure 1. The CO line image was generated by subtracting from the visibility data a Clean-Component model of the continuum emission made from the off-line data. The peak surface brightness for the continuum emission is $0.24\pm0.035$ mJy beam$^{-1}$, with a total flux density $0.41\pm0.07$ mJy. The corresponding numbers for the line emission are $0.183\pm0.035$ mJy beam$^{-1}$ and $0.39\pm0.09$ mJy. The latter corresponds to a velocity-integrated line flux of $0.22\pm0.05$ Jy km/sec, consistent with $0.15\pm0.045$ Jy km/sec deduced for the inner $\sim1$arcsec by Papadopoulos et al. (2001)
The continuum emission is extended north-south, with a position angle and spatial extent as expected based on the three component optical gravitational lens. An attempt was made to decompose the continuum map into three point-like images, centred upon the positions derived from the analysis of HST images (Ibata et al. 1999). Given the image resolution, it was not possible to cleanly separate the north-most images (A+C) which have a separation of $\sim$0.15 arcsecs. The relative flux of (A+C)/B$\sim$1.5, quite similar to the optical ratio presented by Ibata et al. (2001). The line emission, on the other hand, shows distinct curvature away from the continuum position angle, and at the 3$\sigma$ surface brightness level it appears as an almost complete ring with a diameter of about 0.6 arcsec. While the map of the line emission is of low signal to noise, the ring-like structure is apparent in each of several days worth of observations. Also, its size is in agreement with that of the CO(4-3) and CO(9-8) emission (Downes et al. 1999) and the CO(2-1) emission (Papadopoulos et al. 2001). Hence, these observations reveal the gross features of the CO(1-0) emission, although more observations are required to uncover the finer details.
The observed CO(1-0) brightness temperature of the ring (averaged over $\sim$575 km/sec) is 1.4$\pm$0.8 K, which corresponds to an emitted brightness temperature of $T_b\sim$7 K at z$\sim$3.9. A lower limit for the magnification factor of the CO(1-0) emission can be derived assuming the warm gas emitting the high-J CO lines (Downes et al. 1999) to be also emitting the J=1-0. This gas phase is optically thick with $T_{kin}\sim200K$, in agreement with the inferred dust temperatures (Lewis et al. 1998), which then yields a velocity-averaged filling factor of $f\sim7/200=0.035$. Since differential lensing will “boost” a compact warm region at the expense of a more extended and possibly sub-thermally excited gas phase that emits the CO J=1-0, the true value of f will be larger.
Gravitational Lensing {#lensing}
=====================
The lensing galaxy has yet to be identified in [APM08279+5255]{}. Hence, only the quasar positions and magnitudes are available to constrain any lensing model. Other than [APM08279+5255]{}, all other lens systems possess an even-number of images. Theoretically, any non-singular mass distribution should produce an odd number of lensed images (Burke 1981), and the ubiquity of even numbers of images has been used to limit the core radius in lensing systems, as small cores result in the demagnification of one of the images. Due to the brightness of the central source, however, models for [APM08279+5255]{} have required the opposite, a very circular model with a large core / shallow cusp (Lewis et al. 1999; Egami et al. 2000; Munoz et al. 2001). We further explore the lens model of [APM08279+5255]{}in light of the observations presented here.
The concordance between the optical images of [APM08279+5255]{} and the radio continuum at 3.6cm (Ibata et al. 1999) and 23GHz demonstrates they come from coincident regions with a similar scale size. Associated with the active core of [APM08279+5255]{}, these regions are smaller than the gravitational lensing caustics. The quite different morphology displayed in the CO image, however, indicates that this emission arises in a larger region, distinct from the continuum radiation. With its extended nature, the CO emitting region can lie under different parts of the caustic network. The CO source, however, can not be significantly more extended than the caustic network, as the resulting image would not show the ring structure seen in Figure \[fig1\]. Therefore, for a given model, the size of the ring image provides a probe of the scale size of the emitting region.
To form ring-like images, the source must be extended and cover a substantial fraction of the inner caustic structure of the lens; the resulting image is appears as a ring, following the outer critical line in the image plane (see Kochanek, Keeton & McLeod 2001). Examining the critical line structure in all previously published models for [APM08279+5255]{} (Ibata et al. 1999; Egami et al. 2000; Munoz et al 2001), the resulting structure for the Einstein ring of a source centred upon the quasar nucleus should be quite circular, with a radius of $\sim0.2$ arcsecs, passing through the two brighter quasar images; such images for extended source can be seen in the models of Egami et al. (2000). This is quite different to the CO structure displayed in Figure \[fig1\], with the CO emission clearly showing a roughly east-west extension, with the hole in the ring occurring $\sim0.5$ arcsecs from image A. Using the model derived from the HST data (Ibata et al. 1999), we explored the image configurations a range of sizes and shapes, centred upon the quasar source, for the CO emitting region. None reproduced the observed image structure and we reject this previous model.
Naked cusps occur in highly elliptical systems, such as flattened disks, when the inner diamond caustic extends outside the elliptical caustic. A source inside this extension produces three roughly colinear images of similar brightness (Maller, Flores & Primack 1997; Bartelmann & Loeb 1998; Keeton & Kochanek 1998; Moller & Blain 1998; Blain, Moller & Maller 1999; Bartelmann 2000). While gravitational lensing by spiral galaxies has been observed (e.g. B 1600+434; Koopmans, Bruyn & Jackson 1998) no observed lensed quasar system has so far been associated with a naked cusp.
To examine the possibility that the observed image configuration is due to gravitational lensing by a naked cusp, we constructed a simple mass model. Following Batelmann and Loeb (1998), this consists of a truncated flattened disk in a spherical halo. [APM08279+5255]{} has brightened considerably since its discovery (Lewis, Robb & Ibata 1999; E. Ofek priv. comm., see [http://wise-obs.tau.ac.il/$\sim$eran/LM]{}), potentially due to the effects of gravitational microlensing. Comparing the images obtained in 1998 (Ibata et al. 1999; Egami et al. 2000) and those obtained in 1999 (Munoz et al. 2001), the relative image brightness have changed appreciably, with image B changing from being 78% of image A to only 50% in the latter epoch; such behaviour is extremely suggestive of the action of gravitational microlensing, although longer term monitoring is required to confirm this. Hence, the relative image brightnesses cannot be used to constrain any mass model. Given the sparsity of constraints, we choose to find a model that can recover the image positions, while providing a reasonable description of the observed CO emission. As can be imagined, the parameter space is large, so a range of models that reproduced the quasar positions were chosen and then modeling of the CO emission was undertaken by-eye. With this, therefore, we do not claim that the model presented here is unique, only consistent with the general form of data.
In our chosen model, the disk is highly inclined, presenting a projected axis ratio of 0.25, and the disk truncated at 8$h^{-1}$kpc [^2], and possess a core radius of 0.065$h^{-1}$kpc. The rotation velocity of the disk is 200km$s^{-1}$. Figure \[fig2\] presents the source and image plane for this model, with the elliptical and diamond caustic, and corresponding critical lines apparent. In this model, the quasar images are not substantial magnified, with a total magnification of $\sim7$, with the intrinsic source of [APM08279+5255]{} being correspondingly luminous, $L_{bol}\sim7\times10^{14}L_\odot$. While extreme, this value is not necessarily outrageous as the unlensed quasar HD1946+7658 possess an intrinsic luminosity of $\sim4\times10^{14}L_\odot$ (Hagan et al. 1992), and [APM08279+5255]{} may be a member of this very luminous class of quasars. It must be conceded, however, that the non-uniqueness of the lens model translates into uncertainty in the model magnification and a true determination of the intrinsic properties of [APM08279+5255]{} require models derived from better observational constraints.
The CO source is taken to be have an circular surface brightness distribution centred upon the quasar position. One important aspect of the results presented herein is that [APM08279+5255]{} becomes the second system for which the gravitational lens can be used to study structure on sub-kpc scales in the molecular gas associated with the AGN host galaxy, the first system being the Clover Leaf quasar, H1413+117 (Yun et al. 1998; Kneib, Alloin, & Pello, R. 1998). For [APM08279+5255]{}, the observations and lens model require the CO to be distributed on a scale covering a substantial fraction of the caustics in the image plane, but not too large to lose the ring structure. The lower limit to the CO source size based on the modeling is $\sim400h^{-1}$ pc, while a rough upper limit is $\sim$1 kpc. With this model, the CO(1-0) has been magnified by a factor of $\sim2.5-3$. Like the Clover Leaf, we find that the spatial extent and mass of the molecular gas in [APM08279+5255]{}are comparable to those seen in nearby nuclear starburst galaxies (Sanders and Mirabel 1996; Downes and Solomon 1998).
Downes et al. (1999) determined a CO source size of $\sim(80-135)h^{-1}$pc for the estimated magnification factors of $\sim20-7$. This size is much smaller than the one calculated above, while their magnification factors are larger. Their analysis is based upon modeling of CO emission in the gravitationally lensed ultraluminous galaxy IRAS F10214+4724 (Downes, Solomon & Radford 1995), whose image is clearly an extended arc-like feature which possesses an essentially linear magnification. Such a simple model is probably a poor representation of the lensing in [APM08279+5255]{}. Additionally, Downes et al. (1999) assumed that the velocity filling factor is unity, substantially larger than the value derived in Section \[observations\]; as the intrinsic source radius in their model scales inversely with this value and the magnification factor is proportional to it. For f$\sim$0.35, a value well within our estimated range, the Downes et al. (1999) model yields an upper limit for the intrinsic source size and a lower limit for the magnification factor that our similar to ours.
Accounting for the influence of gravitational lensing, the velocity-integrated CO(1-0) flux density implies that the nuclear content of molecular gas in [APM08279+5255]{} is $\sim10^{10} h^{-2} M_\odot$, assuming a CO-to-H$_2$ conversion factor of ${\rm\sim 1 (M_\odot\ km\
sec^{-1}\ pc^2)^{-1}}$ which is typical for starburst/kinematically violent, UV-intense environments of gas-rich, IR-ultraluminous systems (Downes & Solomon 1998). Assuming the CO is in a rotating disk, the dynamical mass can be calculated from the radius of $\sim500$ pc set by the lens modeling, and using a rotational velocity of 350 km sec$^{-1}$ set by the observed of line velocity HWHM = 250 km sec$^{-1}$ and assuming a disk inclination angle of 45$^o$ (Downes et al. 1999). The implied dynamical mass is $1.5\times 10^{10}$ M$_\odot$ within $\sim$500 pc of the nucleus, consistent with the value derived from the CO flux. Hence it appears likely that the molecular gas mass makes a significant, and perhaps dominant, contribution to the total mass within a few hundred parsecs of the nucleus in [APM08279+5255]{}, unless the nuclear CO disk is close to face-on. A similar conclusion has been reached for most nearby nuclear starburst galaxies (Downes and Solomon 1998).
Conclusions
===========
This paper has presented resolved images of nuclear CO(1-0) emission in the gravitationally lensed BAL quasar [APM08279+5255]{}. While the continuum emission is found to be well aligned with the optical quasar images, the CO(1-0) is more extended, with a broken ring-like appearance. Such a structure is consistent with the action of gravitational lensing, with the continuum emission occurring on the scale of the quasar core, while the CO(1-0) arises from a larger region and is differentially magnified. The three-image nature of [APM08279+5255]{} has posed a problem for lens modeling, as an extremely large, flat core is required to produce the central image. Such three image configurations are a nature consequence of gravitational lensing by a flattened potential which can produce naked cusps. Modeling of the CO(1-0) emission supports this hypothesis, although a deficit in constraints implies that the model is not unique. An immediate prediction of this model is that the lensing galaxy, whose position could be revealed by observing below the Lyman limit for this system ${\rm ({\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}4400\AA)}$, hence removing the glare from the quasars, should be offset $\sim0.5$arcsec from the quasar image, rather than lying behind the quasar images.
Currently, our CO images of [APM08279+5255]{} are of limited signal-to-noise. However, with further integration a detailed map of the CO image can be made. As this region will be free from the effects of microlensing, and as its extended nature provides many more constraints (Kochanek, Keeton & McLoed 2001), such imaging has the potential to provide a more accurate model of the lensing in [APM08279+5255]{} than from the quasar images.
[DUM]{}
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[^1]: The VLA is operated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.
[^2]: $\Omega=1$ and $\Lambda=0$ is assumed throughout
|
---
abstract: 'Novel phases of two dimensional electron systems resulting from new surface or interface modified electronic structures have generated significant interest in material science. We utilize photoemission spectroscopy to show that the near-surface electronic structure of a bulk insulating iridate Sr$_3$Ir$_2$O$_7$ lying near metal-Mott insulator transition exhibit weak metallicity signified by finite electronic spectral weight at the Fermi level. The surface electrons exhibit a unique spin structure resulting from an interplay of spin-orbit, Coulomb interaction and surface quantum magnetism, distinct from a topological insulator state. Our results suggest the experimental realization of a novel quasi two dimensional interacting electron surface ground state, opening the door for exotic quantum entanglement and transport phenomena in iridate-based oxide devices.'
author:
- Chang Liu
- 'Su-Yang Xu'
- Nasser Alidoust
- 'Tay-Rong Chang'
- Hsin Lin
- Chetan Dhital
- Sovit Khadka
- Madhab Neupane
- Ilya Belopolski
- Gabriel Landolt
- 'Horng-Tay Jeng'
- 'Robert S. Markiewicz'
- 'J. Hugo Dil'
- Arun Bansil
- 'Stephen D. Wilson'
- 'M. Zahid Hasan'
title: 'Spin correlated electronic state on the surface of a spin-orbit Mott system'
---
Strongly correlated electronic behavior can be modified near surfaces and interfaces of transition metal oxides, leading to novel quantum phenomena [@Hwang; @Syro_Nature; @Thiel_Science]. Surfaces are known to be dramatically modified in topological band insulators and in spin-orbit coupled Rashba semiconductors [@Moore_Nature; @Suyang_NPhys; @BiTeI_1]. These unusual surface effects not only reflect novel physics but also hold potential for future devices where such effects are amplified by nano-structuring, which leads to the enhancement of surface-to-bulk ratio [@Paglione]. Recently, attention has focused on materials in which Mott physics and strong spin-orbit interaction may coexist in the bulk. Iridium oxides (iridates) have been identified to be one of such promising classes of materials [@Cao_327; @BJKim_Rotenberg_PRL; @Moon_PRL; @Dessau; @Baumberger; @Wojek; @Jackeli; @Shitade; @Sakakibara_Nature; @Damascelli]. So far, research on the iridates has largely focused on their bulk properties. Theoretical models suggest the possibility of realizing exotic phenomena in the iridates, such as the arced semimetal [@Wan_Weyl], topological insulator [@YBKim_TI], and high temperature superconductivity [@Senthil], none of which has yet been found experimentally. Here we report a different route to look for exotic surface phenomena in iridates and identify the surface electronic and spin ground state. Surface phenomena are often expected to be enhanced in correlated systems near the bulk metal-Mott insulator transition at which Coulumb interaction, spin-orbit coupling and the often-frustrated magnetic moments compete in determining the ground state. Near such metal-insulator bulk criticality, surface modification is likely to occur since the narrow-gap bulk states are sensitive to changes and relaxations of surrounding crystal potential near the surface. The iridate we focus here, Sr$_3$Ir$_2$O$_7$, belongs to the Ruddlesden-Popper series whose bulk electronic structure lies in between a Mott insulator ($n = 1$) and a correlated metal ($n = \infty$) [@Moon_PRL; @Cao_327], in the vicinity of a bulk criticality. Evidently, it is critically important to identify the surface ground state of such exotic iridates if they differ from the bulk.
We report a systematic study of spin integrated and spin resolved angle resolved photoemission spectroscopy (ARPES) to critically (and thoroughly) investigate the near-surface electronic structure of Sr$_3$Ir$_2$O$_7$. While the bulk exhibits strongly insulating transport properties in this compound, our photoemission results reveal finite spectral weight at the Fermi level and a gap-like suppression for quasiparticles within 30-40 meV of the Fermi level. In addition, the low energy electrons exhibit strong left-right imbalanced modulation related to surface spin polarization, as well as a unique spin fine structure which reveals an in-plane Rashba-like spin polarization induced by onsite Coulomb interaction on the surface. Such a rich character of the surface ground state is not expected within the calculated and predicted bulk electronic structure. We show that this material exhibits a critical interplay of spin-orbit coupling, antiferromagnetism, and surface termination. Our results point toward the experimental realization of a new type of correlated surface electron system on the boundary of a bulk material lying in the critical ($U \sim W$) regime.
Fig. \[Fig1\] shows the ARPES band dispersion data within the $k_x$-$k_y$ plane \[(001) plane\]. At low temperatures, Sr$_3$Ir$_2$O$_7$ changes from a paramagnetic phase to an antiferromagnetic (AF) phase, where the Ir moments point along the $c$-axis [@JWKim] with an in-plane commensurate Néel vector [@Boseggia]. In Fig. \[Fig1\]**a** we plot the raw and differentiated in-plane resistivity as a function of temperature (adapted from Ref. ). The ordering temperature $T_{\mathrm{AF}} \sim 280$ K is clearly shown as a sharp peak in the $\frac {\partial \mathrm{log} \rho}{\partial (1/T)}$ vs. $T$ curve [@Wilson_new]; a drastic upturn of the $\rho$ vs. $T$ curve at low temperatures confirms its bulk insulating transport. Figs. \[Fig1\]**b** and **c** summarize the in-plane electronic structure obtained from ARPES (Figs. \[Fig1\]**d**-**f**). In surprising contrast with the transport data (Fig. \[Fig1\]**a**), two bands ($\alpha$, blue; $\beta$, green) evolve to a close vicinity of the Fermi level. The shape and dispersive pattern of these low-lying bands are rather complicated. Fig. \[Fig1\]**d** shows a typical ARPES Fermi mapping obtained with 35 eV photons. Finite spectral weight is present at the Fermi level ($E_F$), indicative of a nearly conductive ground state. This finite intensity at $E_F$ can be due to possible band bending effect close to the sample surface. Such a band bending effect - strong enough so that $E_F$ descends to near the bottom of the Mott gap at a depth greater than the electron escape depth - will give rise to an ARPES signal dominated by the more conductive surface layer. The fact that this conducting channel is not detected by conventional transport measurements may be due to a small surface/bulk volume ratio and difference in mobility between the bulk- and surface-originated charge carriers. Nonetheless, this is a surface modified effect. At 0.15 eV binding energy (bottom panel of Fig. \[Fig1\]**e**), the $\alpha$-band decomposes into segments; the $\beta$-band shrinks in size despite maintaining intact, signifying its electronlike nature. It is important to point out here that the $\beta$ band is not observed in previous ARPES works on this material [@Dessau; @Baumberger; @Wojek], possibly because of different sample quality and our choice of measuring the second Brillouin zone instead of the first, where no sign of the $\beta$ band is seen. In Fig. \[Fig1\]**f** we show three $k$-$E$ maps along directions shown in Fig. \[Fig1\]**b**. It should be noted from Cuts 1 and 2 that, despite the finite intensity at $E_F$, the quasiparticle structure is gapped at $E_F$ (Fig. \[Fig1\]**g**). The low energy quasiparticles experience a gap-like spectral weight suppression (SWS), signified by a gradual decrease of ARPES intensity as the bands approaching the Fermi level, similar to other correlated oxides [@Chuang_Science]. The second derivative analysis of Cut 1 (Ref. ) (bottom panels) reveals that, along the $\Gamma$-$X$ direction, the $\alpha$ band bends horizontally to form a van Hove-like flat portion (Supplementary Information), while the $\beta$ band loses its intensity. Fig. \[Fig1\]**g** shows the existence of the SWS at the $X$ point where the $\alpha$-band evolves closest to $E_F$ (i.e., top of the flat portion). The exact location of the Fermi level is obtained by fitting the polycrystalline gold data with the Fermi distribution function; the EDC at the $X$ point is then symmetrized with respect to $E_F$. The two peak and valley lineshape of the symmetrized EDC proves the presence of the SWS, which is about 38 meV in size. This value is smaller than the full insulating gap value obtained from optical measurements ($\sim 250$ meV, Ref. ), indicative of a different $E_F$ at the crystal surface resulting from effective band bending. We choose to study the least resistive samples here since (1) these are the only samples who show the presence of the $\beta$ band, which is the main focus of our spin resolved ARPES study, (2) these samples do not show charging effect, and (3) the 38 meV gap observed in these samples does not change with temperature (Supplementary Information). Note that concurrent STS results [@Yoshi] on the same samples used in our studies consistently supports our observation that the surface Fermi level of Sr$_3$Ir$_2$O$_7$ always lies close to the top of the valence bands. To sum up our results in Fig. \[Fig1\], we observe a gap-like suppression of spectral weight for the spin-orbit correlated electrons at the surface of Sr$_3$Ir$_2$O$_7$ within a narrow energy window of $E_F$, as well as an electronlike Fermi contour surrounding $\Gamma$ (the $\beta$ band) which has not been observed in previous studies [@Dessau; @Baumberger; @Wojek].
In Fig. \[Fig2\] we present the spin resolved (SR) ARPES measurements of Sr$_3$Ir$_2$O$_7$. A spin-integrated ARPES $k$-$E$ map along the $X$-$\Gamma$-$X$ direction (see also Fig. \[Fig1\]**f**) is shown in Fig. \[Fig2\]**a** for clarification of momentum space positions and band notations. The important observations of this data set are (1) the strong left-right imbalance of the SR-ARPES signal, and (2) the Rashba-like spin fine structure of the $\beta$ band. We summarize our results in Fig. \[Fig2\]**b**, where the in-plane spin polarization angles at different $k$-points are presented [@Hugo] (numbers in Fig. \[Fig2\]**b**). It is seen from Fig. \[Fig2\]**b** that both the $\alpha$ and the $\beta$ band are spin polarized. The integrated spin for the $\alpha$ band near the left and the right $X$ momenta point to opposite directions. The $\beta$ band consists of two close-by rings with Rashba-like in-plane spin helicity. In Figs. \[Fig2\]**c**-**e** we show the spin polarization analysis for a MDC cut along the $X$-$\Gamma$-$X$ direction at $\sim50$ meV binding energy, within the same energy window as the SWS observed in Fig. \[Fig1\] (see Supplementary Information for more details). The spin direction under study is along $[1,-1,0]$, which is tangential to the $\beta$ contour. First, multiple peaks are present in the total intensity curve (blue circles). The spin-up and spin-down components show strong antisymmetry with respect to the zone center $\Gamma$. Since this behavior contradicts the Kramers’ theorem, one possible reason for its occurrence is the presence of surface antiferromagnetism (AFM) which explicitly breaks time reversal symmetry, although we cannot rule out other many-body effects which give rise to a collective net spin polarization at the surface. These effects introduce a spin polarized background (blue curve in the inset of Fig. \[Fig2\]**d**); spin polarized signals from individual bands sit on top of this background. Second, the spin fine structure of the $\beta$ band is observed in the raw polarization curve (Fig. \[Fig2\]**d**). More specifically, the $\beta$ band consists of two close-by rings with Rashba-like opposite in-plane spin helicity. At around $k = 0.5$ $\mathrm{\AA}$$^{-1}$, the $P_{[1,-1,0]}$ polarization curve first increase and then decrease within a narrow $k$ range (red ellipse). The inner upturn ($k\sim 0.48$ $\mathrm{\AA}$$^{-1}$) originates from a higher spin-up intensity, while the outer downturn ($k\sim 0.55$ $\mathrm{\AA}$$^{-1}$) originates from a higher spin-down intensity (inset of Fig. \[Fig2\]**d**). Momentum splitting for the two rings is determined to be $\Delta k_{[110]} \sim 0.063$ $\mathrm{\AA}$$^{-1}$. Combined with their effective mass $m^*\sim4.8 m_e$, we estimated the Rashba coefficient to be $\alpha_\mathrm{R} = \hbar^2\Delta k_{[110]}/2m^* \sim 5 \times 10^{-12}$ eV m, which is about 5 times smaller than that in the Au(111) surface state [@LaShell]. Although the $\beta$ band spin splitting to the left of $\Gamma$ is not apparent in the raw polarization curve, standard analysis (Supplementary Information) reveal the in-plane spin direction for the inner contour, the results of which are shown as numbers in Fig. \[Fig2\]**b**.
The unique spin fine structure resolved for the $\beta$ band agrees qualitatively with a theoretical model where finite Coulomb $U$ and surface termination give rise to Rashba-like in-plane spin texture (see Supplementary Information for details). In the case of $U = 0$, due to spin orbit coupling (SOC) and inversion symmetry breaking, both the $\alpha$ and $\beta$ bands develop their surface counterpart which exhibit similar bulk band dispersion but finite spin splitting and a small degree of spin polarization. The directions of the spin are strictly in plane ($S_z = 0$) and obey $E(k,\uparrow) = E(-k,\downarrow)$ due to time reversal symmetry. When $U$ is turned on, AFM order is obtained self-consistently with finite staggered magnetic moments along the $z$-direction. The time reversal symmetry is then broken and a finite $S_z$ component is obtained for each state (Supplementary Information). In our SR-ARPES measurements, we observed the in-plane Rashba-like spin texture and signature for non-zero $S_z$ component of the $\beta$ band (Supplementary Information). The presence of spin polarization reconfirms the surface-dominated signal of the ARPES data, since this splitting is not expected in the bulk electronic structure. All aspects of the data taken together, our experimental observation and first principles calculations suggest an interplay between SOC and the (bulk and/or surface) AFM order. Final state photoelectron spin effects reported in strong spin-orbit coupled topological insulators are minimized here, since the spin-orbit splitting and effective coupling in this system is even weaker than that in gold.
Such unique surface spin modulation is previously unobserved for a system with both strong SOC and onsite Coulumb interaction. These spin correlated surface electronic states are not observed in the high-$T_c$ superconducting cuprates, which indicates strong dissimilarity of the surface modified electronic states between the high-$T_c$ superconductors and the layered iridates, unlike what is predicted from theory [@Senthil]. Our finding adds an additional variable (or tunability) to the electronic phase diagram of the iridates. Since such surface related behavior are not observed in Sr$_2$IrO$_4$ or SrIrO$_3$, it is likely that the proximity to the bulk criticality contributes to the surface modification.
In Fig. \[Fig3\] we further show $k_z$ dispersion analysis for Sr$_3$Ir$_2$O$_7$ with photon energies ranging from 25 eV to 80 eV. From the raw dispersive pattern and the associated momentum distribution curves (MDCs) (Figs. \[Fig3\]**a**-**b**), we find that the resolved bands show little $k_z$ dispersion, as both the $\alpha$ and $\beta$ band form nearly vertical lines in the $k_{\|}$-$k_z$ plane along both $\Gamma$-$X$ and $\Gamma$-$M$ directions, which is evident for a quasi two dimensional electronic structure. On the other hand, our detailed analysis shown in Figs. \[Fig3\]**c**-**d** reveals that the $\alpha$ band does show some extent of periodic $k_z$ dispersive pattern, which is required by the symmetry of the AF Brillouin zone. In Fig. \[Fig3\]**d**, we show that the $Y$-$X$-$Y$ segment of the AF zone edge changes from a vertical line at $k_z = 0$, to a single dot $P$ at $k_z = \pi/c$, and finally to a horizontal line at $k_z = 2\pi/c$. As a result, any band close to this segment will have to change from a $k_{[1,-1,0]}$ elongated shape at $k_z = 0$ to a $k_{[110]}$ elongated shape at $k_z = 2\pi/c$. This is consistent with our ARPES results shown in Fig. \[Fig3\]**c**, where the Fermi mappings done with $h\nu = 35$, 50 and 25 eV roughly correspond to the situation at $k_z=0$, $\pi/c$ and $2\pi/c$, respectively. To clarify our observations from Figs. \[Fig1\] and \[Fig3\], a schematic constant energy map close to the top of the $\alpha$ band is presented in Fig. \[Fig3\]**f** associated with the AF Brillouin zone (Fig. \[Fig3\]**e**). From this figure, one can see that although some weak $k_z$ dispersion is discernable, the electronic structure of Sr$_3$Ir$_2$O$_7$ is mostly two dimensional. It is very important to note here that such finite $k_z$ dispersion is *not excluded* for surface-related bands, especially for systems with small insulating gaps, since the surface bands can in principle penetrate deeper into the bulk and thus respect the symmetry of bulk electrons. The *spin textured behavior* together with the observed weak $k_z$ dispersion in these states rules out the possibility that the $\alpha$ and $\beta$ bands are purely bulk bands.
From the data presented in Fig. \[Fig2\], we have experimentally investigated the presence of surface spin fine structure in Sr$_3$Ir$_2$O$_7$. The physical significance of the observed spin texture can be understood by comparing it to the well-known surface state found in topological insulators (TIs), as shown in Fig. \[Fig4\]**a**. At the surface of a TI, a single gapless spin helical two dimensional electron gas (2DEG) presents as long as time reversal symmetry preserves in the system. Intrinsic interaction between the Dirac fermions and lattice phonons is found to be weak [@Batanouny]. At the surface of an AF-ordered correlated TMO, there exists a two dimensional electronic state (2DES) showing a unique spin fine structure. It is instructive to propose (see Fig. \[Fig4\]**b**, adapted from Ref. ) that these spin polarized states reside only in the close vicinity of the MIT, where the bulk band gap is so small such that it is unstable at the crystal surface where lattice relaxation and band bending occur, which then give rise to an effective electric field along the $z$-direction. The spin helical surface state in TI and the spin-textured surfaces in TMOs near bulk criticality are examples of novel surface electronic structures that signify distinct phases of two dimensional condensed matter. In order to harness its exotic behavior, surface-to-bulk ratio can be increased by nano-structuring the sample in future layer-by-layer MBE growth techniques. In such thin films, anomalous quantum and Hall transport are expected in a strongly correlated setting.
In Fig. \[Fig4\]**c**-**e** we present a first-principles GGA + $U$ band calculations that reveal a bulk electronic structure agreeing reasonably well with our measurements for high binding energies (detailed in Supplementary Information). In order to achieve such agreement, we set the Hubbard $U = 1.5$ eV and $\lambda_{\mathrm{SO}} = 1.7$ times the self consistent value in the calculation. Therefore our data places constraints on the magnitude of Coulumb $U$ and spin-orbit coupling experimentally realized in the material under study. The 12$^{\circ}$ rotation of the IrO$_6$ octahedra (Supplementary Information) gives rise to a Jahn-Teller type gap locating at 0.7 - 1.3 eV above $E_F$. The combined effect of $U$ and $\lambda_{\mathrm{SO}}$ causes the opening of a partial gap at $E_F$ close to $\Gamma$; this gap enlarges and becomes a $\sim180$ meV complete gap once AFM is added to the scenario, hence the term “AFM gap”. One important difference between the theoretical bulk calculation and the experimentally observed electronic states lies around the $M$ point where the $\alpha$ and $\beta$ bands are observed to evolve up approaching $E_F$, making the circular electronlike shape of the $\beta$ band, while no bands are present within $E_b < 0.4$ eV in the bulk calculation (red arrow in Fig. \[Fig4\]**c**). This critical difference enables the unique spin texture of the complete $\beta$ contours, indicating strong surface modification of the bands resolved by ARPES. Interestingly, our calculation optimized by fitting with experimental data on the real material also suggests that the real material possess a distinct, well-isolated surface state which energetically lies within the Jahn-Teller gap above the experimental Fermi level. This distinct surface state is robust even for a large variation within the parameter space in our calculation (Supplementary Information). Therefore, based on our experimentally optimized band calculation, we further predict the existence of an isolated correlated surface state in the highly $n$-doped Sr$_3$Ir$_2$O$_7$.
Although the layered nature of Sr$_3$Ir$_2$O$_7$ and the surface band dispersion (at higher binding energies) is similar to that of the bulk bands, surface effect can take place due to the residual interactions between two Ir$_2$O$_4$ blocks. The observed spin polarization is due to the surface effect since the bulk bands are spin degenerate. At the surface, the inversion symmetry is broken; the spin degeneracy is lifted due to the mutual effects of SOC and inversion symmetry breaking, resulting in one-to-one spin and momentum locking for the surface states in a band specific manner. Moreover, the time reversal symmetry is broken due possibly to AFM order so that $E(k,\uparrow)=E(-k,\downarrow)$ no longer holds. The spin splitting cannot be explained within the bulk band structure scenario since the bulk crystal structure of Sr$_3$Ir$_2$O$_7$ possess inversion symmetry, bulk bands must be spin degenerate even if they are quasi two dimensional. The surface effects are thus critically important to the interpretation of our data reflecting the near-surface ground state of iridate.
The surface electronic ground state of Sr$_3$Ir$_2$O$_7$, the $n = 2$ member of the Ruddlesden-Popper iridate series Sr$_{n+1}$Ir$_n$O$_{3n+1}$, is thus distinct from that in Sr$_2$IrO$_4$ which realizes a $J = 1/2$ Mott insulating state, and that of a topological insulator. Finite density of states is found at the Fermi level in Sr$_3$Ir$_2$O$_7$, with a gap-like spectral weight suppression on the order of 40 meV. Spin resolved ARPES data reveals a strong left-right imbalanced modulation on the surface, and finds a unique spin fine structure in one of the bands, which results from onsite Coulomb interaction and bulk and/or surface antiferromagnetism. These observations are evident for a strong interplay between spin-orbit coupling, bandwidth, long range magnetic order as well as surface formation in Sr$_3$Ir$_2$O$_7$. Our results provide unique insight for the quasiparticle interactions near the surface of this system, pointing toward the experimental realization of a novel two dimensional correlated electronic state, as well as paves the way for utilizing the observed surface electronic structure for future nano-structured quantum devices. The multiband nature and unique spin texture realized in this strong spin-orbit coupled material near the surface further suggests that the photon polarization-tuned emission of electron from the correlated surface can lead to harnessing the quantum entanglement of surface wavefunction for designing opto-spintronics [@Damascelli2] using the photon-based control of correlated electronic spin.
Work at Princeton and Princeton-led synchrotron-based measurements and the related theory at Northeastern University are supported by the Office of Basic Energy Sciences, US Department of Energy (grants DE-FG-02-05ER46200, AC03-76SF00098 and DE-FG02-07ER46352), and benefited from the allocation of supercomputer time at NERSC and Northeastern University’s Advanced Scientific Computation Center. Work at Boston College is supported by NSF CAREER DMR-1056625. T.-R. C. is supported by the National Science Council and Academia Sinica, Taiwan and would like to thank NCHC, CINC-NTU, and NCTS, Taiwan for technical support. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The Stanford Synchrotron Radiation Lightsource is supported by the U.S. Department of Energy under Contract No. DE-AC02-76SF00515. The Synchrotron Radiation Center is primarily funded by the University of Wisconsin-Madison with supplemental support from facility users and the University of Wisconsin-Milwaukee. We gratefully thank Sung-Kwan Mo, Jonathan D. Denlinger, Donghui Lu and Mark Bissen for instrumental support; Vidya Madhavan for fruitful discussion. C. L. acknowledges Peng Zhang, Takeshi Kondo, and Adam Kaminski for provision of data analysis software. M. Z. H. acknowledges Visiting Scientist support from LBNL and additional support from the A. P. Sloan Foundation.
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abstract: 'Mortality is an instrument of natural selection. Evolutionary motivated theories imply its irreversibility and life history dependence. This is inconsistent with mortality data for protected populations. Accurate analysis yields mortality law, which is specific for their evolutionary unprecedented conditions, yet universal for species as evolutionary remote as humans and flies. The law is exact, instantaneous, reversible, stepwise, and allows for a rapid (within less than two years for humans) and significant mortality decrease at any (but very old) age.'
author:
- 'Mark Ya. Azbel’'
title: 'Why humans die — an unsolved biological problem'
---
[^1]
Mortality is an instrument of natural selection. In the wild, competition for sparce resources is fierce, and only relatively few genetically fittest animals survive to their evolutionary “goal” - reproduction. Even human life expectancy at birth was around 40 - 45 years just a century ago (e.g., 38.64 years for males in 1876 Switzerland [@database]). Evolutionary motivated biological theories [@Charlesworth] imply that during and beyond reproductive age mortality irreversibly increases and strongly depends on the life history. This disagrees with the demographic observation (see, e. g., [@Vaupel]) that mortality is highly plastic; that different mortalities in Eastern and Western Germanies converged few years after their unification; that Norwegian females born in 1900 at 57 years reversed to mortality they had 36 years younger (see later). Recent experiments [@Mair] prove that the life-prolonging effect of diet in fruit flies is independent of their past, starts immediately and is lost when the dieting stops. Thus, in agreement with theoretical predictions [@Azbel1], mortality in protected populations of species as remote as humans and flies has short memory, is reversible and little depends on life history. Such dominance of nurture over nature is inconsistent with any evolutionary mechanism of mortality. But then, protected populations are indeed evolutionary unprecedented. To unravel the biologically unanticipated mechanism of their mortality, note that similar situation triggered all breakthroughs in physics via analysis of experimental data, which disregarded common wisdom and preceded rather than followed any models. Luckily, accurate forecasts of human mortality, and the resulting age structure of the population, are important for economic, taxation, insurance, etc, etc, purposes. That is why quantitative studies of mortality were started [@Halley] long before Darvin, in 1693, by the famous astronomer Halley (the discoverer of the Halley comet) and followed in 1760 by the great mathematician Euler. To better estimate and forecast mortality, demographers developed over 15 mortality approximations [@Coale]. Total mortality depends on a multitude of unquantified factors which describe all kinds of relevant details about the population and its environment, from conception to the age of death: genotypes; life history; acquired components; age specific factors; even the month of death and the possibility of death being the late onset genetic decease [@Charlesworth; @Doblhammer]. Demographic approximations prove that in a given country all these factors with remarkable accuracy reduce to few parameters only. During the last century, mortality rate in Western Europe at 0, 10 and 40 years decreased correspondingly 50, 100, and 10 times. In contrast to such mortality decrease (primarily due to improving living conditions, medical ones included), the difference between mortality rates at the same age in the same calendar year is rarely more than twofold in different countries. However, demographers do not present universal mortality approximations. They are interested in the most accurate approximation of the most important and specific mortality rate in a given country or its part. To a physicist relative proximity of such approximations in different countries suggests [@Azbel3] certain universality. Mortality in a population is uncontrollably heterogeneous (e.g., $1891/1900$ female infant mortality is almost twice higher in Stockholm than in the rural area [@stat]), and changes with time. It affects different mortality characteristics in a different way. “Additive” variables, whose values in a heterogeneous population are the averages over its different groups of the same age, are invariant to any such averaging. The less heterogeneous the population and its living conditions are, the more accurate their mortality approximations are. I conjecture that a certain fraction of mortality (denote it as “canonic”) accurately yields the law, which is the same (universal) in any population where heterogeneity of additive mortality variables is restricted to accurately quantified universal limits. Such universality is sufficient to establish its law. Any heterogeneous population consists of several “restricted heterogeneity” groups. Its mortality reduces to the universal law and fractions of the population in each of the restricted heterogeneity groups. Mortality in different countries allows one to determine the universal law parameters, and to verify its predictions.
For males and females who died in a given country in a given calendar year demographic life tables list, in particular, “period” probabilities $q(x)$ (for survivors to $x$) and $d(x)$ (for live newborns) to die between the ages $x$ and $(x + 1)$ \[note that $d(0) = q(0)$\]; the probability $l(x)$ to survive to $x$ for live newborns; the life expectancy $e(x)$ at $x$. The tables also present [@database] the data and procedure which allow one to calculate $q(x)$, $d(x)$, $l(x)$, $e(x)$ for human cohorts, which were born in a given calendar year.
Consider a heterogeneous population, consisting of the groups with the number $N^{G}(x)$ of survivors to $x$ in a group $G$. The total number of survivors $N(x)$ is the sum of $N^{G}(x)$ over all $G$. Thus, $l(x) = N(x) / N(0) = \sum c_{G} \ell^{G}(x) =
\langle \ell^{G}(x) \rangle$ is the average of $\ell^{G}(x)$ over all groups, with $c_{G}$ and $\ell^{G}(x)$ being the ratio of the population and the survivability to $x$ in the group $G$. Similarly, $d(x) = \ell(x) - \ell(x + 1)$ reduces to its average over the groups of the same age. In contrast, $q(x) = 1 - \ell(x +
1) / \ell(x)$ reduces to $q^{G}(y)$ in the groups of all ages $y$ from 0 to $x$, since the probability $\ell(x)$ to survive to $x$ equals $p(0) p(1) \ldots p(x - 1)$, where $p(y) = 1 - q(y)$ is the probability to survive from $y$ to $(y + 1)$. The most age specific additive variable is $d(x)$. (Naturally, it allows one to calculate all other mortality characteristics, e. g., $q(x)$). The most time specific one is $d(0) =
q(0)$ – it depends on the time span less than 2 years (from conception to 1 year). Thus, the most specific relation between two additive variables is the relation between $d(x)$ and infant mortality $q(0)$. A universal restriction on the heterogeneity of $q(0)$ is $q_{j} <
q^{G}(0) < q_{j + 1}$, where $q_{j}$, $q_{j + 1}$ determine the universal for all humans boundaries of the $j$-th interval. Universal law implies that the relation between canonic $d(x)$ and $q(0)$ in any universal interval is the same as the relation between their values in any of the restricted heterogeneity groups, i.e. that $d(x) = f_{x}[q(0)]$; $d^{G}(x) =
f_{x}[q^{G}(0)]$, where $f_{x}$ is a universal function. (Here and further on $d$, $q$, etc denote canonic quantities). Since additive $d(x) = \langle d^{G}(x) \rangle$, $q(0) = \langle
q^{G}(0) \rangle$, so $\left\langle f_{x}[q^{G}(0)] \right\rangle
= f_{x} \left[ \langle q^{G}(0) \rangle \right]$. According to a simple property of stochastic variables [@Azbel3], if the average of a function is equal to the function of the average, then the function is linear. So, $$\label{eq1}
d(x) = a_{j}(x) q(0) + b_{j}(x) \quad \text{if} \quad
q_{j} < q(0) < q_{j + 1},$$ where parameters $a_{j}$, $b_{j}$ for a given $x$ are universal constants. (Here and on I skip their argument $x$). When canonic infant mortality $q(0)$ reaches an interval boundary (\[eq1\]), it must homogenize to the boundary value. Since $d(x)$ at all ages reduce to infant mortality, they simultaneously reach the interval boundary and, together with $q(0)$, homogenize there. \[Two such “ultimate” boundaries are well known: $q(x) = 0$ when nobody dies, and $q(x) = 1$ when nobody survives at the age $x$\]. At different intervals linear relations are different. Thus, the universal law implies piecewise linear $d(x)$ vs $q(0)$ with simultaneous at all ages intersections. Any heterogeneous population is distributed at the universal intervals. Their occupation and Eq. (\[eq1\]) determine piecewise linear, but non-universal relation between $d(x)$ and $d(0) = q(0)$. At a given non-universal linear segment $$\label{eq2}
d(x) = aq(0) + b$$ For a given $x$ non-universal $a$ and $b$ reduce to the universal law parameters and to the non-universal fractions of the population in each of its linear intervals. Heterogeneity of mortality in some groups in a given country may be sufficiently low to fit into a single universal interval. Then they yield the universal law and allow for a comprehensive study of canonic mortality. Suppose that a heterogeneous population is distributed at two, e.g., the 1-st and 2-nd, universal intervals with the concentrations $c_{1}$ and $c_{2} = 1 - c_{1}$ correspondingly. Then $q(0) = c_{1} q_{1}(0) + (1 - c_{1}) q_{2}(0)$; $d(x) =
c_{1} d_{1}(x) + (1 - c_{1}) d_{2}(x)$. Thus, by Eq. (\[eq1\], \[eq2\]), $q_{1}(0) = \alpha_{1} q(0)$, $q_{2}(0) = \alpha_{2}
q(0)$, where $c_{1} = (b_{2} - b) / (b_{2} - b_{1})$, and $\alpha_{1} = (a - a_{2}) / [c_{1}(a_{1} - a_{2})]$; $\alpha_{2} =
(a_{1} - a) / [(1 - c_{1})(a_{1} - a_{2})]$. The crossover to the next non-universal segment occurs when, e.g., $q_{1}(0)$ reaches the intersection $q^{U}(0) = (b_{2} - b_{1}) / (a_{1} - a_{2})$ of the first and second universal segments. Then $q_{1}(0) =
q^{U}(0)$ implies $d^{I}(x) = a_{1} q^{I}(x) + b_{1}$. (A subscript $I$ denotes an intersection). So, this non-universal intersection belongs to the first universal linear segment or its extension. Inversely, such universality is the criterion of the population distributed at two universal segments. The set of such “extended” universal segments yields the universal law, while its non-universal intersections determine the fractions of the populations at the universal intervals. A general case (when the population is distributed at more than two universal segments) is more complicated, but also reduces to the universal law and the population fractions at the segments.
To quantitatively verify the universal law, consider the number $D(x)$ of, e.g., Swiss female deaths at a given age $x$ in each calendar year from 1876 to 2001[@database]. At 10 years $D(10)$ decreases (together with mortality) from 126 in 1876 to 1 in 2001. At 80 years $D(80)$ increases (together with the life expectancy) from $231$ to $951$. It depends on the size of the population, e.g. in 1999 Japan at 80 years it is $13, 061$. Its stochastic error is $\sim 1 / D^{1/2}$. If demographic fluctuations in mortality are consistent with this (minimal for a stochastic quantity) generic error for the same age, denote the corresponding mortality as “regular”. Otherwise, denote it as “irregular”. Remarkably, mortality is irregular only during, and few years after, major wars, epidemics, food and water contamination, etc., when its change within few years is not relatively small.
To verify and determine the universal law, I approximate regular empirical $d(x)$ vs $d(0)$ with the minimal number of linear segments which yields their statistical accuracy for each given age $x$. \[Figure \[fig1\]
![The probability $d(80)$ for newborn females in 1898-2001 France (diamonds) and 1950-1999 Japan (triangles) to die between 80 and 81 years of age vs the same calendar year infant mortality $q(0)$. Empty diamonds correspond to 1918 flu pandemic and World Wars. They are disregarded in the linear regressions (straight lines), which minimize the mean linear deviations from black signs to statistical $5 \%$. When Japanese $q(0) = 0$, the corresponding linear regression yields $d(80) = 0$.[]{data-label="fig1"}](fig1.eps){width="8cm"}
presents the examples of $d(80)$ vs. $q(0)$ for Japanese and French females.\] Demographic data demonstrate that all non-universal intersections in most developed countries fall on the universal straight lines (see examples in Fig. \[fig2\]).
![Universal law of the canonic $d(80)$ and $d(60)$ (upper and lower curves, thick lines) vs $q(0)$. Note that both $d(80) = 0$ and $d(60) = 0$ when $q(0) = 0$. Diamonds and squares represent non-universal intersections for (from left to right) England (two successive intersections), France, Italy and Japan, Finland, Netherland, Norway, Denmark, France, England correspondingly. Thin lines extend the universal linear segments.[]{data-label="fig2"}](fig2.eps){width="8cm"}
Thus, the population in each such country reduces to 2 restricted heterogeneity groups (which change at the non-universal intersections). Then non-universal intersections determine the universal law (see Fig. \[fig2\]). The law may be refined by accounting for a (relatively small) contribution from more than two restricted heterogeneity groups.
Consider other implications and predictions of the universal law. The extrapolation of the Japanese piecewise linear dependence in Fig. \[fig1\] to $q(0) = 0$ yields $d(80) = 0$, i.e. zero mortality at (and presumably prior to) 80 years. Similarly, $d(60)$ and $d(80)$ in Fig. \[fig2\] universally $\rightarrow 0$ when $q(0) \rightarrow 0$. This is consistent with, e.g., the dependencies of the life expectancies $e(0)$ at birth and $e(80)$ at 80 years on the values of the same calendar year birth mortality $q(0)$ for Japanese and French females. If nobody dies until 80, then $e(0) = 80 + e(80)$. In fact, the extrapolated $e(0) = 93$ years, $e(80) = 16$. Thus, $e(80) + 80 = 96$ years is just $3 \%$ higher than $e(0)$. Vanishing mortality may have already been observed. In 2001 Switzerland less than 17 females died in any age group from 1 till 26 years, 43 died at 40 years. In Japan, where the population is 18 times larger, 50 girls died at 10 years in 1999 (cf. Fig. \[fig3\]). Such values of a stochastic quantity are consistent with zero mortality in the lowest mortality groups.
The universal law reduces the period canonic mortality at any age to the infant mortality $q(0)$. So, together with $q(0)$, at any age regular mortality rate may be rapidly reduced and reversed to its value at a much younger age. This agrees with the mortality of Norwegian and Swedish female cohorts, born in 1900 and 1920 correspondingly (Fig. \[fig3\]).
![Mortality rates vs age in Norwegian 1900 (diamonds, the large one for 1918 year of flu pandemic in Europe), Swedish 1920 (squares), Japanese 1989 (circles), Swiss 1990 (triangles) female cohorts.[]{data-label="fig3"}](fig3.eps){width="8cm"}
Both countries were neutral in the World Wars. In both infant mortality $q(0)$ is higher than mortality $q(80)$ at 80 years. In Sweden $q(0)$ decreases 63-fold to $q(11)$, then increases 2.3-fold to $q(24)$. Thereafter it decreases to the same value at 34 years as 23 years earlier, at 11, and only at 45 years reaches almost the same value as 30 years younger, at 15. In Norway $q(0)$ decreases 24.5-fold to $q(9)$, doubles at $x = 21$, halves back to the minimal value at $x = 34$, and then slowly changes, until at 57 years it restores the mortality it had at 21, i.e. 36 years younger [@Azbel4].
Rapid crossovers in mortality (see Fig. \[fig1\]) expose several modes of the universal mechanism, which switch simultaneously for all ages (see Fig. \[fig2\]). The changes amplify significant declines of old age mortality in the second half of the 20-th century [@Wilmoth], with its spectacular medical progress. Predicted homogenization of the mortality at the intersections was noticed for male and female mortality [@Azbel1].
The non-universal law determines the non-universal population fractions in different universal intervals for a given country, sex, and calendar year. The fractions depend on genetics, life history, mutation accumulation and other factors. The difference between the total mortality and its piecewise linear approximation may be partitioned into stochastic fluctuations (which yield the Gaussian distribution), singular “irregular fluctuations” (related to, e.g., 1918 flu pandemic and world wars), and systematic deviations (related to evolutionary mechanisms [@Charlesworth; @Azbel1] and all unaccounted for factors). Depending on age, from 3 till 95 years the latter contribute from $2 \%$ to $10 \%$ of the total mortality. The approach may be refined if one considers several additive parameters. Preliminary study suggests this little increases the accuracy.
Regular mortality also dominates in protected populations of flies. The relations between their additive mortality variables are also piecewise linear [@Azbel1]. Thus, their mortality is also predominantly universal. The (properly scaled) law, which is universal for species as remote as humans and well protected populations of flies [@Azbel1] despite their different evolution, may be considered a conservation law in biology and evolution. One wonders how, why and when a law, which is specific for evolutionary unprecedented (protected) populations, could evolutionary emerge. It suggests a possibility of similar “evolutionary unmotivated” laws for other biological characteristics.
To summarize. Universal law of mortality specifies the groups whose infant mortality heterogeneity is restricted by the universal limits. In any population of any age, the law rapidly (on the scale of two years for humans) adjusts a dominant fraction of the total mortality to infant mortality and the fractions of the population in these groups. This implies that, in contrast to mortality in the wild, mortality in well protected populations is dominated by a genetically programmed intrinsic mechanism which provides its unprecedented rapid adaptation to current living conditions. The universal form and relatively high accuracy (mostly on the scale of mortality fluctuations) of the law make it a universal and biologically explicit demographic approximation of the total mortality. The law provides certain clues to its possible physiology and physical model.
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[^1]: Permanent address
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abstract: |
We provide a novel interpretation of optimal transport problem as an optimisation of certain linear functional over the set of all Choquet representations of pairs of probability measures and reprove the Kantorovich duality formula. We also provide such formulation and a proof for multimarginal optimal transport.
We also consider case of martingale optimal transport. Here we provide a novel formulation of the dual problem. We compute the set of extreme points of multidimensional probability measures in convex order. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.
We introduce notion of martingale triangle inequality and prove that if it is satisfied by a cost function, then in the dual problem to martingale optimal transport one may restrict the supremum to a narrower class of pairs of functions that are equal to each other. We prove that any such function on a convex subset of Euclidean space admits extension to the whole space.
author:
- 'Krzysztof J. Ciosmak [^1]'
bibliography:
- 'biblio.bib'
title: Optimal transport and Choquet theory
---
\[firstpage\]
Introduction
============
In this note we explore a link of optimal transport problem, see [@Villani1; @Villani2] for an extensive account, with Choquet theory, see e.g. [@Phelps; @Alfsen]. Suppose we are given two probability Borel measures $\mu, \nu$ on topological spaces $X, Y$ respectively and a measurable cost function $c\colon X\times Y\to\mathbb{R}$. The optimal transport problem, proposed by Monge [@Monge], is concerned with finding an optimal map $T\colon X\to Y$ such that it pushes $\mu$ forward to $\nu$, $T_{\#}\mu=\nu$, i.e. for any Borel set $A\subset Y$ there is $\nu(A)=\mu(T^{-1}(A))$, and such that the integral $$\int_X c(x,T(x))d\mu(x)$$ is minimal. Kantorovich [@Kantorovich; @Kantorovich2] proposed a relaxed version of the problem. Namely, instead of looking for an optimal map, one seeks for a coupling $\pi$, that is a Borel probability measure on $X\times Y$ such that it’s marginal distributions are $\mu$ and $\nu$ respectively, that minimises the integral $$\label{eqn:min}
\int_{X\times Y} cd\pi.$$ Kantorovich also provided a dual formulation of the problem, where one looks for continuous functions $u\colon X\to\mathbb{R}$ and $v\colon Y\to\mathbb{R}$, such that for all $x\in X,y\in Y$ there is $u(x)+v(y)\leq c(x,y)$, and that maximise $$\label{eqn:max}
\int_X u d\mu+\int_Y vd\nu.$$ It has been proven, see e.g. [@Villani2 Theorem 1.3], that the minimal value of (\[eqn:min\]) and the maximal value of (\[eqn:max\]) coincide, under the assumption that $c$ is lower-semicontinuous. We reprove this result using tools of Choquet theory and Strassen’s theorem, see Theorem \[thm:opti\] and Corollary \[col:opticoma\]. Namely, the observation is as follows. Suppose $u$ and $v$ are functions such that for all $x\in X$ and $y\in Y$ there is $u(x)+v(y)\leq c(x,y)$. Consider the set $\mathcal{P}$ of pairs of probability measures $(\mu,\nu)$ such that $u, v$ maximise the integral (\[eqn:max\]). Strassen’s theorem allows us to show that the extreme points of $\mathcal{P}$ is equal to the set of pairs $(\delta_x,\delta_y)$ with $x\in X$ and $y\in Y$ are such that $u(x)+v(y)=c(x,y)$. Choquet’s theorem and identification of extreme points of $\mathcal{P}$ yields existence of a Borel probability measure $\pi$ on $X\times Y$ such that $$(\mu,\nu)=\int_{X\times Y} (\delta_x,\delta_y) d\pi(x,y)$$ and $\pi$ is supported on the set of points $(x,y)$ such that $u(x)+v(y)=c(x,y)$. It follows that the marginals of $\pi$ are $\mu$ and $\nu$ respectively and that $$\int_X u d\mu+\int_Y v d\nu=\int_{X\times Y} \big(u(x)+v(y)\big)d\pi (x,y)=\int_{X\times Y} c d\pi.$$ Therefore the existence of maximisers $u,v$ implies the Kantorovich duality. Note that when the cost function $c$ is Lipschitz such existence may be proven with help of so-called $c$-convexification. If $c$ is lower-semicontinuous, then it may be suitably approximated by Lipschitz functions in such a way that the duality follows.
Similar reasoning may be applied as well in the context of multimarginal optimal transport, see Theorem \[thm:optimany\].
Recently, great attention has been paid to the problem of martingale optimal transport in multidimensional setting, see e.g. [@Ghoussoub; @DeMarch1; @DeMarch2; @DeMarch3; @Siorpaes]. Suppose we are given two probability measures $\mu,\nu$ on $\mathbb{R}^n$ with finite first moments that are in convex order, that is for any convex function $f\colon\mathbb{R}^n\to\mathbb{R}$ $$\int_{\mathbb{R}^n}fd\mu\leq \int_{\mathbb{R}^n}fd\nu.$$ Then a theorem of Strassen [@Strassen] implies that there exists a coupling $\pi$ on $\mathbb{R}^n\times\mathbb{R}^n$ with marginals $\mu$ and $\nu$ such that if $(X,Y)$ is distributed according to $\pi$, then $\mathbb{E}(Y|X)=X$, i.e. the pair $(X,Y)$ is a one-step martingale. The problem is to find such coupling $\pi$ that minimises $$\int_{\mathbb{R}^n\times\mathbb{R}^n}cd\pi$$ for a given measurable cost function $c\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$. The dual problem is to find maximal value of $$\int_{\mathbb{R}^n}u d\mu-\int_{\mathbb{R}^n}vd\nu$$ among all pairs $u,v\colon\mathbb{R}^n\to\mathbb{R}$ of continuous functions such that there exists $\gamma\colon\mathbb{R}^n\to\mathbb{R}^n$ satisfying $$u(x)-v(y)+\langle \gamma(x),y-x\rangle\leq c(x,y)\text{ for all }x,y\in\mathbb{R}^n$$ We prove, see Theorem \[thm:equalityc\] and Corollary \[col:col\], that the set of such functions is equal to the set of pairs $u,v\colon\mathbb{R}^n\to\mathbb{R}$ such that for all $x_1,\dotsc,x_{n+1}\in\mathbb{R}^n$ and all non-negative $t_1,\dotsc,t_{n+1}$ that sum up to one there is $$u\Big(\sum_{i=1}^{n+1}t_ix_i\Big)-\sum_{i=1}^{n+1}t_i v(x_i)\leq \sum_{j=1}^{n+1}t_jc\Big(\sum_{i=1}^{n+1}t_ix_i,x_j\Big).$$ These results complement standard knowledge about convex functions, which follows by taking $u=v$ and $c$ to be equal to zero. The proofs work also in case of general convex sets $K\subset\mathbb{R}^n$.
In the course of the proof of these facts we also characterise the extreme points of the set of probability measures in convex order as the set of pairs of the form $$\Big(\delta_x,\sum_{i=1}^{d+1}t_i\delta_{x_i}\Big)\text{ with }x=\sum_{i=1}^{d+1}t_ix_i$$ for some positive $t_1,\dotsc,t_{d+1}$ summing up to one, some $x_1,\dotsc,x_{d+1}\in\mathbb{R}^n$ in general position and $d\leq n$. This is the assertion of Theorem \[thm:ex\].
We introduce notion of *martingale triangle inequality*, see Definition \[defin:martingale\], and prove that if the cost function $c$ satisfies this inequality and vanishes on the diagonal, then the value of the dual problem will not be changed if we restrict ourselves to functions $u,v$ that satisfy $v=u$, see Theorem \[thm:clc\]. Then for all $x_1,\dotsc,x_{n+1}\in\mathbb{R}^n$ and all non-negative $t_1,\dotsc,t_{n+1}$ that sum up to one there is $$\label{eqn:intor}
u\Big(\sum_{i=1}^{n+1}t_ix_i\Big)-\sum_{i=1}^{n+1}t_i u(x_i)\leq \sum_{j=1}^{n+1}t_jc\Big(\sum_{i=1}^{n+1}t_ix_i,x_j\Big).$$ Martingale triangle inequality demands that for any points $x,x_1,\dotsc,x_{n+1}\in\mathbb{R}^n$ and any non-negative $t_1,\dotsc,t_{n+1}$ that sum up to one there is $$\sum_{i=1}^{n+1}t_i c(x,x_i)-c\Big(x,\sum_{i=1}^{n+1}t_ix_i\Big)\leq \sum_{i=1}^{n+1}t_i c\Big(\sum_{j=1}^{n+1}t_jx_j,x_i\Big).$$ Examples of functions that satisfy martingale triangle inequality are metrics, non-negative functions concave in the second variable and any conical combination of such functions.
We prove that for any continuous function $u\colon K\to\mathbb{R}$ such that (\[eqn:intor\]) is satisfied may be extended to $\mathbb{R}^n$ in such a way that (\[eqn:intor\]) is still fulfilled. The result, Corollary \[col:extend\], provides a formula for the extension similar to the formula of McShane [@McShane] for Lipschitz functions. This also provides an extensions of results of [@Ivan]. We refer the reader to [@Ciosmak] for discussion of similar problem for $1$-Lipschitz maps.
Possible future applications of these finding lie in investigation of cyclical monotonicity principle for martingale optimal transport, see e.g. [@Beiglbock], akin to characterisation of the classical optimal transport of Gangbo, McCann [@Gangbo].
As an application, see Theorem \[thm:uni\], we provide a characterisation of uniformly convex and uniformly smooth functions that complements results of Azè and Penot [@Aze].
Outline of the article
======================
In Section \[s:cones\] we recall necessary definitions and theorems that our results are based on.
In Section \[s:transport\] we provide proofs of Kantorovich duality based on Strassen’s theorem in the two-marginal case.
In Section \[s:Kant-Rub\] we prove the Kantorovich-Rubinstein duality, i.e. the duality result for cost function given by a metric.
In Section \[s:multi\] we provide a proof of Kantorovich duality in the multimarginal setting.
In Section \[s:martingale\] we characterise extreme points of pairs of probability measures in convex order and prove a duality result for martingale optimal transport provided that there exists a maximiser of the dual problem.
In Section \[s:duality\] we investigate class of functions that appear in the dual problem to the martingale optimal transport. We provide an intrinsic characterisation of such functions, which includes, for example, characterisation of convex functions as the functions that lie above its tangent lines.
In Section \[s:uniform\] we apply the result of the previous section and obtain a novel characterisation of uniformly convex and uniformly smooth functions.
In Section \[s:triangle\] we introduce the martingale triangle inequality and prove that if it is satisfied by the cost function, then the maximisation may be restricted to pairs of functions such that $u=v$ in the dual problem to the martingale optimal transport. Moreover, we study extensions properties of such functions and show that they admit similar behaviour to that of Lipschitz functions.
Convex cones {#s:cones}
============
Here we shall recall Strassen’s theorem [@Strassen]. We refer the reader also to [@Meyer] and [@Aliprantis].
Suppose $\Omega$ is a Polish space. Let $\mathcal{C}(\Omega)$ denote the space of continuous functions on $\Omega$ equipped with the supremum norm and let $\mathcal{M}(\Omega)$ denote the space of signed Borel measures on $\Omega$ normed by total variation. Let $\mathcal{F}$ be a closed convex cone in $\mathcal{C}(\Omega)$. Then the dual cone $\mathcal{F}^*\subset\mathcal{M}(\Omega)$ consists of Borel signed measures $\mu$ on $\Omega$ such that $$\int_{\Omega}fd\mu\geq 0\text{ for all }f\in \mathcal{F}.$$ We define a partial order $\prec_{\mathcal{F}}$ on $\mathcal{M}(\Omega)$. We write $\mu\prec_{\mathcal{F}}\nu$ if $\nu-\mu\in \mathcal{F}^*$. Consider the set $$\mathcal{P}_{\mathcal{F}}=\big\{(\mu,\nu)\in\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)\big| \mu\prec_{\mathcal{F}}\nu\big\}.$$ Here $\mathcal{P}(\Omega)$ denotes the set of all Borel probability measures on $\Omega$. We would like to use an approach originating in the Strassen’s paper to compute the set of extreme points of $\mathcal{P}_{\mathcal{F}}$. We claim that for extreme points are of the form $(\delta_x,\nu)$ for some probability measure $\nu$ and a point $x\in \Omega$. Examples of the interesting cones are:
i) the cone of convex functions on $\mathbb{R}^n$,
ii) the cone $\big\{\lambda(f-g)\big|\lambda\geq 0, g\colon X\to\mathbb{R}\text{ is }1\text{-Lipschitz}\big\}$ and $f\colon X\to\mathbb{R}$ is a fixed $1$-Lipschitz map, $X$ is a metric space.
The first example corresponds to the martingale optimal transport problem. The second example corresponds to the optimal transport problem for measures $\mu,\nu$ such that $f$ is the optimal $1$-Lipschitz potential. The idea is to treat the optimal transport problems as optimisations of Choquet’s representations. Let us recall the theorem proven by Strassen [@Strassen].
We say that a map $h\colon X\to\mathbb{R}$ is a support function if it is subadditive and positively homogenous. It is continuous if and only if $${\lVerth\rVert}=\sup\big\{{\lverth(x)\rvert}|{\lVertx\rVert}\leq 1\big\}<\infty.$$
\[thm:strassen\] Let $X$ be a separable Banach space, let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\omega\mapsto h_{\omega}$ be a map from $\Omega$ to continuous support functions on $X$, which is weakly measurable, that is, for every $x\in X$ the map $\omega\mapsto h_{\omega}(x)$ is $\Sigma$-measurable. Suppose that $$\int_{\Omega} {\lVerth_{\omega}\rVert}d\mu(\omega)<\infty.$$ Set $$h(x)=\int_{\Omega}h_{\omega}(x)d\mu(\omega).$$ For a functional $x^*\in X^*$ the following conditions are equivalent:
i) $x^*\leq h$,
ii) there exists a map $\omega\mapsto x_{\omega}^*$ from $\Omega$ to $X^*$ which is weakly measurable in the sense that $\omega\mapsto x_{\omega}^*(x)$ is measurable for any $x\in X$ such that $x_{\omega}^*\leq h_{\omega}$ for any $\omega\in\Omega$ and for all $x\in X$ $$x^*(x)=\int_{\Omega}x_{\omega}^*(x)d\mu(\omega).$$
If $(\Omega,\Sigma)$ and $(\Xi, \Theta)$ are measurable spaces, then a Markov kernel $P$ from $\Omega$ to $\Xi$ is a real function on $\Theta\times \Omega$ such that for any point $\omega\in\Omega$, $P(\cdot,\omega)$ is a probability measure on $\Theta$ and for any $A\in \Theta$, $P(A,\cdot)$ is $\Sigma$-measurable.
If $\mu$ is a probability measure on $\Sigma$, then we define $P\mu$ to be a probability measure on $\Theta$ such that $$P\mu(A)=\int_{\Omega} P(A,\omega)d\mu(\omega)\text{ for all }A\in\Theta.$$ If $z$ is a bounded $\Theta$-measurable function on $\Xi$, then we define $zP$ to be a bounded $\Sigma$-measurable function on $\Omega$ defined by $$zP(\omega)=\int_{\Xi} z(r) dP(r,\omega).$$
We will assume below that the implicit $\sigma$-algebra $\Sigma$ is complete.
\[thm:extr\] Let $\mathcal{F}$ be a closed convex cone in $\mathcal{C}(\Omega)$ that contains constant functions and is closed under maxima. Then for any $(\mu,\nu)$ in the set $$\mathcal{P}_{\mathcal{F}}=\big\{(\mu',\nu')\in\mathcal{P}(\Omega)\times\mathcal{P}(\Omega)\big| \mu'\prec_{\mathcal{F}}\nu'\big\}.$$ there exists a Markov kernel $P$ form $\Omega$ to $\Omega$ such that $\nu=P\mu$ and such that for any $\omega\in\Omega$ $$(\delta_{\omega}, P(\cdot,\omega))\in\mathcal{P}_{\mathcal{F}}.$$ Moreover, the set of extreme points of $\mathcal{P}_{\mathcal{F}}$ is contained in the set of measures of the form $(\delta_{\omega},\eta)\in\mathcal{P}_{\mathcal{F}}$ for some $\omega\in \Omega$.
Set $X=\mathcal{C}(\Omega)$ to be the Banach space of all continuous bounded functions on $\Omega$. Let $x^*=\nu$ be an element of $\mathcal{M}(\Omega)$. Set for $\omega\in\Omega$ $$h_{\omega}(x)=\inf\big\{y(\omega)| y\in-\mathcal{F}, y\geq x\big\}.$$ Then, as $\mathcal{F}$ is a cone, $h_{\omega}$ is subadditive and positively homogenous. By definition $x\leq h_{\omega}(x)$. Moreover, as $\mathcal{F}$ contains constants, we have $h_{\omega}(x)\leq {\lVertx\rVert}$, so that $$-{\lVertx\rVert}\leq x\leq h_{\omega}(x)\leq {\lVertx\rVert}.$$ As $\mathcal{C}(\Omega)$ is separable, so is its subset $$\big\{y| y\in-\mathcal{F}, y\geq x\big\}.$$ Hence, $\omega\mapsto h_{\omega}(x)$ is measurable and integrable and is a pointwise limit of a sequence $(y_k)_{k=1}^{\infty}$ of functions in $-\mathcal{F}$. By the assumption that $\mathcal{F}$ is closed under maxima and contains constants, we may assume also that for all $k\in\mathbb{N}$ there is ${\lverty_k\rvert}\leq {\lVertx\rVert}$. Therefore by the dominated convergence theorem, $$x^*(x)=\int_{\Omega} xd\nu \leq \int_{\Omega} h_{\omega}(x)d\nu(\omega)\leq\int_{\Omega}h_{\omega}(x)d\mu(\omega).$$ By Theorem \[thm:strassen\] there is a weakly measurable function $\omega\mapsto x_{\omega}^*$ that satisfies $x_{\omega}^*\leq h_{\omega}$ and such that $$\label{eqn:rep}
x^*(x)=\int_{\Omega}x_{\omega}^*(x) d\mu(\omega).$$ As ${\lverth_\omega\rvert}\leq {\lVert\cdot\rVert}$, it follows that $x_{\omega}^*$ is of norm at most one, and by (\[eqn:rep\]), and the fact that $\mu$ and $\nu$ are probability measures, for $\mu$-almost every $\omega$, $x_{\omega}^*$ is a probability measure. Now, $P(\cdot,\omega)=x_{\omega}^*$ defines a Markov kernel. By (\[eqn:rep\]), we see that $\nu=P\mu$. Observe that if $f\in \mathcal{F}$, then by the definition of $h_{\omega}$, $$h_{\omega}(-f)=-f\text{ hence }
x_{\omega}^*(-f)\leq - f(\omega),$$ so that $(\delta_{\omega}, x_{\omega}^*)\in\mathcal{P}_{\mathcal{F}}$. We have $$(\mu,\nu)=\int_{\Omega} (\delta_{\omega},x_{\omega}^*)d\mu(\omega),$$ so the claim about the extreme points follows.
Optimal transport {#s:transport}
=================
The next corollary extends the results of the previous section to the case of pair $(\mu,\nu)$ of measures on two, possibly distinct, compact Hausdorff spaces $X$ and $Y$.
\[col:twospaces\] Let $X,Y$ be two compact Hausdorff spaces. Suppose $\mu\in\mathcal{P}(X)$ and $\nu\in\mathcal{P}(Y)$ are two Borel probability measures. Let $\mathcal{F}$ be a closed convex cone of functions on $\mathcal{C}(X\cup Y)$ that contains constants and is closed under maxima. Suppose that for any $\phi\in\mathcal{F}$ we have $$\int_X \phi d\mu\leq \int_Y\phi d\nu.$$ Then there exists a Markov kernel $P$ from $X$ to $Y$ such that $\nu=P\mu$ and such that for any $x\in X$ and any $\phi\in\mathcal{F}$ $$\phi(x)\leq \int_Y \phi(y) P(dy,x).$$ Moreover, the set of extreme points of the set $$\mathcal{P}_{\mathcal{F}}=\big\{(\mu,\nu)\in\mathcal{P}(X)\times\mathcal{P}(Y)| \int_X \phi d\mu\leq\int_Y \phi d\nu\text{ for all }\phi\in\mathcal{F},\mu,\nu \big\}$$ is contained in the set of pairs of the form $(\delta_x,\eta)\in\mathcal{P}_{\mathcal{F}}$ for some $x\in X$ and some probability measure $\eta$ on $Y$.
Let $\Omega$ be disjoint union of $X$ and $Y$. Let $\tilde{\mu},\tilde{\nu}$ be probability Borel measures in $\mathcal{M}(\Omega)$ that are extensions of $\mu$ and $\nu$ respectively. Then $\tilde{\mu}\prec_{\mathcal{F}}\tilde{\nu}$ with the order induced by $\mathcal{F}$. Whence, by Theorem \[thm:extr\], there exists a Markov kernel $\tilde{P}$ from $\Omega$ to $\Omega$ such that $$\tilde{\nu}=\tilde{P}\tilde{\mu}$$ and $$\phi(\omega)\leq \int_{\Omega} \phi(\omega')\tilde{P}(d\omega',\omega)\text{ for all }\omega\in \Omega\text{ and all }\phi\in\mathcal{F}.$$ That is for any Borel set $A$ in $Y$ we have $$\nu(A)=\int_X P(A,x)d\mu(x).$$ Here $P$ is the restriction of $\tilde{P}$ to the Markov kernel from $X$ to $Y$. Then also we have for all $y\in Y$ $$\phi(y)\leq \int_X \phi(x) P(dx,y).$$ The claim on the extreme points follows readily.
In the following theorem Corollary \[col:twospaces\] is employed to prove the Kantorovich duality. The theorem below also provides a reinterpretation of the Kantorovich problem as minimisation of a linear functional $\int_{\mathcal{E}_{\mathcal{F}}} cd\pi$ over all Choquet’s representation of pair of measures $(\mu,\nu)\in\mathcal{P}$.
\[thm:opti\] Let $X,Y$ be two compact metric spaces and let $c\colon X\times Y\to\mathbb{R}$ be a Lipschitz function. Let $\mu$ and $\nu$ be Borel probability measures on $X$ and $Y$ respectively. Then $$\sup\Big\{\int_X \phi d\mu-\int_Y \psi d\nu \big| \phi(x)-\psi(y)\leq c(x,y)\text{ for all }x\in X,y\in Y\Big\}$$ is equal to $$\inf\Big\{\int_{X\times Y} c(x,y)d\pi(x,y)\big| \pi\in \Gamma (\mu,\nu)\Big\},$$ where $\Gamma(\mu,\nu)$ stands for the set of all Borel probability measures on $X\times Y$ such that its marginals are $\mu$ and $\nu$ respectively. Moreover, both supremum and infimum are attained.
We shall first prove that the supremum is attained. By the Arzelà-Asoli theorem, it is enough to show that it may be taken over a uniformly continuous and uniformly bounded subset of functions. This follows from the fact that if $\phi,\psi$ satisfy $$\phi(x)-\psi(y)\leq c(x,y)\text{ for all }x\in X,y\in Y$$ then $$\phi'(x)=\inf\big\{c(x,y)+\psi(y)| y\in Y\big\}\text{ for all }x\in X$$ and $$\psi'(y)= \sup\big\{\phi'(x)-c(x,y)|x\in X\big\}\text{ for all }y\in Y$$ are both Lipschitz, with Lipschitz constant depending on the Lipschitz constant of $c$, and satisfy $$\phi'(x)-\psi'(y)\leq c(x,y)\text{ for all }x\in X,y\in Y\text{ and }\phi\leq\phi'\text{ and } \psi'\leq \psi.$$ Adding appropriate constant we may assume that $\phi'$ and $\psi'$ are bounded by uniform norm of $c$, see Lemma \[lem:boundedness\] below. Now, take $\phi, \psi$ that maximise $$\int_X\phi d\mu-\int_Y \psi d\nu$$ subject to the condition that $\phi(x)-\psi(y)\leq c(x,y)$ for all $x\in X$ and $y\in Y$. Let $\rho_0\colon X\cup Y\to\mathbb{R}$ be defined so that $\rho_0(x)=-\phi(x)$ and $\rho_0(y)=\psi(y)$ for $x\in X$ and $y\in Y$. Let $\mathcal{G}$ denote the set of continuous functions $\rho$ on $X\cup Y$ such that for $x\in X$ and $y\in Y$ there is $$-\rho(x)+\rho(x)\leq c(x,y).$$ Let now $\mathcal{F}$ denote the set of continuous functions on $X\cup Y$ of the form $\lambda(\rho_0-\rho)$ for $\lambda\geq 0$ and $\rho\in\mathcal{G}$, i.e. $\mathcal{F}$ is the tangent cone at $\rho_0$ to $\mathcal{G}$. Observe that $\mathcal{F}$ is a closed, convex cone that contains constants is closed under maxima, and that $$(\mu,\nu)\in \mathcal{P}_{\mathcal{F}}=\big\{(\mu',\nu')\in\mathcal{P}(X)\times\mathcal{P}(Y)\big| \mu'\prec_{\mathcal{F}}\nu'\big\}.$$ By Corollary \[col:twospaces\] the extreme points of $\mathcal{P}_{\mathcal{F}}$ are contained in the set of pairs of the form $(\delta_x,\eta)\in\mathcal{P}_{\mathcal{F}}$ with $x\in X$ and $\eta$ a probability measure on $Y$. By symmetry, the set of extreme points is equal to the set of pairs of the form $(\delta_x,\delta_y)\in\mathcal{P}_{\mathcal{F}}$ for some $x\in X$ and $y\in Y$. For any such pair there is $-\rho_0(x)+\rho_0(y)=c(x,y)$. Indeed, define $\rho(x')=-c(x',y)$ and set $$\rho(y')=\inf\{ c(x',y')-c(x',y)|x'\in X\}.$$ Then $\rho\in\mathcal{G}$ and $-\rho(x)+\rho(y)=c(x,y)$. Thus also $-\rho_0(x)+\rho_0(y)=c(x,y)$. It follows that the set $\mathcal{E}_{\mathcal{F}}$ of extreme points of $\mathcal{P}_{\mathcal{F}}$ is equal to $$\mathcal{E}_{\mathcal{F}}=\Big\{(\delta_x,\delta_y)\big|- \rho_0(x)+\rho_0(y)=c(x,y)\Big\}.$$ By the Choquet’s theorem there is a probability measure $\pi_0$ on $\mathcal{E}_{\mathcal{F}}$ such that $$(\mu,\nu)=\int_{\mathcal{E}_{\mathcal{F}}} (\xi_1,\xi_2)d\pi_0(\xi).$$ Define $$\pi=\int_{\mathcal{E}_{\mathcal{F}}}\xi_1\otimes \xi_2 d\pi_0(\xi).$$ Then $$\int_{X\times Y}c d\pi=\int_X \phi d\mu-\int_Y \psi d\nu$$ and the proof is complete.
\[col:opticoma\] Let $X,Y$ be two compact metric spaces and let $c\colon X\times Y\to\mathbb{R}$ be a lower semi-continuous function. Let $\mu$ and $\nu$ be Borel probability measures on $X$ and $Y$ respectively. Then $$\sup\Big\{\int_X \phi d\mu-\int_Y \psi d\nu \big| \phi(x)-\psi(y)\leq c(x,y)\text{ for all }x\in X,y\in Y\Big\}$$ is equal to $$\inf\Big\{\int_{X\times Y} c(x,y)d\pi(x,y)\big| \pi\in \Gamma (\mu,\nu)\Big\},$$ where $\Gamma(\mu,\nu)$ stands for the set of all Borel probability measures on $X\times Y$ such that its marginals are $\mu$ and $\nu$ respectively.
If $c$ is a lower semicontinuous function, then it may be written as a supremum of a sequence $(c_n)_{n=1}^{\infty}$ of Lipschitz functions, see e.g. [@Villani2]. For each $c_n$ we know by Theorem \[thm:opti\] that the respective supremum and infimum are equal. Therefore, for each $n$ we see that $$\label{eqn:cinf}
\inf\Big\{\int_{X\times Y} c_n(x,y)d\pi(x,y)\big| \pi\in \Gamma (\mu,\nu)\Big\}$$ is at most $$\sup\Big\{\int_X \phi d\mu-\int_Y \psi d\nu \big| \phi(x)-\psi(y)\leq c(x,y)\text{ for all }x\in X,y\in Y\Big\}.$$ The infima (\[eqn:cinf\]) converge to the corresponding infimum for $c$ and the proof is complete.
The above proof extends also to the case of non-compact Polish spaces in the same way as in [@Villani1].
Kantorovich-Rubinstein duality {#s:Kant-Rub}
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Below we present analogous results to the above for the optimal transport problem with metric cost.
\[lem:lip\] Let $X$ be a bounded metric space. Let $f\colon X\to\mathbb{R}$ be a $1$-Lipschitz map such. Define a cone $\mathcal{F}$ by $$\mathcal{F}=\big\{\lambda (f-g)| \lambda\geq 0, g\colon X\to\mathbb{R}\text{ is }1\text{-Lipschitz}\big\}.$$ Then $(\mu,\nu)\in\mathcal{P}_{\mathcal{F}}$ if and only if $$\int_X f d(\mu-\nu)=\sup\Big\{\int_X g d(\mu-\nu)\big| g\colon X\to\mathbb{R}\text{ is }1\text{-Lipschitz}\Big\}.$$ Moreover, the set of extreme points of $\mathcal{P}_{\mathcal{F}}$ is the set of points of the form $$\label{eqn:forml}
(\delta_x,\delta_y)\text{ where }f(y)-f(x)=d(x,y).$$
The first assertion is trivial. We need to prove the claim on the extreme points of $\mathcal{P}_{\mathcal{F}}$. Observe that $\mathcal{F}$ is closed under maxima, as maximum of two Lipschitz functions is again Lipschitz. By Theorem \[thm:extr\], we know that any extreme point in $\mathcal{P}_{\mathcal{F}}$ has the form $(\delta_x,\nu)$ for some $x\in X$ and a Borel probability measure $\nu$. Let $g\colon X\to\mathbb{R}$ be a $1$-Lipschitz function defined as $$g(y)=d(x,y)\text{ for }y\in X.$$ By the assumption on $\nu$, we know that $$\int_X f(y)-f(x) d\nu(y)\geq \int_X d(x,y) d\nu(y).$$ The $1$-Lipschitzness of $f$ yields that we have equality in this inequality, whence for $\nu$-almost every $y\in X$ we have $f(y)-f(x)=d(x,y)$. Since $$(\delta_x,\nu)=\int_X(\delta_x,\delta_y)d\nu(y),$$ any extreme point of $\mathcal{P}_{\mathcal{F}}$ has to be of the form (\[eqn:forml\]). Any pair $(\delta_x,\delta_y)$ such that $f(y)-f(x)=d(x,y)$ is an extreme point of $\mathcal{P}_{\mathcal{F}}$, as it is an extreme point of a larger set of pairs of probability measures.
Note that the cone considered above is exactly the tangent cone of the set of $1$-Lipschitz functions at the point $f$.
Below we shall reprove the Kantorovich-Rubinstein duality formula using the methods developed above. For Borel probability measures $\mu,\nu$ on $X$ we denote by $\Gamma(\mu,\nu)$ the set of Borel probability measures $\pi$ on $X\times X$ such that the respective marginals of $\pi$ are $\mu$ and $\nu$.
For any Borel probability measures $\mu,\nu$ $$\sup\Big\{\int_X g d(\mu-\nu)\big| g\text{ is }1\text{-Lipschitz}\Big\}=
\inf\Big\{\int_{X\times X} dd\pi \big| \pi\in\Gamma(\mu,\nu)\Big\}.$$ Moreover, both supremum and infimum are attained.
Let $f\colon X\to\mathbb{R}$ be such that $$\int_X f d(\mu-\nu)=\sup\Big\{\int_X g d(\mu-\nu)\big| g\colon X\to\mathbb{R}\text{ is }1\text{-Lipschitz}\Big\}.$$ Let $\mathcal{F}$ be the tangent cone at $f$ to the set of $1$-Lipschitz functions. Then, by the Choquet’s theorem, there exists a Borel probability measure $\pi_0$ on the set of extreme points $\mathcal{E}_{\mathcal{F}}$ of $\mathcal{P}_{\mathcal{F}}$ such that $$(\mu,\nu)=\int_{\mathcal{E}_{\mathcal{F}}}\xi d\pi_0(\xi).$$ By Lemma \[lem:lip\], any extreme point of $\mathcal{P}_{\mathcal{F}}$ is of the form $(\delta_x,\delta_y)$ with $$f(y)-f(x)=d(x,y).$$ Let $\pi$ be a measure on $X\times X$ given by $$\pi=\int_{\mathcal{E}_{\mathcal{F}}}\xi_1\otimes \xi_2d\pi_0(\xi)\text{, where }\xi=(\xi_1,\xi_2).$$ Then $\pi$ is a Borel probability measure and $\pi\in\Gamma(\mu,\nu)$, by definition of $\pi_0$. Moreover, by Lemma \[lem:lip\], $$\int_{X\times X} d(x,y) d\pi(x,y)=\int_{X\times X} \big(f(y)-f(x)\big) d\pi(x,y)=\int_X fd(\mu-\nu).$$
Multimarginal optimal transport {#s:multi}
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Here we generalise our setting to multimarginal optimal transport with finitely many marginals, see [@Kellerer2]. We shall need the following lemma, see also [@Pass].
\[lem:convexification\] Let $X_1,\dotsc,X_k$ be metric spaces. Let $$c\colon X_1\times\dotsc\times X_k\to\mathbb{R}$$ be a Lipschitz function. Suppose that $A_i\subset X_i$ for $i=1,\dotsc,k$ and let $$f_i\colon A_i\to\mathbb{R}\text{ for }i=1,\dotsc,k$$ be such that for all $x_i\in A_i$, $i=1,\dotsc,k$ there is $$\label{eqn:inmany}
\sum_{i=1}^k f_i(x_i)\leq c(x_1,\dotsc,x_k).$$ Then there exists Lipschitz functions $\tilde{f}_i\colon X_i\to\mathbb{R}$, $i=1,\dotsc, k$ such that condition (\[eqn:inmany\]) holds true for all $x_i\in X_i$, $i=1,\dotsc, k$ and $f_i(x_i)\leq \tilde{f}_i(x_i)$ for $x_i\in A_i$ and $i=1,\dotsc, k$. Moreover, $\tilde{f}_i$, $i=1,\dotsc,k$, may be taken so that their Lipschitz constants are each at most the Lipschitz constant of $c$.
We define inductively $\tilde{f}_i(x_i)$ for $i=1,\dotsc,k$ and $x_i\in X_i$ as $$\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j=1}^{i-1}\tilde{f}_j(x_j)-\sum_{j=i+1}^k f_j(x_j) |x_j\in X_j\text{ if }j< i, x_j\in A_j\text{ if }j>i\Big\}.$$ Then $\tilde{f}_i$, $i=1,\dotsc,k$, satisfy $\sum_{i=1}^k\tilde{f}_i(x_i)\leq c(x_1,\dotsc,x_k)$, with $x_i\in X_i$ for all $i=1,\dotsc,k$, and thus $$\tilde{f}_i(x_i)\leq\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j\neq i}\tilde{f}_j(x_j)|x_j \in X_j, j\neq i\Big\}.$$ Moreover $f_i\leq\tilde{f}_i$ for all $i=1,\dotsc,k$ on $A_i$ and thus $\tilde{f}_i$ is at least the infimum on the right-hand side of the above equality. This is to say, for $x_i\in X_i$ and $i=1,\dotsc,k$ $$\tilde{f}_i(x_i)=\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j\neq i}\tilde{f}_j(x_j) |x_j\in X_j, j\neq i\Big\}.$$ If $c$ was Lipschitz, then $\tilde{f}_i$, $i=1,\dotsc,k$ are Lipschitz as infima of Lipschitz functions.
\[rem:infima\] Pick $x_1,\in X_i$, $i=1,\dotsc,k$. Let $f(x_1)=c(x_1,\dotsc,x_k)$ for $x\in X_1$ and let $f(x_i)=0$ for $i=2,\dotsc,k$. Then the assumptions of the above lemma are satisfied with $A_i=\{x_i\}$, $i=1,\dotsc,k$. Therefore we may apply the $c$-convexification procedure described above in the proof, to obtain function $f_c\colon \bigcup_{i=1}^kX_i\to\mathbb{R}$ such that $$\sum_{i=1}^k f_c(y_i)\leq c(y_1,\dotsc,y_k)\text{ for all }y_i\in X_i$$ and moreover $$\sum_{i=1}^kf_c(x_i)=c(x_1,\dotsc,x_k).$$
The following lemma is based on [@Villani2 Remark 1.13].
\[lem:boundedness\] Let $X_1,\dotsc,X_k$ be metric spaces. Let $$c\colon X_1\times\dotsc\times X_k\to\mathbb{R}$$ be a bounded function. Suppose that $f_i\colon X_i\to\mathbb{R}$, $i=1,\dotsc, k$ are such that for all $x_i\in X_i$ and $i=1,\dotsc,k$ $$f_i(x_i)=\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j\neq i}f_j(x_j) |x_j\in X_j, j\neq i\Big\}.$$ Then there exist constants $h_1,\dotsc,h_k\in\mathbb{R}$ that sum up to zero, such that functions $\tilde{f}_i=f_i+h_i$ satisfy $$\label{eqn:takie}
\tilde{f}_i(x_i)=\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j\neq i}\tilde{f}_j(x_j) |x_j\in X_j, j\neq i\Big\}$$ and all of them are bounded by the uniform norm of $c$ times $\max\{k, 3\}$.
Note that for any $h_1,\dotsc,h_k$ that sum up to zero there is $$\inf\Big\{c(x_1,\dotsc,x_k)-\sum_{j\neq i}\tilde{f}_j(x_j) |x_j\in X_j, j\neq i\Big\}=f_i(x_i)-\sum_{j\neq i }h_j=\tilde{f}_i(x_i).$$ Thus the first assertion is proven. Let $M$ denote the uniform norm of $c$. Choose $h_1,\dotsc,h_k$ in such a way that $$\sup\{\tilde{f}_i(x_i)|x_i\in X_i\}=M\text{ for }i=2,\dotsc,k.$$ Note that by (\[eqn:takie\]) it follows that for $i=1,\dotsc,k$ and all $x_i\in X_i$ $$-M-\sum_{j\neq i}\sup\{\tilde{f}_j(x_j)|x_j\in X_j\}\leq\tilde{f}_i(x_i)\leq M-\sum_{j\neq i}\sup\{\tilde{f}_j(x_j)|x_j\in X_j\}.$$ Thus, for all $x_1\in X_1$ $$-kM\leq\tilde{f}_1(x_1)\leq (2-k)M.$$ Now, again from (\[eqn:takie\]) and from the above, we get that for $i=2,\dotsc,k$ and $x_i\in X_i$ $$-M-(k-2)M+(k-2)M\leq \tilde{f}_i(x_i)\leq M-(k-2)M+kM$$ Hence $$-M\leq \tilde{f}_i(x_i)\leq 3M.$$
\[thm:extreme\] Let $X_1,\dotsc,X_k$ be compact metric spaces. Let $$c\colon X_1\times\dotsc\times X_k\to\mathbb{R}$$ be a Borel function. Let $\mathcal{B}$ be the set of functions $f$ in $\mathcal{C}(X_1\cup\dotsc\cup X_k)$ such that for all $x_i\in X_i$, $i=1,\dotsc,k$ we have $$\sum_{i=1}^k f(x_i)\leq c(x_1,\dotsc,x_k).$$ Let $g\in \mathcal{B}$. Then the set of extreme points of the set $\mathcal{P}$ of $k$-tuples of probability measures $(\mu_1,\dotsc,\mu_k)\in\mathcal{P}(X_1)\times\dotsc\times\mathcal{P}(X_k)$ such that $$\sum_{i=1}^k\int_{X_i} f d\mu_i\leq \sum_{i=1}^k\int_{X_i} gd\mu_i\text{ for all }f\in\mathcal{B}$$ is contained in the set consisting of $(\delta_{x_1},\dotsc,\delta_{x_k})$ with $x_i\in X_i$, $i=1,\dotsc,k$. If $c$ is additionally Lipschitz then for any $\mu_i\in\mathcal{P}(X_i)$, $i=1,\dotsc,k$, such $g\in\mathcal{B}$ exists and any extreme point $(\delta_{x_1},\dotsc,\delta_{x_k})$ of $\mathcal{P}$, with $x_i\in X_i$, $i=1,\dotsc,k$, satisfies $$\sum_{i=1}^k g(x_i)=c(x_1,\dotsc,x_k).$$
For any $l\in\{1,\dotsc,k\}$ let $I_l=\{1,\dotsc,l-1,l+1,\dotsc,k\}$ and let $\Omega$ be the disjoint union of all $X_i$, $i=1,\dotsc,k$. Let $(\mu_1,\dotsc,\mu_k)\in\mathcal{P}$. Let $\mu_l$ denote the extension of $\mu_l$ to $\Omega$ and let $\mu_{I_l}$ denote the probability measure on $\Omega$ such that $$\mu_{I_l}(A)=\frac1{k-1}\sum_{i\neq l}\mu_i(A)\text{ if }A\subset X_l^c\text{ and }\mu_{I_l}(A)=0\text{ if }A\subset X_l.$$ Then, for any $f\in \mathcal{B}$ we have $$\int_{\Omega}(g-f)d\mu_l+\int_{\Omega}(k-1)(g-f)d\mu_{I_l}\geq 0.$$ Let $\mathcal{F}_l$ denote the convex cone of functions of the form $\lambda(f-g)+c$ on $X_l$ and $(k-1)\lambda(g-f)+c$ on $X_{I_l}$, for non-negative $\lambda$ and $f\in\mathcal{B}$ and $c\in\mathbb{R}$. Then $\mu_l\prec_{\mathcal{F}_l}\mu_{I_l}$ with the order induced by $\mathcal{F}_l$. Moreover, $\mathcal{F}_l$ is closed, contains constants and is closed under maxima. Hence, by Corollary \[col:twospaces\], the extreme points of the set $$\mathcal{P}_{\mathcal{F}_l}=\Big\{(\mu,\nu)\in\mathcal{P}(X_l)\times \mathcal{P}\big(\bigcup_{i\in I_l}X_i\big)\big|\mu\prec_{\mathcal{F}_l} \nu\Big\}$$ are of the form $(\delta_x,\eta)$ for some probability $\eta\in \mathcal{P}\big(\bigcup_{i\in I_l}X_i\big)$. By the Choquet’s theorem there exists a Borel probability measure $\pi_l$ on the set $\mathcal{E}_{\mathcal{F}_l}$ of extreme points of $\mathcal{P}_{\mathcal{F}_l}$ such that $$(\mu_l,\mu_{I_l})=\int_{\mathcal{E}_{\mathcal{F}_l}}\xi d\pi_l(\xi).$$ Hence for any $i\in I_l$ $$\mu_i=\int_{\mathcal{E}_{\mathcal{F}_l}}(k-1)\xi_2|_{X_i}d\pi_l(\xi).$$ It follows that $(k-1)\xi_2|_{X_i}$ are all probabilities. Hence, any extreme point of $\mathcal{P}$ has to be of the form $$(\eta_1,\dotsc,\eta_{l-1},\delta_{x_l},\eta|_{X_{l+1}},\dotsc,\eta|_{X_k})$$ with $x_l\in X_l$ and some probability measures $\eta_i$ for $i\in I_l$. As this hold for any $l=1,\dotsc,k$, any extreme point of $\mathcal{P}$ has to have the form $(\delta_{x_1},\dotsc,\delta_{x_k})$ with $x_i\in X_i$, $i=1,\dotsc,k$.
Suppose now that $c$ is Lipschitz. Take a sequence of functions $(g_n)_{n=1}^{\infty}\in\mathcal{B}$ be such that $$\sum_{i=1}^k \int_{X_i}g_nd\mu_i$$ approaches $$\sup\Big\{\sum_{i=1}^k \int_{X_i}fd\mu_i\big| f\in\mathcal{B}\Big\}.$$ Then by Lemma \[lem:convexification\] we may assume that $(g_n)_{n=1}^{\infty}$ is uniformly Lipschitz and uniformly bounded, see Lemma \[lem:boundedness\]. The existence of $g\in\mathcal{B}$ such that $$\sum_{i=1}^k \int_{X_i}gd\mu_i=\sup\Big\{\sum_{i=1}^k \int_{X_i}fd\mu_i\big| f\in\mathcal{B}\Big\}$$ follows by the Arzelà-Ascoli theorem.
Take now any extreme point $(\delta_{x_1},\dotsc,\delta_{x_k})$ of $\mathcal{P}$ and let $g\in\mathcal{B}$ be as above. Then for any $f\in\mathcal{B}$ we have $$\label{eqn:cosik}
\sum_{i=1}^kf(x_i)\leq\sum_{i=1}^k g(x_i).$$ Define $f\colon \{x_1,\dotsc,x_k\}\to\mathbb{R}$ by setting $f(x_1)=c(x_1,\dotsc,x_k)$ and $f(x_j)=0$ for $2\leq j\leq k$. Then $$\sum_{i=1}^k f(x_i)=c(x_1,\dotsc,x_k).$$ By Remark \[rem:infima\] there exists a function $\tilde{f}\in\mathcal{B}$ such that $$\sum_{i=1}^k \tilde{f}(x_i)=c(x_1,\dotsc,x_k).$$ By (\[eqn:cosik\]) it follows that also $$\sum_{i=1}^k g(x_i)=c(x_1,\dotsc,x_k).$$ The proof is complete.
The following theorem provides a novel interpretation of Kantorovich problem in the multimarginal setting as minimisation of certain linear functional over the set of all Choquet’s representations of $k$-tuples of probability measures $(\mu_1,\dotsc,\mu_k)\in\mathcal{P}$.
\[thm:optimany\] Let $X_1,\dotsc,X_k$ be compact Hausdorff spaces. Let $$c\colon X_1\times\dotsc\times X_k\to\mathbb{R}$$ be a Lipschitz function. Let $\mu_1,\dotsc,\mu_k$ be Borel probability measures on $X_i$, $i=1,\dotsc,k$ respectively. Then $$\sup\Big\{\sum_{i=1}^k\int_{X_i}f d\mu_i\big| f\colon\bigcup X_i\to\mathbb{R}, \sum_{i=1}^kf(x_i)\leq c(x_1,\dotsc,x_k)\text{ for }x_i\in X_i, i=1,\dotsc,k\Big\}$$ is equal to $$\inf\Big\{\int_{X_1\times\dotsc\times X_k} cd\pi\big| \pi\in \Gamma (\mu_1,\dotsc,\mu_k)\Big\},$$ where $\Gamma (\mu_1,\dotsc,\mu_k)$ stands for the set of all Borel probability measures on $X_1\times\dotsc\times X_k$ such that its marginals on $X_i$ are $\mu_i$ respectively for $i=1,\dotsc,k$. Moreover, both supremum and infimum are attained.
The assertions follow from Theorem \[thm:extreme\] and the Choquet’s theorem. Indeed, if $\pi_0$ is a Borel probability measure on the set of extreme points $\mathcal{E}$ of $\mathcal{P}$ of the previous theorem then an optimal $\pi\in\Gamma(\mu_1,\dotsc,\mu_k)$ is given by the formula $$\pi=\int_{\mathcal{E}}\xi_1\otimes \dotsc\otimes \xi_k d\pi_0(\xi)\text{, where }\xi=(\xi_1,\dotsc,\xi_k)\in \mathcal{E}.$$
Let $X_1,\dotsc,X_k$ be compact Hausdorff spaces. Let $$c\colon X_1\times\dotsc\times X_k\to\mathbb{R}$$ be a lower semicontinuous function. Let $\mu_1,\dotsc,\mu_k$ be Borel probability measures on $X_i$, $i=1,\dotsc,k$ respectively. Then $$\sup\Big\{\sum_{i=1}^k\int_{X_i}f d\mu_i\big| f\colon\bigcup_{i=1}^k X_i\to\mathbb{R}, \sum_{i=1}^kf(x_i)\leq c(x_1,\dotsc,x_k)\text{ for }x_i\in X_i, i=1,\dotsc,k\Big\}$$ is equal to $$\inf\Big\{\int_{X_1\times\dotsc\times X_k} cd\pi\big| \pi\in \Gamma (\mu_1,\dotsc,\mu_k)\Big\},$$ where $\Gamma (\mu_1,\dotsc,\mu_k)$ stands for the set of all Borel probability measures on $X_1\times\dotsc\times X_k$ such that its marginals on $X_i$ are $\mu_i$ respectively for $i=1,\dotsc,k$.
The proof follows analogous lines to that of proof of Corollary \[col:opticoma\].
The above proof extends also to the case of non-compact Polish spaces in the same way as in [@Villani2].
Martingale transport {#s:martingale}
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Here we characterise the set of extreme points of two measures $\mu,\nu$ in convex order. Below $\mathcal{F}$ is the cone of continuous convex functions on convex body $K$. Then $(\mu,\nu)\in\mathcal{P}_{\mathcal{F}}$ if and only if $(\mu,\nu)$ are in convex order. In the proof below we could use a result of Winkler [@Winkler]. Instead we follow more direct way for the sake of completeness and clarity.
\[thm:ex\] Let $K\subset\mathbb{R}^n$ be a convex body. Let $\mathcal{F}$ denote the set of continuous convex functions on $K$. Then the set of extreme points of $\mathcal{P}_{\mathcal{F}}$ is equal to the set of pairs $$\label{eqn:form}
\Big(\delta_x,\sum_{i=1}^{d+1}\lambda_i\delta_{x_i}\Big)$$ where $x=\sum_{i=1}^{d+1}\lambda_ix_i$, $d\leq n$, $\lambda_i>0$ for $i=1,\dotsc,d+1$ and $\sum_{i=1}^{d+1}\lambda_i=1$ and moreover $x_1,\dotsc,x_{d+1}$ are in general position.
By Theorem \[thm:extr\], the any extreme point of $\mathcal{P}_{\mathcal{F}}$ is of the form $(\delta_x,\eta)$ for some probability measure $\eta$ on $K$. Moreover, as any affine function belongs to $\mathcal{F}$, we see that $$x=\int_{\Omega} yd\nu(y).$$ Let us fix $x\in K$. Consider the set $\mathcal{A}$ of all Borel probability measures that have $x$ as their barycentre. To prove the assertion we ought to show that the extreme points of $\mathcal{A}$ are of the form $$\sum_{i=1}^{d+1}\lambda_i\delta_{x_i}\text{ for some }\lambda_1,\dotsc,\lambda_{d+1}>0 \text{ that sum up to one}$$ and $x_1,\dotsc,x_{d+1}$ are in general position and $d=1,\dotsc,n$ and that $$x=\sum_{i=1}^{d+1}\lambda_ix_i.$$ Let us first show that any extreme points $\gamma\in\mathcal{A}$ is supported on at most $n+1$ points. Suppose conversely, that there exist pairwise disjoint non-empty Borel sets $A_1,\dotsc,A_{n+2}$ such that $$K=\bigcup_{i=1}^{n+2}A_i\text{ and }\gamma(A_i)>0\text{ for }i=1,\dotsc,n+2.$$ Then there exist real numbers $t_1,\dotsc,t_{n+2}$, not all of them equal, such that $$0=\sum_{i=1}^{n+2}t_i \int_{A_i} (y-x)d\gamma(y).$$ We may assume that these numbers are all less than one. Set $$\gamma_1=\frac{\sum_{i=1}^{n+2}(1-t_i)\gamma|_{A_i}}{1-\sum_{i=1}^{n+2}t_i\gamma(A_i)}\text{ and }\gamma_2=\frac{\sum_{i=1}^{n+2}(1+t_i)\gamma|_{A_i}}{1+\sum_{i=1}^{n+2}t_i\gamma(A_i)}.$$ Then $\gamma_1,\gamma_2$ belong to $\mathcal{A}$. Moreover $$\gamma=\Big(1-\sum_{i=1}^{n+2}t_i\gamma(A_i)\Big)\gamma_1+\Big(1+\sum_{i=1}^{n+2}t_i\gamma(A_i)\Big)\gamma_2.$$ Thus $(\delta_x,\gamma)$ is not an extreme point of $\mathcal{A}$. The contradiction yields that $\gamma$ is supported on at most $n+1$ points.
Let $d+1$ be the number of points in the support. Let us show that we must necessarily have $$\gamma=\sum_{i=1}^{d+1}\lambda_i\delta_{x_i}\text{ for some }\lambda_1,\dotsc,\lambda_{d+1}>0 \text{ that sum up to one}$$ and $x_1,\dotsc,x_{d+1}$ in general position. Suppose that this is not the case. Then there exist non-negative $\alpha_1,\dotsc,\alpha_{d+1}$, not all of them equal to $\lambda_1,\dotsc,\lambda_{d+1}$, such that $$x=\sum_{i=1}^{d+1}\alpha_ix_i\text{ and }\sum_{i=1}^{d+1}\alpha_i=1.$$ Set $\chi=\sum_{i=1}^{d+1}\alpha_i\delta_{x_i}$. Then $\chi\in\mathcal{A}$. Moreover $$\gamma= \frac12(\gamma-\epsilon \chi)+\frac12(\gamma+\epsilon\chi),$$ for any $\epsilon\in(0,\min\{\lambda_1,\dotsc,\lambda_{d+1}\})$. The contradiction concludes the proof of the fact that any extreme point of $\mathcal{P}_{\mathcal{F}}$ is of the form (\[eqn:form\]).
Let us now show that any pair of that form is indeed an extreme point of $\mathcal{P}_{\mathcal{F}}$. Observe that by Jensen’s inequality any such pair belongs to $\mathcal{P}_{\mathcal{F}}$. If we had $$(\delta_x,\nu)=\frac12(\theta_1,\rho_1)+\frac12(\theta_2,\rho_2)\text{ for some }(\theta_1,\rho_1),(\theta_2,\rho_2)\in\mathcal{P}_{\mathcal{F}}\text{, then }$$ necessarily $\theta_1=\theta_2=\delta_x$ and $\rho_1,\rho_2$ are supported on $x_1,\dotsc,x_{d+1}$. As these points are in general position and $$x=\int_{\Omega} yd \rho_1(y)=\int_{\Omega} y d\rho_2(y),$$ we see that $\rho_1=\rho_2=\nu$.
The reasoning presented in Section \[s:transport\] may be generalised to other transportation problems. Here we discuss the case of martingale optimal transport. In this problem one is given two Borel probability measures $\mu,\nu$ on $\mathbb{R}^n$ in convex order. The task is to find a coupling $\pi$ of $\mu$ and $\nu$ such that it is a distribution of a one-step martingale and that minimises the integral $$\int_{\mathbb{R}^n\times\mathbb{R}^n}cd\pi$$ among all such couplings. Here $c\colon \mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ is a Borel measurable funciton, callled a cost function.
In the theorem below we employ the above characterisation of extreme points to prove a duality result for multidimensional martingale optimal transport, provided that the value of the dual problem is attained.
For results related to duality in the martingale optimal transport problem see [@Beiglbock4; @Beiglbock2].
Below $K$ is a convex body in $\mathbb{R}^n$ and $c\colon K\times K\to\mathbb{R}$ a Lipschitz function. We denote by $\mathcal{C}$ the set of continuous functions $g$ on the disjoint union of $K$ and $K$ such that $$g_1\Big(\sum_{i=1}^{n+1}\lambda_i x_i\Big)-\sum_{i=1}^{n+1}\lambda_ig_2(x_i)\leq \sum_{i=1}^{n+1}c\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big)$$ for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one. Here $g_1$ is the restriction of $g$ to the first copy of $K$ and $g_2$ the restriction of $g$ to the second copy of $K$.
\[thm:extrememart\] Let $K$ be a convex body in $\mathbb{R}^n$ and let $c\colon K\times K\to\mathbb{R}$ be a Lipschitz function. Let $f\in \mathcal{C}$. Let $\mathcal{F}_f$ be the tangent cone to $\mathcal{C}$ at $f$, i.e. the set of all continuous functions on $K\cup K$ of the form $\lambda (f-g)$ for some $\lambda\geq 0$ and $g\in \mathcal{C}$. Then the set of extreme points of the associated set $$\mathcal{P}_f=\big\{(\mu,\nu)\in\mathcal{P}(K)\times\mathcal{P}(K)| \int_K \phi_1 d\nu\leq\int_K \phi_2 d\mu\text{ for all }\phi\in\mathcal{F}_f,\mu,\nu \big\}$$ is of the form $(\delta_x,\sum_{i=1}^{d+1}\lambda_i x_i)$ for some $d\leq n$, $\lambda_1,\dotsc,\lambda_{d+1}$ positive that sum up to one and for $x_1,\dotsc,x_{d+1}\in K$ affinely independent, $x=\sum_{i=1}^{d+1}\lambda_ix_i$ such that $$f_1\Big(\sum_{i=1}^{d+1}\lambda_i x_i\Big)-\sum_{i=1}^{d+1}\lambda_if_2(x_i)= \sum_{i=1}^{d+1}\lambda_ic\Big(\sum_{j=1}^{d+1}\lambda_jx_j,x_i\Big).$$
The cone $\mathcal{F}_f$ is a closed convex cone, contains constants and is closed under maxima. Thus by Corollary \[col:twospaces\] any extreme point of $\mathcal{P}_f$ is of the form $(\delta_x,\eta)$ for some $x\in K$ and a Borel probability measure $\eta$ on $K$. Let $(\delta_x,\eta)\in\mathcal{P}$ be such an extreme point. Let $h$ be a function equal to some convex function $h$ on the first copy of $K$ and equal to the same function $h$ on the other copy of $K$. Then $f+h$ belongs to $\mathcal{C}$. Thus $\eta$ majorises $\delta_x$ in the convex order. Then we know that for any $g\in \mathcal{C}$ we have $$\label{eqn:bm}
\int_K \big(g_1(x)-g_2(y)\big)d\eta(y)\leq\int_K\big( f_1(x)-f_2(y)\big))d\eta(y).$$ By the assumption the right-hand side of the above inequality is bounded by $$\label{eqn:c}
\int_K c(x,y)d\eta(y).$$ Indeed, as $\eta$ majorises $\delta_x$, by Theorem \[thm:ex\], there is $$(\delta_x,\eta)=\int_{\mathcal{E}_{\mathcal{F}}}\xi d\pi(\xi),$$ where $\pi$ is a Borel probability measure on the set of extreme points $\mathcal{E}_{\mathcal{F}}$ of $\mathcal{P}_{\mathcal{F}}$. The assumption on $f$ is that for any such extreme point we have $$\int_K f_1d\xi_1-\int_K f_2d\xi_2\leq \int_K c(z,y)d\xi_2(y).$$ with $\delta_z=\xi_1$. Then the fact that (\[eqn:bm\]) is bounded by (\[eqn:c\]) follows by integration against $\pi$.
By the McShane extension formula (see [@McShane]), we may assume that $c$ is defined and Lipschitz on $\mathbb{R}^n\times\mathbb{R}^n$. Let us now take $x\in K$ and $g$ so that for $y\in K$ we have $g_2(y)=-c(x,y)$ and for $x'\in K$ set $$g_1(x')=\inf\Big\{ \sum_{i=1}^{n+1}\lambda_i \big(c(x',y_i)-c(x,y_i)\big)\big|\Big(\delta_{x'},\sum_{i=1}^{d+1}\lambda_i\delta_{y_i}\Big)\in \mathcal{E}_{\mathcal{F}}\Big\}.$$ Here the infimum is over all $y_1,\dotsc,y_{n+1}\in\mathbb{R}^n$, with barycentre $x'$, and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}\geq 0$ summing up to one. Then $$g_1(x')-\sum_{i=1}^{n+1}\lambda_ig_2(y_i)\leq \sum_{i=1}^{n+1}\lambda_ic(x',y_i)$$ for all $y_1,\dotsc,y_{n+1}\in K$ and all and such that $x=\sum_{i=1}^{n+1}\lambda_iy_i$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}\geq 0$ summing up to one. Moreover, $g_1(x)=0$. Moreover, as $c$ is Lipschitz, $g$ is Lipschitz. Indeed, for any $x',z'\in K$ and any $y_1,\dotsc,y_{n+1}\in\mathbb{R}^n$ and non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one and such that $$x'=\sum_{i=1}^{n+1}\lambda_iy_i$$ we have $$\sum_{i=1}^{n+1}\lambda_i \big(c(z',y_i+z'-x')-c(x,y_i+z'
-x')\big)\leq \sum_{i=1}^{n+1}\lambda_i \big(c(x',y_i))-c(x,y_i)\big)+3{\lVertz'-x'\rVert},$$ where $L$ is the Lipschitz constant of $c$. Thus $$g_1(z')\leq g_1(x')+3L{\lVertz'-x'\rVert}.$$ This is to say, $g_1$ is Lipschitz, hence continuous. Observe that for such $g$, the left-hand side of (\[eqn:bm\]) is equal to (\[eqn:c\]). As $\eta$ majorises $\delta_x$ in the convex order, there is a probability measure on the set of extreme points $\mathcal{E}_{\mathcal{F}}$ of $\mathcal{P}_{\mathcal{F}}$ corresponding to the convex order such that $$(\delta_x,\eta)=\int_{\mathcal{E}_{\mathcal{F}}}\xi d\pi(\xi).$$ It follows that for $\pi$-almost every $\xi$ we have $$\int_K (f_1(x)-f_2(y)\big)d\xi_2(y)=\int_K c(x,y)d\xi_2(y).$$ Any such $\xi$ is belongs to $\mathcal{F}_f$. Therefore any extreme point of $\mathcal{P}_f$ is necessary an extreme point of $\mathcal{P}_{\mathcal{F}}$. The assertion follows readily.
Let $K$ be a convex body in $\mathbb{R}^n$ and let $c\colon K\times K\to\mathbb{R}$ be a Lipschitz function. Let $\mu,\nu$ be two Borel probability measures on $K$ in convex order. Suppose that there exist continuous functions $\phi_1,\phi_2\colon K\to\mathbb{R}$ such that $$\int_K\phi_1 d\mu-\int_K \phi_2 d\nu=\sup\Big\{\int_K \phi'_1 d\mu-\int_K\phi'_2 d\nu\big| \phi'\in\mathcal{C}\Big\},$$ where $\mathcal{C}$ is the set of pairs of continuous functions on disjoint union of two copies od $K$ such that for all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that add up to one and all $x_1,\dotsc,x_{n+1}\in K$ there is $$\phi'_1\Big(\sum_{i=1}^{n+1}\lambda_i x_i\Big)-\sum_{i=1}^{n+1}\lambda_i\phi'_2(x_i)\leq \sum_{i=1}^{n+1}\lambda_i c\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big).$$ Here $\phi_1,\phi_2$ denote restriction of a function $\phi\in \mathcal{C}$ to the first and second copy of $K$ respectively. Then the supremum is equal to $$\inf\Big\{\int_{\mathcal{E}_{\mathcal{F}}}\int_K c\Big(\int_K yd\xi_2(y),z\Big)d\xi_2(z)\big| (\mu,\nu)=\int_{\mathcal{E}_{\mathcal{F}}}\xi d\pi(\xi),\pi \text{ is a probability}\Big\}.$$ Moreover the last infimum is attained. It is also equal to $$\inf\Big\{\int_{K\times K}\ cd\pi\big| \pi \text{ is a martingale coupling between }\mu\text{ and }\nu\Big\}.$$
First part of the theorem follows from Theorem \[thm:extrememart\]. The second one follows by taking $$\pi=\int_{\mathcal{E}_{\mathcal{F}}}\xi_1\otimes \xi_2 d\pi_0(\xi)$$ where $\pi_0$ is optimal probability measure on $\mathcal{E}_{\mathcal{F}}$ for the first minimisation problem.
Duality for martingale optimal transport {#s:duality}
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Martingale optimal transport admits a dual problem, which is to maximise $$\int_{\mathbb{R}^n} f_1 d\mu-\int_{\mathbb{R}^n} f_2 d\nu$$ among all continuous functions $f_1,f_2\colon \mathbb{R}^n\to\mathbb{R}$ that satisfy $$f_1(x)-f_2(y)\leq c(x,y)+\langle \gamma(x),y-x\rangle$$ for some $\gamma\colon \mathbb{R}^n\to\mathbb{R}$. In this section we investigate this class of functions.
Let $K\subset\mathbb{R}^n$ be a convex set. Then $F\subset K$ is a *face* of $K$ if for any $z\in F$ and any $t\in (0,1)$ such that $z=tx+(1-t)y$ for some $x,y\in K$ we have $x,y\in F$.
\[lem:infima2\] Let $K$ be a convex body in $\mathbb{R}^n$. Let $c\colon K\times K\to\mathbb{R}$ be a bounded function. Let $\mathcal{C}$ denote the set of all continuous functions $f$ on the disjoint union $K\cup K$ of two copies of $K$ such that for all $x\in K$ there exists $\gamma(x)\in\mathbb{R}^n$ such that for all $y\in K$ that belong to the maximal face of $K$ that contains $x$ in its relative interior there is $$\label{eqn:def}
f_1(x)-f_2(y)\leq c(x,y)+\langle \gamma(x),y-x\rangle.$$ Here $f_1$ is the restriction of $f$ to the first copy of $K$ and $f_2$ is the restriction of $f$ to the second copy of $K$. Then $\mathcal{C}$ is uniformly closed and is closed under maxima.
Choose $x\in \mathrm{int}K$. Let $f\in \mathcal{C}$ and let $\gamma$ be such that (\[eqn:def\]) holds true. Take $y=x-tv$ with $v$ a unit vector such that $$\langle \gamma,v\rangle={\lVert\gamma\rVert},$$ and $1>t>0$ small enough so that $y\in K$. Then, by (\[eqn:def\]), we have $$\label{eqn:bd}
{\lVert\gamma\rVert}\leq \frac{c(x,x-tv)+f_2(x-tv)-f_1(x)}t\leq \frac{{\lVertc\rVert}+2{\lVertf\rVert}}t.$$ Hence, if $(g_k)_{k=1}^{\infty}\in \mathcal{C}$ is a sequence that uniformly converges to a function $g$, then for $x\in\mathrm{int}K$ consider a sequence $(\gamma_k(x))_{k=1}^{\infty}$ of elements of $\mathbb{R}^n$ that satisfy $$\label{eqn:defi}
g_{k1}(y)-g_{k2}(x)\leq c(x,y)+\langle \gamma_k(x),y-x\rangle\text{ for all }y\in K.$$ By (\[eqn:bd\]), we may assume, passing to a subsequence, that $(\gamma_k(x))_{k=1}^{\infty}$ converges to $\gamma(x)$. Then, passing to the limit in (\[eqn:defi\]), we see that (\[eqn:def\]) is satisfied for $x$.
If the maximal face of $K$ that contains $x$ in its relative interior, then we repeat the above argument with $K$ replaced by the intersection of the face with its affine hull. Note that in this case $\gamma(x)$ may be also chosen to lie in the tangent space to the face.
If now $g^1,g^2\in \mathcal{C}$ satisfy (\[eqn:def\]) with $\gamma_1,\gamma_2$, then for $g=\max\{g^1,g^2\}$ we define $\gamma$ by setting $\gamma(x)=\gamma_1(x)$ if $g^1_1(x)\geq g^2_1(x)$ and $\gamma(x)=\gamma_2(x)$ otherwise. Suppose that $g^1_1(x)\geq g^2_1(x)$. Then we have for all $y\in K$, in the maximal face of $K$ containing $x$ in its relative interior, $$g_1(x)=g^1_1(x)\leq c(x,y)+\langle \gamma(x),y-x\rangle+ g^1_2(y)\leq c(x,y)+\langle \gamma(x),y-x\rangle+ g_2(y).$$ If $g_1^1(x)<g_1^2(x)$ then we obtain analogously such inequality. It follows that $g\in\mathcal{C}$.
\[thm:equalityc\] Let $K$ be a convex body in $\mathbb{R}^n$. Let $c\colon K\times K\to\mathbb{R}$ be Lipschitz. Let $$f\colon K\cup K\to\mathbb{R}$$ be a continuous function on a disjoint union of two copies of $K$. Let $f_1$ and $f_2$ denote the restriction of $f$ to the first and second copy of $K$ respectively. Then the following conditions are equivalent:
i) \[i:fc\] for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$f_1\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)-\sum_{i=1}^{n+1}\lambda_i f_2(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big),$$
ii) \[i:gammac\] for any $x\in K$ there exists $\gamma(x)\in\mathbb{R}^n$ such that for all $y\in K$ in the maximal face of $K$ that contains $x$ in its relative interior we have $$f_1(x)-f_2(y)\leq c(x,y)+\langle \gamma(x), y-x\rangle.$$
Without loss of generality $c$ is bounded below by zero. Let us denote the set of continuous functions on $K$ that satisfy condition \[i:fc\]) by $\mathcal{C}_1$ and the set of continuous functions on $K$ that satisfy condition \[i:gammac\]) by $\mathcal{C}_2$. Observe that $\mathcal{C}_2\subset \mathcal{C}_1$. Moreover both $\mathcal{C}_1$ and $\mathcal{C}_2$ are convex sets that are closed with respect to the uniform norm and closed under maxima. This follows by Lemma \[lem:infima2\]. Suppose that there exists $f\in \mathcal{C}_1\setminus \mathcal{C}_2$. Then by the Hahn-Banach theorem there exists a Borel measure $\eta\in\mathcal{M}(K\cup K)$ such that $$\label{eqn:assuc}
\int_{K\cup K} fd\eta\geq \int_{K\cup K} gd\eta +\epsilon\text{ for all }g\in \mathcal{C}_2\text{ and some }\epsilon>0.$$ Observe that since constant functions belong to $\mathcal{C}_2$, measure $\eta$ may be written as a difference of two non-negative Borel measures of equal masses. Since any continuous function that is negative on the first copy of $K$ and positive on the second belong to $\mathcal{C}_2$, we see that $\eta=\eta_+-\eta_-$, with $\eta_+$ and $\eta_-$ supported on the first and on the second copy of $K$ respectively. Without loss of generality we may assume that these measures are probabilities. Let $\mathcal{F}_f$ denote the convex cone of all functions of the form $\lambda(g-f)$ for $\lambda\geq 0$ and $g\in \mathcal{C}_2$. Consider the set $$\mathcal{P}=\Big\{(\mu,\nu)\in\mathcal{P}(K)\times\mathcal{P}(K)\big|\mu\prec_{\mathcal{F}_f}\nu\Big\}.$$ Then $(\eta_+,\eta_-)\in\mathcal{P}$. Moreover, by Lemma \[lem:infima2\], $\mathcal{F}_f$ satisfies the assumptions of Theorem \[thm:extr\]. Therefore any extreme point of $\mathcal{P}$ has the form $(\delta_x,\sigma)$ for some probability measure $\sigma$. Define $h\colon K\cup K\to\mathbb{R}$ by $h_2(y)=-c(x,y)$ for $y\in K$ and for $x'\in K$ set $$h_1(x')=\inf\{c(x',y)-c(x,y)|y\in K\}.$$ Then $h_1(x)=0$, $h$ is continuous by Lipschitzness of $c$ and thus $h\in \mathcal{C}_2$, with $\gamma$ equal to zero. It follows that there exists $\sigma$ such that $$\label{eqn:bouc}
\int_K c(x,y) d\sigma(y)=\int_K \big(h_1(x)-h_2\big)d\sigma\leq \int_K \big(f_1(x)-f_2\big)d\sigma-\epsilon.$$ Observe that $(\delta_x,\sigma)\in\mathcal{P}$ is ordered in convex order. Then there exists a probability measure $\pi$ on the set $\mathcal{E}$ of extreme points of measures in convex order such that $$(\delta_x,\sigma)=\int_{\mathcal{E}}\xi d\pi(\xi).$$ It follows, by Theorem \[thm:ex\], and the definition of $\mathcal{C}_1$ that $$\int_K \big(f_1(x)-f_2\big)d\sigma \leq \int_K c(x,y)d\sigma(y),$$ contradictory to (\[eqn:bouc\]). The converse inclusion follows readily.
\[col:col\] Let $K$ be a convex set in $\mathbb{R}^n$. Suppose that $c\colon K\times K\to\mathbb{R}$ is Lipschitz. Let $$f\colon K\cup K\to\mathbb{R}$$ be a continuous function on a disjoint union of two copies of $K$. Let $f_1$ and $f_2$ denote the restriction of $f$ to the first and second copy of $K$ respectively. Then the following conditions are equivalent:
i) \[i:fcc\] for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$f_1\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)-\sum_{i=1}^{n+1}\lambda_i f_2(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big),$$
ii) \[i:gammacc\] for any $x\in K$ there exists $\gamma(x)\in\mathbb{R}^n$ such that for all $y\in K$ in the maximal face of $K$ that contains $x$ in its relative interior we have $$f_1(x)-f_2(y)\leq c(x,y)+\langle \gamma(x), y-x\rangle.$$
Without loss of generality we may assume that $\mathrm{int}K$ is non-empty. Choose a increasing sequence $(K_n)_{n=1}^{\infty}$ of closed convex sets in $\mathbb{R}^n$ such that its union is $\mathrm{int}K$. Suppose that $f\colon K\cup K\to\mathbb{R}$ satisfies \[i:fcc\]). Pick $x \in\mathrm{int}K$. Then by Theorem \[thm:equalityc\] for any $n\in\mathbb{N}$ sufficiently large so that $x\in\mathrm{int}K_n$ there exists $\gamma_n$ such that for all $y\in K_n$ there is $$\label{eqn:costam}
f_1(x)-f_2(y)\leq c(x,y)+\langle\gamma_n, y-x\rangle.$$ Let $n$ be so large that $B(x,\epsilon)\subset K_n$, where $B(x,\epsilon)$ denotes the closed ball of radius $\epsilon$ centred at $x$. Suppose that $\gamma_n\neq 0$ and set $y_n=x-\epsilon\frac{\gamma_n}{{\lVert\gamma_n\rVert}}$. Then $y_n\in K_n$ and therefore $${\lVert\gamma_n\rVert}\leq\frac1{\epsilon}\big( c(x,y_n)-f_1(x)+f_2(y_n)\big).$$ As $c$ and $f$ are continuous, the right-hand side of the above inequality is uniformly bounded. Hence, so is the left-hand side. We may therefore pick $\gamma$ that is an accumulation point of the sequence $(\gamma_n)_{n=1}^{\infty}$. From (\[eqn:costam\]) and from continuity of $f$ it follows now that for all $y\in \mathrm{int}K$ $$f_1(x)-f_2(y)\leq c(x,y)+\langle\gamma, y-x\rangle.$$ This is to say, $f$ satisfies also \[i:gammacc\]) if $x\in\mathrm{int}K$.
If the maximal face of $K$ that contains $x$ in its relative interior is of lower dimension, then we repeat the above argument with $K$ replaced by the intersection of the face with its affine hull. Note that in this case $\gamma(x)$ may be also chosen to lie in the tangent space to the face. The converse inclusion is straightforward.
\[col:onefunction\] Let $K$ be a convex set in $\mathbb{R}^n$. Suppose that $c\colon K\times K\to\mathbb{R}$ is Lipschitz. Let $$f\colon K\to\mathbb{R}$$ be a continuous function on $K$. The following conditions are equivalent:
i) \[i:fccc\] for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)- \sum_{i=1}^{n+1}\lambda_i f(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big),$$
ii) \[i:gammaccc\] for any $x\in K$ there exists $\gamma(x)\in\mathbb{R}^n$ such that for all $y\in K$ in the same face of $K$ as $x$ we have $$f(x)-f(y)\leq c(x,y)+\langle \gamma(x), y-x\rangle.$$
Moreover, if $K$ is open and additionally ${\lvertc(x,y)\rvert}\leq\Lambda {\lVertx-y\rVert}$ for all $x,y\in\mathbb{R}^n$ and some constant $\Lambda$, then we may drop the assumption on the continuity of $f$.
Follows from the above corollary. The second part follows by Lemma \[lem:lipm\].
\[lem:onedim\] Suppose that $f\colon [a,b]\to\mathbb{R}$ is such that for all $\lambda\in [0,1]$ and all $x,y\in [a,b]$ there is $$\lambda f(x)+(1-\lambda)f(y)-f(\lambda x+(1-\lambda)y)\leq \lambda c(\lambda x+(1-\lambda)y, x)+(1-\lambda)c(\lambda x+(1-\lambda)y,y).$$ Then for all $a\leq x_1<x_2<x_3\leq b$ we have that $\frac{f(x_3)-f(x_1)}{x_3-x_1}$ is bounded below by $$\frac{f(x_3)-f(x_2)}{x_3-x_2}+\frac{c(x_2,x_3)-c(x_2,x_1)}{x_3-x_1}-\frac{c(x_2,x_3)}{x_3-x_2}$$ and above by $$\frac{f(x_2)-f(x_1)}{x_2-x_1}+\frac{c(x_2,x_3)-c(x_2,x_1)}{x_3-x_1}+\frac{c(x_2,x_1)}{x_2-x_1}.$$
Let $\lambda\in (0,1)$ be such that $x_2=\lambda x_1+(1-\lambda)x_3$, that is $$\lambda=\frac{x_3-x_2}{x_3-x_1}.$$ Then we know that $$\lambda f(x_1)+(1-\lambda)f(x_3)-f(x_2)\leq \lambda c(x_2,x_1)+(1-\lambda)c(x_2,x_3).$$ Hence putting formula for $\lambda$ we obtain that $$\frac{f(x_3)-f(x_2)}{x_3-x_2}+\frac{c(x_2,x_3)-c(x_2,x_1)}{x_3-x_1}-\frac{c(x_2,x_3)}{x_3-x_2}\leq \frac{f(x_3)-f(x_1)}{x_3-x_1}$$ and $$\frac{f(x_3)-f(x_1)}{x_3-x_1}\leq\frac{f(x_2)-f(x_1)}{x_2-x_1}+\frac{c(x_2,x_3)-c(x_2,x_1)}{x_3-x_1}+\frac{c(x_2,x_1)}{x_2-x_1}.$$
\[lem:lipm\] Let $K$ be a convex set in $\mathbb{R}^n$. Suppose that $f\colon K\to\mathbb{R}$ is such that for all $x,y\in K$ and all $\lambda\in [0,1]$ there is $$\lambda f(x)+(1-\lambda)f(y)-f(\lambda x+(1-\lambda)y)\leq \lambda c(\lambda x+(1-\lambda)y, x)+(1-\lambda)c(\lambda x+(1-\lambda)y,y).$$ Suppose that $c$ is $L$-Lipschitz in the second variable and is such that for all $x,y\in K$ there is ${\lvertc(x,y)\rvert}\leq \Lambda{\lVertx-y\rVert}$ for some constant $\Lambda$. Then $f$ is locally Lipschitz in $K$.
Suppose that $n=1$. Then $K=[a,d]$ for some $a<d$. Choose numers $b,c$ so that $a<b<c<d$. Then applying Lemma \[lem:onedim\] four times yields that for any $x,y$ such that $b<x<y<c$ we have $$\frac{f(y)-f(x)}{y-x}\leq \frac{f(b)-f(a)}{b-a}+\frac{c(x,y)}{y-x}+\frac{c(b,a)}{b-a}+\frac{c(b,y)-c(b,a)}{y-a}-\frac{c(x,y)-c(x,a)}{y-a}$$ and $$\frac{f(d)-f(c)}{d-c}-\frac{c(c,x)}{c-x}+\frac{c(c,d)-c(c,x)}{d-x}-\frac{c(y,x)}{y-x}-\frac{c(y,d)-c(y,x)}{d-x}\leq\frac{f(y)-f(x)}{y-x}.$$ In particular on $[b,c]$ function $f$ has Lipschitz constant at most $$\max\Big\{\Big|\frac{f(b)-f(a)}{b-a}+2L+2\Lambda\Big|,\Big|\frac{f(d)-f(c)}{d-c}-2L-2\Lambda\Big|\Big\}.$$ Suppose now that $n>1$ and that, by induction, that the Lemma holds true for all dimensions at most $n-1$. Choose any ball $B$ in $K$ and simplices $X,Y$ such that $B\subset X\subset Y\subset K$ and such that $B$ and the boundaries of $X$ and $Y$ are pairwise disjoint. Then, by the inductive assumption, $f$ is continuous on $X$ and on $Y$, and therefore the function $$\partial X\times \partial Y\ni (x,y)\mapsto\frac{{\lvertf(x)-f(y)\rvert}}{{\lVertx-y\rVert}}\in\mathbb{R}$$ is bounded by a constant $M$. Choose any points $x,y\in B$. Choose a unique line passing through $x$ and $y$. Then there exist unique points $x_1,x_2\in X$ and $y_1,y_2\in Y$ such that the line intersects $\partial X$ in $x_1,x_2$ and $\partial Y$ in $y_1,y_2$ where, without loss of generality, $y_1<x_1<x<y<x_2<y_2$ on the line. By Lemma \[lem:onedim\] we see that $$\frac{{\lvertf(y)-f(x)\rvert}}{{\lVertx-y\rVert}}\leq\max\Big\{\Big|\frac{f(y_2)-f(x_2)}{{\lVerty_2-x_2\rVert}}-2L-2\Lambda\Big|,\Big|\frac{f(x_1)-f(y_1)}{{\lVerty_2-x_2\rVert}}+2L+2\Lambda\Big|\Big\}$$ Therefore $f$ has Lipschitz constant at most $M+2L+2\Lambda$ on $B$.
Uniform convexity and uniform smoothness {#s:uniform}
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In this section we employ results of the previous section to provide a characterisation of uniformly smooth and uniformly convex functions on $\mathbb{R}^n$, or, more generally, on an open, convex set $K\subset\mathbb{R}^n$. We refer the reader to [@Aze]. Let us recall the definitions. Let $\sigma\colon\mathbb{R}\to\mathbb{R}$. A function $f\colon K\to\mathbb{R}$ is called $\sigma$-convex provided that $$f(\lambda x+(1-\lambda)y)+\lambda(1-\lambda)\sigma({\lVertx-y\rVert})\leq \lambda f(x)+(1-\lambda)f(y)$$ for all $\lambda\in [0,1]$ and all $x,y\in K$. A function $g\colon K\to\mathbb{R}$ is called $\sigma$-smooth provided that $$g(\lambda x+(1-\lambda)y)+\lambda(1-\lambda)\sigma({\lVertx-y\rVert})\geq \lambda g(x)+(1-\lambda)g(y)$$ for all $\lambda\in [0,1]$ and all $x,y\in K$.
Another notion of convexity and smoothness is as follows (see [@Aze]). Suppose that $\gamma\in\mathbb{R}^n$. We say that $f\colon K\to\mathbb{R}$ is $\sigma$-uniformly convex at $x\in K$ with respect to $\gamma$ if for all $y\in\ K$ there is $$f(x)+\sigma({\lVerty-x\rVert})+\langle \gamma,y-x\rangle\leq f(y).$$ Likewise, $g\colon K\to\mathbb{R}$ is called $\sigma$-uniformly smooth at $x\in K$ with respect to $\gamma$ if for all $y\in K$ there is $$f(x)+\sigma({\lVerty-x\rVert})+\langle \gamma,y-x\rangle\geq f(y).$$ Note that the condition that for any $x\in K$, $f$ is $\sigma$-uniformly convex at any $x$, with respect to some $\gamma(x)$, is equivalent to condition \[i:gammaccc\]) of Corollary \[col:onefunction\] for the cost function $c(x,y)=-\sigma({\lVerty-x\rVert})$, $x,y\in K$. Similarly, $\sigma$-uniform smoothness of a function $g\colon K\to\mathbb{R}$ is equivalent to condition \[i:gammaccc\]) of Corollary \[col:onefunction\] for $-g$ and the cost function $c(x,y)=\sigma({\lVerty-x\rVert})$, $x,y\in K$.
Now, Corollary \[col:onefunction\] implies the following theorem, which complements the results of [@Aze].
\[thm:uni\] Let $K\subset\mathbb{R}^n$ be an open convex set. Suppose that $\sigma\colon\mathbb{R}\to\mathbb{R}$ is Lipschitz and such that for some constant $\Lambda$ $${\lvert\sigma(t)\rvert}\leq \Lambda t\text{ for all non-negative } t.$$ Let $f\colon K\to\mathbb{R}$. The following conditions are equivalent:
i) for any $x\in K$ the function $f$ is $\sigma$-uniformly convex at $x$ with respect to some $\gamma(x)\in\mathbb{R}^n$,
ii) for any $x_1,\dotsc,x_{n+1}\in K$ and any non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)-\sum_{i=1}^{n+1}\lambda_i f(x_i)\leq -\sum_{i=1}^{n+1}\lambda_i\sigma\Big(\Big\lVert \sum_{j=1}^{n+1}\lambda_jx_j-x_i \Big\rVert\Big).$$
Also, the following conditions are equivalent:
i) for any $x\in K$ the function $f$ is $\sigma$-uniformly smooth at $x$ with respect to some $\gamma(x)\in\mathbb{R}^n$,
ii) for any $x_1,\dotsc,x_{n+1}\in K$ and any non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$\sum_{i=1}^{n+1}\lambda_i f(x_i)-f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)\leq \sum_{i=1}^{n+1}\lambda_i\sigma\Big(\Big\lVert \sum_{j=1}^{n+1}\lambda_jx_j-x_i \Big\rVert\Big).$$
Moreover, any function $f$ that satisfies one of the above conditions is locally Lipschitz in $K$.
Follows from the Corollary \[col:onefunction\]. Lipschitzness follows by Lemma \[lem:lipm\].
Martingale triangle inequality {#s:triangle}
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We introduce *martingale triangle inequality* for functions $c\colon K\times K\to\mathbb{R}$, where $K\subset\mathbb{R}^n$ is convex. We show that if it is satisfied by a cost function $c$, which vanishes on the diagonal, then one may take $f_1=f_2$ in the dual problem to the martingale optimal transport.
\[defin:martingale\] Let $K\subset\mathbb{R}^n$ be a convex set. Let $c\colon K\times K\to\mathbb{R}$. We say that $c$ satisfies *martingale triangle inequality* provided that for all $x,x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$\label{eqn:con}
\sum_{i=1}^{n+1}\lambda_i c(x,x_i)-c\Big(x,\sum_{i=1}^{n+1}\lambda_ix_i\Big)\leq \sum_{i=1}^{n+1} \lambda_i c\Big( \sum_{j=1}^{n+1}\lambda_j x_j,x_i\Big).$$
\[thm:clc\] Let $K$ be a convex body in $\mathbb{R}^n$. Let $c\colon K\times K\to\mathbb{R}$ be a continuous function satisfying martingale triangle inequality and such that for all $x\in K$ there is $c(x,x)=0$.
Let $\mathcal{B}_1$ denote the set of continuous functions $f\colon K\to\mathbb{R}$ such that for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)- \sum_{i=1}^{n+1}\lambda_i f(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big).$$
Let $\mathcal{B}_2$ denote the set of continuous functions $g\colon K\cup K\to\mathbb{R}$ on the disjoint union of $K$ and $K$ such that for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$g_1\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)- \sum_{i=1}^{n+1}\lambda_i g_2(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big),$$ where $g_1$ and $g_2$ denote restrictions of $g$ to, respectively, the first and the second copy of $K$. Then $$\label{eqn:supremum}
\sup\Big\{\int_K f d(\mu-\nu)\big| f\in \mathcal{B}_1\Big\}=\sup\Big\{\int_K g_1 d\mu-\int_K g_2 d\nu\big| g\in \mathcal{B}_2\Big\}$$
Clearly, the supremum on the right-hand side of (\[eqn:supremum\]) is at least the supremum on the left-hand side of (\[eqn:supremum\]), as if $f\in\mathcal{B}_1$, then $g\colon K\cup K\to\mathbb{R}$ defined by $g_1=f$ and $g_2=f$ belongs to $\mathcal{B}_2$.
Suppose that there exists $\epsilon>0$ such that for any function $f\in\mathcal{B}_1$ $$\int_K f d(\mu-\nu)+2\epsilon\leq \sup\Big\{\int_K g_1 d\mu-\int_K g_2 d\nu\big| g\in \mathcal{B}_2\Big\}.$$ It follows that there exists $g\in\mathcal{B}_2$ such that for all $f\in\mathcal{B}_1$ we have $$\label{eqn:eps}
\int_K f d(\mu-\nu)+\epsilon\leq \int_K g_1 d\mu-\int_K g_2 d\nu.$$ Let $\mathcal{F}_g$ denote the convex cone of all functions of the form $\lambda(\tilde{f}-g)$, where $\lambda$ is non-negative, $\tilde{f}\colon K\cup K\to\mathbb{R}$ is defined to be equal to $f$ on the first and on the second copy of $K$. Then $\mathcal{F}_g$ is a closed convex cone that is closed under maxima and contains constants. Moreover $(\mu,\nu)$ belongs to the set $$\mathcal{P}=\Big\{(\mu',\nu')\in\mathcal{P}(K)\times\mathcal{P}(K)\big| \mu'\prec_{\mathcal{F}_g}\nu'\Big\}.$$ Therefore, by Theorem \[thm:extr\], any extreme point of $\mathcal{P}$ is of the form $(\delta_x,\eta)$ for some $x\in K$ and probability measure $\eta\in\mathcal{M}(K)$. Define a function $h\colon K\to\mathbb{R}$ by $h(y)=-c(x,y)$ for $y\in K$. Then, by the martingale triangle inequality, $h\in\mathcal{B}_1$. Hence, by (\[eqn:eps\]), we have for some $\eta\in\mathcal{M}(K)$ such that $(\delta_x,\eta)\in\mathcal{P}$ $$\label{eqn:epsprim}
\int_K c(x,y)d\eta(y)+\epsilon\leq \int_K \big( g_1(x)-g_2(y) \big)d\eta(y)$$ Note that any pair $(\delta_x,\eta)\in\mathcal{P}$ is in convex order. Thus there exists a Borel probability measure $\pi$ on the set $\mathcal{E}$ of extreme points of pairs of measures in covnex order such that $$\label{eqn:repres}
(\delta_x,\eta)=\int_{\mathcal{E}}\xi d\pi(\xi).$$ But, as $g\in\mathcal{B}_2$, for any $\xi\in\mathcal{E}$ we have $\int_K (g_1(x)-g_2(y))d\xi(y)\leq \int_K c(x,y) d\xi(y)$. This, together wih (\[eqn:repres\]) and (\[eqn:epsprim\]), yields a contradiction.
Condition (\[eqn:con\]) is satisfied if $c$ is a distance function with respect to some metric on $K$ and also it is satisfied if $c$ is concave in the second variable and non-negative. Also, this condition defines a closed convex cone. Note that for function given by $c(x,y)={\lVertx-y\rVert}^2$ for $x,y\in K$ we have equality in (\[eqn:con\]).
Let us now see what happends when the cost function satisfies the martingale triangle inequality. Let us note first that inequality $$f\Big(\sum_{i=1}^{d+1}\lambda_i x_i
\Big)-\sum_{i=1}^{d+1}\lambda_if(x_i)\leq \sum_{i=1}^{d+1} c\Big(\sum_{j=1}^{d+1}\lambda_j x_j,x_i\Big)$$ is equivalent to saying that for any martingale $(X_0,X_1)$ there is $$\mathbb{E}\big( f(X_0)-f(X_1)\big)\leq \mathbb{E} c(X_0,X_1).$$ Suppose now that $(X_t)_{t\in [0,1]}$ is a martingale. Then $$\mathbb{E}\big( f(X_0)-f(X_1)\big)\leq \mathbb{E}\sum_{i=0}^{k-1} c(X_{t_i},X_{t_{i+1}})$$ for all numbers $0=t_0<t_1<\dotsc<t_k=1$.
In the case of cost $c(x,y)={\lVertx-y\rVert}^2$, passing to the limit, we would get, on the right-hand side, the quadratic variation of the martingale $X$.
In what follows we shall employ quadratic variation of a martingale. We refer the reader to [@Revuz] for a comprehensive introduction to this and related notions.
\[lem:mar\] Let $K\subset\mathbb{R}^n$ be a convex body. Suppose that $c\colon K\times K\to\mathbb{R}$ is continuous and three times continuously differentiable in the second variable and vanishes on the diagonal. Let $(X_t)_{t\in [0,1]}$ be a continuous martingale with values in $K$. Then $$\mathbb{E}\sum_{i=0}^{k-1}c(X_{t_i},X_{t_{i+1}})$$ converges to $$\frac12 \mathbb{E}\int_0^1\big\langle D_2^2 c(X_s,X_s),d\langle X\rangle_s\big\rangle$$ as the mesh of the partition $0=t_0<t_1<\dotsc<t_{k-1}<t_k=1$ converges to zero.
For any $x,y\in K$ there exists $z\in K$ such that $$\label{eqn:tay}
c(x,y)=D_2 c(x,x)(y-x)+\frac12 D_2^2 c(x,x)(y-x)^2 +\frac16 D_2^3 c(x,z)(y-x)^3.$$ Let $C=\frac16 \sup\big\{ {\lVertD_2^3 c(x,z)\rVert}\big| x\in K,z\in K\big\}$. Note that for any given partition $0=t_0<t_1<\dotsc<t_{k-1}<t_k=1$ with mesh $$\delta=\sup\big\{{\lVertt_{i+1}-t_i\rVert}\big|i=0,\dotsc,k-1\big\}$$ there is $$\mathbb{E} \sum_{i=0}^{k-1}{\lVertX_{t_{i+1}}-X_{t_i}\rVert}^3\leq \mathbb{E} \sup\big\{{\lVertX_s-X_t\rVert}\big| s,t\in K{\lVerts-t\rVert}\leq\delta \big\} \sum_{i=0}^{k-1}{\lVertX_{t_{i+1}}-X_{t_i}\rVert}^2.$$ Now, by continuity of the martingale, the first factor converges to zero almost surely as $\delta$ goes to zero. The second factor, converges to the quadratic variation of $X$ in $L^2(\mathbb{P})$, and also in $L^1(\mathbb{P})$. Therefore, by the dominated convergence theorem, the considered expression converges to zero.
Note now that, by (\[eqn:tay\]), by vanishing of $c$ on the diagonal and by the martingale condition, there is $$\mathbb{E}\sum_{i=0}^{k-1}c(X_{t_i},X_{t_{i+1}})=\frac12 \mathbb{E}\sum_{i=0}^{k-1}D_2^2 c(X_{t_i},X_{t_i})(X_{t_{i+1}}-X_{t_i})^2+M_k,$$ where $M_k$ converges to zero, by the above considerations. By the margingale condition, and definition of the quadratic variation, $$\frac12 \mathbb{E}\sum_{i=0}^{k-1}D_2^2 c(X_{t_i},X_{t_i})(X_{t_{i+1}}-X_{t_i})^2=\frac12 \mathbb{E}\sum_{i=0}^{k-1}\int_{t_{i}}^{t_{i+1}}\big\langle D_2^2c(X_{t_i},X_{t_i}),d\langle X\rangle_s\big\rangle$$ The assertion follows now readily by continuity of the second derivative of $c$.
\[thm:condition\] Let $K\subset\mathbb{R}^n$ be a convex body. Let $c\colon K\times K\to\mathbb{R}$ be continuous, three times continuously differentiable in the second variable and vanishing on the diagonal. Suppose that $f\colon K\to\mathbb{R}$ is twice continuously differentiable. Suppose that for any $x_1,\dotsc,x_{n+1}\in K$ and any non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one there is $$\label{eqn:cod}
f\Big(\sum_{i=1}^{n+1}\lambda_i x_i\Big)-\sum_{i=1}^{n+1}\lambda_if(x_i)\leq\sum_{i=1}^{n+1}\lambda_i c\Big( \sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big).$$ Then for any $x\in K$ there is $-D^2f(x)\leq D_2^2c(x,x)$,
As already noted the assumption on $f$ is equivalent to assuming that for any martingale $(X_0,X_1)$ with values in $K$ there is $$\mathbb{E} f(X_0)-f(X_1)\leq \mathbb{E} c(X_0,X_1).$$ Take any martingale $(X_t)_{t\in [0,1]}$ with continuous paths and with values in $K$. Then, for any partition $0=t_0<t_1<\dotsc<t_{k-1}<t_k=1$ there is $$\mathbb{E}f(X_0)-f(X_1)\leq \mathbb{E}\sum_{i=0}^{k-1} c(X_{t_i},X_{t_{i+1}}).$$ By Lemma \[lem:mar\], and by the Itô formula we infer that for any continuous martingale $(X_t)_{t\in [0,1]}$ with values in $K$ there is $$-\mathbb{E}\int_0^1 \big\langle D^2f(X_s),d\langle X\rangle_s\big\rangle\leq \mathbb{E}\int_0^1 \big\langle D_2^2 c(X_s,X_s),d\langle X\rangle_s\big\rangle.$$ The assertion for $x\in\mathrm{int}K$ follows from Lemma \[lem:bound\] below. For the other points, it follows by continuity.
\[lem:bound\] Let $K\subset\mathbb{R}^n$ be a convex open set. Let $f\colon K\to\mathbb{R}^{n\times n}$ be a continuous function. Then the following conditions are equivalent:
i) \[i:pos\] for any continuous martingale $(X_t)_{t\in [0,1]}$ with values in $K$ there is $$\mathbb{E}\int_0^1 \big\langle f(X_s) ,d\langle X\rangle_s\big\rangle\geq 0,$$
ii) $f$ has values in symmetric and positive semidefinite matrices.
Clearly if $f$ has values in symmetric and positive semidefinitie, then the first condition holds true. Suppose that \[i:pos\]) holds true. Let $S\in \mathbb{R}^{n\times n}$ be any positive semidefinie symmetric matrix. Let $x\in K$, let $\epsilon>0$. Define for $t\in [0,1]$ $$X_t=x+\epsilon S B_t.$$ where $(B_t)_{t\in [0,1]}$ is standard Brownian motion. Then $$\langle X\rangle_t=S^2 t.$$ Therefore, the assumption implies that $$\label{eqn:epsi}
\mathbb{E}\int_0^1\langle f\Big(x+\epsilon S B_{s\wedge \tau} \Big),S^2\rangle ds\geq 0,$$ where $\tau$ is the stopping time, the first time $X$ belongs to the complement of $K$. Hence letting $\epsilon$ to zero we get from (\[eqn:epsi\]) that $$\langle f(x),S^2\rangle\geq 0.$$ As any positive symmetric matrix admits a square root, it follows that for any such matrix $T$ and any $x\in K$ there is $\langle f(x),T\rangle\geq 0$. The conclusion follows, as the cone of symmetric positive semidefinite matrices is self-dual.
We shall denote by $\mathcal{B}$ the set of continuous functions $f\colon K\to\mathbb{R}$ such that for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one, there is $$f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)-\sum_{i=1}^{n+1}\lambda_if(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big).$$
Suppose that $K\subset\mathbb{R}^n$ is a convex body. Let $c\colon K\times K\to\mathbb{R}$ be continuous, three times continuously differentiable in the second variable and vanishing on the diagonal. Suppose that $c$ satisfies martingale triangle inequality. Then for any $x,y\in K$ $$D_2^2 c(x,y)\leq D_2^2c(y,y).$$
Follows from Theorem \[thm:condition\], as $c$ satisfies martingale triangle inequality if and only if for any $x\in K$ the function $f=-c(x,\cdot)$ belongs to $\mathcal{B}$.
Below, if $K$ is a convex body, and $x\in K$, we denote by $K(x)$ the face of $K$ that contains $x$ in its relative interior.
\[col:rep\] Let $K\subset\mathbb{R}^n$ be a convex body. Suppose that $c\colon K\times K\to\mathbb{R}$ satisfies martingale triangle inequality and vanishes on the diagonal. A continuous function $f\colon K\to\mathbb{R}$ belongs to $\mathcal{B}$ if and only if it is of the form $$\label{eqn:forma}
f(x)=\sup\Big\{b(y)+\langle a(y),y-x\rangle-c(y,x)\big| y\in K(x)\Big\}$$ for some functions $b\colon K\to\mathbb{R}$ and $a\colon K\to\mathbb{R}^n$. In particular, the considered class $\mathcal{B}$ of functions is a minimal class containing sums of affine functions and functions of the form $x\mapsto -c(y,x)$ that is closed under suprema.
Take any $x\in K$. We need to show that for any $x_1,\dotsc,x_{n+1}\in K$ and any non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one and such that $$\label{eqn:bary}
x=\sum_{i=1}^{n+1}\lambda_i x_i,$$ there is $$\label{eqn:inequality}
f(x)-\sum_{i=1}^{n+1}\lambda_if(x_i)\leq \sum_{i=1}^{n+1}c(x,x_i).$$ If (\[eqn:bary\]) holds true, then it follows that $x_1,\dotsc,x_{n+1}\in K(x)$. Thus, (\[eqn:inequality\]) will follow if we show that for any $y\in K(x)$, any function on $K(x)$ of the form $$z\mapsto b(y)+\langle a(y),z-y\rangle-c(y,z)$$ belongs to $\mathcal{B}$. This follows by the assumption on $c$.
Conversely, Corollary \[col:onefunction\], implies that if $f$ belongs to $\mathcal{B}$ then there exists $\gamma\colon K\to\mathbb{R}^n$ such that if $x\in K$ then for all $y\in K(x)$ there is $$f(x)-f(y)\leq \langle \gamma(x),y-x\rangle+c(x,y).$$ Thus, $$f(y)\geq \sup\Big\{f(x)-\langle \gamma(x),y-x\rangle-c(x,y)\big| y\in K(x)\Big\}.$$ Putting in the supremum $y=x$ we get equality.
The next corollary tells us the the considered class of functions enjoys the extensions properties alike the class of Lipschitz functions. Below we will abuse notation and denote also by $\mathcal{B}$ the set of continuous functions $f\colon\mathbb{R}^n\to\mathbb{R}$ that satisfy $$f\Big(\sum_{i=1}^{n+1}\lambda_ix_i\Big)-\sum_{i=1}^{n+1}\lambda_if(x_i)\leq\sum_{i=1}^{n+1}\lambda_ic\Big(\sum_{j=1}^{n+1}\lambda_jx_j,x_i\Big)$$ for all $x_1,\dotsc,x_{n+1}\in K$ and all non-negative $\lambda_1,\dotsc,\lambda_{n+1}$ that sum up to one.
Let us note that the corollary below extends the results of [@Ivan] on minimal extensions of convex functions, see also [@Ciosmak] for discussion of similar problems related to $1$-Lipschitz maps.
\[col:extend\] Suppose that $K\subset\mathbb{R}^n$ is a compact convex set. Suppose that $c\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ is Lipschitz in the second variable, is such that there exists a constant $\Lambda$ such that for all $x,y\in\mathbb{R}^n$ there is $${\lvertc(x,y)\rvert}\leq\Lambda{\lVertx-y\rVert},$$ and it satisfies martingale triangle inequality. If $g\colon K\to\mathbb{R}$ is belongs to $\mathcal{B}$, then there exisits $\tilde{g}\colon\mathbb{R}^n\to\mathbb{R}$ that belongs to $\mathcal{B}$ and $\tilde{g}|_K=g$. Moreover, if $f\colon \mathbb{R}^n\to\mathbb{R}$ is another function that belongs to $\mathcal{B}$ and such that $$f\leq g \text{ on }K$$ then $\tilde{g}$ may be taken such that $$f\leq \tilde{g}.$$
Without loss of generality $K$ has non-empty interior. By Corollary \[col:rep\] there exist $b\colon K\to\mathbb{R}$ and $a\colon K\to\mathbb{R}^n$ such that for all $x\in K$ $$g(x)=\sup\Big\{g(y)-\langle \gamma(y),y-x\rangle-c(y,x)\big| y\in K(x)\Big\}.$$ Define for $x\in\mathbb{R}^n$ $$\label{eqn:gie}
\tilde{g}_0(x)=\sup\Big\{g(y)-\langle \gamma(y),y-x\rangle-c(y,x)\big| y\in K\Big\}.$$ Then, clearly $\tilde{g}_0(x)= g(x)$ for all $x\in \mathrm{int}K$ and $\tilde{g}_0$ belongs to $\mathcal{B}$. By continuity, see Lemma \[lem:lipm\], $\tilde{g}_0=g$ on $K$. Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a continuous function belonging to $\mathcal{B}$. Suppose that on $K$ $$f\leq g.$$ Take $\tilde{g}=\tilde{g}_0\vee f$. Then $\tilde{g}$ belongs to $\mathcal{B}$, extends $g$, and moreover on $\mathbb{R}^n$ $$f\leq \tilde{g}$$
[^1]: The financial support of St. John’s College in Oxford is gratefully acknowledged. The author would like to acknowledge the kind hospitality of the Erwin Schödinger International Institute for Mathematics and Physics where parts of this research were developed under the frame of the Thematic Programme on Optimal Transport.
|
****
New patterns of travelling waves in\
the generalized Fisher-Kolmogorov equation
and
We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear [Fisher]{}-[Kolmogorov]{}-[Petrovski]{}-[Piscounov]{} equation for the advance of advantageous genes in biology. Depending on the relation between the [*nonlinear diffusion*]{} and the [*nonsmooth reaction function*]{}, which we quantify precisely, we investigate the shape and asymptotic properties of travelling waves. Our method is based on comparison results for semilinear ODEs.
----------------------- ---------------------------------------
[**Running head:**]{} Travelling waves in the FKPP equation
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------------------- ---------------------------------------------------
[**Keywords:**]{} Fisher-Kolmogorov equation, travelling waves,
nonlinear diffusion, nonsmooth reaction function,
comparison principle
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[**2010 Mathematics Subject Classification:**]{} Primary 35Q92, 35K92;
Secondary 35K55, 35K65
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Introduction {#s:Intro}
============
The purpose of this article is to investigate a very basic pattern formation in a reaction-diffusion model, namely, [*travelling waves*]{}. The model is the favorite [*Fisher-KPP equation*]{} (or [*Fisher-Kolmogorov equation*]{}) derived by [R. A. Fisher]{} [@Fisher] in $1937$ and first mathematically analyzed by [A. Kolmogorov]{}, [I. Petrovski]{}, and [N. Piscounov]{} [@KPP] in the same year. However, these original works ([@Fisher; @KPP]) consider only [*linear*]{} diffusion and (sufficiently) [*smooth*]{} (nonlinear) reaction. In our present work, we allow for both, a [***nonlinear***]{} diffusion operator (with a $(p-1)$-homogeneous quasilinear elliptic part, $1 < p < \infty$) and a [***nonsmooth***]{} reaction function of Hölder class $C^{0,\alpha - 1}({\mathbb{R}})$ with $1 < \alpha < 2$, $$\label{e:FKPP}
\left\{
\begin{aligned}
\frac{\partial u}{\partial t}
& = \frac{\partial }{\partial x}
\left( d(u)
\genfrac{|}{|}{}0{\partial u}{\partial x}^{p-2}
\genfrac{}{}{}0{\partial u}{\partial x}
\right) - f(u) \,,\quad (x,t)\in {\mathbb{R}}\times {\mathbb{R}}_+ \,,
\\
u(x,t)&= U(x-ct) \quad\mbox{ for some speed }\, c\in {\mathbb{R}}\,.
\end{aligned}
\right.$$ Here, ${\mathbb{R}}_+{\stackrel{{\mathrm {def}}}{=}}[0,\infty)$, $1 < p < \infty$, $d{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is a positive continuous function, and $f{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is a continuous, but not necessarily smooth function of a “generalized” Fisher-KPP type (specified below). For $p=2$, this equation describes a deterministic version of a stochastic model for the spatial spread of a favored gene in a population, suggested by [R. A. Fisher]{} [@Fisher].
Closely related situations and solution ideas have already been explored, e.g., in [P. Drábek]{}, [R. F. Manásevich]{}, and [P. Takáč]{} [@DrabManTak] and in [Y. Sh. Il’yasov]{} and [P. Takáč]{} [@Ilyasov-Tak], where existence, uniqueness, and stability of [*phase transition solutions*]{} in a Cahn-Hilliard-type model are investigated. More precisely, we study the [***interaction***]{} between the (nonlinear) diffusion and the (nonsmooth) reaction; in paticular, we explore their influence on the formation and the shape of a [*travelling wave*]{} connecting two stable (spatially constant) steady states. Our main result, Theorem \[thm-Main\], contains the [*existence*]{} and [*uniqueness*]{} of such a family of travelling waves (parametrized by a spatial shift). This result is crucial for establishing long-time [*front propagation*]{} (convergence) in nonlinear parabolic equations towards a travelling wave; see, e.g., [D. G. Aronson]{} and [H. F. Weinberger]{} [@AronWein], [P. C. Fife]{} and [J. B. Mc[L]{}eod]{} [@Fife-McLeod], and [F. Hamel]{} and [N. Nadirashvili]{} [@Hamel-Nadira] for $p=2$ (the original semilinear Fisher-KPP equation), [E. Feireisl]{} et [al.]{} [@FeHiPeTa] for any $p\in (1,\infty)$, and [Q. Yi]{} and [J.-N. Zhao]{} [@Yi-Zhao] for $p\in (2,\infty)$ only.
In order to prove Theorem \[thm-Main\] we use a phase plane transformation (cf. [J. D. Murray]{} [@Murray-I], [§]{}13.2, pp. 440–441) to investigate a nonlinear, first order ordinary differential equation with an unknown parameter $c\in {\mathbb{R}}$, see . Since this differential equation is supplemented by homogeneous Dirichlet boundary conditions at both end-points, this boundary value problem is overdetermined. We find a unique value of $c$ for which this problem has a unique positive solution. The nonlinearity in this differential equation does not satisfy a local Lipschitz condition, so, due to the lack of uniqueness of a solution, the classical shooting method cannot be applied directly. The novelty of our approach is to overcome this difficulty: We take advantage of monotonicity properties of $f$ and treat the differential equation in problem as an initial (at $-1$) or terminal (at $+1$) value problem. We thus derive various comparison principles that compensate for the lack of uniqueness in the shooting method. Finally, we obtain new shapes for travelling waves by asymptotic analysis near the end-points $\mp 1$.
Since the seminal paper by [A. M. Turing]{} [@Turing] has appeared in $1952$, the (linear) diffusion has been known to have a [*destabilizing effect*]{} on stable steady states in a reaction-diffusion equation with a smooth reaction function. This effect leads to the formation of new, more complicated patterns, for instance, in morphology used in [Turing]{}’s work [@Turing]. Among other things, we will demonstrate that the analogues of [Turing]{}’s findings do [*not*]{} apply universally to our nonlinear problem setting. (Our situation is similar to, but not identical with that treated in [@Turing].) We will determine a simple, exact relation between the constants $p$ and $\alpha$ ($1 < p < \infty$ and $1 < \alpha < 2$), when a destabilizing effect occurs ($1 < \alpha < p < 2$) and when it does [*not*]{} occur ($1 < p\leq \alpha < \infty$). Loosely speaking, this effect depends on the product $$\mbox{ (``diffusion'') }\times \mbox{ (``smoothness'') }
\equiv \frac{1}{p}\cdot \alpha < (\geq)\; 1 \,.$$ Although our results bear resemblance to [Turing]{}’s observations [@Turing] made in a branching (bifurcation) setting, his destabilizing effect occurs precisely for $p = \alpha = 2$. In this case, we do [*not*]{} have any branching phenomenon in our model in the sense that the values of our travelling wave stay in the open interval $(-1,1)$. In contrast, we speak of a branching phenomenon exactly when the travelling wave attains one of the values $\mp 1$ at a (finite) spatial point $x_{\mp 1}\in {\mathbb{R}}$.
In his monograph [@Murray], [§]{}14.9, pp. 424–430, [J. D. Murray]{} studies *“nonexistence of stable spatial patterns for scalar equations in one dimension with zero flux boundary conditions”. In particular, the following interesting conclusion is derived on p. 426: *Large diffusion prevents spatial patterning in reaction diffusion mechanisms with zero flux boundary conditions. Drawing an analogue of this conclusion to our setting, we will show (Theorem \[thm-Main\]) that this is the case if $p\leq \alpha$. To be more precise, we investigate [*monotone*]{} travelling waves in the degenerate (or singular) second-order parabolic problem of a “generalized” Fisher-KPP type . For $p=2$ and $f$ smooth, extensive studies of travelling waves can be found in [P. C. Fife]{} and [J. B. Mc[L]{}eod]{} [@Fife-McLeod] and [J. D. Murray]{} [@Murray-I], [§]{}13.2, pp. 439–444.**
Recall that $d{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is continuous and positive. The function $f{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is assumed to be continuous, such that $f(\pm 1) = f(s_0) = 0$ for some $-1 < s_0 < 1$, together with $f(s) > 0$ for every $s\in (-1,s_0)$, $f(s) < 0$ for every $s\in (s_0,1)$, and $$\label{int:f(u)}
G(r){\stackrel{{\mathrm {def}}}{=}}\int_{-1}^r d(s)^{1/(p-1)}\, f(s) \,\mathrm{d}s > 0
\quad\mbox{ whenever }\, -1 < r < 1 \,.$$ Typical examples for the diffusion coefficient $d(s) > 0$ are
1. currentlabel[[**d1**]{}]{}\[exam\_1:d(s)\] $\;$ $d(s)\equiv 1$ for all $s\in {\mathbb{R}}$, which yields the $p$-Laplacian in eq. , $1 < p < \infty$.
2. currentlabel[[**d2**]{}]{}\[exam\_2:d(s)\] $\;$ $p=2$ and $d(s) = \varphi'(s) > 0$ for all $s\in {\mathbb{R}}$, which yields the porous medium operator $u\mapsto \frac{\partial^2}{\partial x^2} \varphi(u)$ on the right-hand side of eq. , where the function $\varphi{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is assumed to be monotone increasing and continuous.
An important special case of the reaction function $f$ is $f(s) = F'(s)$ where $F{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is the “generalized” double-well potential $F(s)\equiv F_{\alpha}(s) = \frac{1}{2\alpha} |s^2 - 1|^{\alpha}$ for $s\in {\mathbb{R}}$, where $\alpha\in (1,\infty)$; hence, $s_0 = 0$ and $$\label{e:f_alpha}
f(s)\equiv f_{\alpha}(s)
= |s^2 - 1|^{\alpha - 2} (s^2 - 1) s
\quad\mbox{ for }\, s\in {\mathbb{R}}\,.$$ Notice that, if $1 < \alpha < 2$ then the function $f(s)$ is only $(\alpha - 1)$-Hölder continuous at the points $s = \pm 1$, but certainly not Lipschitz continuous. If the diffusion coefficient $d{:\,}{\mathbb{R}}\to {\mathbb{R}}$ (already assumed to be a positive continuous function) is also even about zero, that is, $d(-s) = d(s)$ for all $s\in {\mathbb{R}}$, then forces $$\label{int:f(u):u=1}
G(1){\stackrel{{\mathrm {def}}}{=}}\int_{-1}^1 d(s)^{1/(p-1)}\, f(s) \,\mathrm{d}s = 0$$ in formula .
This article is organized as follows. Our main results are collected in Section \[s:Main\]. In the next section (Section \[s:Prelim\]), the travelling wave problem for the Fisher-KPP equation , the quasilinear ODE , is transformed into an equivalent problem for an expression $y{:\,}(-1,1)\to {\mathbb{R}}$, the semilinear ODE with the boundary conditions , cf. problem . The substituted unknown expression $y$ is a simple function of the travelling wave $U(x)$ and its derivative $U'(x)$, thus yielding a simple differential equation for the travelling wave $U$, on one hand. On the other hand, in eq. , the unknown function $y$ depends solely on $U\in (-1,1)$ as an independent variable, i.e., $y = y(U)$. This problem, with a monotone, non-Lipschitzian nonlinearity, is solved gradually in Sections \[s:Exist\] (existence) and \[s:Unique\] (uniqueness), respectively. Finally, an important special case (with a Fisher-KPP-type reaction function), which is a slight generalization of , is treated in Section \[s:Asympt\].
Preliminaries {#s:Prelim}
=============
Assuming that the travelling wave takes the form $u(x,t) = U(x-ct)$, $(x,t)\in {\mathbb{R}}\times {\mathbb{R}}_+$, with $U{:\,}{\mathbb{R}}\to {\mathbb{R}}$ being strictly monotone decreasing and continuously differentiable with $U'< 0$ on ${\mathbb{R}}$, below, we are able to find a [*first integral*]{} for the second-order equation for $U$: $$\label{eq:FKPP}
\frac{\mathrm{d} }{\mathrm{d}x}
\left( d(U)
\genfrac{|}{|}{}0{\mathrm{d}U}{\mathrm{d}x}^{p-2}
\genfrac{}{}{}0{\mathrm{d}U}{\mathrm{d}x}
\right)
+ c\, \frac{\mathrm{d}U}{\mathrm{d}x} - f(U)
= 0 \,,
\quad x\in {\mathbb{R}}\,.$$ Following the standard idea of phase plane transformation $(U,V)$ for the $p$-Laplacian (cf. [@EGSanchez Sect. 1]), we make the substitution $$V{\stackrel{{\mathrm {def}}}{=}}{}-
d(U)
\genfrac{|}{|}{}0{\mathrm{d}U}{\mathrm{d}x}^{p-2}
\genfrac{}{}{}0{\mathrm{d}U}{\mathrm{d}x}
> 0 \,,$$ whence $$\label{e:dU/dx}
\genfrac{}{}{}0{\mathrm{d}U}{\mathrm{d}x}
= {}-
\genfrac{(}{)}{}0{V}{d(U)}^{1/(p-1)} < 0 \,,$$ and consequently look for $V = V(U)$ as a function of $U\in (-1,1)$ that satisfies the following differential equation obtained from eq. : $${}- \frac{\mathrm{d}V}{\mathrm{d}U}\cdot \frac{\mathrm{d}U}{\mathrm{d}x}
+ c\, \frac{\mathrm{d}U}{\mathrm{d}x} - f(U)
= 0 \,,
\quad x\in {\mathbb{R}}\,,$$ that is, $$\label{eq:FKPP:V(U)}
\frac{\mathrm{d}V}{\mathrm{d}U}\cdot
\genfrac{(}{)}{}0{V}{d(U)}^{1/(p-1)}
- c\, \genfrac{(}{)}{}0{V}{d(U)}^{1/(p-1)}
- f(U) = 0 \,,
\quad U\in (-1,1) \,.$$ Finally, we multiply the last equation by $d(U)^{1/(p-1)}$, make the substitution $y{\stackrel{{\mathrm {def}}}{=}}V^{p'} > 0$, where $p'= p/(p-1)\in (1,\infty)$, and write $r$ in place of $U$, thus arriving at $$ \frac{1}{p'}\cdot \frac{\mathrm{d}y}{\mathrm{d}r}
- c\, y^{1/p} - d(r)^{1/(p-1)}\, f(r)
= 0 \,,
\quad r\in (-1,1) \,.$$ This means that the unknown function $y{:\,}(-1,1)\to (0,\infty)$ of $r$, $$\label{e:y=V^p'}
y = V^{p'} = d(U)^{p'}\,
\genfrac{|}{|}{}0{\mathrm{d}U}{\mathrm{d}x}^{p}
> 0 \,,$$ must satisfy the following differential equation: $$\label{eq:FKPP:y(r)}
\frac{\mathrm{d}y}{\mathrm{d}r}
= p'\left( c\, (y^{+})^{1/p} + g(r) \right) \,,
\quad r\in (-1,1) \,,$$ where $y^{+} = \max\{ y,\, 0\}$ and $g(r){\stackrel{{\mathrm {def}}}{=}}d(r)^{1/(p-1)}\, f(r)$ satisfies the same hypotheses as $f$:
- $g{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is a continuous, but not necessarily smooth function, such that $g(\pm 1) = g(s_0) = 0$ for some $-1 < s_0 < 1$, together with $g(s) > 0$ for every $s\in (-1,s_0)$, $g(s) < 0$ for every $s\in (s_0,1)$, and , i.e., $$\label{int:g(r)}
G(r){\stackrel{{\mathrm {def}}}{=}}\int_{-1}^r g(s) \,\mathrm{d}s > 0
\quad\mbox{ whenever }\, -1 < r < 1 \,.$$
Since we require that $U = U(x)$ be sufficiently smooth, at least continuously differentiable, with $U'(x)\to 0$ as $x\to \pm\infty$, the function $y = y(r)$ must satisfy the boundary conditions $$\label{bc:FKPP:y(r)}
y(-1) = y(1) = 0 \,.$$ The following remark on the value of $G(1)$ (${}\geq 0$) is in order.
\[rem-sol:G(r)\]
Main Results {#s:Main}
============
We assume that $d{:\,}{\mathbb{R}}\to {\mathbb{R}}$ is a positive continuous function. In fact, we need only $d{:\,}[-1,1]\to {\mathbb{R}}$ to be continuous and positive. Let us recall that the function $g(r) = d(r)^{1/(p-1)}\, f(r)$ of $r\in [-1,1]$ is assumed to satisfy the hypotheses formulated in the previous section (Section \[s:Prelim\]) before Remark \[rem-sol:G(r)\], in particular, ineq. .
Our main result concerning the [*travelling waves*]{} $u(x,t) = U(x-ct)$, $(x,t)\in {\mathbb{R}}\times {\mathbb{R}}_+$, $c\in {\mathbb{R}}$, for problem is as follows.
\[thm-Main\] Let $G(1) > 0$. Then there exists a unique number $c^{\ast}\in {\mathbb{R}}$ such that problem with $c = c^{\ast}$ possesses a travelling wave solution $u(x,t) = U(x - c^{\ast} t)$, $(x,t)\in {\mathbb{R}}\times {\mathbb{R}}_+$, where $U = U(\xi)$ is a monotone decreasing and continuously differentiable function on ${\mathbb{R}}$ taking values in $[-1,1]$. Furthermore, we have $c^{\ast} < 0$ and the set $ \{ \xi\in \RR\colon U'(\xi) < 0\}$ is a nonempty open interval $(x_1, x_{-1})\subset {\mathbb{R}}$ with $$\label{e:U(x),x_+-1}
\lim_{\xi\to (x_1)+} U(\xi) = 1 \qquad\mbox{ and }\qquad
\lim_{\xi\to (x_{-1})-} U(\xi) = -1 \,.$$
[[*Proof.* ]{}]{}The existence and uniqueness of $c^{\ast}\in {\mathbb{R}}$ follow from the transformation of eq. into eq. with the boundary conditions and subsequent application of Theorem \[thm-FKPP:y(r)\] below to the boundary value problem , . We get also $c^{\ast} < 0$.
Inserting $y = y_{ c^{\ast} }{:\,}(-1,1)\to {\mathbb{R}}$ into eq. we obtain $$\left\{
\begin{aligned}
& V(r) = y_{ c^{\ast} }(r)^{1/p'} = y_{ c^{\ast} }(r)^{(p-1)/p} > 0
\quad\mbox{ for }\, r\in (-1,1) \,;
\\
& V(-1) = V(1) = 0 \,.
\end{aligned}
\right.$$ We insert $V = V(U)$ as a function of $U\in (-1,1)$ into eq. , thus arriving at $$ \genfrac{}{}{}0{\mathrm{d}U}{\mathrm{d}x}
= {}-
\genfrac{(}{)}{}0{V(U)}{d(U)}^{1/(p-1)} < 0
\quad\mbox{ for }\, U\in (-1,1) \,.$$ Separation of variables above yields $$ \mathrm{d}x
= {}-
\genfrac{(}{)}{}0{d(U)}{V(U)}^{1/(p-1)} \,\mathrm{d}U
\quad\mbox{ for }\, U\in (-1,1) \,.$$ Finally, we integrate the last equation to arrive at $$\label{int:dx=-dU/U}
x(U)
= x(0) - \int_0^U
\genfrac{(}{)}{}0{d(r)}{V(r)}^{1/(p-1)} \,\mathrm{d}r
\quad\mbox{ for }\, U\in (-1,1) \,,$$ where $x(0) = x_0\in {\mathbb{R}}$ is an arbitrary constant. We remark that $V(r)\to 0$ as $r\to \pm 1$.
Consequently, both monotone limits below exist, $$\label{e:x(U),U=+-1}
x_{-1}{\stackrel{{\mathrm {def}}}{=}}\lim_{U\to (-1)+} x(U)
\qquad\mbox{ and }\qquad
x_1 {\stackrel{{\mathrm {def}}}{=}}\lim_{U\to (+1)-} x(U) \,,$$ and satisfy $-\infty\leq x_1 < x_0 < x_{-1}\leq +\infty$. Thus, the function $x{:\,}(-1,1)\to (x_1, x_{-1})\subset {\mathbb{R}}$ is a diffeomorphism of the open interval $(-1,1)$ onto $(x_1, x_{-1})$ satisfying $\frac{\mathrm{d}x}{\mathrm{d}U} < 0$ in $(-1,1)$ together with . This implies the remaining part of our theorem, especially the limits in .
------------------------------------------------------------------------
It depends on the asymptotic behavior of the function $g = g(r)$, $r\in (-1,1)$, near the points $\pm 1$ whether $U'< 0$ holds on the entire real line ${\mathbb{R}}= (-\infty, +\infty)$ or else $U'< 0$ on a nonempty open interval $(x_1, x_{-1})\subset {\mathbb{R}}$ and $U'= 0$ on its nonempty complement ${\mathbb{R}}\setminus (x_1, x_{-1})$ with $x_1 > -\infty$ and/or $x_{-1} < +\infty$. More precisely, we assume that $g{:\,}{\mathbb{R}}\to {\mathbb{R}}$ has the following asymptotic behavior near $\pm 1$:
There are constants $\gamma^{\pm}, \gamma_0^{\pm}\in (0,\infty)$ such that $$\label{e:g(r),r=+-1}
\lim_{r\to 1-} \frac{g(r)}{ (1-r)^{\gamma^{+}} }
= {}- \gamma_0^{+} \qquad\mbox{ and }\qquad
\lim_{r\to -1+} \frac{g(r)}{ (1+r)^{\gamma^{-}} }
= \gamma_0^{-} \,.$$ For instance, if $g(s){\stackrel{{\mathrm {def}}}{=}}d(s)^{1/(p-1)}\, f(s)$ for $s\in (-1,1)$ is as in Remark \[rem-sol:G(r)\] and $f(s)\equiv f_{\alpha}(s)$ is defined by , then we have $\alpha = 1 + \gamma^{\pm}$.
We have the following conclusions for the limits $x_{\mp 1}{\stackrel{{\mathrm {def}}}{=}}\lim_{ U\to (\mp 1)\pm } x(U)$, where $x = x(U)$ is a solution of $$\label{e:dx/dU}
\genfrac{}{}{}0{\mathrm{d}x}{\mathrm{d}U}
= {}-
\genfrac{(}{)}{}0{d(U)}{V(U)}^{1/(p-1)} < 0
\quad\mbox{ for }\, -1 < U < 1 \,,$$ cf. eq. , given explicitely by formula . Notice that, in this notation, $x{:\,}(-1,1)\to (x_1, x_{-1})$ is the inverse function of the restriction of $U$ to the interval $(x_1, x_{-1})$ in which $U'< 0$, i.e., $x(r) = U^{-1}(r)$ for $-1 < r < 1$.
\[thm-x\_+-1\] Assume that the limits in eq. hold. Then,
[(i)]{}$\;$ in case $1 < p\leq 2$ we have $x_1 = -\infty$ $(x_{-1} = +\infty)$ if and only if $\gamma^{+}\geq p-1$ ($\gamma^{-}\geq p-1$, respectively); whereas
[(ii)]{}$\;$ in case $2 < p < \infty$ we have $x_1 = -\infty$ $(x_{-1} = +\infty)$ if $\gamma^{+}\geq p-1$ ($\gamma^{-}\geq p-1$, respectively).
This theorem is an easy combination of Theorem \[thm-Main\] above with Corollary \[cor-power<g(r)\] and Remark \[rem-power<g(r)\] in Section \[s:Asympt\].
The following corollary to Theorem \[thm-x\_+-1\] for the linear diffusion case ($p=2$) is obvious:
\[cor-x\_+-1\] Let $p=2$. Then $x_1$ $(x_{-1})$ is finite if $\gamma^{+} < 1$ ($\gamma^{-} < 1$, respectively) and $x_1 = -\infty$ $(x_{-1} = +\infty)$ if $\gamma^{+}\geq 1$ ($\gamma^{-}\geq 1$, respectively).
The results stated above hinge upon the existence and uniqueness results for eq. with the Dirichlet boundary conditions , i.e., for the following boundary value problem: $$\label{BVP:FKPP:y(r)}
\left\{
\begin{aligned}
& \frac{\mathrm{d}y}{\mathrm{d}r}
= p'\left( c\, (y^{+})^{1/p} + g(r) \right) \,,
\quad r\in (-1,1) \,;
\\
& y(-1) = y(1) = 0 \,,
\end{aligned}
\right.$$ with the parameter $c\in {\mathbb{R}}$ to be determined. We recall that eq. has been obtained from eq. by means of the substitution in .
The following result for problem is of independent interest (cf. [@FeHiPeTa; @Yi-Zhao]).
\[thm-FKPP:y(r)\] Let $G(1) > 0$. Then there exists a unique number $c^{\ast}\in {\mathbb{R}}$ such that problem with $c = c^{\ast}$ has a unique solution $y = y_{ c^{\ast} }{:\,}(-1,1)\to {\mathbb{R}}$. Furthermore, we have $c^{\ast} < 0$ and $y_{ c^{\ast} } > 0$ in $(-1,1)$.
[[*Proof.* ]{}]{}The existence and uniqueness of $c^{\ast}\in {\mathbb{R}}$ follow from Corollary \[cor-uniq\_c\^\*\] (with the existence established before in Corollary \[cor-c<c\^\*\]). We get also $c^{\ast} < 0$.
------------------------------------------------------------------------
Existence Result for $c^{\ast}$ {#s:Exist}
===============================
In order to verify that only the case $c<0$ can yield a travelling wave for problem , we prove the following simple lemma for the boundary value problem .
\[lem-sol:G(u)\] Let $c\in {\mathbb{R}}$ and $y_0(r) = p'\, G(r)$ for $-1\leq r\leq 1$.
[(i)]{}$\;$ Let $c = 0$. Then the function $y_0(r) = p'\, G(r)$ of $r\in [-1,1]$ is a solution to problem if and only if $G(1) = 0$.
[(ii)]{}$\;$ If $c>0$ $($$c\leq 0$, respectively$)$ then every solution $y{:\,}[-1,1]\to {\mathbb{R}}$ to the initial value problem for eq. with the initial condition $y(-1) = 0$ satisfies $y > y_0$ $($$y\leq y_0$$)$ throughout $(-1,1]$.
[[*Proof.* ]{}]{}Part [(i)]{} is a trivial consequence of combined with the properties of $g$.
Part [(ii)]{}: First, let $c>0$. Then, clearly, $y'\geq y_0' = p'\, g$ throughout $(-1,1)$, together with $y'\not\equiv y_0'$ in $(-1,r)$ for every $r\in (-1,1)$, which forces $y > y_0$ throughout $(-1,1]$, by a simple integration of eq. over the interval $[-1,r]$.
The case $c\leq 0$ is analogous, by reversing the (nonstrict) inequalities.
------------------------------------------------------------------------
For $c\geq 0$, Lemma \[lem-sol:G(u)\] has the following obvious consequence.
\[cor-sol:G(u)\] If [either]{} [(i)]{} $c=0$ and $G(1) > 0$ [or else]{} [(ii)]{} $c>0$ and $G(1)\geq 0$, then eq. possesses [no]{} solution $y{:\,}[-1,1]\to {\mathbb{R}}$ satisfying the boundary conditions $y(-1) = 0$ and $y(1)\leq 0$.
We conclude that it suffices to investigate the case $c<0$ for finding a travelling wave to problem .
\[rem-sol:G(u)\]
More detailed results concerning existence, uniqueness, monotone dependence on certain parameters, and other qualitative properties of solutions to the initial value problem for eq. with the initial condition $y(-1) = 0$ will be established in Corollary \[cor-y\_c:monot\].
In view of Remark \[rem-sol:G(r)\] and Corollary \[cor-sol:G(u)\] above, from now on we assume $$\label{int>0:f(u):u=1}
G(1){\stackrel{{\mathrm {def}}}{=}}\int_{-1}^1 d(s)^{1/(p-1)}\, f(s) \,\mathrm{d}s > 0$$ in formula . Equivalently, we have $y_0(1) = p'\, G(1) > 0$. We will show that there is a unique constant $c < 0$ such that the (unique) solution $y_c{:\,}[-1,1]\to {\mathbb{R}}$ to the initial value problem for eq. with the initial condition $y_c(-1) = 0$ satisfies also the terminal condition $y_c(1) = 0$. To find this constant, let us define $$\label{def:c^*}
c^{\ast}{\stackrel{{\mathrm {def}}}{=}}\inf\left\{ c\in {\mathbb{R}}{:\,}y_c(1) > 0\right\} \,.$$ As expected, we will show that precisely $c^{\ast}$ is the desired value of the constant $c$; see Corollaries \[cor-c<c\^\*\] and \[cor-uniq\_c\^\*\].
We need the following two technical lemmas.
\[lem-y\_c>0\] Let $c\in {\mathbb{R}}$ be arbitrary.
[(i)]{}$\;$ There is some $\delta\equiv \delta(c)\in (0,2)$ such that $y_c(r) > 0$ holds for all $r\in (-1, -1+\delta)$.
[(ii)]{}$\;$ If $y_c(1)\geq 0$ then we have also $y_c(r) > 0$ for all $r\in (-1,1)$.
[[*Proof.* ]{}]{}Part [(i)]{}: On the contrary, if no such number $\delta\in (0,2)$ exists, then there is a sequence $\{ r_n\}_{n=1}^{\infty} \subset (-1,1)$ such that $r_n\searrow -1$ as $n\nearrow \infty$, and $y_c(r_n)\leq 0$ for all $n\in {\mathbb{N}}$. Given any fixed $n\in {\mathbb{N}}$, we cannot have $y_c(s)\leq 0$ for all $s\in (-1,r_n]$ since, otherwise, by eq. , the function $y_c$ would satisfy $$\label{eq:FKPP:y_c(r)}
\frac{\mathrm{d}y_c}{\mathrm{d}r} = p'\, g(r) \,,
\quad r\in (-1,r_n) \,,$$ that is, $y_c = y_0 > 0$ throughout $[-1,r_n]$, by a simple integration of eq. over the interval $[-1,r]$. This conclusion would then contradict our choice $y_c(r_n)\leq 0$. Consequently, there is a point $r_n'\in (-1,r_n)$ such that $y_c(r_n') > 0$. It follows that there is another point $r_n''\in (r_n',r_n]$ such that $y_c(r_n'')\leq 0$ and $y_c'(r_n'')\leq 0$. Since $r_n\searrow -1$ as $n\nearrow \infty$, we have also $r_n''\to -1$ as $n\to \infty$. But now, substituting $r = r_n''$ in eq. , we arrive at $g(r_n'')\leq 0$ for every $n\in {\mathbb{N}}$. A contradiction with our hypothesis on $g$, namely, $g(s) > 0$ for every $s\in (-1,s_0)$, is obtained for every $n\in {\mathbb{N}}$ sufficiently large, such that $r_n''\in (-1,s_0)$.
We have proved the existence of the number $\delta$.
Part [(ii)]{}: On the contrary, let us assume that the set $\{ r\in (-1,1){:\,}y_c(r)\leq 0\}$ is nonempty. We denote by $\overline{r}\in (-1,1)$ the smallest number $r\in (-1,1)$ such that $y_c(r) = 0$. Hence, $-1 < -1+\delta\leq \overline{r} < 1$ and $y_c(r) > 0$ holds for every $r\in (-1,\overline{r})$. We have $y_c(\overline{r}) = 0$ and, therefore, $ p'\, g(\overline{r}) = y_c'(\overline{r})\leq 0 \,,$ by eq. . Our hypothesis on the nodal point $s_0$ of the function $g$ thus forces $\overline{r}\geq s_0$. Hence, we must have $y_c'(r) = p'\, g(r) < 0$ for all $r\in (\overline{r}, 1)\subset (s_0,1)$. We conclude that $y_c(r) < 0$ holds for all $r\in (\overline{r}, 1]$, which contradicts our hypothesis $y_c(1)\geq 0$.
We have proved that $y_c(r) > 0$ holds for all $r\in (-1,1)$ as desired.
------------------------------------------------------------------------
\[lem-r\_0>s\_0\] Assume that $c < 0$ and $G(1) > 0$. Let $y{:\,}[-1,1]\to {\mathbb{R}}$ be a solution to eq. satisfying $y(-1) = y(r_0) = 0$ for some $r_0\in (-1,1)$ and $y(r) > 0$ for all $r\in (-1,r_0)$. Then we have $r_0\geq s_0$ and $y(r) < 0$ holds for all $r\in (r_0,1]$.
[[*Proof.* ]{}]{}On the contrary, suppose that $-1 < r_0 < s_0$. Then $y'(r_0) = p'\, g(r_0) > 0$ holds by our hypotheses on $g$. However, this is contradicts our hypothesis on $y$ requiring $y(r) > 0 = y(r_0)$ for all $r\in (-1,r_0)$ and, thus, forcing $y'(r_0)\leq 0$. This proves $r_0\geq s_0$.
Consequently, we have $g(r) < 0$ for all $r\in (r_0,1)$ which entails $y'(r) = p'\, g(r) < 0$ for all $r\in (r_0,1)$. Integration over the interval $[r_0,r]$ yields the desired result, i.e., $y(r) < 0$ holds for all $r\in (r_0,1]$.
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\[prop-c\^\*\] We have $-\infty < c^{\ast} < 0$ and $y_{c^{\ast}}(1) = 0$.
[[*Proof.* ]{}]{}First, let us recall that the terminal value $y_c(1)\in {\mathbb{R}}$ is a monotone increasing, continuous function $c\mapsto y_c(1){:\,}{\mathbb{R}}_-\to {\mathbb{R}}$ of the parameter $c\in {\mathbb{R}}_-$. These properties yield immediately $-\infty\leq c^{\ast} < 0$ and, if $c^{\ast} > -\infty$ then also $y_{c^{\ast}}(1) = 0$.
Thus, it remains to prove $c^{\ast} > -\infty$. By contradiction, let us suppose that there is a sequence of numbers $c_n\in (-\infty,0)$, such that $y_{c_n}(1) > 0$ for every $n=1,2,3,\dots$ and $c_n\searrow -\infty$ as $n\nearrow \infty$. Then $y_{c_n}(r) > 0$ must hold for all $r\in (-1,1]$, by Lemma \[lem-y\_c>0\], Part [(ii)]{}.
Next, let $z_n{:\,}[s_0,1]\to {\mathbb{R}}$ be the (unique) solution of the initial value problem $$\label{eq:FKPP:z(r)}
\frac{\mathrm{d}z_n}{\mathrm{d}r}
= p' c_n\, (z_n^{+})^{1/p} \,,\quad r\in (s_0,1) \,;\qquad
z_n(s_0) = y_{c_n}(s_0) \,.$$ The monotone (increasing) dependence of the solution $y_c{:\,}[-1,1]\to {\mathbb{R}}$ on the right-hand side of eq. (essentially due to [E. Kamke]{}; see, e.g., [M. W. Hirsch]{} [@Hirsch p. 425] or [W. Walter]{} [@Walter Chapt. III, [§]{}10], Comparison Theorem on p. 112) guarantees the following comparison result: $0 < y_{c_n}(r)\leq z_n(r)$ for all $r\in [s_0,1]$. However, separating variables in eq. and integrating, we obtain $$(z_n(r))^{1/p'} = (z_n(s_0))^{1/p'} + c_n (r-s_0)
\quad\mbox{ for all }\, r\in [s_0,1] \,.$$ Consequently, by the comparison result, we have also $$0 < (y_{c_n}(r))^{1/p'}
\leq (y_{c_n}(s_0))^{1/p'} + c_n (r-s_0)
\quad\mbox{ for all }\, r\in [s_0,1] \,.$$ Setting $r=1$ and recalling $y_{c_n}(1) > 0$, we observe that $(y_{c_n}(s_0))^{1/p'} > - c_n (1-s_0)$ holds for every $n=1,2,3,\dots$. Since also $y_{c_n}\leq y_0$ holds throughout the interval $[-1,1]$, by Lemma \[lem-sol:G(u)\], Part [(ii)]{}, we conclude that $- c_n (1-s_0) < (y_0(s_0))^{1/p'}$, a contradiction to $-c_n\nearrow \infty$ as $n\nearrow \infty$. We have proved $c^{\ast} > -\infty$ as desired.
------------------------------------------------------------------------
Lemma \[lem-r\_0>s\_0\] and Proposition \[prop-c\^\*\] entail the following obvious corollary.
\[cor-c<c\^\*\] There is some number $c\in {\mathbb{R}}$, say, $c = c^{\ast}$ $(< 0)$, such that problem possesses a solution $y{:\,}[-1,1]\to {\mathbb{R}}$. This solution is unique, given by $y = y_{c^{\ast}}$, and it satisfies $y(r) > 0$ for all $r\in (-1,1)$.
For this particular speed $c = c^{\ast}$, the uniqueness of $y_{c^{\ast}}$ has been discussed in Remark \[rem-sol:G(u)\]. The sole existence of $y_{c^{\ast}}$ can be found also in [R. Enguiça]{}, [A. Gavioli]{}, and [L. Sanchez]{} [@EGSanchez Theorem 4.2, p. 182] with a sketched proof.
Uniqueness Result for $c^{\ast}$ {#s:Unique}
================================
For the sake of reader’s convenience, in this section we establish a few comparison results (weak and strong) which provide important technical tools. These results are essentially due to [E. Kamke]{}, as we have already mentioned in the proof of Proposition \[prop-c\^\*\] above. They will enable us to establish the uniqueness of $c^{\ast}$ specified in Corollary \[cor-c<c\^\*\] (see Corollary \[cor-uniq\_c\^\*\] below).
\[def-sub/super\]
\[rem-sub/super\]
We begin with the following comparison result.
\[prop-y:sub/sup\] Let $c\in {\mathbb{R}}$, $-1\leq a < b\leq 1$, and assume that $\underline{y}, \overline{y}{:\,}[a,b]\to {\mathbb{R}}$ is a pair of continuously differentiable functions, such that $\underline{y}$ ($\overline{y}$) is a subsolution $($supersolution, respectively$)$ to equation on $(a,b)$.
- If $c\leq 0$ and $\underline{y}(a)\leq \overline{y}(a)$ then we have also $\underline{y}(r)\leq \overline{y}(r)$ for all $r\in [a,b]$.
- If $c\geq 0$ and $\underline{y}(b)\geq \overline{y}(b)$ then we have also $\underline{y}(r)\geq \overline{y}(r)$ for all $r\in [a,b]$.
Recall our abbreviations ${\mathbb{R}}_+{\stackrel{{\mathrm {def}}}{=}}[0,\infty)$ and ${\mathbb{R}}_-{\stackrel{{\mathrm {def}}}{=}}(-\infty,0]$.
[[*Proof.* ]{}]{}We prove only [Part (a)]{}; [Part (b)]{} is proved analogously.
Hence, assume that $c\leq 0$. We subtract ineq. from , thus arriving at $$ \frac{\mathrm{d}}{\mathrm{d}r}\, (\underline{y} - \overline{y})
\leq p'c
\left( (\underline{y}^{+})^{1/p} - (\overline{y}^{+})^{1/p} \right) \,,
\quad r\in (a,b) \,.$$ Now let us multiply this inequality by $(\underline{y} - \overline{y})^{+}$, thus obtaining $$\label{sub-sup:y(-1)}
(\underline{y} - \overline{y})^{+}\,
\frac{\mathrm{d}}{\mathrm{d}r}\, (\underline{y} - \overline{y})
\leq p'c
\left( (\underline{y}^{+})^{1/p} - (\overline{y}^{+})^{1/p} \right)
(\underline{y} - \overline{y})^{+}
\leq 0 \,,\quad r\in (a,b) \,,$$ thanks to $c\leq 0$ combined with the montonicity of the functions $\xi\mapsto \xi^{+}$ and $\xi\mapsto (\xi^{+})^{1/p}$ from ${\mathbb{R}}$ to ${\mathbb{R}}_+$. The last inequality entails $$ \frac{1}{2}\,
\frac{\mathrm{d}}{\mathrm{d}r}
\left( (\underline{y} - \overline{y})^{+} \right)^2
\leq 0 \quad\mbox{ for almost every }\, r\in (a,b) \,,$$ which shows that $w{\stackrel{{\mathrm {def}}}{=}}\left( (\underline{y} - \overline{y})^{+} \right)^2$ is a monotone nonincreasing, nonnegative function on $[a,b]$. Since $w(a) = 0$ by our hypothesis, it follows that $w(r) = 0$ holds for every $r\in [a,b]$. This completes our proof of [Part (a)]{}.
------------------------------------------------------------------------
The following corollary on the uniqueness and the monotone dependence on the parameter $c\in {\mathbb{R}}_-$ of the solution $y\equiv y_c{:\,}[-1,1]\to {\mathbb{R}}$ to the initial value problem for eq. with the initial condition $y(-1) = 0$ is an easy direct consequence of Proposition \[prop-y:sub/sup\].
\[cor-y\_c:monot\] [(i)]{}$\;$ Given any $c\in {\mathbb{R}}_-$, the initial value problem for eq. with the initial condition $y(-1) = 0$ possesses a unique $($continuously differentiable$)$ solution $y\equiv y_c{:\,}[-1,1]\to {\mathbb{R}}$.
[(ii)]{}$\;$ If $-\infty < c_1\leq c_2\leq 0$ then $y_{c_1}\leq y_{c_2}$ holds pointwise throughout $[-1,1]$. In particular, the function $c\mapsto y_c(1){:\,}{\mathbb{R}}_-\to {\mathbb{R}}$ is monotone increasing.
[[*Proof.* ]{}]{}[Part (i)]{} (the uniqueness claim) follows trivially from Proposition \[prop-y:sub/sup\], [Part (a)]{}.
[Part (ii)]{}: Let $-\infty < c_1\leq c_2\leq 0$ and choose any constant $c\in [c_1,c_2]$; hence, $c\leq 0$. Then $y_{c_1}, y_{c_2}{:\,}[-1,1]\to {\mathbb{R}}$ is a pair of sub- and supersolutions to equation on $(-1,1)$, respectively, with $c$ chosen above ($c_1\leq c\leq c_2$) and the initial condition $y_{c_1}(-1) = y_{c_2}(-1) = 0$. The pointwise ordering $y_{c_1}\leq y_{c_2}$ throughout $[-1,1]$ now follows from Proposition \[prop-y:sub/sup\], [Part (a)]{}, again.
------------------------------------------------------------------------
We complement Proposition \[prop-y:sub/sup\] with another comparison result.
\[prop-y(1):sub/sup\] Let $c\in {\mathbb{R}}$, $a\in [s_0,1)$, and assume that $\underline{y}, \overline{y}{:\,}[a,1]\to {\mathbb{R}}$ is a pair of continuously differentiable functions, such that $\underline{y}$ ($\overline{y}$) is a subsolution $($supersolution, respectively$)$ to equation on $(a,1)$. Moreover, assume that
- $\underline{y}(r) > 0$ and $\overline{y}(r) > 0$ for all $r\in [a,1)$, together with $\overline{y}(1) = 0\leq \underline{y}(1)$.
Then we have also
- $\overline{y}(r)\leq \underline{y}(r)$ for all $r\in [a,1]$.
- More precisely, if $\overline{y}(1) = 0 < \underline{y}(1)$ then $\overline{y}(r) < \underline{y}(r)$ holds for all $r\in [a,1]$. If $\overline{y}(1) = \underline{y}(1) = 0$ then there is some number $a'\in [a,1]$ such that $\overline{y}(r) < \underline{y}(r)$ for all $r\in [a,a')$ and $\overline{y}(r) = \underline{y}(r)$ for all $r\in [a',1]$.
The comparison result in Part [(b)]{} takes care of the lack of uniqueness in the terminal value problem for eq. with the terminal condition $y(1) = 0$ and $c<0$, as announced in Remark \[rem-sol:G(u)\].
[*Proof of*]{} Proposition \[prop-y(1):sub/sup\]. Part [(a)]{}: We begin by substituting the variable $t = 1-r$ with $0\leq t\leq 1-a$ $(\leq 1 - s_0)$ and the functions $$\underline{z}(t){\stackrel{{\mathrm {def}}}{=}}(\underline{y}(1-t))^{1/p'}
\qquad\mbox{ and }\qquad
\overline{z}(t) {\stackrel{{\mathrm {def}}}{=}}(\overline{y}(1-t))^{1/p'}$$ with the derivatives (recall that $\frac{1}{p} + \frac{1}{p'} = 1$) $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}\, \underline{z}(t)
& = {}- \frac{1}{p'}\, (\underline{y}(1-t))^{-1/p}\cdot
\frac{\mathrm{d}}{\mathrm{d}t}\, \underline{y}(1-t)
\quad\mbox{ and }\quad
\\
\frac{\mathrm{d}}{\mathrm{d}t}\, \overline{z}(t)
& = {}- \frac{1}{p'}\, (\overline{y}(1-t))^{-1/p}\cdot
\frac{\mathrm{d}}{\mathrm{d}t}\, \overline{y}(1-t)
\quad\mbox{ for }\, 0 < t\leq 1-a \,.\end{aligned}$$ Hence, inequalities and , respectively, are equivalent with $$\label{sub:FKPP:z(1)}
\frac{ \mathrm{d} \underline{z} }{\mathrm{d}t}
\geq {}- c - g(1-t)\, ( \underline{z}(t) )^{-1/(p-1)} \,,
\quad t\in (0,1-a] \,,$$ and $$\label{sup:FKPP:z(1)}
\frac{ \mathrm{d} \overline{z} }{\mathrm{d}t}
\leq {}- c - g(1-t)\, ( \overline{z}(t) )^{-1/(p-1)} \,,
\quad t\in (0,1-a] \,.$$ We subtract ineq. from , thus arriving at $$\label{sub-sup:FKPP:z(1)}
\frac{\mathrm{d}}{\mathrm{d}t}\, (\overline{z} - \underline{z})
\leq {}- g(1-t)
\left[ ( \overline{z}(t) )^{-1/(p-1)}
- ( \underline{z}(t) )^{-1/(p-1)}
\right] \,,
\quad t\in (0,1-a] \,.$$ Now let us multiply ineq. by $(\overline{z} - \underline{z})^{+}$, thus obtaining $$\label{sub-sup:z(1)}
(\overline{z} - \underline{z})^{+}\,
\frac{\mathrm{d}}{\mathrm{d}t}\, (\overline{z} - \underline{z})
\leq {}- g(1-t)
\left[ ( \overline{z}(t) )^{-1/(p-1)}
- ( \underline{z}(t) )^{-1/(p-1)}
\right]
(\overline{z} - \underline{z})^{+}
\leq 0$$ for all $t\in (0,1-a]$, thanks to $g(1-t)\leq 0$ for $0 < t\leq 1-a\leq 1 - s_0$. The last inequality entails $$ \frac{1}{2}\,
\frac{\mathrm{d}}{\mathrm{d}t}
\left( (\overline{z} - \underline{z})^{+} \right)^2
\leq 0 \quad\mbox{ for almost every }\, r\in (0,1-a) \,,$$ which shows that $w{\stackrel{{\mathrm {def}}}{=}}\left( (\overline{z} - \underline{z})^{+} \right)^2$ is a monotone nonincreasing, nonnegative function on $[0,1-a]$. Since $w(0) = 0$ by our hypothesis, it follows that $w(t) = 0$ holds for every $t\in [0,1-a]$. Equivalently, $\overline{y}(r)\leq \underline{y}(r)$ holds for all $r\in [a,1]$, which proves Part [(a)]{}.
Part [(b)]{}: We return to ineq. . The difference on the right-hand side takes the form $$\label{est:z^-z_}
\begin{aligned}
& 0\leq
( \overline{z}(t) )^{-1/(p-1)}
- ( \underline{z}(t) )^{-1/(p-1)}
=
\\
& {}- \frac{1}{p-1}
\left( \int_0^1
\left[ (1-\theta) \overline{z}(t) + \theta \underline{z}(t)
\right]^{-p'} \,\mathrm{d}\theta
\right)
\left( \overline{z}(t) - \underline{z}(t) \right)
\\
& \leq L\left( \underline{z}(t) - \overline{z}(t) \right)
\quad\mbox{ for all }\, t\in [1-b,1-a] \,,
\end{aligned}$$ where $b\in (a,1)$ is arbitrary and the Lipschitz constant $L\equiv L(b)\in (0,\infty)$ depends on $b$. We may take $$L = \frac{1}{p-1}
\left( \min_{ t\in [1-b,1-a] } \overline{z}(t) \right)^{-p'}
\in (0,\infty) \,,$$ by the inequality $$\int_0^1
\left[ (1-\theta) \overline{z}(t) + \theta \underline{z}(t)
\right]^{-p'} \,\mathrm{d}\theta
\leq ( \overline{z}(t) )^{-p'} \,,
\quad t\in (0,1-a] \,.$$ Consequently, we can estimate the right-hand side of ineq. as follows: $$\label{sub-sup:z^-z_}
\frac{\mathrm{d}}{\mathrm{d}t}\, (\overline{z} - \underline{z})
\leq \hat{L}
\left( \underline{z}(t) - \overline{z}(t) \right) \,,
\quad t\in [1-b,1-a] \,,$$ where $$\hat{L}\equiv \hat{L}(b){\stackrel{{\mathrm {def}}}{=}}L\cdot \sup_{[-1,1]} |g| \,,\qquad
\hat{L}\in (0,\infty) \,.$$ Ineq. is equivalent with $$\label{exp:z^-z_}
\frac{\mathrm{d}}{\mathrm{d}t}
\left[ \mathrm{e}^{\hat{L} t}
\left( \overline{z}(t) - \underline{z}(t) \right)
\right] \leq 0
\quad\mbox{ for every }\, t\in [1-b,1-a] \,.$$ By integration over any compact interval $[t_1,t_2]\subset [1-b,1-a]$, we get $$\label{a<b:z^-z_}
\overline{z}(t_2) - \underline{z}(t_2)
\leq \mathrm{e}^{- \hat{L} (t_2 - t_1)}
\left( \overline{z}(t_1) - \underline{z}(t_1) \right)
\leq 0 \,.$$ First, assume that $\overline{y}(1) = 0 < \underline{y}(1)$, that is, $\overline{z}(0) = 0 < \underline{z}(0)$. Then, by continuity, also $\overline{z}(t_1) < \underline{z}(t_1)$ holds provided $t_1\in (0,1-a)$ is chosen to be small enough. From ineq. we deduce that also $\overline{z}(t) < \underline{z}(t)$ for all $t\in [t_1,1-a]$. We have proved $\overline{z}(t) < \underline{z}(t)$ for every $t\in [0,1-a]$. The desired inequality $\overline{y}(r) < \underline{y}(r)$ for every $r\in [a,1]$ follows immediately.
Now assume $\overline{y}(1) = \underline{y}(1) = 0$, that is, $\overline{z}(0) = \underline{z}(0) = 0$. Let $t'\in [0,1-a]$ be the greatest number $t\in [0,1-a]$, such that $\overline{z}(t) - \underline{z}(t) = 0$ holds for every $t\in [0,t']$. Equivalently, $\overline{y}(r) - \underline{y}(r) = 0$ holds for every $r\in [a',1]$, where $a'= 1 - t'\in [a,1]$. If $t'= 1-a$ then we are done. Thus, assume $0\leq t'< 1-a$. Then there is a sequence $\{ \tau_n\}_{n=1}^{\infty} \subset (t',1-a)$, such that $\tau_n\searrow t'$ as $n\nearrow \infty$ and $\overline{z}(\tau_n) - \underline{z}(\tau_n) < 0$ for every $n = 1,2,3,\dots$. Again, from ineq. we deduce that also $\overline{z}(t) < \underline{z}(t)$ for all $t\in [\tau_n,1-a]$. We have proved $\overline{z}(t) < \underline{z}(t)$ for every $t\in (t',1-a]$. The desired inequality $\overline{y}(r) < \underline{y}(r)$ for every $r\in [a,a')$ follows immediately.
The proof is complete.
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Corollary \[cor-c<c\^\*\], Corollary \[cor-y\_c:monot\], and Proposition \[prop-y(1):sub/sup\] entail the following important result.
\[cor-uniq\_c\^\*\] There is a unique number $c\in {\mathbb{R}}$ such that problem possesses a solution $y{:\,}[-1,1]\to {\mathbb{R}}$, namely, $c = c^{\ast}$ $(< 0)$ defined in eq. . This solution is given by $y = y_{c^{\ast}}$ and satisfies $y(r) > 0$ for all $r\in (-1,1)$.
[[*Proof.* ]{}]{}The existence of $c^{\ast}$ $(< 0)$ has been established in Corollary \[cor-c<c\^\*\]. On the contrary to uniqueness, suppose that there is another number $c$, say, $c = c'\in {\mathbb{R}}$, $c'\neq c^{\ast}$, such that $y = y_{c'}$ satisfies the boundary conditions $y(-1) = y(1) = 0$. We have $y_{c'}(r) > 0$ and $y_{c^{\ast}}(r) > 0$ for all $r\in (-1,1)$, by Lemma \[lem-y\_c>0\], Part [(ii)]{}. By our definition of $c^{\ast}$ in eq. , we must have $c'< c^{\ast}$ $(< 0)$. Fixing any constant $c\in [c', c^{\ast}]$, we observe that $y_{c'}$ ($y_{c^{\ast}}$, respectively) is a subsolution (supersolution) to equation on $(-1,1)$ (with this $c$).
Corollary \[cor-y\_c:monot\], Part [(ii)]{}, implies $y_{c'}\leq y_{c^{\ast}}$ throughout $[-1,1]$. In contrast, Proposition \[prop-y(1):sub/sup\], Part [(b)]{}, forces $y_{c^{\ast}}\leq y_{c'}$ on $[s_0,1]$. We conclude that $y_{c'}\equiv y_{c^{\ast}}$ on $[s_0,1]$. Consequently, both, $y_{c'}$ and $y_{c^{\ast}}$, verify eq. on the interval $(s_0,1)$ for both values of $c$, $c = c'$ and $c = c^{\ast}$. Since also $y_{c'}(r) = y_{c^{\ast}}(r) > 0$ for every $r\in [s_0,1)$, eq. with $c = c'$ and $c = c^{\ast}$ forces $c'= c^{\ast}$. We have proved also the uniqueness.
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Asymptotic Behavior {#s:Asympt}
===================
In this section we investigate the [*asymptotic behavior*]{} of the (unique) solution $y = y_{c^{\ast}}{:\,}[-1,1]\to {\mathbb{R}}$ to problem with $c = c^{\ast}$. Of course, we assume $G(1) > 0$ throughout the present section. We recall that the existence and uniqueness of both, $c^{\ast}$ $(< 0)$ and $y_{c^{\ast}}$, have been obtained in Corollaries \[cor-c<c\^\*\] and \[cor-uniq\_c\^\*\].
In fact, the asymptotic estimates for $r\in (-1,1)$ near the endpoints $\pm 1$ obtained in this section remain valid in the following more general situation with an arbitary value of the parameter $c\leq 0$ near the left endpoint $-1$: We assume that $y = y_c{:\,}[-1,1]\to {\mathbb{R}}$ is the (unique) solution to eq. satisfying the initial condition $y(-1) = 0$.
We assume that $g(r){\stackrel{{\mathrm {def}}}{=}}d(r)^{1/(p-1)}\, f(r)$ satisfies also the following hypothesis:
- There are constants $\gamma, \gamma_0\in (0,\infty)$ such that $$\label{lim:g(r)}
\lim_{r\to -1}\, \frac{g(r)}{ (1+r)^{\gamma} } = \gamma_0 \,.$$
Equivalently, we have $$\nonumber
\tag{\ref{lim:g(r)}$'$}
g(r) = ( \gamma_0 + \eta(r) ) (1+r)^{\gamma}
\quad\mbox{ for }\, r\in [-1,1] \,,$$ where $\eta{:\,}[-1,1]\to {\mathbb{R}}$ is a continuous function with $\eta(0) = 0$.
We abbreviate the differential operator $$\label{def:A(r)}
(\mathcal{A}y)(r){\stackrel{{\mathrm {def}}}{=}}\frac{\mathrm{d}y}{\mathrm{d}r}
- p'\left( c\, (y^{+})^{1/p} + g(r) \right) \,,
\quad r\in (-1,1) \,,$$ for any function $y\in C^1([-1,1])$. We are interested in constructing sub- and supersolutions to equation as defined in Section \[s:Unique\]. Let $\kappa\in (0,\infty)$ be a constant (to be determined later) and let us consider the (nonnegative) function $$\label{e:w_kappa}
w_{\kappa}(r){\stackrel{{\mathrm {def}}}{=}}\kappa\, (1+r)^{1+\gamma}
\quad\mbox{ for }\, r\in [-1,1] \,.$$ Then we have $$\label{e:Aw_kappa}
\begin{aligned}
& (\mathcal{A}w_{\kappa})(r)
= \frac{ \mathrm{d}w_{\kappa} }{\mathrm{d}r}
- p'\left( c\, (w_{\kappa})^{1/p} + g(r) \right)
\\
& = \kappa (1 + \gamma) (1+r)^{\gamma}
- p'\left( c\, \kappa^{1/p} (1+r)^{(1+\gamma)/p}
+ ( \gamma_0 + \eta(r) ) (1+r)^{\gamma}
\right)
\\
& = \left[ \kappa (1 + \gamma) - p'\, ( \gamma_0 + \eta(r) )
\right] (1+r)^{\gamma}
- p' c\, \kappa^{1/p} (1+r)^{(1+\gamma)/p}
\end{aligned}$$ for $r\in (-1,1)$. Letting $r\to -1$ we observe that the first expression, with the power $(1+r)^{\gamma}$, dominates the second one, with the power $(1+r)^{(1+\gamma)/p}$, if and only if $\gamma\leq (1+\gamma)/p$ holds, that is, $\gamma\leq p'/p = 1/(p-1)$. Consequently, if this is the case, then, for every $r\in (-1,1)$ sufficiently close to $-1$ we have:
- $(\mathcal{A}w_{\kappa})(r) < 0$ provided $\kappa > 0$ is small enough, and
- $(\mathcal{A}w_{\kappa})(r) > 0$ provided $\kappa > 0$ is large enough.
We can state these simple facts about $w_{\kappa}$ being a sub- or supersolution to problem on some interval $(-1,-1+\varrho)$ as follows ($0 < \varrho < 2$):
\[lem-power<g(r)\] Let $c\in {\mathbb{R}}$ and assume that $g{:\,}[-1,1]\to {\mathbb{R}}$ satisfies with $0 < \gamma\leq 1/(p-1)$. Then there exist numbers $\varrho\in (0,2)$ and $0 < \underline{\kappa} < \overline{\kappa} < \infty$, such that
1. if $0 < \kappa\leq \underline{\kappa}$ then $(\mathcal{A}w_{\kappa})(r) < 0$ holds for all $r\in (-1,-1+\varrho)$; and
2. if $\overline{\kappa}\leq \kappa < \infty$ then $(\mathcal{A}w_{\kappa})(r) > 0$ holds for all $r\in (-1,-1+\varrho)$.
We remark that all three numbers $\varrho$, $\underline{\kappa}$, and $\overline{\kappa}$ may depend on $c$, in general.
Now, let us fix $c = c^{\ast}$ given by eq. and recall Corollary \[cor-uniq\_c\^\*\]. We combine Proposition \[prop-y:sub/sup\], Part [(a)]{}, with Lemma \[lem-power<g(r)\] to conclude that $$\label{ineq:y_c*}
w_{\underline{\kappa}}(r) = \underline{\kappa}\, (1+r)^{1+\gamma}
\leq y_{c^{\ast}}(r)\leq
w_{\overline{\kappa}}(r) = \overline{\kappa}\, (1+r)^{1+\gamma}$$ for all $r\in (-1,-1+\varrho)$. We recall that $V(U) = \left[ y_{c^{\ast}}(r)\right]^{1/p'}$ with $U=r$, by eq. . Consequently, inequalities above read $$\label{ineq:V_c*}
(\underline{\kappa})^{1/p'}\, (1+U)^{(1+\gamma) / p'}
\leq V(U)\leq
(\overline{\kappa})^{1/p'}\, (1+U)^{(1+\gamma) / p'}$$ for all $U\in (-1,-1+\varrho)$. In eq. , that is $$\nonumber
\tag{\ref{e:dx/dU}}
\genfrac{}{}{}0{\mathrm{d}x}{\mathrm{d}U}
= {}-
\genfrac{(}{)}{}0{d(U)}{V(U)}^{1/(p-1)} < 0 \,,\quad
-1 < U < 1 \,,$$ for the unknown function $x = x(U)$, we need the following equivalent form of , $$\nonumber
\tag{\ref{ineq:V_c*}$'$}
(\underline{\kappa})^{1/p}\, (1+U)^{(1+\gamma) / p}
\leq V(U)^{1/(p-1)}\leq
(\overline{\kappa})^{1/p}\, (1+U)^{(1+\gamma) / p}$$ for all $U\in (-1,-1+\varrho)$.
It remains to investigate the case $\gamma > (1+\gamma)/p$, that is, $\gamma > p'/p = 1/(p-1)$, in which the second expression in eq. , with the power $(1+r)^{(1+\gamma)/p}$, dominates the first one, with the power $(1+r)^{\gamma}$. Here, we assume $c < 0$. Then, given any $\kappa\in (0,\infty)$, we have (thanks to $c < 0$)
- $(\mathcal{A}w_{\kappa})(r) > 0$ for every $r\in (-1,1)$ sufficiently close to $-1$.
We can state this simple fact about $w_{\kappa}$ being a supersolution to problem on some interval $(-1,-1+\varrho)$ as follows ($0 < \varrho < 2$):
\[lem-power>g(r)\] Let $c < 0$ and assume that $g{:\,}[-1,1]\to {\mathbb{R}}$ satisfies with $\gamma > 1/(p-1)$. Then, given any $\kappa\in (0,\infty)$, there exists a number $\varrho\in (0,2)$, such that
- $(\mathcal{A}w_{\kappa})(r) > 0$ holds for all $r\in (-1,-1+\varrho)$.
We remark that, again, the number $\varrho$ may depend on $\kappa$ and $c$, in general.
Now, let us fix $c = c^{\ast}$ given by eq. and recall Corollary \[cor-uniq\_c\^\*\]. We combine Proposition \[prop-y:sub/sup\], Part [(a)]{}, with Lemma \[lem-power<g(r)\] to conclude that $$\label{inequ:y_c*}
y_{c^{\ast}}(r)\leq w_{\kappa}(r) = \kappa\, (1+r)^{1+\gamma}$$ for all $r\in (-1,-1+\varrho)$. We recall that $V(U) = \left[ y_{c^{\ast}}(r)\right]^{1/p'}$ with $U=r$, by eq. . Consequently, ineq. above reads $$\label{inequ:V_c*}
V(U)\leq \kappa^{1/p'}\, (1+U)^{(1+\gamma) / p'}$$ for all $U\in (-1,-1+\varrho)$. In eq. , that is $$ \genfrac{}{}{}0{\mathrm{d}x}{\mathrm{d}U}
= {}-
\genfrac{(}{)}{}0{d(U)}{V(U)}^{1/(p-1)} < 0 \,,\quad
-1 < U < 1 \,,$$ for the unknown function $x = x(U)$, we need the following equivalent form of , $$\nonumber
\tag{\ref{inequ:V_c*}$'$}
V(U)^{1/(p-1)}\leq \kappa^{1/p}\, (1+U)^{(1+\gamma) / p}$$ for all $U\in (-1,-1+\varrho)$.
\[cor-power<g(r)\] Let $c = c^{\ast}$ $(< 0)$ and assume that $g{:\,}[-1,1]\to {\mathbb{R}}$ satisfies . Then we have the following conclusions for the limit $x_{-1}{\stackrel{{\mathrm {def}}}{=}}\lim_{U\to -1} x(U)$ $({}\leq +\infty)$ of the solution $x = x(U)$ to the differential equation :
1. In case $1 < p\leq 2$, the limit $x_{-1}$ is finite if and only if $0 < \gamma < p-1$. $($Equivalently, $x_{-1} = +\infty$ if and only if $\gamma\geq p-1$.$)$
2. In case $2 < p < \infty$, the limit $x_{-1} = +\infty$ if $\gamma\geq p-1$.
Consequently, we have $x_{-1} = +\infty$ whenever $0 < p-1\leq \gamma < \infty$.
The [*proof*]{} of Part [(i)]{} (Part [(ii)]{}, respectively) follows directly from eq. combined with ineq. $'$ (ineq. $'$).
In the linear diffusion case, i.e., for $p=2$, $x_{-1}$ is finite if $\gamma < 1$, whereas $x_{-1} = +\infty$ if $\gamma\geq 1$.
\[rem-power<g(r)\]
Acknowledgments {#acknowledgments .unnumbered}
---------------
The work of Pavel Drábek was supported in part by the Grant Agency of the Czech Republic (GAČR) under Grant [\#]{}$13-00863$S, and the work of Peter Takáč by a grant from Deutsche Forschungsgemeinschaft (DFG, Germany) under Grant [\#]{} TA 213/16–1.
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|
=1000 cyracc.def =wncyss10
epsf
In Einstein’s gravitational theory, the [*metric*]{} $g$ of spacetime is the fundamental field variable. The metric governs temporal and spatial distances and angles. All other geometrical notions can be derived from the metric. In particular, the affine properties of spacetime, i.e. those related to parallel transfer and represented by the linear [*connection*]{} $\Gamma$, are subordinate to the metric. In Riemannian spacetime, the connection $\Gamma$ can be exclusively expressed in terms of the metric, and the same is true for the curvature $R\sim curl\, \Gamma$.
In [*gauge*]{}-theoretical approaches to gravity, the connection $\Gamma$ becomes an independent field variable. In the viable Einstein-Cartan theory of gravity, e.g., it is Lorentz algebra valued and couples to the spin of matter. In fact, in such approaches, the coframe $\vartheta$ and the connection $\Gamma$ appear as fundamental variables. The metric is sort of absorbed by a priori choosing the coframe to be pseudo-orthonormal and the connection to be Lorentz algebra valued.
In any case, the dichotomy of metric and affine properties of spacetime and attempts to understand it runs through much of present-day theorizing on the fundamental structure of spacetime. Already Eddington, e.g., tried to derive $g$ from (a symmetric) $\Gamma$ by choosing a suitable gravitational Lagrangian $\sim |\det
{\rm Ric}_{ij}(\Gamma,\partial\Gamma)|^{1/2}$, see [@Schro]; here and in future $i,j,k\dots=0,1,2,3$ are coordinate indices. Then $\Gamma$ would be fundamental and $g$ be a derived concept.
Thinking more from the point of view of a quantum substructure supposedly underlying classical spacetime, arguments were advanced [@Sach] that the metric is some kind of effective field which “froze out” during a cosmic phase transition in the early universe, cf. also [@Fink]. In other words, also here the metric would be a secondary structure comparable to the strain field in a solid.
The aim of this letter is, however, more modest: We formulate classical electrodynamics in the so-called metric-free version, see [@Truesdell; @Post62; @Stachel], by taking recourse to the conservation laws of electric charge and magnetic flux and to the existence of a Lorentz force density. To complete the apparatus of the field equations, we have eventually to specify the constitutive relation between field strength $F=(E,B)$ and excitation $H=({\cal D},{\cal H})$.
We choose a [*linear*]{} law $H_{ij}=\kappa_{ij}{}^{kl}F_{kl}/2$ with $21$ independent constitutive functions $\kappa_{ij}{}^{kl}(x)$. The linear law can be interpreted as a new kind of duality operation $\#$ mapping 2-forms into 2-forms: $H\sim{}^\#F$. Let us stress that [*no*]{} metric has been used so far.
We impose a constraint on the duality operator, namely $^{\#\#}=-1$ (for Euclidean signature $+1$). This, together with two formulas of Urbantke [@Urban] which had been used in a Yang-Mills context, allows us to derive from $\kappa_{ij}{}^{kl}$ a metric with pseudo-Euclidean signature. In this way we recognize of how closely the concept of a metric is connected with the electromagnetic properties of spacetime itself or of material media embedded therein – a fact which, perhaps, doesn’t come as a surprise in view of the principle of the constancy of the velocity of light.
\(1) [*Three axioms of electrodynamics:*]{} Spacetime is assumed to be a 4-dimensional differentiable manifold which allows a foliation into 3-dimensional submanifolds which can be numbered with a monotonically increasing parameter $\sigma$. The existence of an electric current (3-form) $J=-j\wedge d\sigma+\rho$ is postulated which, by axiom 1, is conserved: $$\label{axiom1}
\oint\limits_{C_3}J=0\,,\quad
\partial C_3=0\,.$$ Here $C_3$ is an arbitrary closed 3-dimensional submanifold of the 4-manifold. By de Rham’s theorem, the inhomogeneous Maxwell equation is a consequence therefrom, $$J=dH\,,$$ with $H=H_{ij}dx^i\wedge dx^j/2= -{\cal
H}\wedge d\sigma+{\cal D}$. The current $J$, together with a force density $f_\alpha$, originating from mechanics, allow to formulate the Lorentz force density as axiom 2: $$\label{axiom2}
f_\alpha= (e_\alpha\rfloor F) \wedge J\,.$$ Greek indices $\alpha,\beta,\dots=0,1,2,3$ are anholonomic or frame indices and $e_\alpha$ is the local frame, the interior product is denoted by $\rfloor$. This axiom introduces the electromagnetic field strength (2-form) $F=F_{ij}dx^i\wedge
dx^j/2=E\wedge d\sigma+B$ as a new concept. In axiom 3, the corresponding magnetic flux is assumed to be conserved, $$\label{axiom3}
\oint\limits_{C_2}F=0\,,\quad \partial C_2=0\,,$$ for any closed submanifold $C_2$. As a consequence, we find the homogeneous Maxwell equation $$dF=0\,.$$
These equations are all diffeomorphism invariant and don’t depend on metric or connection. This is also true for the exterior [*and*]{} the interior product. Therefore these equations are valid in special and general relativity likewise and in non-Riemannian spacetimes of gauge theories of gravity, see [@Punt]. Electric charges and, under favorable conditions, also magnetic flux quanta can be counted. This is why the metric is dispensible under those circumstances.
\(2) [*Constitutive law as axiom 4:*]{} The simplest constitutive law is, of course, a linear law. If it is additionally isotropic, it yields in particular vacuum electrodynamics. However, isotropy can only be formulated if a metric is available which is not the case under the present state of our discussion. Therefore we assume only linearity: $$H_{ij}={1 \over 2}\,\kappa_{ij}{}^{kl}(x)\,F_{kl}\,.\label{kappa}$$ Since $H$ is an odd and $F$ an even form, the constitutive matrix $\kappa$ is odd. Therefore we split off the Levi-Civita symbol $$\label{chi}
\tilde{\chi}_{ijkl}=\frac{1}{2}\,\kappa_{ij}{}^{mn}\,
\epsilon_{mnkl}=f{\chi}_{ijkl}\,,$$ where $f$ is a dimensionful scalar function such that $\chi_{ijkl}$ is dimensionless. The tensor density $\chi_{ijkl}$ carries the weight $-1$. The Lagrangian of the theory is quadratic in $F$. Thus we find $\chi_{ijkl}=\chi_{klij}$, i.e., $21$ independent functions. Since $H_{ij}$ and $F_{kl}$ can be measured independently, the constitutive functions $\chi_{ijkl}$ can be experimentally determined.
It is convenient to write $H$ and $F$ as row vectors $H_I=(H_{01},
H_{02},H_{03},H_{23},H_{31},H_{12})$, where $I$ runs from $1$ to $6$, etc. Then the constitutive law reads $$\label{chiHF1}
H_I=\kappa_I{}^KF_K=\chi_{IM}\,\epsilon^{MK}fF_K\,\>{\rm
with}\>\chi_{IM}=\chi_{MI}\,.$$ Furthermore, in local coordinates, the basis of the 2-forms is represented by the six 2-forms $dx^i\wedge dx^j$. They can be put into the column vector Cyrillic $B$, namely $\cb^I$. Then $H=H_I\cb^I$ and $F=F_I\cb^I$.
\(3) [*Duality operator $\#$ and its closure:*]{} We can define, by means of the linear constitutive law (\[chiHF1\]), a new [*duality operator*]{} mapping 2-forms into 2-forms. Accordingly, we require for the 2-form basis $$\label{sharp0}
{}^\#\cb ^I=\left(\chi_{KM}\,\epsilon^{MI}\right)\cb ^K\,,$$ i.e., the duality operator incorporates the constitutive properties specified in (\[chiHF1\]). In particular, we have for the electromagnetic field two-forms $H=f{}^\# F$.
A duality operator, applied twice, should, up to a sign, lead back to the identity. By such a postulate we can constrain the number of independent components of $\chi$ without using, say, a metric: $$\label{closure}
{}^{\#\#}=-1\,.$$ We concentrate here on the minus sign; the rule $^{\#\#}= +1$ would lead to Euclidean signature.
It is convenient to write the $6\times 6$ matrices, which define the duality operator, in terms of $3\times 3$-matrices: $$\label{xixi1}
\chi_{IK}=\chi_{KI}= \left(\begin{array}{cr}A& C \\C^{{\rm T}} &B
\end{array}\right)\,,\> \epsilon^{IK} =\epsilon^{KI}=
\left(\begin{array}{cr}0& {\mathbf 1} \\{\mathbf 1}
&0\end{array}\right)$$ Here $A=A^{\rm T}\,,B=B^{\rm T}$, the superscript ${}^{\rm T}$ denoting transposition. The general non-trivial solution of (\[closure\]) is given by $$\label{xixi3}
\chi_{IK}= \left(\begin{array}{cc}pB^{-1} + qN& B^{-1}K \\ -KB^{-1}
&B \end{array}\right)\,,$$ Here $B$ is a nondegenerate arbitrary [*symmetric*]{} matrix (6 independent components) and $K$ an arbitrary [*antisymmetric*]{} matrix (3 components). Furthermore, we construct the symmetric matrix $N$ as a solution of the homogeneous system $KN=NK=0$. With $K_{ab}=\epsilon_{abc}
k^c$, we have explicitly: $$N = \left(\begin{array}{ccc}
(k^1)^2& k^1 k^2 & k^1 k^3 \\ k^1k^2&(k^2)^2& k^2 k^3 \\
k^1 k^3 & k^2 k^3 &(k^3)^2\end{array}\right)\,.\label{Ndia}$$ Finally $q = - 1/{\det B}$, $p = [{\rm tr}(NB)/\det B] - 1$.
\(4) [*Selfduality and a triplet of 2-forms:*]{} With our new duality operator we can define the selfdual (s) and the anti-selfdual (a) of a 2-form. For the 2-form basis $\cb$ we have $${\stackrel {({\rm s})} \cb}:={\frac 1
2}(\cb\,-i\,^\#\!\cb),\label{scb}\,,\quad {\stackrel {({\rm a})}
\cb}:={\frac 1 2}(\cb\,+i\,^\#\!\cb)\,,\label{acb}$$ with $^\#{\stackrel {({\rm s})} \cb}= i{\stackrel {({\rm s})} \cb}$, $^\#{\stackrel {({\rm a})} \cb}= -i{\stackrel {({\rm a})} \cb}$. Thus the 6-dimensional space of 2-forms decomposes into two 3-dimensional invariant subspaces corresponding to the eigenvalues $\pm i$ of the duality operator. With the decomposition into two 3-dimensional row vectors, $$\label{cb3x2}
\cb^I = \left(\begin{array}{c} \beta^a \\ \gamma^b
\end{array}\right)\,,\quad a,b, \dots = 1,2,3\,,$$ we take care of this fact also in the 2-form basis.
One of the 3-dimensional invariant subspaces can be spanned by, say, ${\stackrel {({\rm s})} \gamma}$, whereas ${\stackrel {({\rm s})}
\beta}$ is obtained from it by means of the linear transformation ${\stackrel {({\rm s})} \beta}=(i + B^{-1}K)B^{-1}{\stackrel {({\rm
s})} \gamma}$. Therefore ${\stackrel {({\rm s})} \gamma}$ subsumes the properties of this invariant subspace and so does the triplet of 2-forms $$\label{triplet}
S^{(a)}:= -(B^{-1})^{ab}\,{\stackrel{({\rm s})}{ \gamma}}{}^b\,.$$ Hereafter, $(B^{-1})^{ab}$ and $B_{cd}$ denote the matrix elements of $B^{-1}$ and $B$, respectively. Incidentally, the anti-self dual ${\stackrel {({\rm a})} \gamma}$ spans the other invariant subspace. The whole information of the linear constitutive law (\[chiHF1\]) is now encoded into the triplet of 2-forms $S^{(a)}$. A direct calculation demonstrates that they satisfy the so-called completeness condition: $$S^{(a)}\wedge S^{(b)} = {\frac 1 3}\,
(B^{-1})^{ab}\,(B)_{cd}\,S^{(c)}\wedge S^{(d)}.\label{complSB}$$
\(5) [*Extracting the metric:*]{} Urbantke [@Urban] (see also the discussions in [@Harnett; @Tert]) was able to derive, within $SU(2)$ Yang-Mills theory, a 4-dimensional metric $g_{ij}$ (with $i,j,
\dots = 0,1,2,3$) from a triplet of 2-forms which are related to 2-plane elements of spacetime with certain distinguished properties. Since the completeness condition (\[complSB\]) is fulfilled, the Urbantke’s formulas $$\begin{aligned}
\sqrt{{\det}\,g}\,g_{ij} & =& -\,{\frac 2 3}\,\sqrt{\det
B}\,\epsilon_{abc}\, \epsilon^{klmn}\,S^{(a)}_{ik}S^{(b)}_{lm}
S^{(c)}_{nj}\,,\label{urbantke1}\\ \sqrt{{\det}\,g} & =& -\,{\frac 1
6}\,\epsilon^{klmn}\,B_{cd}\,
S^{(c)}_{kl}S^{(d)}_{mn}\,,\label{urbantke2}\end{aligned}$$ are also applicable in our case. Here the $S^{(a)}_{ij}$ are the components of the 2-form triplet according to $S^{(a)} = S^{(a)}_{ij}
dx^i\wedge dx^i/2$.
If we express ${\stackrel {({\rm s})} \gamma}$ in terms of $\beta$ and $\gamma$ and then substitute (\[triplet\]) into (\[urbantke1\]), (\[urbantke2\]), we can, after a very involved computation, display the metric explicitly: $$\label{gij}
g_{ij} = {\frac 1 {\sqrt{\det B}}}\left(\begin{array}{c|c}\det B &
-\,k_a \\ \hline -\,k_b & -\,B_{ab} + (\det B)^{-1}\,k_a\,k_b
\end{array}\right)\,.$$ Here we used the abbreviation $k_a:=B_{ab}\,k^b = B_{ab}\,$ $ \epsilon
^{bcd}K_{cd}/2$. The $3\times 3$ matrix $B_{ab}$ can have any signature. Nevertheless, Eq.(\[gij\]) always yields a metric with Minkowskian signature.
This representation (\[gij\]) of the metric is our basic result. Since the triplet $S^{(a)}$ is defined up to an arbitrary scalar factor, our procedure in general defines a [*conformal*]{} class of metrics rather than a metric itself.
As the most simple example, we will construct the Minkowski metric. Recall that (\[gij\]) depends on the symmetric $3\times 3$ matrix $B$ (not to be confused with the magnetic field $B$) and the antisymmetric $3\times 3$ matrix $K$. If we choose $f^2 =
\varepsilon_0/\mu_0$, $B = (\varepsilon_0\mu_0)^{-1/2}\,{\mathbf 1}$ and $K = 0$, then, according to (\[chi\]), this translates into the conventional vacuum relations $$\vec{\cal D}=\varepsilon_0\vec{E}\quad{\rm and}\quad
\vec{\cal H}=\vec{B}/\mu_0. \label{vacuum}$$ On the other hand, if we substitute it into (\[gij\]), we immediately find (denoting $c:=1/\sqrt{\varepsilon_0\mu_0}$) $$g_{ij} = {\frac 1 {\sqrt{c}}}\left(\begin{array}{crrr}c^2 & 0 & 0 & 0
\\ 0 & -1 & 0 & 0\\ 0& 0 & -1 & 0\\ 0& 0& 0& -1\end{array}\right),$$ i.e. the Minkowski metric of special relativity (including its signature) has been derived from the conventional vacuum relation (\[vacuum\]) between electromagnetic excitation $H=({\cal D},{\cal H})$ and field strength $F=(E,B)$.
\(6) [*Outlook:*]{} In this letter we have demonstrated that the metric-free formulation of classical electrodynamics (in the spirit of the old Kottler-Cartan-van Dantzig approach, cf.[@Post62]) naturally leads to the reconstruction of the spacetime [*metric from the constitutive law*]{}. It seems worthwhile to remind ourselves that the constitutive (or material) relation is a postulate which arises not from a pure mathematical considerations but rather from the analysis of experimental data [@Truesdell]. A development of an alternative axiomatics of the Maxwell theory on the basis of the postulate of the well-posedness of Cauchy problem [@Laemm] (see also a related discussion in [@Stachel]) gives good reasons to assume that the crucial closure condition (\[closure\]) is tantamount to a postulate of a well-posed Cauchy problem for Maxwell’s equations.
We are grateful to Claus Lämmerzahl for helpful discussions on Maxwell’s theory and its axiomatic basis. This work was partially supported by the grant INTAS-96-842 of the European Union (Brussels).
[*Note added in proof:*]{} After this work was completed, we have learned that Schönberg [@schoen], in a not widely available journal, had already developed the approach to the spacetime metric on the basis of the constitutive relation (\[kappa\]) and the “closure” relation (\[closure\]). We are grateful to Helmuth Urbantke and Ted Jacobson for corresponding remarks and to José W. Maluf for sending us a copy of Schönberg’s paper and his curriculum vitae. However, our derivation of the general solution (\[xixi3\]) of the “closure" relation, and the complete explicit construction of the metric (\[gij\]) are new. Moreover, even earlier, Peres [@peres] investigated related structures, and thus he can be considered as a forerunner of Schönberg. We thank Asher Peres for pointing out to us the relevance of his paper. In the meantime Guillermo Rubilar has checked our formula (\[xixi3\]) by means of computer algebra. For further discussions of our preprint we would like to thank Ted Frankel, Yakov Itin, Bahram Mashhoon, and Eckehard Mielke.
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abstract: 'This study numerically investigates the linear response of two-dimensional frictional granular materials under an oscillatory shear. The storage modulus $G''$ and the loss modulus $G''''$ are dependent on the initial strain amplitude of the oscillatory shear before the measurement. The shear jammed state (satisfying $G''>0$) can be observed at an amplitude greater than the critical initial strain amplitude. The fragile state is defined by the coexistence of a liquid-like state and a solid-like state in initial shear. In this state, the observed $G''$ after the reduction of the strain amplitude depend on the phase of the external shear strain. The loss modulus $G''''$ exhibits a discontinuous jump corresponding to the discontinuous shear thickening in the fragile state.'
author:
- Michio Otsuki
- Hisao Hayakawa
title: ' Shear jamming, discontinuous shear thickening, and fragile states in dry granular materials under oscillatory shear '
---
[*Introduction.–*]{} The amorphous materials comprising repulsive and dissipative particles including randomness such as granular materials, colloidal suspensions, foams, and emulsions can form solid-like jammed states. Since Liu and Nagel suggested that jammed states exist only above the critical packing fraction (the jamming point) [@Liu], jamming transitions have attracted much attention among physicists [@Hecke; @Behringer18]. Jammed states have manifested in several numerical simulations of frictionless grains, which exhibit continuous pressure transitions but discontinuous coordination-number transitions [@OHern02; @OHern03; @Wyart05]. Other researchers have reported various critical scaling laws of rheological quantities near the jamming point for the frictionless particles under steady shears [@Olsson; @Hatano07; @Hatano08; @Tighe; @Hatano10; @Otsuki08; @Otsuki09; @Otsuki10; @Nordstrom; @Olsson11; @Vagberg; @Otsuki12; @Ikeda; @Olsson12; @DeGiuli; @Vagberg16; @Boyer; @Trulsson; @Andreotti; @Lerner; @Vagberg14; @Kawasaki15; @Suzuki; @Rahbari] and oscillatory shears [@Tighe11; @Otsuki14].
Regardless, the mutual friction between adjacent granular particles is inevitable in granular systems. Zhang et al. [@Bi11] suggested that the jamming process qualitatively differs between frictional and frictionless grains; in frictional systems, the shear flows induce jammed states even below the friction-dependent critical fraction $\phic$. Such transitions, known as shear jamming, have been extensively studied both experimentally [@Zhang08; @Zhang10; @Wang18] and numerically [@Sarkar13; @Sarkar16]. Zhang et al. [@Bi11] further proposed the existence of a fragile state in a system under pure shear, characterized by the percolation of a force chain only in the compressive direction [@Footnote1]. In contrast, the force chains in the shear jammed state percolate in all the directions. However, the definition of a fragile state in Ref. [@Bi11] is non-quantitative and inapplicable to other systems, necessitating a quantitative definition.
The mutual friction between grains causes a drastic rheological transition [@Otsuki11; @Chialvo; @Brown; @Seto; @Fernandez; @Heussinger; @Bandi; @Ciamarra; @Mari; @Grob; @Kawasaki14; @Wyart14; @Grob16; @Hayakawa16; @Hayakawa17; @Peters; @Fall; @Sarkar; @Singh; @Kawasaki18; @Thomas] known as discontinuous shear thickening (DST). DST is used in industrial applications such as protective vests, robotic manipulators, and traction controls [@Brown14; @Brown10]. Several studies have investigated the relation between DST and shear jamming in suspensions of frictional grains under steady shear [@Peters; @Fall; @Sarkar; @Singh]. In stress-controlled experiments, DST can be observed over a wide region of the phase diagram [@Peters]; however, in rate-controlled experiments, DST can be observed only as a broad line between shear jamming and continuous shear thickening in the phase diagram [@Fall]. Because these results seem to be inconsistent, the relation between shear jamming and DST is not yet clarified.
To resolve the aforementioned problems, we numerically measure the complex shear modulus in two-dimensional frictional grains near the jamming point under oscillatory shear. Therefore, we apply the discrete element method (DEM). In this Letter, we clarify the relations among the shear jammed state, the fragile state, and the DST by controlling the initial strain amplitude $\gamma_0^{\rm (I)}$ and the area fraction $\phi$.
[*Setup of our simulation.–*]{} Let us consider the two-dimensional assembly of $N$ frictional granular particles having the identical density $\rho$ confined in a square box of linear size $L$. The inter-particle interactions are modeled as linear springs with normal and tangential spring constants of $k^{\rm (n)}$ and $k^{\rm (t)}$, respectively, a Coulomb friction constant $\mu$, and a restitution constant $e$ [@Cundall]. DEM is detailed in Supplemental Material [@Supple]. To avoid crystallization, we constructed a bidispersed system with an equal number of grains of two diameters ($d_0$ and $d_0/1.4$). We also set the number of particles $N$ to $4000$, $k^{\rm (n)}=0.2k^{\rm (t)}$, $\mu = 1.0$, and $e = 0.043$.
To suppress the shear bands, we apply an oscillatory shear along the $x$-direction under the Lees–Edwards boundary condition using the Sllod method [@Evans]. Initially, the disks are randomly distributed throughout the system with an area fraction of $\phi_{\rm I}=0.75$. Further, the system is slowly compressed until the area fraction reached a specified value $\phi$ [@Otsuki17]. Note that we estimate the jamming point $\phic=0.821$ for $\mu=1.0$; however, the jamming point depends on the initial preparation [@Luding; @Kumar]. See Ref. [@Supple] for the initial preparation details and the determination and $\mu$-dependence of $\phic$. We further apply the shear strain $\gamma (t) = \gamma_0
\left \{ \cos\theta - \cos(\omega t + \theta) \right \}$ in the $x$-direction of the compressed system, where $\gamma_0$, $\omega$, and $\theta$ denote the strain amplitude, the angular frequency, and the initial phase, respectively. Over the initial $N_c^{\rm (I)}=10$ cycles, we assume the initial strain amplitude as $\gamma_0 = \gamma_0^{\rm (I)}$. After $N_c^{(I)}$ cycles, we reduce the strain amplitude to $\gamma_0 = \gamma_0^{\rm (F)}=1.0 \times 10^{-4}$ and apply $N_c^{\rm (F)}=10$ cycles of oscillatory shear to measure the storage modulus $G'$ and the loss modulus $G''$ in the linear response region. Here, $G'$ and $G''$ are, respectively defined by [@Doi] $$\begin{aligned}
G' & = & - \frac{\omega}{\pi } \int_{0}^{2 \pi/\omega} \ dt \
\sigma(t) \cos(\omega t + \theta)/\gamma_0^{\rm (F)}, \\
G'' & = & \frac{\omega}{\pi} \int_{0}^{2 \pi/\omega} \ dt \
\sigma(t) \sin(\omega t + \theta)/\gamma_0^{\rm (F)}.\end{aligned}$$ The moduli $G'$ and $G''$ are measured in the final cycle. The shear stress $\sigma$ in the above expressions is given by $$\begin{aligned}
\sigma = - \frac{1}{2L^2} \sum_{i} \sum_{j>i}
\left ( r_{ij,x} F_{ij,y} + r_{ij,y} F_{ij,x} \right ),
\label{sigma}\end{aligned}$$ where $F_{ij,\alpha}$ and $r_{ij,\alpha}$ denote the $\alpha$ components of the interaction force $\Vect{F}_{ij}$ and the relative position vector $\Vect{r}_{ij}$ between two grains $i$ and $j$, respectively. The contributions of the kinetic part of $\sigma$ and the coupled stress (i.e., the asymmetric part of the shear stress) are ignored because they are less than $1 \%$ of $\sigma$. Note that when $\omega \le 10^{-2} t_0^{-1}$ and $\gamma_0^{\rm (F)}\le 10^{-3}$, $G'$ and the dynamic viscosity $\eta(\omega) \equiv G''(\omega)/\omega$ corresponding to the apparent viscosity are almost independent of $\omega$ and $\gamma_0^{\rm (F)}$ with $\gamma_0^{\rm (I)} \le 1.0$ and $t_0 = \sqrt{m_0/k^{\rm (n)}}$ being the characteristic time scale with the mass $m_0$ for a grain of the diameter $d_0$ [@Otsuki17]. Thus, we investigate only the effects of $\gamma_0^{\rm (I)}$, $\theta$, and $\phi$ on the shear modulus, fixing $\omega = 10^{-4} t_0^{-1}$ and $\gamma_0^{\rm (F)} = 10^{-4}$. We have also confirmed that $G'$ is almost independent of $N_c^{\rm (I)}$ and $N_c^{\rm (F)}$ when $N_c^{\rm (I)}\ge 10$ and $N_c^{\rm (F)}\ge10$ [@Supple]. We adopt the leapfrog algorithm with time step $\Delta t = 0.05 t_0$.
[*Mechanical response.–*]{} Figure \[F\_SJ\] plots the force chains immediately after the reduction of the strain amplitude for $\phi=0.820<\phic$ and $\theta = 0$. Here, $\gamma_0^{\rm (I)}$ is varied as $0.1, 0.12$, and $1.0$. When the initial strain amplitude is small ($\gamma_0^{\rm (I)}=0.1$), the system remains in a liquid-like state with no percolating force chains. Under high initial strains ($\gamma_0^{\rm (I)} = 0.12$ and $1.0$), the system develops anisotropic percolating force chains.
[c]{}
![ The snapshots of grains (circles) and force chains (lines) for $\phi=0.820$ and $\theta = 0$ immediately after the initial strain amplitudes (a) $\gamma_0^{\rm (I)} = 0.1$, (b) $0.12$, and (c) $1.0$ are reduced to $\gamma_0^{(F)}=1.0\times 10^{-4}$. Panels (a), (b), and (c) correspond to the unjammed, fragile, and shear jammed states, respectively. The color and width of each line are dependent on the absolute value of the interaction force between the grains. []{data-label="F_SJ"}](F_SJ1.eps){width="1.0\linewidth"}
![ The snapshots of grains (circles) and force chains (lines) for $\phi=0.820$ and $\theta = 0$ immediately after the initial strain amplitudes (a) $\gamma_0^{\rm (I)} = 0.1$, (b) $0.12$, and (c) $1.0$ are reduced to $\gamma_0^{(F)}=1.0\times 10^{-4}$. Panels (a), (b), and (c) correspond to the unjammed, fragile, and shear jammed states, respectively. The color and width of each line are dependent on the absolute value of the interaction force between the grains. []{data-label="F_SJ"}](F_SJ2.eps){width="1.0\linewidth"}
![ The snapshots of grains (circles) and force chains (lines) for $\phi=0.820$ and $\theta = 0$ immediately after the initial strain amplitudes (a) $\gamma_0^{\rm (I)} = 0.1$, (b) $0.12$, and (c) $1.0$ are reduced to $\gamma_0^{(F)}=1.0\times 10^{-4}$. Panels (a), (b), and (c) correspond to the unjammed, fragile, and shear jammed states, respectively. The color and width of each line are dependent on the absolute value of the interaction force between the grains. []{data-label="F_SJ"}](F_SJ3.eps){width="1.0\linewidth"}
Figure \[G\_ga\_theta\] plots $G'$ versus $\gamma_0^{\rm (I)}$ for $\theta=0$ and $\pi/2$ with $\phi=0.820$. The shear induces the transition from a liquid-like to a solid-like state. $G'$ strongly depends on $\theta$ near the critical strain amplitude (shaded region of Fig. \[G\_ga\_theta\]). The inset of Fig. \[G\_ga\_theta\] plots $G'$ versus $\theta$ for $\phi=0.82$ and $\gamma_0^{\rm (I)}=0.12$. The storage modulus $G'$ peaks at $n \pi$ and falls to $0$ near $(n + 1/2) \pi$, where $n$ is an integer.
![ Plots of the storage modulus $G'$ versus $\gamma_0^{\rm (I)}$ for $\phi=0.82$ with $\theta = 0$ and $\pi/2$. The shaded region highlights the fragile state. Inset: Storage modulus $G'$ versus $\theta$ for $\phi=0.82$ with $\gamma_0^{\rm (I)}=0.12.$ []{data-label="G_ga_theta"}](G_ga_theta.eps){width="1.0\linewidth"}
Figure \[s\_ga\_nu0.82000\] plots $\sigma(t)$ versus the strain $\gamma(t)$ in the last cycle of the initial oscillation with $\gamma_0^{\rm (I)}=1.2$ and $\phi=0.820$ at $\theta = 0$ and $\pi/2$. When $\theta=0$, the shear stress $\sigma$ can be fitted by the linear functions of the strain $\gamma$ near the maximum and minimum values of $\sigma$ but remains $0$ over $0.03<\gamma<0.2$ (left panel of Fig. \[s\_ga\_nu0.82000\]). The linear response after the reduction of the strain amplitude is consistent with that observed in the solid-like state (i.e., $G'>0$ near $\gamma \approx 0$ at $\theta=0$). Setting $\theta = \pi/2$ shifted the stress–strain curve of the initial oscillation without significantly changing its shape from that of $\theta = 0$ (see the right panel of Fig. \[s\_ga\_nu0.82000\]). In this case, the linear response after the reduction of the strain amplitude denotes a liquid-like state near $\gamma\approx 0$ (i.e., $G'=0$). These results explain the $\theta$-dependence of $G'$ in Fig. \[G\_ga\_theta\].
![ Plots of shear stress $\sigma$ versus strain $\gamma$ in the last cycle of the initial oscillatory shear with $\gamma_0^{\rm (I)}=0.12$ and $\phi=0.820$ at $\theta = 0$ (left) and $\pi/2$ (right). The solid squares indicate the positions of the linear response measurements after the strain amplitude is reduced. []{data-label="s_ga_nu0.82000"}](s_ga_nu0.82000.eps){width="1.0\linewidth"}
Figure \[G\_ga\_MC\] plots the storage modulus $G'$ versus $\gamma_0^{\rm (I)}$ for various $\phi$ at $\theta=0$. When $\phi>\phic$, $G'$ is finite for $\gamma_0^{\rm (I)}=0$ but depends on $\gamma_0^{\rm (I)}$. When $\phi>0.84$, $G'$ is a decreasing function of $\gamma_0^{\rm (I)}$, consistent with the softening observed in glassy materials under steady-shear conditions[@Fan]. In $0.82 < \phi < 0.84$, $G'$ is minimized at intermediate values of $\gamma_0^{(I)}$. Shear jamming is observed in $\phisj <\phi < \phic$, where $\phisj=0.795$ (as determined in Ref. [@Supple]). We also observe re-entrant behavior at $\phi=0.824$, where $G'$ changes from $G'>0$ to $G' \simeq 0$ and reverts to $G'>0$ at higher $\gamma_0^{(I)}$.
![ Plots of the storage modulus $G'$ versus $\gamma_0^{\rm (I)}$ for various $\phi$ at $\theta = 0$. []{data-label="G_ga_MC"}](G_ga_MC.eps){width="0.8\linewidth"}
Figure \[G2\_ga\] plots the dimensionless dynamic viscosity versus $\gamma_0^{\rm (I)}$ for $\theta=0$ and various $\phi$. The viscosity $\eta$ is almost independent of $\gamma_0^{\rm (I)}$ when $\phi$ exceeds $\phic$, but jumps from a negligibly small value to a large value in $\phisj < \phi < \phic$. This discontinuity, which takes place at a critical amplitude of the initial strain $\gadst$, corresponds to the DST under a steady shear.
![ Plots of dynamic viscosity $\eta$ versus the initial strain amplitude $\gamma_0^{\rm (I)}$ for $\theta=0$ and various $\phi$.[]{data-label="G2_ga"}](G2_ga_MC.eps){width="0.8\linewidth"}
[*Phase diagram.–*]{} Figure \[Phase\_ga\_MC\] depicts the phase diagram on the $\gamma_0^{\rm (I)}$ versus $\phi$ plane. Here, we have introduced the shear storage modulus with no initial oscillatory shearing as $G'_0(\phi) \equiv \lim_{\gamma_0^{\rm (I)} \to 0} G'\left(\phi,\gamma_0^{\rm (I)}\right)$. We then define the jammed (J) state in which $G'_0(\phi)>G_{\rm th}$ and $G'\left(\phi, \gamma_0^{\rm (I)}\right) > G_{\rm th}$ for any $\theta$ with a sufficiently small threshold $G_{\rm th} = 10^{-4} k^{\rm (n)}$. Note that the phase diagram is unchanged by setting $G_{\rm th} = 10^{-5} k^{\rm (n)}$. The unjammed (UJ) state is defined as $G'\left(\phi, \gamma_0^{\rm (I)}\right) < G_{\rm th}$ for any $\theta$, and the shear jammed (SJ) state is defined as $G'_0(\phi)<G_{\rm th}$ and $G'\left(\phi, \gamma_0^{\rm (I)}\right) > G_{\rm th}$ for any $\theta$. Finally, in the fragile (F) state, whether the state is solid-like with $G'\left(\phi, \gamma_0^{\rm (I)}\right) > G_{\rm th}$ or liquid-like with $G'\left(\phi, \gamma_0^{\rm (I)}\right) < G_{\rm th}$ depends on the value of $\theta$ (see Fig. \[G\_ga\_theta\], inset). In Fig. \[Phase\_ga\_MC\], the SJ state exists in the range $\phisj < \phi < \phic$ and $\gamma_0^{\rm(I)} > 0.1$. Remarkably, the UJ phase exists even when $\phi>\phic$, and the J state at large $\gamma_0^{(I)}$ and $\phi>\phic$ (located above the bay-like unjammed state) may be regarded as an SJ-like state; however, this state differs from the SJ state defined as the memory effect of the initial strain as introduced above. We have also confirmed the fragile state between the UJ and SJ states.
![ Phase diagram on the $\phi$ versus $\gamma_0^{\rm (I)}$ plane. Circles, triangles, squares, and crosses represent the J, SJ, F, and UJ states, respectively. The thick black line, thin blue line, and thin red line represent the critical strain amplitudes $\gadst$ at $\theta=0$, $\pi/4$, and $\pi/2$, respectively. []{data-label="Phase_ga_MC"}](Phase_ga_MC.eps){width="0.8\linewidth"}
Figure \[Phase\_ga\_MC\] also plots the critical strain amplitude $\gadst$ at which the DST-like behavior emerges, where the viscosity $\eta$ exceeds the threshold $10^{-3} \sqrt{m_0 k^{\rm (n)}}$. Note that at $\gadst$, $G'$ simultaneously changes from $0$ to a finite value. When $\theta$ is $0$, the critical strain amplitude $\gadst$ resides on the boundary between the UJ and fragile states, whereas at other $\theta$, it resides in the fragile state. This suggests that the fragile state exhibits DST-like behavior at least when $\gamma_0^{(I)}$ is not excessively large.
[*Discussion and concluding remarks.–*]{} Let us now discuss our results. Recent numerical simulations [@Kumar; @Jin; @Urbani; @Jin18; @Bertrand; @Baity; @Chen18; @VinuthaN; @VinuthaJ; @VinuthaA] indicated that shear jamming occurs even in frictionless systems. In our simulation, the SJ state disappears at $\mu=0$ (see Ref. [@Supple]). Nevertheless, the re-entrant process in the range $\phic<\phi<0.826$ of our system seems to be related with the SJ states in frictionless systems [@Kumar; @Jin; @Urbani; @Jin18; @Bertrand; @Baity; @Chen18].
The fragile state was originally defined by the anisotropic percolation of force chains under a quasi-static pure shear process [@Bi11]. Because no compressive direction or quasi-static operations are imposed in our system, we cannot apply the original argument based on percolation networks (Fig. \[F\_SJ\](b)). Regardless, the stress anisotropy $\tau/P$ [@Sarkar16; @Thomas; @Chen18] is maximized in the fragile state and remained constant in the SJ state, as shown in Fig. \[tau\_P\_ga\_theta\]. In this figure, $\tau=(\sigma_1-\sigma_2)/2$ and $P=-(\sigma_1+\sigma_2)/2$, where $\sigma_1$ and $\sigma_2$ denote the maximum and minimum principal stresses, respectively. This behavior is unchanged under fabric anisotropy [@Supple] and is qualitatively similar to the experimentally observed behavior [@Sarkar16]. It is possibly explained by a phenomenology based on the probability distribution of sliding forces [@DeGiuli17]. The mutual relation between the fragile state and the anisotropy requires further careful investigation.
![ Stress anisotropy $\tau/P$ versus $\gamma_0^{\rm (I)}$ for $\phi=0.820$ with $\theta = 0$ and $\pi/2$. The shaded region highlights the fragile state. []{data-label="tau_P_ga_theta"}](tau_P_ga_theta.eps){width="0.8\linewidth"}
In conclusion, we have numerically studied the frictional granular systems under oscillatory shearing. By controlling the strain amplitude of the oscillatory shear before the measurement, we have observed that shear jamming is a memory effect of the initial shear. We have also defined a fragile state in which the linear response can be solid-like or liquid-like depending on the initial phase of the oscillation. In this state, the solid-like and liquid-like states coexist under initial shearing with a large strain amplitude. This protocol has also detected DST-like behavior, manifesting a remarkable discontinuity in the viscosity versus the initial strain plot. The region of DST-like behavior is almost identical with that of the fragile state.
The authors thank R. Behringer, B. Chakraborty, T. Kawasaki, C. Maloney, C. S. O’Hern, K. Saitoh, S. Sastry, S. Takada, and H. A. Vinutha for fruitful discussions. We would like to dedicate this paper to the memory of R. Behringer who has passed away in July, 2018. This work is partially supported by the Grant-in-Aid of MEXT for Scientific Research (Grant No. 16H04025, No. 17H05420, and No. 19K03670). One of the authors (M.O.) appreciates the warm hospitality of Yukawa Institute for Theoretical Physics at Kyoto University during his stay there supported by the Program No. YITP-T-18-03 and YITP-W-18-17.
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**Supplemental Materials:**
Introduction
============
This Supplemental Material describes some details which are not written in the main text. In Sec. \[Model\], we explain the details of our simulation model (DEM) and the initial preparation. In Sec. \[Sec:s\], we explain how the shear jammed state appears in the stress-strain curve of the initial oscillation. In Sec. \[mu\], we discuss the dependence of the transition points of the jamming and the shear jamming on the friction coefficient $\mu$. In Sec. \[Rs\], we show the fabric anisotropy of the contact network in our simulation. The dependence of the phase diagram on the number of the oscillatory shear is discussed in Sec \[Nc\].
Details of our DEM and the preparation of the initial configuration {#Model}
===================================================================
In this section, we present the detail of our DEM. We also explain how to prepare the initial configuration of our system.
Equation of motion of grain $i$ (the mass $m_i$, the position $\bm{r}_i=(x_i,y_i)$, and the diameter $d_i$) is written as $$m_i \frac{d^2}{dt^2}{\bm{r}}_i=\bm{F}_{i}.
\label{S1}$$ The total force $\bm{F}_{i}$ acting on the grain is given by $$\begin{aligned}
\bm{F}_{i} & = & \sum_{j \neq i}
\left (F_{ij}^{\rm (n)}\Vect{n}_{ij} + F_{ij}^{\rm (t)}\Vect{t}_{ij} \right )
\nonumber \\
& = &
\sum_{j \neq i}
\left(
\begin{array}{cc}
\cos \alpha_{ij} & -\sin \alpha_{ij} \\
\sin \alpha_{ij} & \cos \alpha_{ij} \\
\end{array}
\right)
\left(
\begin{array}{c}
F_{ij}^{\rm (n)}\\
F_{ij}^{\rm (t)}\\
\end{array}
\right)\end{aligned}$$ with the normal contact force $F_{ij}^{\rm (n)}$, the tangential contact force $F_{ij}^{\rm (n)}$, the normal unit vector $\Vect{n}_{ij}$, and the tangential unit vector $\Vect{t}_{ij}$ between two grains $i$ and $j$. $\Vect{n}_{ij}$ and $\Vect{t}_{ij}$, respectively, satisfy $\Vect{n}_{ij}=(\cos \alpha_{ij},\sin \alpha_{ij})$ and $\Vect{t}_{ij}=(-\sin \alpha_{ij},\cos \alpha_{ij})$ with $\alpha_{ij} = \tan^{-1}((y_i-y_j)/(x_i-x_j))$. The normal contact force $F_{ij}^{\rm (n)}$ is given by $F_{ij}^{\rm (n)}
= -\left ( k^{\rm (n)} u_{ij}^{\rm (n)}
+ \zeta^{\rm (n)} v_{ij}^{\rm (n)} \right )
\Theta(d_{ij}-r_{ij})$ with the normal displacement $u_{ij}^{\rm (n)} = r_{ij}-d_{ij}$, $d_{ij}=(d_i+d_j)/2$, $r_{ij}=|\bm{r}_{ij}|=|\bm{r}_i-\bm{r}_j|$, the normal velocity $v_{ij}^{\rm (n)} = ({\Vect{v}}_i - {\Vect{v}}_i)
\cdot \Vect{n}_{ij}$, the velocity ${\Vect{v}}_i$ of grain $i$, the normal spring constant $k^{\rm (n)}$, and the normal damping constant $\zeta^{\rm (n)}$. $\Theta(x)$ is the Heviside step function satisfying $\Theta(x)=1$ for $x\ge 0$ and $\Theta(x)=0$ otherwise. The tangential force is given by $F_{ij}^{\rm (t)} =
{\rm min}\left( |\tilde F_{ij}^{\rm (t)}|,\mu F_{ij}^{\rm (n, el)}\right )
{\rm sgn}\left( \tilde F_{ij}^{\rm (t)} \right )
\Theta(d_{ij}-r_{ij})$, where ${\rm min}(a,b)$ selects the smaller one between $a$ and $b$, ${\rm sgn}(x)=1$ for $x\ge 0$ and ${\rm sgn}(x)=-1$ otherwise, and $\tilde F_{ij}^{\rm (t)}$ is given by $\tilde F_{ij}^{\rm (t)} = -k^{\rm (t)} u_{ij}^{\rm (t)} -
\zeta^{\rm (t)} v_{ij}^{\rm (t)}$ with the tangential spring constant $k^{\rm (t)}$ and the tangential damping constant $\zeta^{\rm (t)}$. The tangential velocity $v_{ij}^{\rm (t)}$ and the tangential displacement $u_{ij}^{\rm (t)}$, respectively, satisfy $v_{ij}^{\rm (t)}
= ({\Vect{v}}_i - {\Vect{v}}_i)
\cdot \Vect{t}_{ij} - (d_i \omega_i+d_j \omega_j)/2$ with the angular velocity $\omega_i$ of grain $i$ and $\dot{u}_{ij}^{\rm (t)}=v_{ij}^{\rm (t)}$ for $|\tilde F_{ij}^{\rm (t)}| < \mu F_{ij}^{\rm (n, el)}$. If $|\tilde F_{ij}^{\rm (t)}| \ge \mu F_{ij}^{\rm (n, el)}$, $u_{ij}^{\rm (t)}$ remains unchanged. We note that $u_{ij}^{\rm (t)}$ set to be zero if the grains $i$ and $j$ are detached. We adopt $k^{\rm (t)} = 0.2 k^{\rm (n)}$ and $\zeta^{\rm (t)} = \zeta^{\rm (n)}= \sqrt{m_0 k^{\rm (n)}}$ in this Letter. This set of parameters corresponds to the constant restitution coefficient $$e = \exp \left ( - \frac{\pi}{\sqrt{2 k^{\rm (n)}m_0/\zeta^{\rm (n)} - 1 }}\right ) \simeq 0.043$$ for the grain with the diameter $d_0$.
At the beginning of our simulation, the frictional disks are randomly placed with the initial area fraction $\phi_{\rm I}=0.75$, which is much lower than the jamming fraction $\phic = 0.821$ for $\mu=1.0$, and we slowly compress the system until the area fraction reaches a designated value $\phi$. In each step of the compression process, we increase the area fraction by $\Delta \phi = 10^{-4}$ with the affine transformation, and relax the grains to the mechanical equilibrium state where the kinetic temperature $T < T_{\rm th}=10^{-8}(k^{\rm (n)}d_0^2)$. We have confirmed that the shear modulus after the compression are insensitive to the values of $T_{\rm th}$ and $\Delta \phi$ for $T_{\rm th} \le 10^{-8}(k^{\rm (n)}d_0^2)$ and $\Delta \phi \le 10^{-4}$.
Note that some of the shear jammed states for frictionless systems disappear in the thermodynamic limit [@BertrandS; @BaityS; @Chen18S]. However, we have confirmed that the shear jammed state in our frictional system is stable and the shear modulus is almost independent of $N$ for $N\ge4000$.
Initial stress-strain curve and the shear jamming {#Sec:s}
=================================================
In this section, we explain how the shear jamming in the linear response regime is related to the initial stress-strain curve for large strain amplitudes. We also explain the reason why the liquid-like response can be observed if the initial strain amplitude is sufficiently small.
In Fig. \[s\_ga\], we plot the shear stress $\sigma$ versus the strain $\gamma$ for $\gamma_0^{\rm (I)}=0.2$, $\phi=0.820$, and $\theta=0$. Note that $\gamma_0^{\rm (I)}=0.2$ for this area fraction corresponds to the shear jammed state. The stress $\sigma$ follows a stress-strain loop once $\gamma$ exceeds $\gamma \simeq 0.02$. Even after the reduction of the strain amplitude, there is finite gradient of $\sigma$ against $\gamma$ around $\gamma=0$ which is equivalent to $G'>0$. Note that the red filled square in Fig. \[s\_ga\] is the measurement point. This emergence of $G'>0$ is regarded as the occurrence of the shear jamming.
![ The plot of the shear stress $\sigma$ versus the strain $\gamma$ for $\gamma_0^{\rm (I)}=0.2$, $\phi=0.820$, and $\theta = 0$. The triangle and the square indicate the states before and after the initial oscillatory shear, respectively. The arrows indicate the direction of time evolution in the stress-strain curve. []{data-label="s_ga"}](s_ga_MC.eps){width="0.8\linewidth"}
Figure \[s\_ga\] is useful to understand the reason why we observe the liquid-like response if $\gamma_0^{(I)}$ is small for $\phi=0.82$ and $\theta=0$. Indeed, $\sigma$ remains almost zero for $\gamma\le 0.01$ in this figure. Then, if we reduce $\gamma_0$ to $\gamma_0^{(F)}=1.0\times 10^{-4}$, we only obtain $G'=0$ for $\gamma_0^{(I)}\le 0.01$.
Determination of transition points and their dependence on $\mu$ {#mu}
================================================================
In this section, we first explain how to determine $\phic$ for the jamming and $\phisj$ for the shear jamming. We also discuss the $\mu$-dependence of these transition points.
For a given set of $\gamma_0^{\rm (I)}$ and $\theta$, the storage modulus $G'$ exhibits a transition from $G'=0$ to $G'>0$ at a transition point $\phith(\gamma_0^{\rm (I)},\theta)$. In Fig. \[phic\_ga\], we plot the transition point $\phith(\gamma_0^{\rm (I)},\theta)$ versus $\gamma_0^{\rm (I)}$ for $\theta=0$ and $\mu=1.0$. The transition point increases with $\gamma_0^{(I)}$ for $\gamma_0^{(I)}<0.04$, and decreases with $\gamma_0^{(I)}$ for $ \gamma_0^{(I)} > 0.04$. A similar dependence of the transition point for frictionless grains is reported in Ref. [@KumarS]. Then, we define the jamming point without shear as $$\phic \equiv \lim_{\gamma_0^{\rm (I)}\to0}\phith(\gamma_0^{\rm (I)},\theta),$$ which is independent of $\theta$ by definition. We also define the transition point for the shear jamming as $$\phisj \equiv \min_{\gamma_0^{\rm (I)}, \theta} \phith(\gamma_0^{\rm (I)},\theta).
\label{phisj}$$ Within our observation, $\phith(\gamma_0^{(I)},\theta)$ takes its smallest value at $\theta=0$ and seems to converge for sufficiently large $\gamma_0^{(I)}$. We, thus, evaluate $\phisj$ as $\phith(\gamma_0^{\rm (I)}=4.0,\theta=0)$, which is the transition point at the largest initial strain amplitude we apply in our simulation.
![ The plot of the transition point $\phith$ versus $\gamma_0^{\rm (I)}$ for $\theta=0$ and $\mu=1.0$. The solid thin line parallel to the horizontal axis represents $\phic$. []{data-label="phic_ga"}](phic_ga.eps){width="\linewidth"}
In the main text, we have presented the data only for $\mu=1.0$, but we show the $\mu$-dependence of the critical points $\phic$ and $\phisj$ in Fig. \[phic\]. Note that the shear jamming in terms of Eq. is observed only for $\phisj \le \phi \le \phic$. As shown in Fig. \[phic\], the difference between $\phic$ and $\phisj$ decreases as $\mu$ decreases. Then, the shear jamming based on our definition disappears in the frictionless limit.
![ Plots of the transition points $\phic$ and $\phisj$ versus $\mu$. []{data-label="phic"}](phic.eps){width="0.8\linewidth"}
The fabric anisotropy of the contact network {#Rs}
============================================
In this section, we present the result of the fabric anisotropy of the contact network in the fragile and the shear jammed states. Let us introduce the contact fabric tensor $R_{\alpha \beta}$ as [@BiS] $$\begin{aligned}
R_{\alpha \beta} = \frac{1}{N} \sum_{i} \sum_{j>i}
\frac{r_{ij,\alpha} r_{ij,\beta}}{r_{ij}^2}\Theta(d_{ij} - r_{ij}).
\label{R}\end{aligned}$$ Figure \[rho\] plots the fabric anisotropy $R_1 - R_2$ versus $\gamma_0^{\rm (I)}$ for $\phi=0.820$ with $\theta=0$ and $\pi/2$, where the maximum and the minimum eigenvalues of $R_{\alpha \beta}$ are denoted as $R_1$ and $R_2$, respectively. The fabric anisotropy takes the maximum in the fragile state and keeps constant in SJ, which corresponds to the stress anisotropy in Fig. 7 of the main text.
![ Plots of the fabric anisotropies $R_1 - R_2$ versus $\gamma_0^{\rm (I)}$ for $\phi=0.820$ with $\theta=0$ and $\pi/2$. The shaded region corresponds to the fragile state. []{data-label="rho"}](rho_ga.eps){width="0.7\linewidth"}
The dependence of the phase boundaries on $N_c^{\rm (I)}$ {#Nc}
=========================================================
![ Plots of $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ versus $\phi$ for various $N_c^{\rm (I)}$. []{data-label="gc"}](gc.eps){width="0.8\linewidth"}
In this section, we show the dependence of the phase diagram on the number $N_c^{\rm (I)}$ of the initial cycles in the oscillatory shear. Here, we introduce the minimum strain amplitude $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ for SJ, where $G'(\phi,\gamma_0^{\rm (I)})>G_{\rm th}$ for any $\theta$ if $\gamma_0^{\rm (I)} > \gamma_{0,{\rm min}}^{\rm (I)}(\phi)$. It should be noted that $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ gives the boundary between SJ and F in Fig. 6 of the main text. In Fig. \[gc\], we plot $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ versus $\phi$ for various $N_c^{\rm (I)}$, where $\gamma_{0,{\rm min}}^{\rm (I)}(\phi=0.82)$ slightly increase with $N_c^{\rm (I)}$, though $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ is insensitive to $N_c^{\rm (I)}$ for $\phi \le 0.81$. Therefore, we safely state that $\gamma_{0,{\rm min}}^{\rm (I)}(\phi)$ converges for $N_c^{\rm (I)} \ge 10$ and arbitrary $\phi$.
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|
---
abstract: 'We investigate a relationship network of humans located in a metric space where relationships are drawn according to a distance-dependent probability density. The obtained spatial graph allows us to calculate the average separation of people in a very simple manner. The acquired results agree with the well-known presence of the small-world phenomenon in human relationships. They indicate that this feature can be understood merely as a consequence of the probability composition. We also examine how this phenomenon evolves with the development of human society.'
address: |
Department of Theoretical Physics and Physics Education,\
Mlynsk' a dolina, 842 48 Bratislava, Slovak Republic
author:
- 'Mat'' uš Medo'
title: 'Distance-dependent connectivity: Yet Another Approach to the Small World Phenomenon'
---
Small-world phenomenon, random networks, spatial graphs, convolution.
Introduction
============
In the 1960’s, the american social psychologist Stanley Milgram examined how people know each other and introduced a quantity named the degree of separation $D$. It is the number of people needed to bind two chosen persons via a chain of acquaintances. E. g., if persons $A$ and $B$ do not know each other, but they have a common friend $C$, their degree of separation is $D(A,B)=1$. Milgram measured the mean degree of separation between people in USA and found a surprisingly small value, ${\langle D\rangle}=6$. This gave another name to this phenomenon – “six degrees of separation”.
Nowadays, if a network has small average distance between its vertices together with a large value of the average clustering coefficient, we say that small world phenomenon is present and such a network is called small world network (SWN). We often encounter SWP in random networks.
Clearly “To be an acquaintance” is a somewhat vague statement. There are various possible definitions – e. g. shaking the other’s hands, talking to each other for at least one hour, etc. Fortunately, results do not depend significantly on the specific choice, SWN was observed in all of those cases. However, the number six in the name of the phenomenon can not be taken literally. Actually it is just an expression for the number, which is very small compared to the size of the investigated population, which is taken to be $6\,400$ millions (the approximate number of people on the Earth) in this article.
Nowadays, SWP is a well known feature of various natural and artificial random graphs [@Watts]. Article citations, World Wide Web, neural networks and other examples exhibit this feature [@Dorogo-Mendes; @Albert-Bar].
There are many ways to construct a SWN. Some models are rather mathematical and do not examine the mechanism of the origin of a network. They impose some heuristic rules (e. g. [@Erd-Ren1; @Erd-Ren2; @Watts-Strog]) instead. Other models look for the reasons for the introduced rules. This is much more satisfactory from the physicist’s point of view. The first such model is known as “preferential linking” [@Bar-Albert]. It is quite reasonable for cases like the growth of the WWW, where sites with many links to them are well known and in the future will presumably attract more links than poorly linked pages.
In this work we focus on the random network of human relationships. It evolves in a very complicated manner, therefore it is very hard to impose some well accepted rules for its growth. Hence, we do not look for the time evolution of human acquaintances. Instead, we inquire a static case with the random network already developed.
If the acquaintance between $A$ and $B$ is present, we link them with an edge. We obtain the random graph of human relationships in this way. We can introduce a metric to this network by assigning a fixed position in the plane to every person. In order to obtain analytical results, we assume a constant population density. In particular, we suppose that people–vertices are distributed regularly and form a square lattice in the plane. With proper rescaling, the edges of unit squares in this lattice have length $1$. Further we assume that the probability that two people know each other, depends on their distance by means of some distribution function. This model should keep some basic features of the real random network of human relationships.
The Mathematical Model
======================
Let’s have an infinite square lattice where squares have sides equal to $1$ and there is one person in every vertex. We label the probability that two people with distance $d$ know each other $Q(d)$. We assume homogeneity of the population, therefore this probability function is the same for every pair.
Summation of $Q(d)$ through all vertices leads to the average number of acquaintances for any person which we denote $N_{\mathrm A}$. Next, we assume that the function $Q(d)$ is changing slowly on the scale of $1$. Therefore, we can change summation to integration and obtain $$\label{normalization}
N_{\mathrm A}=\int_{-\infty}^{\infty}\dd x
\int_{-\infty}^{\infty}\dd y\,Q\big(\sqrt{x^2+y^2}\big).$$ Our aim is to quantify the average degree of separation ${\langle D\rangle}$ for couples with the same geometrical distance equal to $b$. To achieve this, we choose two such people and label them $A$ and $B$ with positions $\vec{r}_A=[0,0]$, $\vec{r}_B=[b,0]$ (this particular choice will not affect our results substantially).
An Analytical Solution
======================
Every person in the lattice can be located by its coordinates $[x,y]$. We will denote the distance between $X$ and $Y$ as $d_{XY}$. Let us introduce a symbol $\sim$ for the relation of acquaintances. This is a binary relation which is symmetric but not transitive. The probability that $X$ knows $Y$ is then $P(X\sim Y)=Q(d_{XY})\equiv Q_{XY}$.
We name $P(D)$ the probability that the degree of separation for $A$ and $B$ with distance $b$ is equal to the number $D$.If we want to find out the average degree of separation in our present network, ${\langle D\rangle}$, we need to know the probabilities $P(D)$ for all different values of $D$. At the moment, only $P(0)$ is known, since apparently $P(0)=Q(d_{AB})=Q(b)$. $$\includegraphics[scale=1.2]{sw_figs.1}$$ Let’s examine the degree of separation $D=2$. This means that there are two other persons on the path between $A$ and $B$. We denote their coordinates as $\vec{r}_1=(x_1,y_1)$ and $\vec{r}_2=(x_2,y_2)$. For the presence of such a track, edges $A1$, $12$ and $2B$ are needed together with edges $A2$, $1B$ and $AB$ missing (see picture above). Since their presence is independent, we have $$\begin{aligned}
\label{first}
P(2)&=&\sum_{1,2}Q_{A1}Q_{12}Q_{2B}\big(1-Q_{A2}\big)
\big(1-Q_{1B}\big)\big(1-Q_{AB}\big)\approx\notag\\
&\approx&\iint\limits_{1,2}Q_{A1}Q_{12}Q_{2B}\big(1-Q_{A2}\big)
\big(1-Q_{1B}\big)\big(1-Q_{AB}\big)\,\dd\vec{r}_1\dd\vec{r}_2.\end{aligned}$$ where the summation runs through various placements of persons $1$ and $2$. The change of summation to integration is possible due to the fact that $Q(d)$ is changing slowly on the scale of $1$.
Here we utilized the fact that in probabilities addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ we can neglect the last term since probabilities $P(A)$, $P(B)$ are small and $P(A\cap B)$ is of the higher order of smallness. Unfortunately due to this approximation we apparently reach “probabilities” $P(D)$ higher than $1$ for high enough value of $D$. Though probabilities $P(D)$ small with respect to $1$ can be considered accurate. This implies that obtained results cannot be used to evaluate the exact value of the average degree of separation for nodes $A$ and $B$ because in such a calculation we would need value of $P(D)$ for every $D$. Still from the growth of $P(D)$ we can easily see for which $D^*$ it reaches relevant values, e. g. $P(D^*)=1/3$. This $D^*$ then characterizes the mean degree of separation of $A$ and $B$.
We can compute the first approximation to (\[first\]), getting $$\label{1approx}
P(2)^{(0)}=\iint\limits_{1,2}Q_{A1}Q_{12}Q_{2B}
\,\dd\vec{r}_1\,\dd\vec{r}_2.$$ As $Q_{A1}=Q(x_1-0,y_1-0)$, $Q_{12}=Q(x_2-x_1,y_2-y_1)$, and $Q_{2B}=Q(b-x_2,0-y_2)$ we notice that (\[first\]) is a double convolution of the function $Q(d)$ enumerated at point $(b,0)$. Thus we can write $$P(2)^{(0)}=\big[Q\ast Q\ast Q\big](b,0)\implies
P(D)^{(0)}=Q^{[D]}(b,0).$$ For the Fourier transformation of the convolution, the following equation holds: $$\mathscr{F}\big\{Q^{[D]}\big\}=\big(\mathscr{F}\{Q\}\big)^D.$$ Using this formula we can write $P(D)^{(0)}$ in the form $$\label{outcome}
P(D)^{(0)}=\mathscr{F}^{-1}
\Big\{\big(\mathscr{F}[Q]\big)^D\Big\}(b,0).$$
The mean clustering coefficient ${\langle C\rangle}$ is the probability that two acquaintances of $A$ know each other. It can be evaluated in a way very similar to the calculation of $P(D)$, the corresponding graph is on the picture below. $$\includegraphics[scale=1.2]{sw_figs.2}$$ In order to write down an expression for ${\langle C\rangle}$ it is straightforward to rewrite (\[first\]). We obtain the number of connected triples $A12$ with node $A$ fixed by this integration. We just have to avoid double counting of every track (interchanging positions of $1$ and $2$) – this brings an additional factor of $1/2$. The average number of acquaintances for every vertex is $N_{\mathrm A}$, therefore the average number of possible triples is $N_{\mathrm A}(N_{\mathrm A}-1)/2\approx N_{\mathrm A}^2/2$. The mean clustering coefficient is the ratio of the average number of triples to the average number of possible triples. That is, $$\begin{aligned}
\label{cc}
{\langle C\rangle}&=&\frac1{N_{\mathrm A}^2}
\iint\limits_{1,2}Q_{A1}Q_{12}Q_{2A}
\,\dd\vec{r}_1\,\dd\vec{r}_2=
\frac1{N_{\mathrm A}^2}\big[Q\ast Q\ast Q\big](0,0)=\notag\\
&=&\frac1{N_{\mathrm A}^2}\mathscr{F}^{-1}
\Big\{\big(\mathscr{F}[Q]\big)^3\Big\}(0,0).\end{aligned}$$
Equations (\[outcome\]) and (\[cc\]) are solutions of the problem. Unfortunately, the relevant functions $Q(d)$ (see next section) do not have an analytical form of their forward and inverse Fourier transformation. Therefore we have to calculate the values of ${\langle C\rangle}$ and $P(D)$ numerically. Equation (\[outcome\]) requires a very high calculation precision. This makes the evaluation of $P(D)$ very slow and even with some clever treatment (see Appendix A) it is in practise impossible for high values of $b$. This is just our case, because we are interested in $b=50\,000$. Thus some other (approximate) approach is needed. First we have to find more about the nature of function $Q(d)$.
An Empirical Entries
====================
In the present, there are approximately $6\,400$ millions people on the Earth. It means that length of the assumed square lattice side is $2L=80\,000$. In order to obtain a numeric results we choose $b=50\,000$ and the average number of acquaintances $N_{\mathrm A}=1\,000$.
To get some insight on the distribution $Q(d)$, some analysis is needed. First it is clear that $Q(d)$ should be decreasing with $d$. Moreover, closely living people know each other almost certainly. That is $$\label{limit}
\lim_{d\to0} Q(d)=1.$$ Together with (\[normalization\]) we now have two requirements for $Q(d)$. Indeed, there are many functions satisfying them, e. g. we can choose $Q(d)=C\exp[-r/a]$.
The last quantity we can compute is the average number of [*distant people*]{} every person in the lattice know, $N_{\mathrm d}$. Here [*distant*]{} means that people’s distance from the chosen fixed person (node) is greater than $L/2$. This is a simple analogy to the number of people we know on the other side of the Earth. So we have $$N_{\mathrm d}=N_{\mathrm A}-2\pi\int\limits_0^{L/2}rQ(r)\,\dd r.$$ If exponential distribution discussed above satistfies (\[normalization\]) and (\[limit\]) it folllows that $N_{\mathrm d}\approx10^{-13}$. This is in a clear contradiction to the fact that there are people who have very distant friends. Still we can improve $N_{\mathrm d}$ if we use stretched exponential $Q(d)=\exp[-K\,d^a]$ with exponent $a$ between $0.2$ and $0.3$.[^1] However, if we check $Q(1)$ (probability to know our closest person) it is well below $0.3$. Stretched exponentials therefore satisfy condition (\[normalization\]) just formally and we will not it discuss it later. Moreover, mean clustering coefficient is then very small (from $2\cdot 10^{-4}$ to $3\cdot10^{-3}$).
Now it’s clear that distribution $Q(d)$ can’t decrease so fast as exponential functions, wide tails are inevitable in our model. This leads us to power-law distributions $1/x^a$. According to (\[limit\]) we demand $$\label{powerlaw}
Q_a(d)=\frac1{1+bd^a},$$ where $b$ is fixed by (\[normalization\]). Number of far friends now ranges from $N_{\mathrm d}\approx0.01$ ($a=3.5$) to $N_{\mathrm d}\approx 17$ ($a=2.5$). This range of exponents gives us reasonable range for values of $N_{\mathrm d}$.
In this article we also show results for the normal distribution $Q_{\mathrm n}(d)=\exp[-ad^2]$ ($N_{\mathrm d}\approx0$) and the uniform distribution within fixed radius $Q_{\mathrm u}(d)=\vartheta(R_{\mathrm A}-d)$ ($N_{\mathrm d}=0$).
With regard to the fact that all used distributions $Q(d)$ approach to zero for large values of $d$ it is almost certain that the shortest chain of acquaintances between chosen $A$ and $B$ do not run out of the examined lattice with side $80\,000$. Therefore it doesn’t matter if we have integration (summation) bound in infinity or $\pm L=\pm 40\,000$. This allows us to use all results derived for infinite lattice in the real case of finite lattice.
An Approximate Solution for Power-law Distributions
===================================================
To demonstrate the calculation we take $P(2)$ as an example again. In the previous section we found out that power-law distributions are especially important in our model. Their joint probability $Q(r_1)Q(b-r_1)$ has sharp maximum for $r_1=0$ and low minimum for $r_1=b/2$. Their ratio is $$\frac{Q(b/2)^2}{Q(b)Q(0)}\approx\Big(\frac{4}{b}\Big)^a$$ where $a$ is the exponent in (\[powerlaw\]). This implies that in (\[first\]) we can constrain summation to $r_{A1},r_{A2}\ll b$ or $r_{B2},r_{B1}\ll b$ or $r_{A1},r_{B2}\ll b$ (see picture below). $$\includegraphics[scale=1.2]{sw_figs.3}$$ Here we obtained three different diagrams. Let’s examine first one in detail.
Since edges $AB$ and $B1$ are long we can write $$P(2)\approx\iint\limits_{1{,}2}
Q_{A1}Q_{12}Q_{2B}\big(1-Q_{A2}\big)
\,\dd\vec{r}_1\,\dd\vec{r}_2.$$ It is easy to show that for power-law distributions $Q(b-r_1)\approx Q(b)\equiv Q_{AB}$ when $r_1\ll b$. Thus we have (for corresponding diagram see picture below) $$\begin{aligned}
P(2)&\approx\iint\limits_{1{,}2}
Q_{A1}Q_{12}Q_{AB}\big(1-Q_{A2}\big)
\,\dd\vec{r}_1\,\dd\vec{r}_2=\\
&=Q(b)\iint\limits_{1{,}2}
Q_{A1}Q_{12}\,\dd\vec{r}_1\,\dd\vec{r}_2-
Q(b)\iint\limits_{1{,}2}
Q_{A1}Q_{12}Q_{A2}\,\dd\vec{r}_1\,\dd\vec{r}_2.\end{aligned}$$ $$\includegraphics[scale=1.2]{sw_figs.4}$$ Both integrals are easy to compute. Second one brings average clustering coefficient ${\langle C\rangle}$ into account. The result is $$P(2)\approx Q(b)N_{\mathrm A}^2\big(1-{\langle C\rangle}\big).$$ Remaining two diagrams for $P(2)$ can be evaluated in the same way.
In the computation of $P(D)$ for higher values of $D$ we encounter products of kind $(1-Q_{13})(1-Q_{24})\ldots$ even after neglecting probabilities $Q_{ij}$ for long edges $ij$. Here we can make first order approximation $$(1-Q_{13})(1-Q_{24})\approx 1-Q_{13}-Q_{24}$$ which is valid almost everywhere except small spatial region that do not contributes substantially (see section Results and discussion). Moreover, second approximation considering terms $Q_{13}Q_{24}$ would increase evaluated probabilities. Therefore first approximation results will be some lower bound estimates of $P(D)$.
Higher values of $D$ introduce long closed loops of kind $A12\ldots nA$ ($n\leq D$). Corresponding integrals can be carried out in the same way like it was presented in the derivation of (\[cc\]). Finally we obtain $$\label{cn}
C_n\equiv\frac{1}{N_{\mathrm A}^n}\iint\limits_{1{,}2}
Q_{A1}Q_{12}\cdots Q_{nA}\,\dd\vec{r}^n=
\frac{1}{N_{\mathrm A}^n}\mathscr{F}^{-1}
\Big\{\big(\mathscr{F}[Q]\big)^n\Big\}(0,0).$$ This helps us to find values of $C_n$ for any $n$. Clearly $C_2={\langle C\rangle}$. With the use of such a closed loop integrals we can write $$\label{final}
\begin{aligned}
P(0)&=Q(b),\\
P(1)&=Q(b)N_{\mathrm A}2,\\
P(2)&=Q(b)N_{\mathrm A}^2\big(3-2C_2\big),\\
P(3)&=Q(b)N_{\mathrm A}^3\big(4-6C_2-2C_3\big),\\
P(4)&=Q(b)N_{\mathrm A}^4\big(5-12C_2-6C_3-2C_4\big),\\
P(5)&=Q(b)N_{\mathrm A}^5
\big(6-20C_2-12C_3-6C_4-2C_5\big),\,\ldots
\end{aligned}$$
Results and Discussion
======================
In this section we summarize results for various distributions $Q(d)$ ranging from the uniform $Q_{\mathrm u}$ and normal $Q_{\mathrm n}$ to power-law distributions $Q_{3.5}$–$Q_{2.5}$ (see (\[powerlaw\])) and flat distribution $Q_{\mathrm ER}$. This list is sorted according to the quantity of long shortcuts in such networks of relationships.
Flat Distribution {#flat-distribution .unnumbered}
-----------------
If we have flat distribution $Q_{\mathrm ER}$, every pair of vertices is connected with the same probability $p$. It is shown in [@Erd-Ren1] that in the network consisting of $N$ vertices holds $${\langle l\rangle}\approx\frac{\ln N}{\ln pN}.$$ Here $pN$ is the average number of acquaintances for a person in the network, $pN=N_{\mathrm A}$. We have $N_{\mathrm A}=1\,000$ and $N=6\,400$ millions thus $D^*_{\mathrm ER}={\langle l\rangle}-1\approx 2.3$ and ${\langle C\rangle}\approx0$.
Uniform Distribution Within Fixed Radius {#uniform-distribution-within-fixed-radius .unnumbered}
----------------------------------------
We can discuss such a case where every person knows just $N_{\mathrm A}$ closest neighbors. This leads us to the distribution $Q_{\mathrm u}(d)=\vartheta(R_{\mathrm A}-d)$ where distance $R_{\mathrm A}$ is fixed by (\[normalization\]). It gives us $R_{\mathrm A}=\sqrt{N_{\mathrm A}/\pi}$ and therefore $$D^*_{\mathrm u}\approx b\,\sqrt{\frac{\pi}{N_{\mathrm A}}}.$$ It’s worth to note that we don’t have any randomness in this model thus $D^*_{\mathrm u}={\langle D_{\mathrm u}\rangle}$.[^2]
Normal Distribution {#normal-distribution .unnumbered}
-------------------
The only distribution which allows us to evaluate (\[outcome\]) analytically is normal distribution $Q_{\mathrm n}$. The result is $$P(D)=\frac{N_{\mathrm A}^D}{D+1}\exp
\bigg[-\frac{\pi b^2}{N_{\mathrm A}(D+1)}\bigg].$$ It was argued before that a solution of the equation $P(D^*_{\mathrm n})=1/3$ characterizes value of the mean degree of separation ${\langle D\rangle}_{\mathrm n}$. For $N_{\mathrm A}=1\,000$ and $b=50\,000$ we can use some approximations which lead us to $$D^*_{\mathrm n}\approx n_{\mathrm n}\approx
b\,\sqrt{\frac{\pi}{N_{\mathrm A}\ln N_{\mathrm A}}}=
\frac{D^*_{\mathrm u}}{\sqrt{\ln N_{\mathrm A}}}.$$ The actual value of $D^*_{\mathrm n}$ is about one third of $D^*_{\mathrm u}$ (this is due to the existence of some longer connections in the network, although it is extremely suppressed by the exponential decay). We can note that both $D^*_{\mathrm u}$ and $D^*_{\mathrm n}$ scale with $b^1$. This clearly differs from $\ln b$ scaling of the Erd" os-R' enyi model. The clustering coefficient ${\langle C\rangle}$ can be evaluated easily both for normal and uniform distribution. We obtain high values of ${\langle C\rangle}$ (see graph below) in both cases. This agrees with our expectations.
Power-law Distributions {#power-law-distributions .unnumbered}
-----------------------
Numerical computation of coefficients $C_2,\ldots,C_5$ with (\[cn\]) is rather fast – their values are shown in the table below.
$a=2.5$ $a=3.0$ $a=3.5$
------- --------- --------- ---------
$C_2$ 0.068 0.154 0.233
$C_3$ 0.030 0.090 0.153
$C_4$ 0.016 0.059 0.109
$C_5$ 0.009 0.042 0.084
Substituting these values into (\[final\]) leads us to values of mean degree of separation which are marked in the Fig. \[fig:graf\]. We see that power-law distributions $Q(d)$ results in high values of mean clustering coefficients ${\langle C\rangle}=C_2$ together with small values of $D^*$ (from $6$ to $4$). Thus small world phenomenon is clearly present in these networks.
![Graphs of mean degree of separation and clustering coefficients for various distribution functions.[]{data-label="fig:graf"}](sw_figs.5 "fig:") ![Graphs of mean degree of separation and clustering coefficients for various distribution functions.[]{data-label="fig:graf"}](sw_figs.6 "fig:")
One can also ask for some comparison with the well-known Barabasi-Albert model. Mean vertex degree is then ${\langle k\rangle}=2m$ and mean clustering coefficient is ${\langle C\rangle}=(m-1)\ln^2N/8N$ (here $m$ is degree of just added vertices, see [@FFH]). With respect to our choice $N_{\mathrm A}=1\,000$, $N=6.4\cdot 10^9$ it follows that $m=500$ and ${\langle C\rangle}\approx 10^{-11}$. This is certinaly nonrealistic value, our model gives better estimation of ${\langle C\rangle}$.
For presented values of coefficients $C_i$ expressions in parentheses in (\[final\]) do not fall close to zero for quite wide range of values of $D$. Therefore we can (very approximately) write $$P(D)\approx Q(b)DN_{\mathrm A}^D.$$ Solution of the equation $P(D^*)=1/3$ is approximately $D^*\approx a\ln b/\ln N_{\mathrm A}$. With $b$ this scales as $\ln b$. This is very different from $b^1$ scaling of ${\langle D\rangle}$ for the uniform a normal distribution. Such a scaling is similar to the scaling in the Erd" os-R' enyi model, though values of clustering coefficient are kept high as we demanded in the introduction.
Probability $P(2)$ can be evaluated also by straightforward summation in accordance with (\[first\]) although it takes huge amount of computer time. Obtained values agree very well with results presented above for all examined exponents but $2.5$ – this case requires more computer time than it was given. Computation of $P(3)$ in the same way exceeds our computer possibilities for every exponent but we do not regard it necessary.
Time Evolution and Some Limitations {#time-evolution-and-some-limitations .unnumbered}
-----------------------------------
Human relationships in modern world are much more widespread than it was in the past. One can think of slowly changing exponent of the power-law distribution function $Q(d)$ from large values to smaller (perhaps resulting to almost flat distribution in the future – internet helps to bridge the distances). According to the Fig. \[fig:graf\] we see that this would affect exact value of clustering coefficient. However it would remain high enough for wide range of exponents. Similarly changes of mean degree of separation are not important at all – it remains very small compared to the size of human population.
Finally it has to be noted that in the described model we do not consider presence of some organized hierarchic structures in human society. E. g. chief of the firm knows his employees, but he also knows another chiefs who know their employees, etc. Amount of people involved in the hierarchical tree grows exponentially with the number of its levels. Such an arrangement therefore introduces additional way how to know each other with small resulting degree of separation. In presented calculation we didn’t include this effect. Yet there is one important insight. If we proved the degree of separation being small without considering of the hierarchies, their presence would even decrease it.
Conclusion
==========
We have examined the mean degree of separation and the clustering coefficient for a random network of human relationships in this article. We were able to compute these quantities in our model. For a power-law decay of probability $Q(d)$, we obtained a small mean degree of separation compared to the size of the network, along with a large value of the mean clustering coefficient. Both of these features are typical for small world networks. Thus we have shown that the small world phenomenon can be understood as a simple consequence of additivity of probabilities.
We saw that the style of calculation depends on the used distribution $Q(d)$. The computation was finished analytically for some special cases. In other cases, thanks to some approximations, we utilized the advantage of (\[cc\]) where $b$ do not enter the inverse Fourier transformation, making it easy to evaluate numerically.
It’s worth to note that the model solved herein is similar to the Watts and Strogatz model [@Watts-Strog] where long shortcuts were introduced by a random rewiring procedure. In our model long shortcuts are present thanks to wide tails of power-law distributions. This model model brings two basic advantages. First, the derivation and the resulting relations for $C$ and $D^*$ are more simple. Moreover, our model has more realistic foundations. Nevertheless, the typical behavior of this model is the same as in previous models. The introduction of long shortcuts to the system decreases the average degree of separation rapidly, but also keeps the clustering coefficient high enough for the so called small world phenomenon to appear.
Numerics of the Fourier Transformation
======================================
The Fourier integrals encountered in the solution of presented problem can not be solved analytically thus numerical techniques have to be used. In the inverse Fourier transform this is especially awkward because we meet rapidly oscillating term $\exp[\mathrm{i}\,bu]$. Here $b$ is the distance between chosen persons $A$ and $B$, by assumption big number ($b=50\,000$). Therefore we have to compute Fourier transformation of $f(d)$ very accurately. In order to make computation less demanding on the computer time, it is convenient to find some approximation in the computing of the inverse Fourier transformation. We will continue with this derivation in the onedimensional case for the sake of simplicity.
The Fourier transformation of the even function $f(x)$ is an even real function. According to the (\[outcome\]) we are looking for the inverse Fourier transformation of its $n$-th power, we will denote it $\hat{g}(u)$. It is also even real function. Therefore its inverse Fourier transformation is real function (sine-proportional terms vanish). Thus $$g(b)=\frac1{2\pi}\int\limits_{-\infty}^{\infty}\hat{g}(u)
\mathrm{e}^{\mathrm{i}bu}\,\dd u=\frac1{2\pi}
\int\limits_{-\infty}^{\infty}\hat{g}(u)\cos[bu]\,\dd u.$$ This integral can be expressed as the sum of contributions from all periods of the $\cos[bu]$ function, $I_n={\langle 2\pi n/b,2\pi(n+1)/b\rangle}$ (here $n\in\mathbb{N}$) $$g(b)=\sum_{n=-\infty}^{\infty} S_n(b),\quad
S_n(b)=\frac1{2\pi}\int\limits_{I_n}\hat{g}(u)\cos[bu]\,\dd u.$$ In the integrand of previous equation we can make Taylor expansion of $\hat{g}(u)$ around $\xi_n=2\pi(n+1/2)/b$. Thereafter terms of kind $u^m\cos[bu]$ emerge ($m\in\mathbb{N}$). Such integrals are easy to compute – first two terms of resulting expansion are then $$S_n(b)=\frac1{b^3}
\frac{\dd^2\hat{g}}{\dd u^2}\bigg\rvert_{\xi_n}+
\frac{\pi^2-6}{6b^5}
\frac{\dd^4\hat{g}}{\dd u^4}\bigg\rvert_{\xi_n}.$$ Finally we have $$\label{priblizenie}
g(b)=\frac1{b^3}\sum_{n=-\infty}^{\infty}
\frac{\dd^2\hat{g}}{\dd u^2}\bigg\rvert_{\xi_n}+
\frac{\pi^2-6}{6b^5}\sum_{n=-\infty}^{\infty}
\frac{\dd^4\hat{g}}{\dd u^4}\bigg\rvert_{\xi_n}.$$ This helps us to speed up inverse Fourier transformation – we do not have to know so many values of $\hat{g}(u)$. For every range $I_n$ evaluation of $\hat{g}(u)$ in three points (for numerical calculation of second derivative in the leading term of (\[priblizenie\])) is enough. We just have to keep in mind that these points have to be close enough (with respect to $2\pi/b$), otherwise we can obtain evidently incorrect results (e. g. $g(b)=0$ when border points have distance $2\pi/b$).
The author would like to thank to staff of his department, especially to Martin Mojžiš and Vladim' ir Čern' y for valuable conversations and to Mari' an Klein for computer time. Acknowledgement belongs also to J' an Boďa for introduction to the field, Miška Sonlajtnerov' a for her enthusiastic encouragement and my parents for their support.
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Erd" os P. and R' enyi A., On random graphs, [*Publications Mathematicae*]{} [**6**]{} (1959), 290.
Erd" os P. and R' enyi A., On the evolution of random graphs, [*Publ. Math. Inst. Gung. Acad.*]{} [**5**]{} (1960), 17.
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[^1]: Approximate solution presented in next section can be used also for this distribution.
[^2]: Randomness can be introduced by random placement of vertices. Hence we obtain so called random geometric graphs discussed in [@Penrose]. This approach is complementary to presented one where vertex placement is fixed but their connecting is due to some probability distribution.
|
---
abstract: 'Grunewald and O’Halloran conjectured in 1993 that every complex nilpotent Lie algebra is the degeneration of another, non isomorphic, Lie algebra. We prove the conjecture for the class of nilpotent Lie algebras admitting a semisimple derivation, remaining open for the class of characteristically nilpotent Lie algebras. In dimension 7, where the first characteristically nilpotent Lie algebras appear, we prove the conjecture and we also exhibit explicit nontrivial degenerations to every 7-dimensional nilpotent Lie algebra.'
address: 'CIEM-FaMAF, Universidad Nacional de Córdoba, Argentina'
author:
- 'Joan Felipe Herrera-Granada and Paulo Tirao'
date: 'October 31, 2013'
title: |
The Grunewald-O’Halloran conjecture\
for nilpotent Lie algebras of rank $\ge 1$
---
Introduction
============
The study of the algebraic varieties of Lie algebras, solvable, and nilpotent Lie algebras of dimension $n$ turned out to be a very hard subject. The theory of deformations of algebras started with a series of papers by Gerstenhaber, the first being [@G]. Since then a lot of efforts has been done (see for instance [@NR1; @R; @NR2; @V; @C1; @K]), however many natural questions remain unsolved. For example, the determination of the irreducible components of the variety of nilpotent Lie algebras seems today out of reach.
Among the open questions there are two conjectures about nilpotent Lie algebras. One, due to Grunewald and O’Halloran [@GO2], states that every complex nilpotent Lie algebra is the degeneration of another, non isomorphic, Lie algebra. The other one, known as Vergne’s conjecture, states that there are no rigid complex nilpotent Lie algebras in the algebraic variety ${\mathcal{L}}_n$ of complex Lie algebras of dimension $n$. Meaning that there are no nilpotent Lie algebras with open orbit in ${\mathcal{L}}_n$, that is such that their isomorphisms classes are open in ${\mathcal{L}}_n$. The first conjecture is a priori stronger than the second one. In this short paper we address the Grunewald-O’Halloran conjecture.
It is well known that, over fields of characteristic zero, geometric rigidity is equivalent to formal rigidity, the latest meaning that all formal deformations are trivial [@GS]. However, this does not imply that the Grunewald-O’Halloran conjecture and Vergne’s conjecture are equivalent. If so, it would also imply that every non geometrically rigid Lie algebra is the degeneration of another non isomorphic Lie algebra, which is not true already in dimension $n=3$. In fact the only complex rigid Lie algebra of dimension 3 is the simple Lie algebra ${\mathfrak{sl}}_2({\mathbb{C}})$ and, for instance, the solvable (non nilpotent) Lie algebra $\mathfrak{r}+{\mathbb{C}}$, where $\mathfrak{r}$ is the 2-dimensional solvable Lie algebra, is on top of the Hasse diagram of degenerations, and in particular it is not the degeneration of any other Lie algebra (see [@CD] and [@BSt]).
Complex Lie algebras and nilpotent Lie algebras of small dimension are classified and in this cases all the degenerations among them and also which are rigid is known. All degenerations that occur among complex Lie algebras of dimension $\le 4$ are given in [@St] and [@BSt]. In [@GO1] and [@Se] all degenerations for complex nilpotent Lie algebras of dimension 5 and 6 are given and more recently, in [@B], some degenerations for some 5-step and 6-step complex nilpotent Lie algebras of dimension 7 are given. Results on the different varieties and on rigidity in low dimensions may be found in [@CD; @C2]. In [@AG] and [@AGGV] the components of the varieties of nilpotent Lie algebras of dimension 7 and 8 are given.
Carles [@C1] investigated the structure of rigid Lie algebras over algebraically closed fields of characteristic zero. In particular he proved that nilpotent Lie algebras of rank $\ge 1$ are never rigid and moreover nilpotent Lie algebras with a codimension 1 ideal of rank $\ge 1$ are also never rigid. That is, Vergne’s conjecture holds for this class, remaining open for characteristically nilpotent Lie algebras for which all its ideals of codimension 1 are also characteristically nilpotent.
In the paper [@GO2], the authors constructed nontrivial linear deformations for large classes of nilpotent Lie algebras and left open the question of which of those deformations correspond to degenerations. Their construction of linear deformations of a given Lie algebra ${\mathfrak{g}}$, relies on the existence of a codimension 1 ideal ${\mathfrak{h}}$ of ${\mathfrak{g}}$ with a semisimple derivation $D\in {\operatorname{Der}}({\mathfrak{h}})$, and applies not only to nilpotent Lie algebras. In general, the deformations constructed do not correspond to a degeneration. A fixed ideal ${\mathfrak{h}}$ may produce many non equivalent deformations, some of which may correspond to a degeneration and some may not.
We prove two things. On the one hand we prove that the Grunewald-O’Halloran conjecture holds for nilpotent Lie algebras of rank $\ge 1$, leaving it open for characteristically nilpotent Lie algebras. On the other hand, we prove that the conjecture holds for 7-dimensional nilpotent Lie algebras and moreover and interesting for us we exhibit explicit degenerations to each 7-dimensional nilpotent Lie algebra.
More precisely, we show that if the semisimple derivation $D$ of ${\mathfrak{h}}$ is the restriction to ${\mathfrak{h}}$ of a semisimple derivation of ${\mathfrak{g}}$, then the associated deformation does correspond to a degeneration. Then we are able to prove the following.
\[thm:1\] If ${\mathfrak{n}}$ is a complex nilpotent Lie algebra with a nontrivial semisimple derivation, then ${\mathfrak{n}}$ is the degeneration of another, non isomorphic, Lie algebra.
The first characteristically nilpotent Lie algebras appear in dimension 7. Hence, by Theorem \[thm:1\], the Grunewald-O’Halloran conjecture holds in dimension $<7$. Complex nilpotent Lie algebras of dimension 7 are classified: there are infinitely many isomorphism classes and infinitely many of them are characteristically nilpotent. We shall refer to the classification by Magnin [@M]. We work out this family on a case by case basis, by considering linear deformations constructed after choosing suitable codimension 1 ideals and particular derivations of them, proving the following result.
\[thm:2\] Every complex nilpotent Lie algebra of dimension $\le 7$, is the degeneration of another, non isomorphic, Lie algebra.
We note that the variety of complex nilpotent Lie algebras of dimension 7 has two components, each of which is the closure of the orbit of a family of Lie algebras [@AG]. Being degeneration transitive, to proof Theorem \[thm:2\] it is enough to find nontrivial degenerations to these two families. In the case of dimensions $<7$ this argument reduces the proof to finding a nontrivial degeneration to a single algebra. This is easy to do and for completeness we do it in dimension 6.
In this paper all Lie algebras will be over the complex numbers.
Linear deformations and degenerations
=====================================
Let ${\mathcal{L}}_n$ be the algebraic variety of complex Lie algebras of dimension $n$, that is the algebraic variety of Lie brackets $\mu$ on ${\mathbb{C}}^n$ (${\mathcal{L}}_n \subseteq {\mathbb{C}}^{n^3}$). Given a complex Lie algebra ${\mathfrak{g}}=({\mathbb{C}}^n,\mu)$, we shall refer to it indistinctly by ${\mathfrak{g}}$, $({\mathfrak{g}},\mu)$ or $\mu$. The group $GL_n=GL_n({\mathbb{C}})$ acts on ${\mathcal{L}}_n$ by ‘change of basis’: $$g\cdot \mu (x,y)=g(\mu(g^{-1}x,g^{-1}y)), \qquad g\in GL_n.$$ Thus the orbit ${\mathcal{O}}(\mu)$ of $\mu$ in ${\mathcal{L}}_n$, is the isomorphism class of $\mu$.
A Lie algebra $\mu$ is said to degenerate to a Lie algebra $\lambda$, denoted by $\mu \rightarrow_{{\operatorname{deg}}} \lambda$, if $\lambda\in\overline{{\mathcal{O}}(\mu)}$, the Zariski closure of ${\mathcal{O}}(\mu)$. If $\lambda\not\simeq\mu$, then $\lambda$ is in the boundary of the orbit ${\mathcal{O}}(\mu)$ but outside it. Since the Zariski closure of ${\mathcal{O}}(\mu)$ coincides with its closure in the relative topology of ${\mathbb{C}}^{n^3}$, if $g:{\mathbb{C}}^\times\rightarrow GL_n$, $t\mapsto g_t$, is continuous and $\lim_{t\mapsto 0}g_t\cdot \mu=\lambda$, then $\mu\rightarrow_{{\operatorname{deg}}}\lambda$. The degeneration $\mu\rightarrow_{{\operatorname{deg}}}\lambda$ is said to be realized by a 1-PSG, if $g_t$ is a 1-parameter subgroup as a morphism of algebraic groups. Recall that if $g_t$ is a 1-PSG, then $g_t$ is diagonalizable with eigenvalues $t^{m_i}$ for some integers $m_i$.
A linear deformation of a Lie algebra $\mu$ is, for the aim of this paper, a family $\mu_t$, $t\in {\mathbb{C}}^\times$, of Lie algebras such that $$\mu_t=\mu + t\phi,$$ where $\phi$ is a skew-symmetric bilinear form on ${\mathbb{C}}^n$. It turns out that $\mu_t$ is a linear deformation of $\mu$ if and only if $\phi$ is a Lie algebra bracket which in addition is a 2-cocycle of $\mu$.
If a given a linear deformation $\mu_t$ of $\mu$ is such that $\mu_t\in{\mathcal{O}}(\mu_1)$ for all $t\in{\mathbb{C}}^\times$, then $\mu_1\rightarrow_{{\operatorname{deg}}}\mu$. In fact, for each $t\in{\mathbb{C}}^\times$ there exist $g_t\in GL_n$ such that $g_t^{-1}\cdot \mu_1=\mu_t$, then $\lim_{t\mapsto 0}g_t^{-1}\cdot \mu_1=\lim_{t\mapsto 0}\mu_t=\mu$. Hence, in order to show that $\mu_1\rightarrow_{{\operatorname{deg}}}\mu$, one only needs to prove that for each $t\in{\mathbb{C}}^\times$ there exist $g_t\in GL_n$ such that $$\label{eqn:degeneration}
\mu_1(g_t(x),g_t(y)))=g_t(\mu_t(x,y)), \quad\text{for all $x,y\in{\mathbb{C}}^n$}.$$
Construction of linear deformations
-----------------------------------
We recall now the construction of linear deformations in [@GO2].
Let $({\mathfrak{g}},\mu)$ be a given Lie algebra of dimension $n$ and let ${\mathfrak{h}}$ be a codimension 1 ideal of ${\mathfrak{g}}$ with a semisimple derivation $D$. For any element $X$ of ${\mathfrak{g}}$ outside ${\mathfrak{h}}$, ${\mathfrak{g}}=\langle X \rangle \oplus {\mathfrak{h}}$. The bilinear form $\mu_D$ on ${\mathfrak{g}}$ defined by $\mu_D(X,z)= D(z)$ and $\mu_D(y,z)=0$, for $y,z\in{\mathfrak{h}}$, is a 2-cocycle for $\mu$ and a Lie bracket. Hence, $$\label{eqn:linear-def}
\mu_t=\mu + t\mu_D,$$ is a linear deformation of $\mu$. If ${\mathfrak{g}}$ is nilpotent, then $\mu_t$ is always solvable but not nilpotent. In particular, $\mu_t$ is not isomorphic to $\mu$ for all $t\in{\mathbb{C}}^\times$. The construction described above can be carried out also for any derivation $D$, not necessarily semisimple. However, one can not assure that $\mu_t$ is not isomorphic to $\mu$ in this case.
Degenerations from deformations
-------------------------------
Under certain hypothesis on the derivation $D$, the deformation constructed above does correspond to a degeneration.
\[prop:restriction\] Let ${\mathfrak{n}}$ be a nilpotent Lie algebra with an ideal ${\mathfrak{h}}$ of codimension 1 admitting a nontrivial semisimple derivation $D$. If $D$ is the restriction of a semisimple derivation $\tilde D$ of ${\mathfrak{n}}$ such that it is nontrivial on a direct invariant complement of ${\mathfrak{h}}$, then ${\mathfrak{n}}$ is the degeneration of another, non isomorphic, Lie algebra. Moreover, the degeneration can be realized by a 1-PSG.
Let ${\mathfrak{n}}=({\mathfrak{n}},\mu)$. Let $X$ be an eigenvector of $\tilde D$ complementary to ${\mathfrak{h}}$ and let $\lambda_0\ne 0$ be its eigenvalue. We may assume that $\lambda_0=1$ (by considering $\tilde D/\lambda_0$ and $D/\lambda_0$ instead of $\tilde D$ and $D$).
Let $\lambda_1,\dots,\lambda_k$ be the different eigenvalues of $D$ and let ${\mathfrak{h}}={\mathfrak{h}}_{\lambda_1}\oplus\dots\oplus {\mathfrak{h}}_{\lambda_k}$ be the corresponding graded decomposition of ${\mathfrak{h}}$, that is $\mu({\mathfrak{h}}_{\lambda_i},{\mathfrak{h}}_{\lambda_j})\subseteq {\mathfrak{h}}_{\lambda_i+\lambda_j}$.
Hence, $${\mathfrak{n}}=(\langle X\rangle \oplus {\mathfrak{h}},\mu)$$ where both summands of ${\mathfrak{n}}$ are $\tilde D$-invariant and $\mu(X,{\mathfrak{h}}_{\lambda_j})\subseteq {\mathfrak{h}}_{1+\lambda_j}$.
Let $\mu_t=\mu+t\mu_D$ be the linear deformation constructed as in , which is given by $$\begin{aligned}
\mu_{t}(X,y_{j}) &= \mu(X,y_j)+t \lambda_j y_j,\quad \text{if $y_j\in {\mathfrak{h}}_{\lambda_j}$, for $1\le j \le k$}. \\
\mu_{t}(y_i,y_j) &= \mu(y_i,y_j),\quad \text{if $y_i\in{\mathfrak{h}}_{\lambda_i}$ and $y_j\in{\mathfrak{h}}_{\lambda_j}$, for $1\le i,j\le k$}.\end{aligned}$$
Let $g_t\in GL_n$, where $n=\dim {\mathfrak{n}}$, be defined by $$g_t|_{\langle X \rangle}=t I \qquad\text{and}\qquad g_t|_{{\mathfrak{h}}_{\lambda_i}}=t^{\lambda_i} I, \quad\text{for $i=1\dots k$}.$$ It is not difficult to check that is satisfied. In fact, if $y_i\in{\mathfrak{h}}_{\lambda_i}$ and $y_j\in{\mathfrak{h}}_{\lambda_j}$ for $1\le i,j\le k$, then $$\begin{aligned}
g_t(\mu_{t}(X,y_j)) &=& g_t(\mu(X,y_j)+\lambda_j ty_j)=t^{1+\lambda_j}\mu (X,y_j)+\lambda_jt^{\lambda_j+1}y_j, \\
\mu_{1}(g_t(X),g_t(y_j)) &=& \mu_{1}(t X,t^{\lambda_j}y_j)=t^{1+\lambda_j}\mu(X,y_j)+\lambda_jt^{\lambda_j+1}y_j,\\ \end{aligned}$$ and $$\begin{aligned}
g_t(\mu_{t}(y_i,y_j)) &=& g_t(\mu(y_i,y_j))=t^{\lambda_i+\lambda_j}\mu (y_i,y_j), \\
\mu_{1}(g_t(y_i),g_t(y_j)) &=& \mu_{1}(t^{\lambda_i}y_i,t^{\lambda_j}y_j)=t^{\lambda_i+\lambda_j}\mu(y_i,y_j). \\ \end{aligned}$$ Therefore, being $\mu_1$ solvable, $\mu$ is the degeneration of another, non isomorphic, Lie algebra.
In the above proposition the ideal ${\mathfrak{h}}$ is given, but clearly any such ideal will work. Hence, if $\tilde D$ is a derivation of ${\mathfrak{n}}$ that preserves an ideal ${\mathfrak{h}}$ and such that its restriction to ${\mathfrak{h}}$ is semisimple, we get for ${\mathfrak{n}}$ the same conclusion of Proposition \[prop:restriction\]. This is the statement in Theorem \[thm:1\].
The semisimple derivation $D$ of ${\mathfrak{n}}$ preserves the (characteristic) ideal $[{\mathfrak{n}},{\mathfrak{n}}]$. Let $V$ be a $D$-invariant complement of $[{\mathfrak{n}},{\mathfrak{n}}]$ and let $\{X_1,\dots,X_r\}$ be a basis of $V$ formed by eigenvectors of $D$. Since $V$ generates ${\mathfrak{n}}$ as a Lie algebra (see for instance [@J], page 29) and $D$ is nontrivial, $D$ is nontrivial on $V$ and we may assume that $X_1$ is an eigenvector with nonzero eigenvalue. Now let ${\mathfrak{h}}=\langle X_2,\dots,X_r \rangle \oplus [{\mathfrak{n}},{\mathfrak{n}}]$. Clearly ${\mathfrak{h}}$ is an ideal of ${\mathfrak{n}}$ of codimension 1, $D$ preserves ${\mathfrak{h}}$, $D|{\mathfrak{h}}$ is semisimple and $D$ is nontrivial on $X_1$. Therefore, by Proposition \[prop:restriction\], ${\mathfrak{n}}$ is the degeneration of a Lie algebra non isomorphic to ${\mathfrak{n}}$.
The conjecture in dimension 7
=============================
All nilpotent Lie algebras of dimension $<7$ have semisimple derivations. Therefore the Grunewald-O’Halloran conjecture holds in this case.
Moreover, in dimensions 2, 3, 4, 5 and 6 all nilpotent Lie algebras (finite number of isomorphism classes) are the degeneration of a single one [@GO1; @Se]. Hence, an algebra degenerating to it degenerates to all the others as well.
By considering different linear deformations, we found that each nilpotent Lie algebra of dimension $<7$ is the degeneration of many others, non isomorphic, Lie algebras. Many of those degenerations can be realized by a 1-PSG, but others can not.
The 6-dimensional nilpotent Lie algebra $12346_E$ in [@Se], that we rename $\mu$, defined by $$\begin{aligned}
\label{ec}
\mu(e_1,e_2)&=e_3, & \quad \mu(e_1,e_3)&=e_4, & \quad \mu(e_1,e_4)&=e_5, \\
\mu(e_2,e_3)&=e_5, & \quad \mu(e_2,e_5)&=e_6, & \quad \mu(e_3,e_4)&=-e_6,\nonumber\end{aligned}$$ degenerates to all other nilpotent Lie algebras of dimension 6 [@Se].
We now construct a solvable linear deformation of $\mu$ that degenerates to it, and therefore to all other 6-dimensional nilpotent Lie algebras. To this end consider the ideal ${\mathfrak{h}}=\langle e_2,e_3,e_4,e_5,e_6 \rangle$ and the derivation $D$ of ${\mathfrak{h}}$ defined by $$D(e_2)=e_2, \quad D(e_4)=2e_4, \quad D(e_5)=e_5, \quad D(e_6)=2e_6.$$ This produces the 2-cocycle $\mu_{D}$, defined by $$\mu_{D}(e_1,e_2)=e_2, \quad \mu_{D}(e_1,e_4)=2e_5, \quad \mu_{D}(e_1,e_5)=e_5, \quad \mu_{D}(e_1,e_6)=2e_6.$$ The corresponding deformation of $\mu$, $\mu_t=\mu+t\mu_{D}$, is then given by $$\begin{aligned}
\mu_{t}(e_1,e_2)&=e_3+te_2, \quad & \mu_{t}(e_1,e_3)&=e_4, \quad & \mu_{t}(e_1,e_4)&= e_5+2te_4,\\
\mu_{t}(e_1,e_5)&=te_5, \quad & \mu_{t}(e_1,e_6)&=2te_6, \quad & \mu_{t}(e_2,e_3)&=e_5,\\
\mu_{t}(e_2,e_5)&=e_6, \quad & \mu_{t}(e_3,e_4)&=-e_6,\end{aligned}$$ and in particular $\mu_{1}$ is given by $$\begin{aligned}
\mu_{1}(e_1,e_2)&=e_3+e_2, \quad & \mu_{1}(e_1,e_3)&=e_4, \quad & \mu_{1}(e_1,e_4)&=e_5+2e_4,\\
\mu_{1}(e_1,e_5)&=e_5, \quad & \mu_{1}(e_1,e_6)&=2e_6, \quad & \mu_{1}(e_2,e_3)&=e_5,\\
\mu_{1}(e_2,e_5)&=e_6, \quad & \mu_{1}(e_3,e_4)&=-e_6.\end{aligned}$$ Let $g_t\in GL_6$ be the 1-PSG given by $$g_t=\begin{pmatrix} t \\ & t^2 \\ & & t^3 \\ & & & t^4 \\ & & & & t^5 \\ & & & & & t^7 \end{pmatrix}.$$ It is easy to verify that, for all $t\ne 0$, $g_t^{-1}\cdot \mu_{1} =\mu_{t}$ and thus $\mu_1\rightarrow_{{\operatorname{deg}}} \mu$.
The variety of nilpotent Lie algebras of dimension 7 has two irreducible components, each of which is the closure of the orbits of two families $\mu_\alpha^1$ and $\mu_\alpha^2$, with $\alpha\in{\mathbb{C}}$ [@AG Main Theorem]. The first family is made of nilpotent Lie algebras of rank $\ge 1$, while the second family is made entirely of characteristically nilpotent algebras.
By Theorem \[thm:1\] and being degeneration transitive, to prove Theorem \[thm:2\] it suffices to find for each algebra in the second family another non isomorphic Lie algebra degenerating to it. All algebras $\mu_\alpha^2$ are indecomposable, because in dimension 7 all the decomposables are of rank $\ge 1$. In what follows we refer to the classification by Magnin [@M]. Here the (indecomposable) characteristically nilpotent Lie algebras are given as a continuous family and seven isolated algebras: $${\mathfrak{g}}_{7,0.1}\quad {\mathfrak{g}}_{7,0.2}\quad {\mathfrak{g}}_{7,0.3}\quad {\mathfrak{g}}_{7,0.4(\lambda)}\quad {\mathfrak{g}}_{7,0.5}\quad {\mathfrak{g}}_{7,0.6}\quad {\mathfrak{g}}_{7,0.7}\quad {\mathfrak{g}}_{7,0.8}$$ Without identifying the algebras $\mu_\alpha^2$ within this classification, Theorem \[thm:2\] follows if we are able to construct for each of these another Lie algebra degenerating to it. Notice that by doing this, we are exhibiting for each 7-dimensional characteristically nilpotent Lie algebra, another one degenerating non-trivially to it, something that we find interesting in itself.
We start by considering the family ${\mathfrak{g}}_{7,0.4(\lambda)}$, which is defined by $$\begin{aligned}
\mu(e_{1},e_{2})&=e_{3}, \quad \mu(e_{1},e_{3})=e_{4}, \quad \mu(e_{1},e_{4})=e_{6}+\lambda e_{7}, \\
\mu(e_{1},e_{5})&=e_{7}, \quad \mu(e_{1},e_{6})=e_{7}, \quad \mu(e_{2},e_{3})=e_{5}, \\
\mu(e_{2},e_{4})&=e_{7}, \quad \mu(e_{2},e_{5})=e_{6}, \quad \mu(e_{3},e_{5})=e_{7}.\end{aligned}$$ Take the ideal ${\mathfrak{h}}=\langle e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\rangle$ and $D\in{\operatorname{Der}}({\mathfrak{h}})$ defined by $$D(e_{2})=e_{2}\quad D(e_{5})=e_{5}\quad D(e_{6})=2e_{6},\quad D(e_{7})=e_{7}.$$ The corresponding 2-cocycle $\mu_{D}$ is given by $$\mu_{D}(e_{1},e_{2})=e_{2},\quad \mu_{D}(e_{1},e_{5})=e_{5},\quad \mu_{D}(e_{1},e_{6})=2e_{6},\quad \mu_{D}(e_{1},e_{7})=e_{7},$$ and the corresponding deformation $\mu_{t}=\mu + t\mu_{D}$ of $\mu$ is given by $$\begin{aligned}
\mu_{t}(e_{1},e_{2})&=e_{3}+te_{2}, \quad & \mu_{t}(e_{1},e_{3})&=e_{4}, \quad & \mu_{t}(e_{1},e_{4})&=e_{6}+\lambda e_{7}, \\
\mu_{t}(e_{1},e_{5})&=e_{7}+te_{5}, \quad & \mu_{t}(e_{1},e_{6})&=e_{7}+2te_{6}, & \quad \mu_{t}(e_{1},e_{7})&=te_{7}, \\
\mu_{t}(e_{2},e_{3})&=e_{5}, \quad & \mu_{t}(e_{2},e_{4})&=e_{7}, \quad & \mu_{t}(e_{2},e_{5})&=e_{6},\\
\mu_{t}(e_{3},e_{5})&=e_{7}.\end{aligned}$$ Consider now $g_t=g_{t}(\lambda)\in GL_7$ given by $$g_t=\left(\begin{smallmatrix}
t & 0 & 0 & 0 & 0 & 0 & 0\\[0.1cm]
0 & 1 & 0 & 0 & 0 & 0 & 0\\[0.1cm]
0 & 0 & t & 0 & 0 & 0 & 0\\[0.1cm]
0 & 0 & 0 & t^{2} & 0 & 0 & 0\\[0.1cm]
\frac{1}{4}\left(\frac{t^{2}-1}{t}\right) & \left(1-\lambda+\frac{\lambda}{t}-\frac{1}{t^{2}}\right) & 0 & 0 & t & 0 & 0\\[0.1cm]
0 & 0 & \frac{1}{4}\left(\frac{1-t^{2}}{t}\right) & \frac{1}{2}\left(1-t^{2}\right) & 0 & t & 0\\[0.1cm]
0 & 0 & \left(t-\lambda t+\lambda-\frac{1}{t}\right) & \left(\frac{1}{2}t^{2}-\lambda t^{2}+\lambda t-\frac{1}{2}\right)& \left(\lambda t-t-\lambda + \frac{1}{t}\right) & 0 & t^{2}
\end{smallmatrix}\right).$$ The calculations below show that $g_t^{-1}\cdot\mu_1=\mu_t$ and thus $\mu_{1}\rightarrow_{deg}\mu$.
g\_[t]{}\_[t]{}(e\_[1]{},e\_[2]{})&=g\_[t]{}(e\_[3]{}+te\_[2]{})&\
&=te\_[3]{}+()e\_[6]{}+(t-t+-)e\_[7]{}+te\_[2]{}+t(1-+-)e\_[5]{}&\
&=te\_[2]{}+te\_[3]{}+(t-t+-)e\_[5]{}+()e\_[6]{}+(t-t+-)e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[2]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},e\_[2]{}+(1-+-)e\_[5]{})&\
&=t(e\_[3]{}+e\_[2]{})+t(1-+-)(e\_[7]{}+e\_[5]{})-()e\_[6]{}&\
&=te\_[2]{}+te\_[3]{}+(t-t+-)e\_[5]{}+()e\_[6]{}+(t-t+-)e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[1]{},e\_[3]{})&=g\_[t]{}e\_[4]{}&\
&=t\^[2]{}e\_[4]{}+(1-t\^[2]{})e\_[6]{}+(t\^[2]{}-t\^[2]{}+t-)e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[3]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},te\_[3]{}+()e\_[6]{}+(t-t+-)e\_[7]{})&\
&=t\^[2]{}e\_[4]{}+t()(e\_[7]{}+2e\_[6]{})+t(t-t+-)e\_[7]{}-(t\^[2]{}-1)e\_[7]{}&\
&=t\^[2]{}e\_[4]{}+(1-t\^[2]{})e\_[6]{}+(-t\^[2]{}+t\^[2]{}-t\^[2]{}+t-1-t\^[2]{}+ )e\_[7]{}&\
&=t\^[2]{}e\_[4]{}+(1-t\^[2]{})e\_[6]{}+(t\^[2]{}-t\^[2]{}+t-)e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[1]{},e\_[4]{})&=g\_[t]{}(e\_[6]{}+e\_[7]{})&\
&=te\_[6]{}+t\^[2]{}e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[4]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},t\^[2]{}e\_[4]{}+(1-t\^[2]{})e\_[6]{}+(t\^[2]{}-t\^[2]{}+t-)e\_[7]{})&\
&=t\^[3]{}(e\_[6]{}+e\_[7]{})+t(1-t\^[2]{})(e\_[7]{}+2e\_[6]{})+t(t\^[2]{}-t\^[2]{}+t-)e\_[7]{}&\
&=(t\^[3]{}+t-t\^[3]{})e\_[6]{}+(t\^[3]{}+t-t\^[3]{}+t\^[3]{}-t\^[3]{}+t\^[2]{}-t)e\_[7]{}&\
&=te\_[6]{}+t\^[2]{}e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[1]{},e\_[5]{})&=g\_[t]{}(e\_[7]{}+te\_[5]{})&\
&=t\^[2]{}e\_[7]{}+t\^[2]{}e\_[5]{}+t(t-t-+)e\_[7]{}&\
&=t\^[2]{}e\_[5]{}+(t\^[2]{}-t+1)e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[5]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},te\_[5]{}+(t-t-+)e\_[7]{})&\
&=t\^[2]{}(e\_[7]{}+e\_[5]{})+t(t-t-+)e\_[7]{}&\
&=t\^[2]{}e\_[5]{}+(t\^[2]{}+t\^[2]{}-t\^[2]{}-t+1)e\_[7]{}&\
&=t\^[2]{}e\_[5]{}+(t\^[2]{}-t+1)e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[1]{},e\_[6]{})&=g\_[t]{}(e\_[7]{}+2te\_[6]{})&\
&=t\^[2]{}e\_[7]{}+2t\^[2]{}e\_[6]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[6]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},te\_[6]{})&\
&=t\^[2]{}(e\_[7]{}+2e\_[6]{})&\
&=t\^[2]{}e\_[7]{}+2t\^[2]{}e\_[6]{}&
\
g\_[t]{}\_[t]{}(e\_[1]{},e\_[7]{})&=g\_[t]{}(te\_[7]{})&\
&=t\^[3]{}e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[1]{},g\_[t]{}e\_[7]{})&=\_[1]{}(te\_[1]{}+()e\_[5]{},t\^[2]{}e\_[7]{})&\
&=t\^[3]{}e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[2]{},e\_[3]{})&=g\_[t]{}e\_[5]{}&\
&=te\_[5]{}+(t-t-+)e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[2]{},g\_[t]{}e\_[3]{})&=\_[1]{}(e\_[2]{}+(1-+-)e\_[5]{},te\_[3]{}+()e\_[6]{}+(t-t+-)e\_[7]{})&\
&=te\_[5]{}-t(1-+-)e\_[7]{}&\
&=te\_[5]{}+(t-t-+)e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[2]{},e\_[4]{})&=g\_[t]{}e\_[7]{}&\
&=t\^[2]{}e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[2]{},g\_[t]{}e\_[4]{})&=\_[1]{}(e\_[2]{}+(1-+-)e\_[5]{},t\^[2]{}e\_[4]{}+(1-t\^[2]{})e\_[6]{}+(t\^[2]{}-t++)e\_[7]{})&\
&=t\^[2]{}e\_[7]{}&
\
g\_[t]{}\_[t]{}(e\_[2]{},e\_[5]{})&=g\_[t]{}e\_[6]{}&\
&=te\_[6]{}&\
\_[1]{}(g\_[t]{}e\_[2]{},g\_[t]{}e\_[5]{})&=\_[1]{}(e\_[2]{}+(1-+-)e\_[5]{},te\_[5]{}+(t-t-+)e\_[7]{})&\
&=te\_[6]{}&
\
g\_[t]{}\_[t]{}(e\_[3]{},e\_[5]{})&=g\_[t]{}e\_[7]{}&\
&=t\^[2]{}e\_[7]{}&\
\_[1]{}(g\_[t]{}e\_[3]{},g\_[t]{}e\_[5]{})&=\_[1]{}(te\_[3]{}+()e\_[6]{}+(t-t+-)e\_[7]{},te\_[5]{}+(t-t-+)e\_[7]{})&\
&=t\^[2]{}e\_[7]{}&
\
For the remaining seven algebras $\mu$ we give, in the table below, the ideal ${\mathfrak{h}}$ of codimension 1, the semisimple derivation $D\in Der({\mathfrak{h}})$ that we choose to construct the linear deformation $\mu_t$, and the family $g_{t}\in GL_{7}$ satisfying $g_t^{-1}\cdot\mu_1=\mu_t$, which is not difficult to check by hand. Therefore $\mu_1\rightarrow_{{\operatorname{deg}}}\mu$ and the proof is complete.
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[**Acknowledgements.**]{} This paper is part of the PhD. thesis of the first author. He thanks CONICET for the Ph.D. fellowship awarded that made this possible. We thank Oscar Brega, Leandro Cagliero and Edison Fernández-Culma for their comments that helped us improved the presentation of this paper.
[AGGV]{}
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---
abstract: 'We study the problem of distributed adaptive estimation over networks where nodes cooperate to estimate physical parameters that can vary over both *space* and *time* domains. We use a set of basis functions to characterize the space-varying nature of the parameters and propose a diffusion least mean-squares (LMS) strategy to recover these parameters from successive time measurements. We analyze the stability and convergence of the proposed algorithm, and derive closed-form expressions to predict its learning behavior and steady-state performance in terms of mean-square error. We find that in the estimation of the space-varying parameters using distributed approaches, the covariance matrix of the regression data at each node becomes rank-deficient. Our analysis reveals that the proposed algorithm can overcome this difficulty to a large extent by benefiting from the network stochastic matrices that are used to combine exchanged information between nodes. We provide computer experiments to illustrate and support the theoretical findings.'
author:
- 'Reza Abdolee$^*$, Benoit Champagne, and , [^1] [^2][^3] [^4]'
title: 'Estimation of Space-Time Varying Parameters Using a Diffusion LMS Algorithm'
---
Diffusion adaptation, distributed processing, parameter estimation, space-varying parameters, sensor networks, interpolation.
introduction
============
previous studies on diffusion algorithms for adaptation over networks, including least-mean-squares (LMS) or recursive least squares (RLS) types, the parameters being estimated are often assumed to be *space-invariant* [@cattivelli2008diffusion; @cattivelli2010diffusion; @chen2012diffusion; @chouvardas2011adaptive; @tu2012diffusion; @sayed2012diffusion]. In other words, all agents are assumed to sense and measure data that arise from an underlying physical model that is represented by fixed parameters over the spatial domain. Some studies considered particular applications of diffusion strategies to data that arise from potentially different models [@tu2012adaptive; @di2012decentralized]. However, the proposed techniques in these works are not immediately applicable to scenarios where the estimation parameters vary over space across the network. This situation is encountered in many applications, including the monitoring of fluid flow in underground porous media [@lee1987estimation], the tracking of population dispersal in ecology [@holmes1994partial], the localization of distributed sources in dynamic systems [@alpay2000model], and the modeling of diffusion phenomena in inhomogeneous media [@van1988diffusion]. In these applications, the space-varying parameters being estimated usually result from discretization of the coefficients of an underlying partial differential equation through spatial sampling.
The estimation of spatially-varying parameters has been addressed in several previous studies, including [@chung1988identification; @richter1981numerical; @isakov2000identification; @demetriou2007process; @demetriou2009estimation]. In these works and other similar references on the topic, the solutions typically rely on the use of a central processing (fusion) unit and less attention is paid to distributed and in-network processing solutions. Distributed algorithms are useful in large networks when there is no powerful fusion center and when the energy and communication resources of individual nodes are limited. Many different classes of distributed algorithms for parameter estimation over networks have been proposed in the recent literature, including incremental method[@bertsekas1997new; @Nedia2001Incremental; @rabbat2005quantized; @lopes2007incremental; @li2010distributed], consensus methods [@tsitsiklis1986distributed; @xiao2006space; @stankovic2007decentralized; @braca2008running; @sardellitti2010fast; @aysal2009broadcast; @boyd2006randomized; @dimakis2010gossip; @srivastava2011distributed; @kar2011convergence; @di2011bio; @hu2010adaptive], and diffusion methods[@lopes2008diffusion; @cattivelli2010diffusion; @chen2012diffusion; @sayed2012diffusion; @sayed2013DiffusionMagazine; @zhao2012performance]. Incremental techniques require to set-up a cyclic path between nodes over the network and are therefore sensitive to link failures. Consensus techniques require doubly-stochastic combination policies and can cause network instability in applications involving continuous adaptation and tracking [@tu2012diffusion]. In comparison, diffusion strategies demonstrate a stable behavior over networks regardless of the topology and endow networks with real-time adaptation and learning abilities [@sayed2012diffusion; @sayed2013DiffusionMagazine; @tu2012diffusion].
Motivated by these considerations, in this paper, we develop a distributed LMS algorithm of the diffusion type to enable the estimation and tracking of parameters that may vary over both *space* and *time*. Our approach starts by introducing a linear regression model to characterize space-time varying phenomena over networks. This model is derived by discretizing a representative second-order partial differential equation (PDE), which can be useful in characterizing many dynamic systems with spatially-varying properties. We then introduce a set of basis functions, e.g., shifted Chebyshev polynomials, to represent the space-varying parameters of the underlying phenomena in terms of a finite set of space-invariant expansion coefficients. Building on this representation, we develop a diffusion LMS strategy that cooperatively estimates, interpolates, and tracks the model parameters over the network. We analyze the convergence and stability of the developed algorithm, and derive closed-form expressions to characterize the learning and convergence behavior of the nodes in mean-square-error sense. It turns out that in the context of space-time varying models, the covariance matrices of the regression data at the various nodes can become rank deficient. This property influences the learning behavior of the network and causes the estimates to become biased. We elaborate on how the judicious use of stochastic combination matrices can help alleviate this difficulty.
The paper is organized as follows. In Section \[sec.:SpaceDependentLinearRegression\], we introduce a space-varying linear regression model which is motivated from a physical phenomenon characterized by a PDE, and formulate an optimization problem to find the unknown parameters of the introduced model. In Section \[sec.:AlgorithmDevelopment\], we derive a diffusion LMS algorithm that solves this problem in a distributed and adaptive manner. We analyze the performance of the algorithm in Section \[sec.:performance\_analysis\], and present the numerical results of computer simulations in Section \[sec.:results\]. The concluding remarks appear in .
*Notation:* Matrices are represented by upper-case and vectors by lower-case letters. Boldface fonts are reserved for random variables and normal fonts are used for deterministic quantities. Superscript $(\cdot)^T$ denotes transposition for real-valued vectors and matrices and $(\cdot)^{\ast}$ denotes conjugate transposition for complex-valued vectors and matrices. The symbol $\E[\cdot]$ is the expectation operator, $\text{Tr}(\cdot)$ represents the trace of its matrix argument and diag$\{\cdot\}$ extracts the diagonal entries of a matrix, or constructs a diagonal matrix from a vector. $I_M$ represents the identity matrix of size $M\times M$ (subscript $M$ is omitted when the size can be understood from the context). The vec$(\cdot)$ operator vectorizes a matrix by stacking its columns on top of each other. A set of vectors are stacked into a column vector by $\col\{\cdot\}$.
Modeling and Problem Formulation {#sec.:SpaceDependentLinearRegression}
================================
In this section, we motivate a linear regression model that can be used to describe dynamic systems with spatially varying properties. We derive the model from a representative second-order one-dimensional PDE that is used to characterize the evolution of the pressure distribution in inhomogeneous media and features a diffusion coefficient and an input source, both of which vary over space. Extension and generalization of the proposed approach, in modeling space-varying phenomena, to PDEs of higher order or defined over two-dimensional space are generally straightforward (see, e.g., Section \[subsec.:Diffusion LMS for Process Estimation\]).
The PDE we consider is expressed as [@van1988diffusion; @mattheij2005partial]: $$\begin{aligned}
\frac{\partial f(x,t)}{\partial t}=\frac{\partial}{\partial x}\Bigg ( \theta(x)\frac{\partial f(x,t)}{\partial x} \Bigg)+q(x,t)
\label{eq.:one_dimensional_pde1}\end{aligned}$$ where $(x,t)\in [0,L]\times [0,T]$ denote the space and time variables with upper limits $L \in {\amsbb R}^+$ and $T \in {\amsbb R}^+$, respectively, $f(x,t):{\amsbb R}^2 \rightarrow {\amsbb R}$, represents the system distribution (e.g., pressure or temperature) under study, $\theta(x){\text :}\, {\amsbb R} \rightarrow {\amsbb R}$ is the space-varying diffusion coefficient and $q(x,t){\text :}\, {\amsbb R}^2 \rightarrow {\amsbb R}$ is the input distribution that includes sources and sinks. PDE (\[eq.:one\_dimensional\_pde1\]) is assumed to satisfy the Dirichlet boundary conditions[^5], $f(0,t)=f(L,t) =0$ for all $t \in [0,T]$. The distribution of the system at $t=0$ is given by $f(x,0) =y(x)\; \text{for} \; x\in[0,\,L]$. It is convenient to rewrite (\[eq.:one\_dimensional\_pde1\]) as: $$\begin{aligned}
\frac{\partial f(x,t)}{\partial t}= \theta(x)\frac{{\partial}^2 f(x,t)}{{\partial x}^2}+\frac{\partial \theta(x)}{\partial x}\frac{\partial f(x,t)}{\partial x}+q(x,t)
\label{eq.:one_dimensional_pde2}\end{aligned}$$ and employ the finite difference method (FDM) to discretize the PDE over the time and space domains [@thomas1995numerical]. For $N$ and $P$ given positive integers, let $\Delta x=L/(N+1)$ and $x_k=k \Delta x$ for $k \in \{0,1,2,\ldots,N+1\}$, and similarly, let $\Delta t=T/P$ and $t_i=i\Delta t$ for $i \in \{0,1,2, \ldots, P\}$. We further introduce the sampled values of the pressure distribution $ f_k(i)\triangleq f(x_k ,t_i)$, input $q_k(i)\triangleq q(x_k,t_i)$, and space-varying coefficient $\theta_k\triangleq\theta(x_k)$. It can be verified that applying FDM to (\[eq.:one\_dimensional\_pde2\]), yields the following recursion: $$f_k(i)={u}_{k,i}h^o_k+\Delta t \, q_k(i-1), \quad k \in \{1,2,\ldots,N\}
\label{eq.:fieldRecursion}$$ where the vectors $h^o_k \in {\amsbb R}^{3\times 1}$ and ${u}_{k,i} \in {\amsbb R}^{1\times 3}$ are defined as $$\begin{aligned}
&{h}^o_k\triangleq[h^o_{1,k},h^o_{2,k},h^o_{3,k}]^T \label{eq.:local_vectors_components}\\
&{u}_{k,i}\triangleq[f_{k-1}(i-1),\, f_k(i-1), \, f_{k+1}(i-1)]\end{aligned}$$ the entries $h^o_{m,k} \in {\amsbb R}$ are: $$\begin{aligned}
h^o_{1,k}& = \frac{\nu}{4}(\theta_{k-1}+4\theta_k-\theta_{k+1}) \label{eq.:local_vectors_relation_with_theta1} \\
h^o_{2,k}&=1-2\nu \, \theta_k
\label{eq.:local_vectors_relation_with_theta2}\\
h^o_{3,k}& =\frac{\nu}{4}(-\theta_{k-1}+4\theta_k+\theta_{k+1}) \label{eq.:local_vectors_relation_with_theta3}\end{aligned}$$ and $\nu={\Delta t}/{\Delta x^2}$. Note that relation (\[eq.:fieldRecursion\]) is defined for $k \in \{1,2,\cdots,N\}$, i.e., no data sampling is required to be taken at $x=\{0,L\}$ because $f_0(i)$ and $f_{N+1}(i)$ respectively correspond to the known boundary conditions $f(0,t)$ and $f(L,t)$. For monitoring purposes (e.g., estimation of $\theta(x)$), sensor nodes collect noisy measurement samples of $f(x,t)$ across the network. We denote these scalar measurement samples by $$\z_k(i)= f_k(i)+ \n_k(i)
\label{eq.:NoisyMeasurement}$$ where $\n_k(i) \in {\amsbb R}$ is random noise term. Substituting (\[eq.:fieldRecursion\]) into (\[eq.:NoisyMeasurement\]) leads to \_k(i)=u\_[k,i]{}h\^o\_k+ \_k(i) \[eq.:state\_dependent\_regression0\] where \_k(i)\_k(i)-t q\_k(i-1) The space-dependent model (\[eq.:state\_dependent\_regression0\]) can be generalized to accommodate higher order PDE’s, or to describe systems with more than one spatial dimension. In the generalized form, we assume that $u_{k,i}$ is random due to the possibility of sampling errors, and therefore represent it using boldface notation $\u_{k,i}$. We also let $h_k^o$ and $\u_{k,i}$ be $M$-dimensional vectors. In addition, we denote the noise more generally by the symbol $\v_k(i)$ to account for different sources of errors, including the measurement noise shown in (\[eq.:NoisyMeasurement\]) and modeling errors. Considering this generalization, the space-varying regression model that we shall consider is of the form: $$\d_k(i)=\u_{k,i}h^o_k+ \v_k(i)
\label{eq.:state_dependent_regression}$$ where $\d_k(i) \in {\amsbb R}, \u_{k,i} \in {\amsbb R}^{1 \times M}, h^o_k \in {\amsbb R}^{M \times 1}$ and $\v_k(i) \in {\amsbb R}$. In this work, we study networks that monitor phenomena characterized by regression models of the form (\[eq.:state\_dependent\_regression\]), where the objective is to estimate the space-varying parameter vectors $h_k^o$ for $k \in \{1,2,\cdots,N\}$. In particular, we seek a distributed solution in the form of an adaptive algorithm with a diffusion mode of cooperation to enable the nodes to estimate and track these parameters over both space and time. The available information for estimation of the $\{h^o_k\}$ are the measurement samples, $\{\d_k(i), \u_{k,i}\}$, collected at the $N$ spatial position $x_k$, which we take to represent $N$ nodes.
Several studies, e.g., [@chung1988identification; @richter1981numerical; @isakov2000identification], solved space-varying parameter estimation problems using [ *centralized*]{} techniques. In centralized optimization, the space-varying parameters $\{h^o_k\}$ are found by minimizing the following global cost function over the variables $\{h_k\}$: $$J(h_1,\ldots,h_N)\triangleq \sum_{k=1}^N J_k(h_k)
\label{eq.:DecomposedCostFunction}$$ where $$J_k(h_k) \triangleq \E| \d_k(i)-\u_{k,i}h_k|^2
\label{eq.:localCostFunction}$$ To find $h^o_k$ using distributed mechanisms, however, preliminary steps are required to transform the global cost (\[eq.:DecomposedCostFunction\]) into a suitable form convenient for decentralized optimization [@cattivelli2010diffusion]. Observe from (\[eq.:local\_vectors\_relation\_with\_theta1\])-(\[eq.:local\_vectors\_relation\_with\_theta3\]) that collaborative processing is beneficial in this case because the $h_k^o$ of neighboring nodes are related to each other through the space-dependent function $\theta(x)$.
Note that if nodes individually estimate their own space-varying parameters by minimizing $J_k(h_k)$, then at each time instant, they will need to transmit their estimates to a fusion center for interpolation, in order to determine the value of the model parameters over regions of space where no measurements were collected. Using the proposed distributed algorithm in Section \[subsec.:diffusion\_strategy\], it will be possible to update the estimates and interpolate the results in a fully distributed manner. Cooperation also helps the nodes refine their estimates and perform more accurate interpolation. $\blacksquare$
Adaptive Distributed Optimization {#sec.:AlgorithmDevelopment}
=================================
In distributed optimization over networked systems, nodes achieve their common objective through collaboration. Such an objective may be defined as finding a global parameter vector that minimizes a given cost function that encompasses the entire set of nodes. For the problem stated in this study, the unknown parameters in (\[eq.:DecomposedCostFunction\]) are node-dependent. However, as we explained in Section \[sec.:SpaceDependentLinearRegression\], these space-varying parameters are related through a well-defined function, e.g., $\theta(x)$ over the spatial domain. In the continuous space domain, the entries of each $h^o_k$, i.e., $\{h^o_{1,k},\cdots,h^o_{M,k}\}$ can be interpreted as samples of $M$ unknown space-varying parameter functions $\{h^o_1(x),\cdots,h^o_M(x)\}$ at location $x=x_k$, as illustrated in .
![. For simplicity in defining the vectors $b_k$ in (\[eq.:b\_k-entries\]), for this example, we assume that the node positions $x_k$ are uniformly spaced, however, generalization to non-uniform spacing is straightforward.[]{data-label="fig.:network-topology-and-parameters"}](./TopoParam){width="7.5" height="5cm"}
We can now use the well-established theory of interpolation to find a set of linear expansion coefficients, common to all the nodes, in order to estimate space-varying parameters using distributed optimization. Specifically, we assume that the unknown space-varying parameter function, $h^o_m(x)$ can be expressed as a unique linear combination of some $N_b$ space basis functions, i.e., $$h^o_m(x)=W_{m,1}b_1(x)+W_{m,2}b_2(x)+\cdots+W_{m,N_b}b_{N_b}(x)
\label{eq.:countinoush(x)}$$ where $\{W_{m,n}\}$ are the unique expansion coefficients and $\{b_{n}(x)\}$ are the basis functions. In the application examples treated in Section \[sec.:results\], we adopt shifted Chebyshev polynomials as basis functions, which are generated using the following expressions [@mason2003chebyshev] $$\begin{aligned}
&b_{1}(x)=1, \qquad b_{2}(x)=2x-1 \\
&b_{n+1}(x)=2(2x-1) b_{n}(x)-b_{n-1}(x),\quad 2<n<N_b
\label{eq.:ShiftedCountinuesChebyshev}\end{aligned}$$ The choice of a suitable set of basis functions $\{b_n(x)\}_{n=1}^{N_b}$ is application-specific and guided by multiple considerations such as representation efficiency, low computational complexity, interpolation capability, and other desirable properties, such as orthogonality. Chebyshev basis functions yield good results in terms of the above criteria and helps avoid the Runge’s phenomenon at the endpoints of the space interval [@mason2003chebyshev]. The sampled version of the space-varying parameter $h^o_m(x)$ in (\[eq.:countinoush(x)\]), at $x=x_k=k\Delta x$, can be written as: $$h^o_{m,k}=W_m^T b_k
\label{eq.:m-th-entryOfSpaceVarPara}$$ where $$\begin{aligned}
W_m&\triangleq [W_{m,1},W_{m,2},\cdots,W_{m,N_b}]^T \\
b_{k}&\triangleq [b_{1,k},\cdots,b_{N_b,k}]^T
\label{eq.:b_k-entries}\end{aligned}$$ and each entry $b_{n,k}$ is obtained by sampling the corresponding basis function at the same location, i.e., b\_[n,k]{} b\_n(x\_k) =b\_n(kx) \[eq.:b-nk\] Collecting the sampled version of all $M$ functions $ h_m^o(x)$ for $m\in \{1,\cdots,M\}$ into a column vector as h\^o\_k=\[h\^o\_[1,k]{},h\^o\_[2,k]{},, h\^o\_[M,k]{}\]\^T and using (\[eq.:m-th-entryOfSpaceVarPara\]), we arrive at: $$h^o_k=W^o b_k
\label{eq.:B-spline function}$$ where $$W^o\triangleq \left [ \begin{array}{cccc}
W^o_{1,1} & W^o_{1,2} & \ldots & W^o_{1,N_b}\\
W^o_{2,1} & W^o_{2,2} & \ldots & W^o_{2,N_b}\\
\vdots & \vdots & \ldots & \vdots\\
W^o_{M,1} & W^o_{M,2} & \ldots & W^o_{M,N_b}
\end{array}\right]$$
Several other interpolation techniques can be used to obtain the basis functions $b_n(x)$, such as the so-called kriging method [@isaaks1989introduction]. The latter is a data-based weighting approach, rather than a distance-based interpolation. It is applicable in scenarios where the unknown random field to be interpolated, in our case $h^o_k$, is wide-sense stationary; accordingly, it requires knowledge about the means and covariances of the random field over space, as employed in [@kim2011cooperative]. If these covariances are not available, then the variogram models, describing the degree of spatial dependence of the random field, are used to generate substitutes for these covariances [@cressie1993statistics]. However, [*a-priori*]{} knowledge about the parameters of variogram models, including nugget, sill, and range, are required to obtain the spatial covariances. In this work, since neither the means and covariances nor the parameters of the variogram models of the random fields are available, we focus on interpolation techniques that rely on distance information rather than the statistics of the random field to be interpolated. $\blacksquare$
Returning to equation (\[eq.:B-spline function\]), it is convenient to rearrange $W^o$ into an $MN_b\times 1$ column vector $w^o$ by stacking up the columns of ${W^o}^T$, i.e., $w^o=\vec({W^o}^T)$, and defining the block diagonal matrix $B_k \in {\amsbb R}^{M\times MN_b}$ as: $$B_k\triangleq I_M \otimes b^T_k
\label{eq.:B_k}$$ Then, relation (\[eq.:B-spline function\]) can be rewritten in terms of the unique parameter vector $w^o$ as: $$h^o_k=B_k w^o
\label{eq.:local_global_relation}$$ so that substituting $h^o_k$ from (\[eq.:local\_global\_relation\]) into (\[eq.:state\_dependent\_regression\]) yields: $$\begin{aligned}
&\d_k(i)= \u_{k,i}B_k w^o+ \v_k(i) \label{eq.:measurement_linear_model2}\end{aligned}$$ Subsequently, the global cost function (\[eq.:DecomposedCostFunction\]) becomes: $$\begin{aligned}
&J(w)=\sum_{k=1}^N \E|\d_k(i)- \u_{k,i}B_k w|^2 \label{eq.:LocalObjectiveFunction}\end{aligned}$$ In the following, we elaborate on how the parameter vector $w^o$ and, hence, the $\{h_{k}^o\}$ can be estimated from the data $\{\d_k(i),\u_{k,i}\}$ using centralized and distributed adaptive optimization.
Centralized Adaptive Solution {#sebsec.:CentralizedSolution}
-----------------------------
We begin by stating the assumed statistical conditions on the data over the network.
\[assm.:regressor assumption\] We assume that $\{\d_k(i), \u_{k,i}, \v_{k}(i)\}$ in model (\[eq.:measurement\_linear\_model2\]) satisfy the following conditions:
1. $\d_k(i)$ and $ \u_{k,i}$ are zero-mean, jointly wide-sense stationary random processes with second-order moments: $$\begin{aligned}
r_{du,k}&=\E[\d_k(i)\u_{k,i}^T] \in {\amsbb R}^{M \times 1}\\
R_{u,k}&=\E[\u_{k,i}^T\u_{k,i}] \in {\amsbb R}^{M \times M}\end{aligned}$$
2. The regression data $\{ \u_{k,i}\}$ are i.i.d. over time, independent over space, and their covariance matrices, $R_{u,k}$, are positive definite for all $k$.
3. The noise processes $\{\v_k(i)\}$ are zero-mean, i.i.d. over time, and independent over space with variances $\{\sigma^2_{v,k}\}$.
4. The noise process $\v_k(i)$ is independent of the regression data $\u_{m,j}$ for all $i,j$ and $k,m$. $\blacksquare$
The optimal parameter $w^o$ that minimizes (\[eq.:LocalObjectiveFunction\]) can be found by setting the gradient vector of $J(w)$ to zero. This yields the following normal equations: (\_[k=1]{}\^N |[R]{}\_[u,k]{}) w\^o = \_[k=1]{}\^N |[r]{}\_[du,k]{} \[eq.CentralizedNormalEquation\] where $\{\bar{R}_{u,k}, \bar{r}_{du,k}\}$ denote the second-order moments of $\u_{k,i} B_{k}$ and $\d_k(i)$: |[R]{}\_[u,k]{}B\_k\^T R\_[u,k]{} B\_k, |[r]{}\_[du,k]{} B\_k\^T r\_[du,k]{} \[eq.:bar\_Ru\] It is clear from (\[eq.CentralizedNormalEquation\]) that when $\sum_{k=1}^N \bar{R}_{u,k}>0$, then $w^o$ can be determined uniquely. If, on the other hand, $\sum_{k=1}^N \bar{R}_{u,k}$ is singular, then we can use its pseudo-inverse to recover the minimum-norm solution of (\[eq.CentralizedNormalEquation\]). Once the global solution is estimated, we can retrieve the space-varying parameter vectors $h^o_k$ by substituting $w^o$ into (\[eq.:local\_global\_relation\]).
Alternatively the solution $w^o$ of (\[eq.CentralizedNormalEquation\]) can be sought iteratively by using the following steepest descent recursion: $$\begin{aligned}
& \w^{(c)}_i= \w^{(c)}_{i-1}+\mu \sum_{k=1}^N \big (\bar{r}_{du,k}-\bar{R}_{u,k} \w^{(c)}_{i-1}\big )\label{eqe:centralized_steepest_eq1}\end{aligned}$$ where $\mu>0$ is a step-size parameter and $ \w^{(c)}_i$ is the estimate of $w^o$ at the $i$-th iteration. Recursion (\[eqe:centralized\_steepest\_eq1\]) requires the centralized processor to have knowledge of the covariance matrices, $\bar{R}_{u,k}$, and cross covariance vectors, $\bar{r}_{du,k}$, across all nodes. In practice, these moments are unknown in advance, and we therefore use their instantaneous approximations in (\[eqe:centralized\_steepest\_eq1\]). This substitution leads to the centralized LMS strategy (\[eq.:centralized-step1\])–(\[eq.:centralized-step2\]) for space-varying parameter estimation over networks.
$$\begin{aligned}
&\w^{(c)}_i= \w^{(c)}_{i-1}+\mu\displaystyle\sum_{k=1}^N B_k^T \u_{k,i}^T \big(\d_k(i)-\u_{k,i} B_k \w^{(c)}_{i-1}\big)
\label{eq.:centralized-step1}\\
&\h_{k,i}=B_k \w^{(c)}_i, \quad k \in \{1,2,\cdots,N\}
\label{eq.:centralized-step2}\end{aligned}$$
In this algorithm, at any given time instant $i$, each node transmits its data $\{\u_{k,i},\d_k(i)\}$ to the central processing unit to update $\w^{(c)}_{i-1}$. Subsequently, the algorithm obtains an estimate for the space-varying parameters, $\h_{k,i}$, by using the updated estimate $\w^{(c)}_i$, and the basis function matrix at location $k$, (i.e., $B_k$). This latter step can also be used as an interpolation mechanism to estimate the space-varying parameters at locations other than the pre-determined locations $\{x_k\}$, by using the corresponding matrix $B(x)$ for some desired location $x$.
Adaptive Diffusion Strategy {#subsec.:diffusion_strategy}
---------------------------
There are different distributed optimization techniques that can be applied to (\[eq.:LocalObjectiveFunction\]) in order to estimate $w^o$ and consequently obtain the optimal space-varying parameters $h^o_k$. Let $\mathcal{N}_k$ denote the index set of the neighbors of node $k$, i.e., the nodes with which node $k$ can share information (including $k$ itself). One possible optimization strategy is to decouple the global cost (\[eq.:LocalObjectiveFunction\]) and write it as a set of constrained optimization problems with local variables $w_k$, [@boyd2011a], i.e., $$\begin{aligned}
&\min_{w_k}\sum_{\ell\in {\cal N}_k} c_{\ell,k}\E| \d_{\ell}(i)- \u_{\ell,i}B_k w_k|^2 \nonumber \\
&\text{subject to} \quad w_k=w
\label{minimizationProblemLagrangien}\end{aligned}$$ where $c_{\ell, k}$ are nonnegative entries of a right-stochastic matrix $C \in \amsbb{R}^{N \times N}$ satisfying: $$\begin{aligned}
&c_{\ell,k}=0 \;\, \text{if} \;\, \ell\notin \mathcal{N}_k \;\;\, \textrm{and} \;\;\; C{\mathbb 1}={\mathbb 1}
\label{eq.sthocastic_matrix_c_conditions}\end{aligned}$$ and ${\mathbb 1}$ is the column vector with unit entries.
The optimization problem (\[minimizationProblemLagrangien\]) can be solved using, for example, the alternating directions method of multipliers (ADMM) [@bertsekas1989parallel; @boyd2011a]. In the algorithm derived using this method, the Lagrangian multipliers associated with the constraints need to be updated at every iteration during the optimization process. To this end, information about the network topology is required to establish a hierarchical communication structure between nodes. In addition, the constraints imposed by (\[minimizationProblemLagrangien\]) require all agents to agree on an exact solution; this requirement degrades the learning and tracking abilities of the nodes over the network. When some nodes observe relevant data, it is advantageous for them to be able to respond quickly to the data without being critically constrained by perfect agreement at that stage. Doing so, would enable information to diffuse more rapidly across the network.
A technique that does not suffer from these difficulties and endows networks with adaptation and learning abilities in real-time is the diffusion strategy [@lopes2008diffusion; @cattivelli2010diffusion; @chen2012diffusion; @sayed2012diffusion; @sayed2013DiffusionMagazine]. In this technique, minimizing the global cost (\[eq.:LocalObjectiveFunction\]) motivates solving the following unconstrained local optimization problem for [@cattivelli2010diffusion]: $$\begin{aligned}
\min_w \Big(&\sum_{\ell\in {\cal N}_k} c_{\ell,k}\E| \d_{\ell}(i)- \u_{\ell,i}B_k w|^2 \nonumber \\
&\hspace{1cm}+\sum_{\ell\in {\cal N}_k \backslash\{k\}} p_{\ell,k}\|w-\psi_{\ell}\|^2\Big) \label{eq.:constrained-local-objectiveFunc.}\end{aligned}$$ where $\psi_{\ell}$ is the available estimate of the global parameter at node $\ell$, ${\cal N}_k\backslash \{k\}$ denotes set ${\cal N}_k$ excluding node $k$, and $\{p_{\ell, k}\}$ are nonnegative scaling parameters. Following the arguments in [@cattivelli2010diffusion; @chen2012diffusion; @sayed2012diffusion], the minimization of (\[eq.:constrained-local-objectiveFunc.\]) leads to a general form of the diffusion strategy described by (\[eq.diff-step1\])–(\[eq.diff-step4\]), which can be specialized to several simpler yet useful forms.
$$\begin{aligned}
&\bphi_{k,i-1}=\displaystyle\sum_{\ell\in \mathcal{N}_k} a^{(1)}_{\ell,k} \w_{\ell,i-1} \label{eq.diff-step1}\\
&{\bpsi}_{k,i}=\bphi_{k,i-1}+\mu_k\displaystyle\sum_{\ell\in \mathcal{N}_k}c_{\ell,k} B_{\ell}^T \u_{\ell,i}^T \big(\d_{\ell}(i)- \u_{\ell,i}B_{\ell}\bphi_{k,i-1}\big)\label{eq.diff-step2}\\
&\w_{k,i}=\displaystyle\sum_{\ell\in \mathcal{N}_k}a^{(2)}_{\ell, k}{\bpsi}_{\ell,i} \label{eq.diff-step3}\\
&\h_{k,i}=B_k \w_{k,i} \label{eq.diff-step4}\end{aligned}$$
In this algorithm, $\mu_k>0$ is the step-size at node $k$, $\{\w_{k,i}, \bpsi_{k,i}, \bphi_{k,i-1}\}$ are intermediate estimates of $w^o$, $\h_{k,i}$ is an intermediate estimate of $h^o_k$, and $\{ a^{(1)}_{\ell,k},a^{(2)}_{\ell,k}\}$ are nonnegative entries of left-stochastic matrices $A_1, A_2 \in \amsbb{R}^{N \times N}$ that satisfy: $$\begin{aligned}
& a^{(1)}_{\ell,k}=a^{(2)}_{\ell,k}=0 \quad \text{if} \, \ell\notin \mathcal{N}_k
\label{eq.sthocastic_matrix_a1a2_conditions1} \\
&\quad A_1^T{\mathbb 1}={\mathbb 1}\quad A_2^T{\mathbb 1}={\mathbb 1}
\label{eq.sthocastic_matrix_a1a2_conditions2}\end{aligned}$$ Each node $k$ in the first combination step fuses $\{\w_{\ell,i-1}\}_{\ell \in {\cal N}_k}$ in a convex manner to generate $\bphi_{k,i-1}$. In the following step, named adaptation, each node $k$ uses its own data and that of neighboring nodes, i.e., $\big \{ \u_{\ell,i}, \d_{\ell}(i)\big\}_{\ell \in {\cal N}_k}$ to adaptively update $\bphi_{k,i-1}$ to an intermediate estimate $\bpsi_{k,i}$. In the third step, which is also a combination, the intermediate estimates $\{ \bpsi_{\ell,i}\}_{\ell \in \mathcal{N}_k}$ are fused to further align the global parameter estimate at node $k$ to that of its neighbors. Subsequently, the desired space-varying parameter $\h_{k,i}$ is obtained from $\w_{k,i}$. Note that each step in the algorithm runs concurrently over the network.
The main difference between Algorithm \[eq.GeneralizedLmsSpaVarParaEst\] and the previously developed diffusion LMS strategies in, e.g., [@lopes2008diffusion; @cattivelli2010diffusion; @sayed2012diffusion] is in the transformed domain regression data $\u_{\ell,i} B_{\ell}$ in (\[eq.diff-step2\]) which now have singular covariance matrices. Moreover, there is an additional interpolation step (\[eq.diff-step4\]). $\blacksquare$
\[Re.:MNandMNbRelation\] The proposed diffusion LMS algorithm estimates $N M$ spatially dependent variables $\{h_k^o\}$ using $N_b M$ global invariant coefficients in $w^o$. From the computational complexity and energy efficiency point of view, it seems this is advantageous when the number of nodes, $N$, is greater than the number of basis functions $N_b$. However, even if this is not the case, using the estimated $N_b M$ global invariant coefficients, the algorithm not only can estimate the space-varying parameters at the locations of the $N$ nodes, but can also estimate the space-varying parameters at locations where no measurements are available. Therefore, even when $N<N_b$, the algorithm is still useful as it can perform interpolation. $\blacksquare$
There are different choices for the combination matrices $\{A_1,A_2,C\}$. For example, the choice $A_1=A_2=C=I$ reduces the above diffusion algorithm to the non-cooperative case where each node runs an individual LMS filter without coordination with its neighbors. Selecting $C=I$ simplifies the adaptation step (\[eq.diff-step2\]) to the case where node $k$ uses only its own data $\{\d_k(i),\u_{k,i}\}$ to perform local adaptation. Choosing $A_1=I$ and $A_2=A$, for some left-stochastic matrix $A$, removes the first combination step and the algorithm reduces to an adaptation step followed by combination (this variant of the algorithm has the Adapt-then-Combine or ATC diffusion structure) [@cattivelli2010diffusion; @sayed2012diffusion]. Likewise, choosing $A_1=A$ and $A_2=I$ removes the second combination step and the algorithm reduces to a combination step followed by adaptation (this variant has the Combine-then-Adapt (CTA) structure of diffusion [@cattivelli2010diffusion; @sayed2012diffusion]). Often in practice, either the ATC or CTA version of the algorithm is used with $C$ set to $C=I$ such as using the following ATC diffusion version described by equations (\[eq.atc-step1\])–(\[eq.atc-step3\]).
$$\begin{aligned}
&{\bpsi}_{k,i}=\w_{k,i-1}+\mu_k B_{k}^T \u_{k,i}^T\big(\d_{k}(i)-\u_{k,i}B_{k}\w_{k,i-1}\big)
\label{eq.atc-step1}\\
&\w_{k,i}=\displaystyle\sum_{\ell\in \mathcal{N}_k}a_{\ell, k}{\bpsi}_{\ell,i}
\label{eq.atc-step2}\\
&\h_{k,i}=B_k \w_{k,i} \label{eq.atc-step3}\end{aligned}$$
Nevertheless for generality, we shall study the performance of Algorithm \[eq.GeneralizedLmsSpaVarParaEst\] for arbitrary matrices $\{A_1,A_2,C\}$ with $C$ right-stochastic and $\{A_1,A_2\}$ left-stochastic. The results can then be specialized to various situations of interest, including ATC, CTA, and the non-cooperative case.
The combination matrices $\{A_1,A_2,C\}$ are normally obtained using some well-known available combination rules such as the Metropolis or uniform combination rules [@xiao2006space; @lopes2008diffusion; @cattivelli2010diffusion]. These matrices can also be treated as free variables in the optimization procedure and used to further enhance the performance of the diffusion strategies. Depending on the network topology and the quality of the communication links between nodes, the optimized values of the combination matrices differ from one case to another[@sayed2012diffusion; @zhao2012imperfect; @takahashi2010diffusion; @abdolee2011diffusion].
Performance Analysis {#sec.:performance_analysis}
====================
In this section, we analyze the performance of the diffusion strategy (\[eq.diff-step1\])-(\[eq.diff-step4\]) in the mean and mean-square sense and derive expressions to characterize the network mean-square deviation (MSD) and excess mean-square error (EMSE). In the analysis, we need to consider the fact that the covariance matrices $\{\bar{R}_{u,k}\}_{k=1}^N$ defined in (\[eq.:bar\_Ru\]) are now rank-deficient since we have $N_b>1$. We explain in the sequel the ramifications that follow from this rank-deficiency.
Mean Convergence
----------------
We introduce the local weight-error vectors \_[k,i]{}w\^o- \_[k,i]{}, \_[k,i]{} w\^o-\_[k,i]{},\_[k,i]{}w\^o-\_[k,i]{} and define the network error vectors: $$\begin{aligned}
&\tilde{\bphi}_i \triangleq \col\{\tilde{\bphi}_{1,i},\ldots,\tilde{\bphi}_{N,i}\}\\
&\tilde{\bpsi}_i \triangleq \col\{\tilde{\bpsi}_{1,i},\ldots,\tilde{\bpsi}_{N,i}\}\\
&\tilde \w_i \triangleq \col\{\tilde \w_{1,i},\ldots,\tilde \w_{N,i}\}\end{aligned}$$ We collect the estimates from across the network into the block vector: $$\begin{aligned}
&\w_i\triangleq \col\{ \w_{1,i},\ldots, \w_{N,i}\}
\label{eq.:bomegai}\end{aligned}$$ and introduce the following extended combination matrices: $$\begin{aligned}
{\mathcal A_1}& \triangleq A_1\otimes I_{MN_b} \\
{\mathcal A_2}& \triangleq A_2\otimes I_{MN_b} \\
{\mathcal C}& \triangleq C\otimes I_{MN_b}\end{aligned}$$ We further define the block diagonal matrices and vectors: $$\begin{aligned}
\boldsymbol{\cal R}_i& \triangleq \text{diag}\Big \{ \small \sum_{\ell \in {\cal N}_k} c_{\ell, k} B_{\ell}^T \u_{\ell,i}^T \u_{\ell,i} B_{\ell}
: k=1,\cdots,N \normalsize \Big \}\\
{\mathcal M}& \triangleq \text{diag}\big \{ \mu_1 I_{MN_b},\ldots, \mu_N I_{MN_b} \big \} \label{eq.:calM}\\
\boldsymbol{t}_i& \triangleq \col\Big \{\sum_{\ell \in {\cal N}_k} c_{\ell, k} B_{\ell}^T \u_{\ell,i}^T \d_{\ell}(i): k=1,\cdots,N \Big \}\\
\boldsymbol{g}_i& \triangleq {\mathcal C}^T \, \col\big \{B_1^T \u_{1,i}^T \v_1(i),\cdots, B_N^T \u_{N,i}^T \v_N(i) \big \}\end{aligned}$$ and introduce the expected values of $\boldsymbol{\cal R}_i$ and $\boldsymbol{t}_i$: $$\begin{aligned}
&{\cal R}\triangleq \E[\boldsymbol{\cal R}_i] =\text{diag}\big \{{R}_1,\cdots,{R}_N\big\}
\label{eq.:Rcal_rank_defficientDiff}\\
&r \triangleq \E[\boldsymbol{{t}}_i]=\col\big \{r_1,\cdots, r_N\big\}
\label{eq.:Rdcal_rank_defficientDiff}\end{aligned}$$ where R\_k \_[\_k]{} c\_[,k]{} |[R]{}\_[u,]{} \[eq.:localR\] r\_[k]{}\_[\_k]{} c\_[,k]{} |[r]{}\_[du,]{} \[eq.:localr\] We also introduce an indicator matrix operator, denoted by $\Ind(\cdot)$, such that for any real-valued matrix $X$ with $(k,j)$-th entry $X_{k,j}$, the corresponding entry of $Y=\Ind(X)$ is: $$Y_{k,j}=\left\{
\begin{array}{l l}
1, \qquad \text{if}\;X_{k,j}>0\\
0, \qquad \text{otherwise}
\end{array} \right.$$ Now from (\[eq.diff-step1\])–(\[eq.diff-step3\]), we obtain: $$\begin{aligned}
&\w_i= \boldsymbol{\mathcal{B}}_i \w_{i-1}+{\mathcal A}^T_2{\mathcal M}\boldsymbol{t}_i
\label{eq.:network_vector_update}\end{aligned}$$ where \_i \_2\^T(I-[M]{}\_i)[A]{}\^T\_1 In turn, making use of (\[eq.:measurement\_linear\_model2\]) in (\[eq.:network\_vector\_update\]), we can verify that the network error vector follows the recursion $$\begin{aligned}
\tilde \w_i=\boldsymbol{\mathcal{B}}_i \tilde \w_{i-1}-{\mathcal A}_2^T{\mathcal M} \boldsymbol{g}_i
\label{eq.:global_error_vector_w}\end{aligned}$$ By taking the expectation of both sides of (\[eq.:global\_error\_vector\_w\]) and using , we arrive at: $$\begin{aligned}
\E[\tilde \w_i]={\cal B}\, \E[\tilde \w_{i-1}]
\label{eq.:mean_perfomance}\end{aligned}$$ where in this relation: $$\begin{aligned}
{\cal B}\triangleq \E[\boldsymbol{\mathcal{B}}_i]={\mathcal A}_2^T(I-{\mathcal M}{\cal R}){\mathcal A}^T_1
\label{eq.:calB}\end{aligned}$$ To obtain (\[eq.:mean\_perfomance\]), we used the fact that the expectation of the second term in (\[eq.:global\_error\_vector\_w\]), i.e., $\E[{\mathcal A}^T_2{\mathcal M} \boldsymbol{g}_i]$, is zero because $ \v_k(i)$ is independent of $ \u_{k,i}$ and $\E[ \v_k(i)]=0$. The rank-deficient matrices $\{\bar{R}_{u,k}\}$ appear inside ${\cal R}$ in (\[eq.:calB\]). We now verify that despite having rank-deficient matrix ${\cal R}$, recursion (\[eq.:mean\_perfomance\]) still guarantees a bounded mean error vector in steady-state.
To proceed, we introduce the eigendecomposition: $$R_k=Q_k\Lambda_k Q_k^T$$ where $Q_k=[q_{k,1},\cdots,q_{k,MN_b}]$ is a unitary matrix with column eigenvectors $q_{k,j}$ and $\Lambda_k=\diag\{\lambda_k(1),\cdots,\lambda_k({MN_b})\}$ is a diagonal matrix with eigenvalues $\lambda_k(j)\geq 0$. For this decomposition, we assume that the eigenvalues of $R_k$ are arranged in descending order, i.e, $\lambda_{\max}(R_k) \triangleq \lambda_k(1) \geq\lambda_k(2)\geq\cdots\geq\lambda_k(MN_b)$, and the rank of $R_k$ is $L_k\leq MN_b$. If we define ${\cal Q}\triangleq \text{diag}\{Q_1,\ldots
,Q_N\}$ and $\Lambda\triangleq \diag \{\Lambda_1,\cdots ,\Lambda_{N}\}$, then the network covariance matrix, ${\cal R}$, given by (\[eq.:Rcal\_rank\_defficientDiff\]) can be expressed as: $${\cal R}={\cal Q}\Lambda {\cal Q}^T
\label{eq.:EigenDecompositionOfD}$$ We now note that the mean estimate vector, $\E[\tilde{\w}_i]$, expressed by (\[eq.:mean\_perfomance\]) will be asymptotically unbiased if the spectral radius of ${\cal B}$, denoted by $\rho({\cal B})$, is strictly less than one. Let us examine under what conditions this requirement is satisfied. Since $A_1$ and $A_2$ are left-stochastic matrices and ${\cal R}$ is block-diagonal, we have from [@sayed2012diffusion] that: $$\rho({\cal B})=\rho\Big({\cal A}^T_2(I-{\cal M} {\cal R}){\cal A}^T_1\Big)\leq \rho\big(I-{\cal M} {\cal R}\big)
\label{eq.:spectral-inequalities}$$ Therefore, if ${\cal R}$ is positive-definite, then choosing ensures convergence of the algorithm in the mean so that $\E[\tilde{\w}_i]\rightarrow 0$ as $i\rightarrow\infty$. However, when ${\cal R}$ is singular, it may hold that $\rho({\cal B})=1$, in which case choosing the step-sizes according to the above bound guarantees the boundedness of the mean error, $\E[\tilde{\w}_i]$, but not necessarily that it converges to zero. The following result clarifies these observations.
\[lemm.:rank\_deficient\_distributed\_lms\_thm1\] If the step-sizes are chosen to satisfy $$0<\mu_k<\frac{2} {\lambda_{\max}(R_k)}
\label{eq.:step_size_difusion_lms_space_varying}$$ then, under Assumption \[assm.:regressor assumption\], the diffusion algorithm is stable in the mean in the following sense: (a) If $\rho({\cal B})<1$, then $\E[\tilde{\w}_i]$ converges to zero and (b) if $\rho({\cal B})=1$ then $$\begin{aligned}
\lim_{i \rightarrow \infty}\big\|\E[{\tilde \w}_i]\big\|_{b,\infty}&\leq\|I-\Ind(\Lambda)\|_{b,\infty} \, \big \|\E[\tilde{\w}_{-1}]\big \|_{b,\infty}
\label{eq.:mean_perfomance5}\end{aligned}$$ where $\|\cdot\|_{b,\infty}$ stands for the block-maximum norm, as defined in [@sayed2012diffusion; @takahashi2010diffusion].
See Appendix \[apex.:Mean Convergence Proof\].
In what follows, we examine recursion (\[eq.:network\_vector\_update\]) and derive an expression for the asymptotic value of $\E[\w_i]$—see (\[eq.:lim-bomegai2\]) further ahead. Before doing so, we first comment on a special case of interest, namely, result (\[eq.:network\_global\_decoupled\_solutionA1A2I\_PSD\]) below.
[*Special case*]{}: Consider a network with ${A}_1={A}_2=I$ and an arbitrary right stochastic matrix $C$ satisfying (\[eq.sthocastic\_matrix\_c\_conditions\]). Using (\[eq.:measurement\_linear\_model2\]) and (\[eq.:localR\])-(\[eq.:localr\]), it can be verified that the following linear system of equations holds at each node $k$: R\_k w\^o = r\_k \[eq.:NodeNormalEquation\] We show in Appendix \[apex.:error-bound-A1A2I\] that under condition (\[eq.:step\_size\_difusion\_lms\_space\_varying\]) the mean estimate of the diffusion LMS algorithm at each node $k$ will converge to: $$\begin{aligned}
\lim_{i\rightarrow \infty}\E[\w_{k,i}]=R_k^{\dag} r_k+\sum_{n={L_k+1}}^{MN_b} q_{k,n}q_{k,n}^T \E[\w_{k,-1}]
\label{eq.:network_global_decoupled_solutionA1A2I_PSD}\end{aligned}$$ where $R_k^{\dag}$ represents the pseudo-inverse of $R_k$, and $\w_{k,-1}$ is the node initial value. This result is consistent with the mean estimate of the stand-alone LMS filter with rank-deficient input data (which corresponds to the situation $A_1=A_2=C=I$)[@mclernon2009convergence]. Note that $R_k^{\dag} r_k$ in (\[eq.:network\_global\_decoupled\_solutionA1A2I\_PSD\]) corresponds to the minimum-norm solution of $R_k w=r_k$. Therefore, the second term on the right hand side of (\[eq.:network\_global\_decoupled\_solutionA1A2I\_PSD\]) is the deviation of the node estimate from this minimum-norm solution. The presence of this term after convergence is due to the zero eigenvalues of $R_k$. If $R_k$ were full-rank so that $L_k=MN_b$, then this term would disappear and the node estimate will converge, in the mean, to its optimal value, $w^o$. We point out that even though the matrices $\bar{R}_{u,\ell}$ are rank deficient since $N_b>1$, it is still possible for the matrices $R_k$ to be full rank owing to the linear combination operation in (\[eq.:localR\]). This illustrates one of the benefits of employing the right-stochastic matrix $C$. However, if despite using $C$, $R_k$ still remains rank-deficient, the second term on the right-hand side of (\[eq.:network\_global\_decoupled\_solutionA1A2I\_PSD\]) can be annihilated by proper node initialization (e.g., by setting $\E[\w_{k,-1}]=0$). By doing so, the mean estimate of each node will then approach the unique minimum-norm solution, $R_k^{\dag} r_k$.
[*General case*]{}: Let us now find the mean estimate of the network for arbitrary left-stochastic matrices $A_1$ and $A_2$. Considering definitions (\[eq.:Rcal\_rank\_defficientDiff\])-(\[eq.:Rdcal\_rank\_defficientDiff\]) and relation (\[eq.:NodeNormalEquation\]) and noting that ${\cal A}^T_1 ({\mathbb 1} \otimes w^o)= {\cal A}^T_2 ({\mathbb 1} \otimes w^o)=({\mathbb 1} \otimes w^o)$, it can be verified that $({\mathbb 1} \otimes w^o)$ satisfies the following linear system of equations: $$\begin{aligned}
(I-{\cal B}) ({\mathbb 1} \otimes w^o) ={\mathcal A}^T_2{\mathcal M}r
\label{eq.:network_estimate_rank_def1}\end{aligned}$$ This is a useful intermediate result that will be applied in our argument.
Next, if we iterate recursion (\[eq.:network\_vector\_update\]) and apply the expectation operator, we then obtain =\^[i+1]{} +\_[j=0]{}\^[i]{}[B]{}\^j[ A]{}\^T\_2[M]{}r \[eq.:w\_iteratedSolutionRank-deficient\] The mean estimate of the network can be found by computing the limit of this expression for $i\rightarrow \infty$. To find the limit of the first term on the right hand side of (\[eq.:w\_iteratedSolutionRank-deficient\]), we evaluate $\lim_{i\rightarrow \infty}{\mathcal{B}}^{i}$ and find conditions under which it converges. For this purpose, we introduce the Jordan decomposition of matrix ${\cal B}$ as [@meyer2000matrix]: $$\begin{aligned}
{\cal B}&={\cal Z}\Gamma{\cal Z}^{-1}
\label{eq.:jordan-decompostion}\end{aligned}$$ where ${\cal Z}$ is an invertible matrix, and $\Gamma$ is a block diagonal matrix of the form $$\begin{aligned}
\Gamma=\diag \Big \{ \Gamma_{1},\Gamma_{2},\cdots,\Gamma_{s}\Big\}\end{aligned}$$ where the $l$-th Jordan block, $\Gamma_{l}\in {\mathbb C}^{m_l \times m_l}$, can be expressed as: \_l=\_l I\_[m\_l]{}+N\_[m\_l]{} \[eq.:block-decomposition\] In this relation, $N_{m_l}$ is some nilpotent matrix of size $m_l\times m_l$. Using decomposition (\[eq.:jordan-decompostion\]), we can express ${\cal B}^i$ as $$\begin{aligned}
{\mathcal{B}}^{i}&={\cal Z}\Gamma^i{\cal Z}^{-1}\end{aligned}$$ Since $\Gamma$ is block diagonal, we have $$\begin{aligned}
\Gamma^i=\diag \Big \{\Gamma_1^i,\Gamma_2^i,\cdots,\Gamma_s^i\Big\}\end{aligned}$$ From this relation, it is deduced that $\lim_{i\rightarrow \infty} {\cal B}^i$ exists if $\lim_{i \rightarrow \infty} \Gamma_l^i$ exists for all $l\, \in\{1,\cdots,s\}$. Using (\[eq.:block-decomposition\]), we can write [@meyer2000matrix]: $$\begin{aligned}
\lim_{i\rightarrow \infty} \Gamma_l^i=\lim_{i\rightarrow \infty} \gamma_l^{i-m_l}\Bigg(\gamma_l^{{\scriptscriptstyle }m_l} I_{ m_{{\scriptscriptstyle }l}}+\sum_{p=1}^{m_l-1} \dbinom{i}{p} \gamma_l^{m_l-p}N_{{\scriptscriptstyle }m_l}^p \Bigg)
\label{eq.:limGammaL}\end{aligned}$$ When ${i\rightarrow \infty}$, $\gamma_l^{i-m_l}$ becomes the dominant factor in this expression. Note that under condition (\[eq.:step\_size\_difusion\_lms\_space\_varying\]), we have $\rho({\cal B})\leq 1$ which in turn implies that the magnitude of the eigenvalues of $\cal B$ are bounded as $0 \leq |\gamma_n|\leq 1$. Without loss of generality, we assume that the eigenvalues of ${\cal B}$ are arranged as $|\gamma_1|\leq \cdots \leq |\gamma_{{\scriptscriptstyle }{L}}| < |\gamma_{{\scriptscriptstyle }{L+1}}|=\cdots=|\gamma_{ s}|=1$. Now we examine the limit (\[eq.:limGammaL\]) for every $|\gamma_l|$ in this range. Clearly for $|\gamma_l|<1$, the limit is zero (an obvious conclusion since in this case $\Gamma_l$ is a stable matrix). For $|\gamma_l|=1$, the limit is the identity matrix if $\gamma_l=1$ and $m_l=1$. However, the limit does not exist for unit magnitude complex eigenvalues and eigenvalues with value , even when $m_l=1$. Motivated by these observations, we introduce the following definition.
[**Definition**]{}: We refer to matrix ${\cal B}$ as *power convergent* if (a) its eigenvalues $\gamma_n$ satisfy $0\leq |\gamma_n| \leq 1$, (b) its unit magnitude eigenvalues are all equal to one, and (c) its Jordan blocks associated with $\gamma_n=1$ are all of size $1\times 1$. $\blacksquare$
[*Example 1*]{}: Assume $N_b=1$, $B_k=I_M$, and uniform step-sizes and covariance matrices across the agents, i.e., $\mu_k\equiv \mu$, $R_{u,k}\equiv R_u$ for all $k$. Assume further that $C$ is doubly-stochastic (i.e., $C^T {\mathbb 1}={\mathbb 1}=C{\mathbb 1})$ and $R_u$ is singular. Then, in this case, the matrix ${\cal B}$ can be written as the Kronecker product ${\cal B}=A_2^TA_1^T\otimes (I_M-\mu R_u)$. For strongly-connected networks where $A_1A_2$ is a primitive matrix, it follows from the Perron-Frobenius Theorem [@horn2003matrix] that $A_1A_2$ has a single unit-magnitude eigenvalue at one, while all other eigenvalues have magnitude less than one. We conclude in this case, from the properties of Kronecker products and under condition (\[eq.:step\_size\_difusion\_lms\_space\_varying\]), that ${\cal B}$ is a power-convergent matrix. $\blacksquare$\
[*Example 2*]{}: Assume $M=2$, $N=3$, $N_b=1$, $B_k=I_M$, and uniform step-sizes and covariance matrices across the agents again. Let $A_2=I=C$ and select A\_1=A=which is not primitive. Let further $R_u=\mbox{\rm diag}\{\beta,0\}$ denote a singular covariance matrix. Then, it can be verified in this case the corresponding matrix ${\cal B}$ will have an eigenvalue with value $-1$ and is not power convergent. $\blacksquare$
Returning to the above definition and assuming ${\cal B}$ is power convergent, then this means that the Jordan decomposition (\[eq.:jordan-decompostion\]) can be rewritten as: =\[\_[Z]{}\] \_ \_[[Z]{}\^[-1]{}]{} \[eq.:cal-B-definition\] where $J$ is a Jordan matrix with all eigenvalues strictly inside the unit circle, and the identity matrix inside $\Gamma$ accounts for the eigenvalues with value one. In (\[eq.:cal-B-definition\]) we further partition ${\cal Z}$ and ${\cal Z}^{-1}$ in accordance with the size of $J$. Using (\[eq.:cal-B-definition\]), it is straightforward to verify that $$\begin{aligned}
\lim_{i \rightarrow \infty} {\cal B}^{i+1}&={\cal Z}_2{\bar{\cal Z}}_2
\label{eq.:Bi-limit}\end{aligned}$$ and if we multiply both sides of (\[eq.:network\_estimate\_rank\_def1\]) from the left by $\bar{\cal Z}_2$, it also follows that $$\begin{aligned}
{\bar{\cal Z}}_2{ \cal A}^T_2{\mathcal M}r&=0
\label{eq.:barCalZ2-Mr}\end{aligned}$$ Using these relations, we can now establish the following result, which describes the limiting behavior of the weight vector estimate.
\[lemm.:mean estimate-general\] If the step-sizes $\{\mu_1,\cdots,\mu_N\}$ satisfy (\[eq.:step\_size\_difusion\_lms\_space\_varying\]) and matrix ${\cal B}$ is power convergent, then the mean estimate of the network given by (\[eq.:w\_iteratedSolutionRank-deficient\]) asymptotically converges to: $$\begin{aligned}
\lim_{i\rightarrow \infty} &\E[\w_i]=({\cal Z}_2{\bar{\cal Z}}_2)\, \E[\w_{-1}]+(I-{\cal B})^{-} {\cal A}^T_2{\mathcal M}r
\label{eq.:lim-bomegai2}\end{aligned}$$ where the notation $X^-$ denotes a (reflexive) generalized inverse for the matrix $X$. In this case, the generalized inverse for $I-{\cal B}$ is given by (I-[B]{})\^[-]{} = [Z]{}\_1 (I-J)\^[-1]{}|[[Z]{}]{}\_1 which is in terms of the factors $\{{\cal Z}_1,\bar{\cal Z}_1,J\}$ defined in (\[eq.:cal-B-definition\]).
See Appendix \[apex.:mean estimate-general\].
We also argue in Appendix \[apex.:mean estimate-general\] that the quantity on the right-hand side of (\[eq.:lim-bomegai2\]) is invariant under basis transformations for the Jordan factors $\{{\cal Z}_1,\bar{\cal Z}_1,{\cal Z}_2,\bar{\cal Z}_2\}$. It can be verified that if $A_1=A_2=I$ then ${\cal B}$ will be symmetric and the result (\[eq.:lim-bomegai2\]) will reduce to (\[eq.:network\_global\_decoupled\_solutionA1A2I\_PSD\]). Now note that the first term on the right hand side of (\[eq.:lim-bomegai2\]) is due to the zero eigenvalues of $I-{\cal B}$. From this expression, we observe that different initialization values generally lead to different estimates. However, if we set $\E[\w_{-1}]=0$, the algorithm converges to: \_[i]{}=(I-[B]{})\^[-]{} [A]{}\^T\_2[M]{}r \[eq.:lim-bomegai3\] In other words, the diffusion LMS algorithm will converge on average to a generalized inverse solution of the linear system of equations defined by (\[eq.:network\_estimate\_rank\_def1\]).
When matrix ${\cal B}$ is stable so that $\rho({\cal B})<1$ then the factorization (\[eq.:cal-B-definition\]) reduces to the form ${\cal B}={\cal Z}_1 J {\bar {\cal Z}}_1$ and $I-{\cal B}$ will be full-rank. In that case, the first term on the right hand side of (\[eq.:lim-bomegai2\]) will be zero and the generalized inverse will coincide with the actual matrix inverse so that (\[eq.:lim-bomegai2\]) becomes $$\begin{aligned}
\lim_{i\rightarrow \infty} &\E[\w_i]=(I-{\cal B})^{-1}{ \cal A}^T_2{\mathcal M}r
\label{eq.:lim-bomegaiIBfullRank2}\end{aligned}$$ Comparing (\[eq.:lim-bomegaiIBfullRank2\]) with (\[eq.:network\_estimate\_rank\_def1\]), we conclude that: $$\begin{aligned}
\lim_{i\rightarrow \infty} &\E[\w_i]={\mathbb 1} \otimes w^o\end{aligned}$$ which implies that the mean estimate of each node will be $w^o$. This result is in agreement with the previously developed mean-convergence analysis of diffusion LMS when the regression data have full rank covariance matrices [@sayed2012diffusion].
Mean-Square Error Convergence {#subsec.:Mean-SquareStability}
-----------------------------
We now examine the mean-square stability of the error recursion (\[eq.:global\_error\_vector\_w\]) in the rank-deficient scenario. We begin by deriving an error variance relation as in [@sayed2008; @al2003transient]. To find this relation, we form the weighted square “norm” of (\[eq.:global\_error\_vector\_w\]), and compute its expectation to obtain: $$\begin{aligned}
\E\|\tilde \w_i\|^2_{\Sigma}=\E\big( \|\tilde \w_{i-1}&\|^2_{{\boldsymbol{\Sigma}}'}\big)+\E[\boldsymbol{g}^T_i {\mathcal M} {\mathcal A_2}\Sigma{\mathcal A}^T_2{\mathcal M}\boldsymbol{g}_i]
\label{variance_relation_1}\end{aligned}$$ where $\|x\|^2_{\Sigma}=x^T \Sigma x$ and $\Sigma\geq0$ is an arbitrary weighting matrix of compatible dimension that we are free to choose. In this expression, $$\begin{aligned}
{\boldsymbol{\Sigma}}'={{\mathcal A_1}(I-{\mathcal M}\boldsymbol{\cal R}_i)^T {\mathcal
A_2}\Sigma{\mathcal A}^T_2(I-{\mathcal M}\boldsymbol{\cal R}_i){\mathcal A}^T_1}
\label{eq.SigmaBold}\end{aligned}$$ Under the temporal and spatial independence conditions on the regression data from Assumption \[assm.:regressor assumption\], we can write: $$\begin{aligned}
\E\big(\|\tilde \w_{i-1}&\|^2_{{\boldsymbol{\Sigma}}'}\big)=\E\|\tilde \w_{i-1}\|^2_{\E[{\boldsymbol{\Sigma}}']}\end{aligned}$$ so that (\[variance\_relation\_1\]) becomes: $$\begin{aligned}
\E\|\tilde \w_i\|^2_{\Sigma}=\E\|\tilde \w_{i-1}&\|^2_{{\Sigma}'}+\Tr[\Sigma {\mathcal A}^T_2 {\mathcal M}{\mathcal
G}{\mathcal M}{\mathcal A_2}]
\label{variance_relation_2}\end{aligned}$$ where ${\mathcal G} \triangleq
\E[{\boldsymbol g}_i{\boldsymbol g}^T_i]$ is given by $$\begin{aligned}
{\mathcal G}={\mathcal C}^T \text{diag} \big \{\sigma^2_{v,1}\bar{R}_{u,1},\ldots,\sigma^2_{v,N}{\bar R}_{u,N}\big \}{\mathcal C}
\label{eq.cal-G}\end{aligned}$$ and $$\begin{aligned}
\Sigma'&\triangleq \E [\boldsymbol{\Sigma}']={\cal B}^T\Sigma {\cal B}+O({\cal M}^2)
\approx{\cal B}^T\Sigma {\cal B}
\label{eq.:Sigma'Aprox}\end{aligned}$$ We shall employ (\[eq.:Sigma’Aprox\]) under the assumption of sufficiently small step-sizes where terms that depend on higher-order powers of the step-sizes are ignored. We next introduce \^T\_2[M]{}[G]{} [M]{}[A]{}\_2 \[eq.:calY\] and use (\[variance\_relation\_2\]) to write: $$\begin{aligned}
\E\|{\tilde \w}_i\|^2_{ \Sigma}=\E\|{\tilde \w}_{i-1}\|^2_{\Sigma'}+ \Tr(\Sigma {\cal Y})
\label{eq.:MSE-stability-analysis-1}\end{aligned}$$ From (\[eq.:MSE-stability-analysis-1\]), we arrive at $$\begin{aligned}
\E\|{\tilde \w}_i\|_{\Sigma}^2 =&\E\|\tilde{\w}_{-1}\|^2_{({\cal B}^T)^{i+1}\Sigma {\cal B}^{i+1}}+ \sum_{j=0}^{i}\Tr\Big(({\cal B}^T)^j \Sigma {\cal B}^j {\cal Y}\Big)
\label{eq.:MSE-stability-analysis-2}\end{aligned}$$ To prove the convergence and stability of the algorithm in the mean-square sense, we examine the convergence of the terms on the right hand side of (\[eq.:MSE-stability-analysis-2\]).
In a manner similar to (\[eq.:barCalZ2-Mr\]), it is shown in Appendix \[apex.:mean-square-derivation\] that the following property holds: |[Z]{}\_2 [Y]{}=0,|[Z]{}\_2\^T=0 Exploiting this result, we can arrive at the following statement, which establishes that relation (\[eq.:MSE-stability-analysis-2\]) converges as $i\rightarrow\infty$ and determines its limiting value.
\[lemm.:mean-square-derivation\] Assume the step-sizes are sufficiently small and satisfy (\[eq.:step\_size\_difusion\_lms\_space\_varying\]). Assume also that ${\cal B}$ is power convergent. Under these conditions, relation (\[eq.:MSE-stability-analysis-2\]) converges to $$\begin{aligned}
\lim_{i \rightarrow \infty }\E\|{\tilde \w}_i\|^2_{\Sigma}&=\E\|\tilde{\w}_{-1}\|^2_{({\cal Z}_2\bar{\cal Z}_2)^T \Sigma {\cal Z}_2\bar{\cal Z}_2} \nonumber \\
&\qquad +\big(\vec({\cal Y})\big)^T (I-{\cal F})^{-1} \vec(\Sigma)
\label{eq.:network-steady-state-mean-square}\end{aligned}$$ where (([Z]{}\_1\_1)(JJ)( [|[Z]{}\_1]{}))\^T and factors $\{{\cal Z}_1,\bar{\cal Z}_1,J\}$ are defined in (\[eq.:cal-B-definition\]).
See Appendix \[apex.:mean-square-derivation\].
In a manner similar to the proof at the end of Appendix \[apex.:mean estimate-general\], the term on the right hand side of (\[eq.:network-steady-state-mean-square\]) is invariant under basis transformations on the factors $\{{\cal Z}_1,\bar{\cal Z}_1,{\cal Z}_2,\bar{\cal Z}_2\}$. Note that the first term on the right hand side of (\[eq.:network-steady-state-mean-square\]) is the network penalty due to rank-deficiency. When the node covariance matrices are full rank, then choosing step-sizes according to (\[eq.:step\_size\_difusion\_lms\_space\_varying\]) leads to $\rho({\cal B})<1$. When this holds, then ${\cal B}={\cal Z}_1J \bar{\cal Z}_1$. In this case, the first term on the right hand side of (\[eq.:network-steady-state-mean-square\]) will be zero, and ${\cal F}=({\cal B}\otimes{\cal B})^T$. In this case, we obtain: $$\begin{aligned}
\lim_{i \rightarrow \infty }\E\|{\tilde \w}_i\|^2_{\Sigma}=\big(\vec({\cal Y})\big)^T (I-{\cal F})^{-1} \vec(\Sigma)
\label{eq.:network-steady-state-mean-square-full-rank}\end{aligned}$$ which is in agreement with the mean-square analysis of diffusion LMS strategies for regression data with full rank covariance matrices given in [@cattivelli2010diffusion; @sayed2012diffusion].
Learning Curves
---------------
For each $k$, the MSD and EMSE measures are defined as: $$\begin{aligned}
\eta_k&=\lim_{i \rightarrow \infty } \E\|\tilde \h_{k,i}\|^2=\lim_{i \rightarrow \infty }\E\|\tilde \w_{k,i}\|^2_{B_k^T B_k} \\
\zeta_k&=\lim_{i \rightarrow \infty } \E\| \u_{k,i}\tilde \h_{k,i-1}\|^2=\lim_{i \rightarrow \infty } \E\|\tilde \w_{k,i-1}\|^2_{{\bar R}_{u,k}}\end{aligned}$$ where $\tilde{\h}_{k,i}=h^o_k- \h_{k,i}$. These parameters can be computed from the network error vector (\[eq.:network-steady-state-mean-square\]) through proper selection of the weighting matrix $\Sigma$ as follows: $$\begin{aligned}
\eta_k=\lim_{i \rightarrow \infty } \E\|{\tilde \w}_i\|^2_{\Sigma_{\msd_k}},
\quad \zeta_k=\lim_{i \rightarrow \infty } \E\|{\tilde \w}_{i-1}\|^2_{\Sigma_{\emse_k}},\end{aligned}$$ where \_[\_k]{}=(e\_k)(B\_k\^T B\_k), \_[\_k]{}=(e\_k) \_[u,k]{} \[eq.:Sigma-MSD-EMSE\] and $\{e_k \}_{k=1}^N$ denote the vectors of a canonical basis set in $N$ dimensional space. The network MSD and EMSE measures are defined as $$\begin{aligned}
\eta_{\net}=\frac{1}{N}\sum_{k=1}^N \eta_k, \qquad \zeta_{\net}=\frac{1}{N}\sum_{k=1}^N\zeta_k
\label{eq.:steady_emse_msd_network}\end{aligned}$$ We can also define MSD and EMSE measures over time as $$\begin{aligned}
\eta_k(i) &=& \E\|\tilde{\h}_{k,i}\|^2=\E\|\tilde{\w}_i\|^2_{\Sigma_{\msd_k}}\\
\zeta_k(i) &=& \E\|\u_{k,i}\tilde{\h}_{k,i-1}\|^2=
\E\|\tilde{\w}_{i-1}\|^2_{\Sigma_{\emse_k}}\end{aligned}$$ Using (\[eq.:MSE-stability-analysis-2\]), it can be verified that these measures evolve according to the following dynamics: $$\begin{aligned}
&\eta_k(i)= \eta_k(i-1)-\|w^o\|_{{\cal H}^i(I- {\cal H}) \sigma_{\msd_k}}+ \alpha^T {\cal H}^i \sigma_{\msd_k}
\label{eq.:transient_msd_nodes}\end{aligned}$$ $$\begin{aligned}
&\zeta_k(i)= \zeta_k(i-1)-\|w^o\|_{ {\cal H}^i(I- {\cal H}) \sigma_{\emse_k}}+ \alpha^T {\cal H}^{i} \sigma_{\emse_k}
\label{eq.:transient_emse_nodes}\end{aligned}$$ where $$\begin{aligned}
{\cal H}&=&({\cal B}\otimes{\cal B})^T \\
\alpha&=&\vec({\cal Y})\\
\sigma_{\msd_k}&=&\vec(\Sigma_{\msd_k})\\
\sigma_{\emse_k}&=&\vec(\Sigma_{\emse_k})\end{aligned}$$ To obtain (\[eq.:transient\_msd\_nodes\]) and (\[eq.:transient\_emse\_nodes\]), we set $\E[\w_{k,-1}]=0$ for all $k$.
Computer Experiments {#sec.:results}
====================
In this section, we examine the performance of the diffusion strategy (\[eq.diff-step1\])-(\[eq.diff-step4\]) and compare the simulation results with the analytical findings. In addition, we present a simulation example that shows the application of the proposed algorithm in the estimation of space-varying parameters for a physical phenomenon modeled by a PDE system over two spatial dimensions.
Performance of the Distributed Solution
---------------------------------------
We consider a one-dimensional network topology, illustrated by Fig. \[fig.:network-topology-and-parameters\], with $L=1$ and equally spaced nodes along the $x$ direction. We choose $A_1$ as the identity matrix, and compute $A_2$ and $C$ based on the uniform combination and Metropolis rules [@cattivelli2010diffusion; @sayed2012diffusion], respectively. We choose $M=2$ and $N_b=5$ and generate the unknown global parameter $w^o$ randomly for each experiment. We obtain $B_k$ using the shifted Chebyshev polynomials given by (\[eq.:ShiftedCountinuesChebyshev\]) and compute the space varying parameters $h^o_k$ according to (\[eq.:local\_global\_relation\]). The measurement data $\d_k(i),\, k \in \{1,2,\cdots,N\}$ are generated using the regression model (\[eq.:state\_dependent\_regression\]). The SNR for each node $k$ is computed as $\text{SNR}_k=\E\| \u_{k,i}{h^o_k}\|^2/\sigma^2_{v,k}$. The noise and the entries of the regression data are white Gaussian and satisfy Assumption \[assm.:regressor assumption\]. The noise variances, $\{\sigma^2_{v,k}\}$, and the trace of the covariance matrices, $\{\Tr(R_{u,k})\}$, are uniformly distributed between $[0.05,0.1]$ and $[1,5]$, respectively.
Figure \[fig.:network-MSD-N=4\] illustrates the simulation results for a network with $N=4$ nodes. For this experiment, we set $\mu_k=0.01$ for all $k$ and initialize each node at zero. In the legend of the figure, we use the subscript $h$ to denote the MSD for ${\tilde \h}_{k,i}$ and the subscript $w$ to refer to the MSD of ${\tilde \w}_{k,i}$. The simulation curves are obtained by averaging over $300$ independent runs. it can be seen that the simulated and theoretical results match well in all cases. To obtain the analytical results, we use expression (\[eq.:network-steady-state-mean-square\]) to assess the steady-state values and expression (\[eq.:transient\_msd\_nodes\]) to generate the theoretical learning curves.
Two important points in Fig. \[fig.:network-MSD-N=4\] need to be highlighted. First, note from the top plot that the network MSD for ${\tilde \w}_{k,i}$ is larger than that for ${\tilde \h}_{k,i}$. This is because \_[k,i]{}\^2=\_[k,i]{}\^2\_[B\_k\^T B\_k]{} \[eq.:msdh-msdw-relation\] so that the MSD of ${\tilde \h}_{k,i}$ is a weighted version of the MSD of ${\tilde \w}_{k,i}$. In this experiment, the weighting leads to a lower estimation error. Second, note from the bottom plot that while the MSD values of $\tilde{\w}_{k,i}$ are largely independent of the node index, the same is not true for the MSD values of $\tilde{\h}_{k,i}$. In previous studies on diffusion LMS strategies, it has been shown that, for strongly-connected networks, the network nodes approach a uniform MSD performance level [@sayed2013DiffusionMagazine]. The result in Fig. \[fig.:MSDTransientN4Nb5\] supports this conclusion where it is seen that the MSD of ${\tilde \w}_{k,i}$ for nodes 2 and 4 converge to the same MSD level. However, note that the MSD of ${\tilde \h}_{k,i}$ is different for nodes 2 and 4. This difference in behavior is due to the difference in weighting across nodes from (\[eq.:msdh-msdw-relation\]).
Comparison with Centralized Solution
------------------------------------
We next compare the performance of the diffusion strategy (\[eq.diff-step1\])-(\[eq.diff-step4\]) with the centralized solution (\[eq.:centralized-step1\])–(\[eq.:centralized-step2\]). We consider a network with $N=10$ nodes with the topology illustrated by Fig. \[fig.:network-topology-and-parameters\]. In this experiment, we set $\mu_k=0.02$ for all $k$, while the other network parameters are obtained following the same construction described for Fig. \[fig.:network-MSD-N=4\]. As the results in Fig. \[fig.:network-MSD-N=10\] indicate, the diffusion and centralized LMS solutions tend to the same MSD performance level in the $w$ domain. This conclusion is consistent with prior studies on the performance of diffusion strategies in the full-rank case over strongly-connected networks [@sayed2013DiffusionMagazine]. However, discrepancies in performance are seen between the distributed and centralized implementations in the $h$ domain, and the discrepancy tends to become larger for larger values of $N$. This is because, in moving from the $w$ domain to the $h$ domain, the inherent aggregation of information that is performed by the centralized solution leads to enhanced estimates for the $h_{k}^o$ variables. For example, if the estimates $\w_{k,i}$ which are generated by the distributed solution are averaged prior to computing the $\h_{k,i}$, then it can be observed that the MSD values of ${\tilde \h}_{k,i}$ for both the centralized and the distributed solution will be similar.
In these experiments, we also observe that if we increase the number of basis functions, $N_b$, then both the centralized and diffusion algorithms will converge faster but their steady-state MSD performance will degrade. Therefore, in choosing the number of basis functions, $N_b$, there is a trade off between convergence speed and MSD performance.
Example: Two-Dimensional Process Estimation {#subsec.:Diffusion LMS for Process Estimation}
-------------------------------------------
In this example, we consider a two-dimensional network with $13 \times 13$ nodes that are equally spaced over the unit square $(x,y) \in [0, 1]\times[0, 1]$ with $\Delta x=\Delta y=1/12$ (see Fig. \[fig.:2DNetworkTopology\]). This network monitors a physical process $f(x,y)$ described by the Poisson PDE: $$\begin{aligned}
\frac{\partial^2 f(x,y)}{\partial x^2}+\frac{\partial^2 f(x,y)}{\partial y^2}=h(x,y)
\label{eq.:TwoDimensionalPoissonProcess}\end{aligned}$$ where $h(x,y): [0,\, 1]^2 \rightarrow \amsbb{R}$ is an unknown input function. The PDE satisfies the following boundary conditions: $$\begin{aligned}
f(x,0)=f(0,y)=f(x,1)=f(1,y)=0 \nonumber\end{aligned}$$ For this problem, the objective is to estimate $h(x,y)$, given noisy measurements collected by $N=N_x \times N_y=11 \times 11$ nodes corresponding to the [*interior points*]{} of the network. To discretize the PDE, we employ the finite difference method (FDM) with uniform spacing of $\Delta x$ and $\Delta y$. We define $x_{k_1}\triangleq k_1\Delta x$, $y_{k_2}\triangleq k_2 \Delta y$ and introduce the sampled values ${f}_{k_1,k_2}\triangleq f(x_{k_1} ,y_{k_2})$ and $h^o_{k_1,k_2}\triangleq h(x_{k_1},y_{k_2})$. We use the central difference scheme [@thomas1995numerical] to approximate the second order partial derivatives: $$\begin{aligned}
&\frac{{\partial}^2 f(x,y,t)}{\partial x^2}\approx\frac{1}{\Delta x^2}[f_{k_1+1,k_2}-2f_{k_1,k_2}+f_{k_1-1,k_2}] \\
&\frac{{\partial}^2 f(x,y,t)}{\partial y^2}\approx\frac{1}{\Delta y^2}[f_{k_1,k_2+1}-2f_{k_1,k_2}+f_{k_1,k_2-1}]
\label{eq.:space_deravative}\end{aligned}$$ This leads to the following discretized input function: $$\begin{aligned}
h^o_{k_1,k_2}=&\frac{1}{\Delta x^2}\big(f_{k_1+1,k_2}+f_{k_1,k_2+1}+f_{k_1-1,k_2}\nonumber \\
&\qquad+f_{k_1,k_2-1}-4f_{k_1,k_2}\big)
\label{eq.:TwoDimensionalPoissonProcess_discretized}\end{aligned}$$ For this example, the unknown input process is h\^o\_[k\_1,k\_2]{}=e\^[-((k\_1-4)\^2+(k\_2-4)\^2)]{}-5e\^[-((k\_1-8)\^2+(k\_2-8)\^2)]{}+1 where $\kappa=(N_x-1)^2/4$.
![Spatial distribution of SNR over the network.[]{data-label="fig.:networkSNR"}](./NetworkSnr){width="4.5cm" height="3.4cm"}
To obtain $f_{k_1,k_2}$, we solve (\[eq.:TwoDimensionalPoissonProcess\]) using the Jacobi over-relaxation method [@bertsekas1989parallel]. Figure \[fig.:f\_distribution\] illustrates the values of $f_{k_1,k_2}$ over the spatial domain. For the estimation of $h_{k_1,k_2}$, the given information are the noisy measurement samples $\z_{k_1,k_2}(i)= f_{k_1,k_2}+{\n}_{k_1,k_2}(i)$. In this relation, the noise process ${\n}_{k_1,k_2}(i)$ is zero mean, temporally white and independent over space. For this network, the two dimensional reference signal is the distorted version of $h^o_{k_1,k_2}$ which is represented by $\d_{k_1,k_2}(i)$. The reference signal is obtained from (\[eq.:TwoDimensionalPoissonProcess\_discretized\]) with $f_{k_1,k_2}$ replaced by their noisy measured samples $\z_{k_1,k_2}(i)$, i.e., $$\begin{aligned}
\d_{k_1,k_2}(i)=&\frac{1}{\Delta x^2} \Big(\z_{k_1+1,k_2}(i)+\z_{k_1,k_2+1}(i)+\z_{k_1-1,k_2}(i) \nonumber \\
&\qquad+\z_{k_1,k_2-1}(i)-4\z_{k_1,k_2}(i)\Big)
\label{eq.:reference-signal2d}\end{aligned}$$
According to (\[eq.:reference-signal2d\]), the linear regression model for this problem takes the following form: $$\begin{aligned}
\d_{k_1,k_2}(i)=&\u_{k_1,k_2}(i) h^o_{k_1,k_2}+{\v}_{k_1,k_2}(i)
\label{eq.:2D-linear-model}\end{aligned}$$ where $\u_{k_1,k_2}(i)=1$. Therefore, in this example, we are led to a linear model (\[eq.:2D-linear-model\]) with [*deterministic*]{} as opposed to random regression data. Although we only studied the case of random regression data in this article, this example is meant to illustrate that the diffusion strategy can still be applied to models involving deterministic data in a manner similar to [@cattivelli2008diffusion; @cattivelli2011distributed].
To represent $h^o_{k_1,k_2}$ as a space-invariant parameter vector, we use two-dimensional shifted Chebyshev basis functions [@mukundan2001image]. Using this representation, $h^o_{k_1,k_2}$ can be expressed as: $$\begin{aligned}
h^o_{k_1,k_2}=\sum_{n=1}^{N_b} w^o_{n}\, p_{n,k_1,k_2}
\label{eq.:2d_parameter_interpolation}\end{aligned}$$ where each element of the two-dimensional basis set is: $$\begin{aligned}
p_{n,k_1,k_2}=b_{n_1,k_1}b_{n_2,k_2}
\label{eq.:2d_chebyshev function}\end{aligned}$$ where $\{b_{n_1,k_1}\}$ and $\{b_{n_2,k_2}\}$ are the one-dimensional shifted Chebyshev polynomials in the $x$ and $y$ directions, respectively–recall (\[eq.:b-nk\]). In the network, each interior node communicates with its four immediate neighbors. We use $A_1=I$ and compute $C$ and $A_2$ by using the Metropolis and relative degree rules [@lopes2008diffusion; @cattivelli2010diffusion; @sayed2012diffusion]. All nodes are initialized at zero and $\mu_k=0.01$ for all $k$. The signal-to-noise ratio (SNR) of the network is uniformly distributed in the range $[20,30]$dB and is shown in Fig. \[fig.:networkSNR\].
Figures \[fig.:trueSourceValue\] and \[fig.:EstimatedSourceValue\] show three dimensional views of the true and estimated input process using the proposed diffusion LMS algorithm after $3000$ iterations. Figure \[fig.:SourceEstimationMSDPerformance\] illustrates the MSD of the estimated source, i.e., .
![Network steady-state MSD performance in dB.[]{data-label="fig.:SourceEstimationMSDPerformance"}](./Msd3DSteadyCounturSim){width="7cm" height="5.5cm"}
Conclusion {#sec.:conclusion}
==========
By combining interpolation and distributed adaptive optimization, we proposed a diffusion LMS strategy for estimation and tracking of space-time varying parameters over networks. The proposed algorithm can find the space-varying parameters not only at the node locations but also at spaces where no measurement is collected. We showed that if the network experiences data with rank-deficient covariance matrices, the non-cooperative LMS algorithm will converge to different solutions at different nodes. In contrast, the diffusion LMS algorithm is able to alleviate the rank-deficiency problem through its use of combination matrices especially since, as shown by (\[eq.:spectral-inequalities\]), $\rho({\cal B})\leq \rho (I-{\cal M} {\cal R})$, where $I-{\cal M} {\cal R}$ is the coefficient matrix that governs the dynamics of the non-cooperative solution. Nevertheless, if these mechanisms fail to mitigate the deleterious effect of the rank-deficient data, then the algorithm converges to a solution space where the error is bounded. We analyzed the performance of the algorithm in transient and steady-state regimes, and gave conditions under which the algorithm is stable in the mean and mean-square sense.
Mean Error Convergence {#apex.:Mean Convergence Proof}
======================
Based on the rank of ${\cal R}=\diag\{R_1,\cdots,R_N\}$, we have two possible cases:
[a) $R_k>0 \; \forall k \in \{1,\cdots,N\}$]{}: As (\[eq.:mean\_perfomance\]) implies, $\E[\tilde \w_i]$ converges to zero if $\rho({\cal B})<1$. In [@sayed2012diffusion], it was shown that when ${\cal R}>0$, choosing the step-sizes according to (\[eq.:step\_size\_difusion\_lms\_space\_varying\]) guarantees $\rho({\cal B})<1$.
[ b) $\exists k \in \{1,\cdots,N\}$ for which $R_k$ is rank-deficient]{}: For this case, we first show that \^[i+1]{} \_[b,]{} (I-[M]{})\^[i+1]{} \_[b,]{} \[eq.:meanError-inequality1\] where $\|\cdot\|_{b,\infty}$ denotes the block-maximum norm for block vectors with block entries of size $MN_b \times 1$ and block matrices with blocks of size $MN_b\times MN_b$. To this end, we note that for the left-stochastic matrices $A_1$ and $A_2$, we have $\|{\cal A}_1^T\|_{b,\infty}=\|{\cal A}_2^T\|_{b,\infty}=1$ [@sayed2012diffusion], and use the sub-multiplicative property of the block maximum norm [@zhao2012imperfect] to write: $$\begin{aligned}
\big \|{\cal B}^{i+1} \big \|_{b,\infty}&\leq \|{\cal A}_2^T\|_{b,\infty} \; \|I-{\cal M}{\cal R}\|_{b,\infty}\; \|{\cal A}_1^T\|_{b,\infty} \times \cdots \nonumber \\
& \qquad \times \|{\cal A}_2^T\|_{b,\infty} \; \|I-{\cal M}{\cal R}\|_{b,\infty}\; \|{\cal A}_1^T\|_{b,\infty}
\nonumber \\
&=\big \|I-{\cal M}{\cal R}\big \|^{i+1}_{b,\infty} \label{eq.:meanError-inequality2}\end{aligned}$$ If we introduce the (block) eigendecomposition of ${\cal R}$ (\[eq.:EigenDecompositionOfD\]) into (\[eq.:meanError-inequality2\]) and consider the fact that the block-maximum norm is invariant under block-diagonal unitary matrix transformations [@sayed2012diffusion; @takahashi2010diffusion], then inequality (\[eq.:meanError-inequality2\]) takes the form: $$\begin{aligned}
\big \|{\cal B}^{i+1} \big \|_{b,\infty} & \leq \big \|I-{\mathcal M}{\Lambda}\big \|^{i+1}_{b,\infty}
\label{eq.:meanError-inequality3}\end{aligned}$$ Using the property $\|X\|_{b,\infty}=\rho(X)$ for a block diagonal Hermitian matrix $X$ [@sayed2012diffusion], we obtain: $$\begin{aligned}
\big \|(I-{\cal M}{\Lambda})^{i+1} \big \|_{b,\infty}=&\rho\Big((I-{\cal M}\Lambda)^{i+1}\Big) \nonumber \\
=&\max_{\substack{1 \leq k \leq N \\ 1 \leq n \leq M N_b}} \Big | \big(1-\mu_k \lambda_k(n)\big)^{i+1} \Big | \nonumber \\
=&\Big (\max_{\substack{1 \leq k \leq N \\ 1 \leq n \leq M N_b}} | 1-\mu_k \lambda_k(n)| \Big )^{i+1} \nonumber \\
=&\Big(\rho(I-{\cal M}{\Lambda})\Big)^{i+1} \nonumber \\
=&\big \|I-{\cal M}{\Lambda}\big \|^{i+1}_{b,\infty} \label{eq.:meanError-inequality4}\end{aligned}$$ Using (\[eq.:meanError-inequality4\]) in (\[eq.:meanError-inequality3\]), we arrive at (\[eq.:meanError-inequality1\]). We now proceed to show the boundedness of the mean error for case (b). We iterate (\[eq.:mean\_perfomance\]) to get: $$\begin{aligned}
\E[\tilde \w_i]={\cal B}^{i+1} \E[\tilde{\w}_{-1}]
\label{eq.:mean_perfomance2}\end{aligned}$$ Applying the block maximum norm to (\[eq.:mean\_perfomance2\]) and using inequality (\[eq.:meanError-inequality1\]), we obtain: $$\begin{aligned}
\lim_{i \rightarrow \infty} \big \| \E[\tilde{\w}_i] \big \|_{b,\infty}&\leq \lim_{i \rightarrow \infty} \big \| (I-{\cal M} \Lambda)^{i+1} \big \|_{b,\infty} \; \big \|\E[\tilde{\w}_{-1}]\big \|_{b,\infty}
\label{eq.:mean_perfomance3}\end{aligned}$$ The value of $\lim_{i\rightarrow \infty} \|(I-{\mathcal M}{\Lambda})^{i+1}\|_{b,\infty}$ can be computed by evaluating the limits of its diagonal entries. Considering the step-sizes as in (\[eq.:step\_size\_difusion\_lms\_space\_varying\]), the diagonal entries are computed as: \_[i]{}(1-\_k \_k(n))\^[i+1]{}={
[l l]{} 1, & \_k(n)=0\
0, &
. Therefore, (\[eq.:mean\_perfomance3\]) reads as: $$\begin{aligned}
\lim_{i \rightarrow \infty}\big \|\E[{\tilde \w}_i]\big\|_{b,\infty}&\leq \|I-\Ind(\Lambda)\|_{b,\infty}
\; \big \|\E[\tilde{\w}_{-1}]\big \|_{b,\infty}
\label{eq.:mean_perfomanc6}\end{aligned}$$
Mean Behavior When ($A_1=A_2=I$) {#apex.:error-bound-A1A2I}
================================
Setting $A_1=A_2=I$ in the diffusion recursions (\[eq.diff-step1\])-(\[eq.diff-step3\]) and subtracting $w^o$ from both sides of (\[eq.diff-step2\]), we get: $$\tilde \w_{k,i}=\tilde \w_{k,i-1}-\mu_k\sum_{\ell \in {\cal N}_k}c_{\ell,k} B^T_{\ell} \u^T_{\ell,i}(\d_{\ell}(i)-\u_{\ell,i}B_{\ell} \w_{k,i-1})$$ Under Assumption \[assm.:regressor assumption\] and using $\d_{\ell}(i)=\u_{\ell,i}B_{\ell}w^o+\v_{\ell}(i)$, we obtain: $$\begin{aligned}
\E[\tilde \w_{k,i}]&=Q_k[I-\mu_k \Lambda_k]Q^T_k \, \E[\tilde \w_{k,i-1}]\end{aligned}$$ We define $ \p_{k,i} \triangleq {Q}_k^T \tilde \w_{k,i}$ and start from some initial condition to arrive at $$\begin{aligned}
\E[\p_{k,i}]=[I-\mu_k \Lambda_k]\E[ \p_{k,i-1}]=[I-\mu_k \Lambda_k]^{i+1} \E[\p_{k,-1}] \nonumber\end{aligned}$$ If we choose the step-sizes according to (\[eq.:step\_size\_difusion\_lms\_space\_varying\]) then we get: $$\begin{aligned}
\lim_{i\rightarrow \infty} \E[ \p_{k,i}]=\big[I-\Ind(\Lambda_k)\big] \E [\p_{k,-1}]
\label{eq.:lms_rank_defficient_weight_error_vector}\end{aligned}$$ Equivalently, this can be written as: $$\begin{aligned}
\lim_{i\rightarrow \infty} \E[ {\tilde \w}_{k,i}]=Q_k \big[I-\Ind(\Lambda_k)\big] Q^T_k \, \E[{\tilde \w}_{k,-1}]
\label{eq.:lms_rank_defficient_weight_error_vector2}\end{aligned}$$ This result indicates that the mean error does not grow unbounded. Now from (\[eq.:NodeNormalEquation\]), we can verify that: $$Q_k \Ind(\Lambda_k) Q_k^T w^o = R_k^\dag r_k$$ Then, upon substitution of $\tilde{\w}_{k,i}=w^o-\w_{k,i}$ into (\[eq.:lms\_rank\_defficient\_weight\_error\_vector2\]), we obtain: $$\begin{aligned}
\lim_{i \rightarrow \infty} \E[\w_{k,i}]
&=\small Q_k \Ind(\Lambda_k) Q_k^T w^o+Q_k [I-\Ind(\Lambda_k)]Q_k^T \E[\w_{k,-1}] \normalsize \nonumber \\
&= R_k^{\dagger}r_k+\sum_{n=L_k+1}^{MN_b} q_{k,n} q_{k,n}^T \E[\w_{k,-1}]\end{aligned}$$
Proof of Lemma \[lemm.:mean estimate-general\] {#apex.:mean estimate-general}
==============================================
From (\[eq.:Bi-limit\]), we readily deduce that $$\begin{aligned}
\lim_{i \rightarrow \infty} {\cal B}^{i+1}\E[\w_{-1}]=({\cal Z}_2{\bar{\cal Z}}_2)\, \E[\w_{-1}]
\label{eq.:limit-first-term}\end{aligned}$$ On the other hand, from (\[eq.:cal-B-definition\]), we have \_[i ]{} \_[j=0]{}\^i [B]{}\^[j]{} [A]{}\^T\_2[M]{}r= \_[i ]{}\_[j=0]{}\^i ([Z]{}\_1 J\^[j]{} |[Z]{}\_1+[Z]{}\_2 |[Z]{}\_2) [A]{}\^T\_2[M]{}r \[eq.:limit-second-term-v1\] Using (\[eq.:barCalZ2-Mr\]), the term involving $\bar{\cal Z}_2$ cancels out and the above reduces to $$\begin{aligned}
\lim_{i \rightarrow \infty}\sum_{j=0}^i {\cal B}^{j}{ \cal A}^T_2{\mathcal M}r&=\lim_{i \rightarrow \infty}\sum_{j=0}^i \big({\cal Z}_1 J^{j}{\bar{\cal Z}_1}\big)
{ \cal A}^T_2{\mathcal M}r \nonumber \\
&={\cal Z}_1(I-J)^{-1}{\bar{\cal Z}_1}{ \cal A}^T_2{\mathcal M}r
\label{eq.:limit-second-term-v1}\end{aligned}$$ since $\rho(J)<1$. We now verify that the matrix X\^[-]{}=[Z]{}\_1(I-J)\^[-1]{}[|[Z]{}\_1]{} is a (reflexive) generalized inverse for the matrix $X=(I-{\cal B})$. Recall that a (reflexive) generalized inverse for a matrix $Y$ is any matrix $Y^{-}$ that satisfies the two conditions [@ben2003generalized]: $$\begin{aligned}
YY^{-}Y&=Y
\label{eq.:generalized-inverse-condition-1}\\
Y^{-}YY^{-}&=Y^{-}
\label{eq.:generalized-inverse-condition-2}\end{aligned}$$ To verify these conditions, we first note from ${\cal Z}{\cal Z}^{-1}=I$ and ${\cal Z}^{-1}{\cal Z}=I$ in (\[eq.:cal-B-definition\]) that the following relations hold: $$\begin{aligned}
&{\cal Z}_1{\bar {\cal Z}}_1+{\cal Z}_2{\bar {\cal Z}}_2=I
\label{eq.:Z-Prop-1}\\
&{\bar {\cal Z}}_1{\cal Z}_2=0 \label{eq.:Z-Prop-2} \\
&{\bar {\cal Z}}_2{\cal Z}_1=0 \label{eq.:Z-Prop-3}\\
&{\bar {\cal Z}}_1{\cal Z}_1=I \label{eq.:Z-Prop-4}\\
&{\bar {\cal Z}}_2{\cal Z}_2=I \label{eq.:Z-Prop-5}\end{aligned}$$ We further note that $X$ can be expressed as: X=(I-[B]{})=[Z]{}\_1(I-J)[|[Z]{}\_1]{} It is then easy to verify that the matrices $\{X,X^{-}\}$ satisfy conditions (\[eq.:generalized-inverse-condition-1\]) and (\[eq.:generalized-inverse-condition-2\]), as claimed. Therefore, (\[eq.:limit-second-term-v1\]) can be expressed as: \_[i ]{}\_[j=0]{}\^i [B]{}\^[j]{}[ A]{}\^T\_2[M]{}r=(I-[B]{})\^[-]{} [ A]{}\^T\_2[M]{}r \[eq.:limit-second-term-v2\] Substituting (\[eq.:limit-first-term\]) and (\[eq.:limit-second-term-v2\]) into (\[eq.:w\_iteratedSolutionRank-deficient\]) leads to (\[eq.:lim-bomegai2\]). Let us now verify that the right-hand side of (\[eq.:lim-bomegai2\]) remains invariant under basis transformations for the Jordan factors $\{{\cal Z}_1,\bar{\cal Z}_1,{\cal Z}_2,\bar{\cal Z}_2\}$. To begin with, the Jordan decomposition (\[eq.:cal-B-definition\]) is not unique. Let us assume, however, that we fix the central term $\mbox{\rm diag}\{J,I\}$ to remain invariant and allow the Jordan factors $\{{\cal Z}_1,\bar{\cal Z}_1,{\cal Z}_2,\bar{\cal Z}_2\}$ to vary. It follows from (\[eq.:cal-B-definition\]) that |[Z]{}\_2[B]{}=|[Z]{}\_2,\_2=[Z]{}\_2 so that the columns of ${\cal Z}_2$ and the rows of $\bar{\cal Z}_2$ correspond to right and left-eigenvectors of ${\cal B}$, respectively, associated with the eigenvalues with value one. If we replace ${\cal Z}_2$ by any transformation of the form ${\cal Z}_2 {\cal X}_2$, where ${\cal X}_2$ is invertible, then by (\[eq.:Z-Prop-5\]), $\bar{\cal Z}_2$ should be replaced by ${\cal X}_2^{-1}\bar{\cal Z}_2$. This conclusion can also be seen as follows. The new factor ${\cal Z}$ is given by
= and, hence, the new ${\cal Z}^{-1}$ becomes \^[-1]{}= which confirms that $\bar{\cal Z}_2$ is replaced by ${\cal X}^{-1}_2\bar{\cal Z}_2$. It follows that the product ${\cal Z}_2\bar{\cal Z}_2$ remains invariant under arbitrary invertible transformations ${\cal X}_2$. Moreover, from (\[eq.:cal-B-definition\]) we also have that |[Z]{}\_1[B]{}=J|[Z]{}\_1,\_1=[Z]{}\_1 J Assume we replace ${\cal Z}_1$ by any transformation of the form ${\cal Z}_1 {\cal X}_1$, where ${\cal X}_1$ is invertible, then by (\[eq.:Z-Prop-4\]), $\bar{\cal Z}_1$ should be replaced by ${\cal X}_1^{-1}\bar{\cal Z}_1$. However, since we want to maintain $J$ invariant, then this implies that the transformation ${\cal X}_1$ must also satisfy \_1\^[-1]{} J [X]{}\_1 = J It follows that the product ${\cal Z}_1 (I-J)^{-1} \bar{\cal Z}_1$ remains invariant under such invertible transformations ${\cal X}_1$, since $$\begin{aligned}
{\cal Z}_1 (I-J)^{-1} \bar{\cal Z}_1 &=&
{\cal Z}_1 {\cal X}_1{\cal X}_1^{-1}(I-J)^{-1}{\cal X}_1{\cal X}_1^{-1} \bar{\cal Z}_1\nonumber\\
&=&{\cal Z}_1 {\cal X}_1(I-{\cal X}_1^{-1} J {\cal X}_1)^{-1}{\cal X}_1^{-1} \bar{\cal Z}_1\nonumber\\
&=&{\cal Z}_1 {\cal X}_1(I-J)^{-1} {\cal X}_1^{-1}\bar{\cal Z}_1\end{aligned}$$
PROOF OF LEMMA \[lemm.:mean-square-derivation\] {#apex.:mean-square-derivation}
===============================================
We first establish that $\bar{\cal Z}_2{\cal Y}$ and ${\cal Y}\bar{\cal Z}^T_2$ are both equal to zero. Indeed, we start by replacing $r$ in (\[eq.:barCalZ2-Mr\]) by its expression from (\[eq.:Rdcal\_rank\_defficientDiff\]) and (\[eq.:localr\]) as $r={\cal C}^T \col\{\bar{r}_{du,1},\cdots,\bar{r}_{du,N}\}$ that leads to: |[Z]{}\_2[A]{}\_2\^T [M]{} [C]{}\^T {|[r]{}\_[du,1]{},,|[r]{}\_[du,N]{}} = 0 By further replacing $\bar{r}_{du,k}$ by their values from (\[eq.:bar\_Ru\]), we obtain: |[Z]{}\_2 [A]{}\_2\^T [M]{} [C]{}\^T {B\_1\^T,,B\_N\^T} {r\_[du,1]{}, , r\_[du,N]{}}= 0 \[eq.orthogonality-Z2mr\] This relation must hold regardless of the cross-correlation vectors $\{r_{du,k}\}$. Therefore, |[Z]{}\_2 [A]{}\_2\^T [M]{} [C]{}\^T {B\_1\^T,,B\_N\^T}=0 \[eq.orthogonality-Z2mr2\] We now define ={\^2\_[v,1]{}I\_[MN\_b]{},,\^2\_[v,N]{}I\_[MN\_b]{}} and rewrite expression (\[eq.:calY\]) as $$\begin{aligned}
{\cal Y}&= {\cal A}^T_2{\cal M} {\cal C}^T \diag\{B_1^T,\cdots,B_N^T\} \diag \{R_{u,1},\cdots,R_{u,N}\}\nonumber \\
&\qquad \times \diag\{B_1,\cdots,B_N\}\, {\cal V}\, {\cal C}{\cal M}{\cal A}_2
\label{eq.:calY-v2}\end{aligned}$$ Multiplying this from the left by $\bar{\cal Z}_2$ and comparing the result with (\[eq.orthogonality-Z2mr2\]), we conclude that |[Z]{}\_2[Y]{} =0 \[eq.orthogonality-Z2Y1\] Noting that ${\cal Y}$ is symmetric, we then obtain: |[Z]{}\^T\_2=0 \[eq.orthogonality-Z2Y2\] Returning to recursion (\[eq.:MSE-stability-analysis-2\]), we note first from (\[eq.:cal-B-definition\]) that ${\cal B}$ can be rewritten as =[Z]{}\_1 J [|[Z]{}\_1]{}+[Z]{}\_2[|[Z]{}]{}\_2 \[eq.:cal-B-definition-2\] Since ${\cal B}$ is power convergent, the first term on the right hand side of (\[eq.:MSE-stability-analysis-2\]) converges to $$\begin{aligned}
\lim_{i \rightarrow \infty}\E\|\tilde{\w}_{-1}&\|^2_{({\cal B}^T)^{i+1} \Sigma {\cal B}^{i+1}}=\E\|{\tilde \w}_{-1}\|^2_{({\cal Z}_2\bar{\cal Z}_2)^T \Sigma {\cal Z}_2\bar{\cal Z}_2}
\label{eq.:addtional-zero-initial-v2}\end{aligned}$$ Substituting (\[eq.:cal-B-definition-2\]) into the second term on the right hand side of (\[eq.:MSE-stability-analysis-2\]) and using (\[eq.orthogonality-Z2Y1\]) and (\[eq.orthogonality-Z2Y2\]), we arrive at $$\begin{aligned}
\lim_{i\rightarrow \infty}\sum_{j=0}^{i}\Tr\Big(({\cal B}^T)^j \Sigma {\cal B}^j {\cal Y}\Big)
=&\Tr\Big(\lim_{i\rightarrow \infty} \sum_{j=0}^{i}({\cal Z}_1 J^j {\bar{\cal Z}_1})^T \nonumber \\
&\times\Sigma ({\cal Z}_1 J^j {\bar{\cal Z}_1}){\cal Y}\Big)
\label{eq:mse-second-term-limit-v2}\end{aligned}$$ If matrices $X_1$, $X_2$ and $\Sigma$ are of compatible dimensions, then the following relations hold[@sayed2012diffusion]: $$\begin{aligned}
\Tr(X_1X_2)=\big(\vec (X_2^T)\big)^T \vec(X_1) \\
\vec(X_1\Sigma X_2)=(X_2^T\otimes X_1)\vec(\Sigma)\end{aligned}$$ Using these relations in (\[eq:mse-second-term-limit-v2\]), we obtain $$\begin{aligned}
\Tr\Big(&\lim_{i\rightarrow \infty}\sum_{j=0}^{i}({\cal B}^T)^j \Sigma {\cal B}^j {\cal Y}\Big)=\Big(\vec({\cal Y}^T)\Big)^T \nonumber \\
&\times \Big(\lim_{i \rightarrow \infty} \sum_{j=0}^{i}({\cal Z}_1 J^j {\bar{\cal Z}_1})^T \otimes ({\cal Z}_1 J^j {\bar{\cal Z}_1})^T \Big)\vec(\Sigma)
\label{eq:mse-second-term-limit-v3}\end{aligned}$$ This is equivalent to: $$\begin{aligned}
\Tr\Big(&\sum_{j=0}^{\infty}({\cal B}^T)^j \Sigma {\cal B}^j {\cal Y}\Big)=\big(\vec({\cal Y})\big)^T \Big(\sum_{j=0}^{\infty}{\cal F}^j\Big)\vec(\Sigma)
\label{eq:mse-second-term-limit-v4}\end{aligned}$$ where =(([Z]{}\_1\_1)(JJ)( [|[Z]{}\_1]{}))\^T Since $\rho(J\otimes J)<1$, the series converges and we obtain: $$\begin{aligned}
\Tr\Big(\lim_{i\rightarrow \infty}&\sum_{j=0}^{i}({\cal B}^T)^j \Sigma {\cal B}^j {\cal Y}\Big)=\Big(\vec({\cal Y})\Big)^T(I-{\cal F})^{-1}\vec(\Sigma)
\label{eq:mse-second-term-limit-v5}\end{aligned}$$ Upon substitution of (\[eq.:addtional-zero-initial-v2\]) and (\[eq:mse-second-term-limit-v5\]) into (\[eq.:MSE-stability-analysis-2\]), we arrive at (\[eq.:network-steady-state-mean-square\]).
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[Reza Abdolee]{} is currently a Ph.D. candidate at the Department of Electrical and Computer Engineering, McGill University, Montreal, Canada. In 2012, he was a research scholar at the Bell Labs, Alcatel-Lucent, Stuttgart, Germany. In 2011, he was a visiting Ph.D. student at the Department of Electrical Engineering, University of California, Los Angeles (UCLA). From 2006 to 2008, Mr. Abdolee worked as a staff engineer at the Wireless Communication Center, University of Technology, Malaysia (UTM), where he implemented a switch-beam smart antenna system for wireless network applications. His research interests include communication theory, statistical signal processing, optimization, and hardware design and integration. Mr. Abdolee was a recipient of several awards and scholarships, including, NSERC Postgraduate Scholarship, FQRNT Doctoral Research Scholarship, McGill Graduate Research Mobility Award, DAAD-RISE International Internship scholarship (Germany), FQRNT International Internship Scholarship, McGill Graduate Funding and Travel award, McGill International Doctoral Award, ReSMiQ International Doctoral Scholarship, Graduate Student Support Award (Concordia University), and International Tuition Fee Remission Award (Concordia University).
[Benoit Champagne]{} received the B.Ing. degree in Engineering Physics from the Ecole Polytechnique de Montréal in 1983, the M.Sc. degree in Physics from the Université de Montréal in 1985, and the Ph.D. degree in Electrical Engineering from the University of Toronto in 1990. From 1990 to 1999, he was an Assistant and then Associate Professor at INRS-Telecommunications, Université du Quebec, Montréal. In1999, he joined McGill University, Montreal, where he is now a Full Professor within the Department of Electrical and Computer Engineering. He also served as Associate Chairman of Graduate Studies in the Department from 2004 to 2007.
His research focuses on the development and performance analysis of advanced algorithms for the processing of information bearing signals by digital means. His interests span many areas of statistical signal processing, including detection and estimation, sensor array processing, adaptive filtering, and applications thereof to broadband communications and speech processing, where he has published extensively. His research has been funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Fonds de Recherche sur la Nature et les Technologies from the Government of Quebec, Prompt Quebec, as well as some major industrial sponsors, including Nortel Networks, Bell Canada, InterDigital and Microsemi.
He has been an Associate Editor for the IEEE Signal Processing Letters, the IEEE Trans. on Signal Processing and the EURASIP Journal on Applied Signal Processing. He has also served on the Technical Committees of several international conferences in the fields of communications and signal processing. He is currently a Senior Member of IEEE.
[Ali H. Sayed]{} is professor and former chairman of electrical engineering at the University of California, Los Angeles, where he directs the UCLA Adaptive Systems Laboratory. An author of over 430 scholarly publications and five books, his research involves several areas including adaptation and learning, information processing theories, statistical signal processing, network science, and biologically-inspired designs. His work has been recognized with several awards including the 2012 Technical Achievement Award from the IEEE Signal Processing Society, the 2005 Terman Award from the American Society for Engineering Education, a 2005 Distinguished Lecturer from the IEEE Signal Processing Society, the 2003 Kuwait Prize, and the 1996 IEEE Donald G. Fink Prize. He has also been awarded several paper awards from the IEEE Signal Processing Society (2002,2005,2012) and is a Fellow of both the IEEE and the American Association for the Advancement of Science (AAAS). He has been active in serving the Signal Processing community in various roles. Among other activities, he served as Editor-in-Chief of the IEEE Transactions on Signal Processing (2003-2005), Editor-in-Chief of the EURASIP J. on Advances in Signal Processing (2006-2007), General Chairman of ICASSP (Las, Vegas, 2008), and Vice-President of Publications of the IEEE Signal Processing Society (2009-2011). He also served as member of the Board of Governors (2007-2011), Awards Board (2005), Publications Board (2003-2005, 2009-2011), Conference Board (2007-2011), and Technical Directions Board (2008-2009) of the same Society and as General Chairman of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2008.
[^1]: Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
[^2]: R. Abdolee and B. Champagne are with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, H3A 2A7 Canada (e-mail: reza.abdolee@mail.mcgill.ca, benoit.champagne@mcgill.ca).
[^3]: A. H. Sayed is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095 USA (e-mail: sayed@ee.ucla.edu).
[^4]: The work of R. Abdolee and B. Champagne was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of A. H. Sayed was supported in part by NSF grant CCF-1011918.
[^5]: Generalization of the boundary conditions to nonzero values is possible as well.
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abstract: 'We show that two graphene layers stacked directly on top of each other (AA stacking) form strong chemical bonds when the distance between planes is 0.156 nm. Simultaneously, C-C in-plane bonds are considerably weakened from partial double-bond (0.141 nm) to single bond (0.154 nm). This polymorphic form of graphene bilayer is meta-stable w.r.t. the one bound by van der Waals forces at a larger separation (0.335 nm) with an activation energy of 0.16 eV/cell. Similarly to the structure found in hexaprismane, C forms four single bonds in a geometry mixing $90^{0}$ and $120^{0}$ angles. Intermediate separations between layers can be stabilized under external anisotropic stresses showing a rich electronic structure changing from semimetal at van der Waals distance, to metal when compressed, to wide gap semiconductor at the meta-stable minimum.'
author:
- 'P.L. de Andres'
- 'R. Ramírez'
- 'J.A. Vergés'
title: Strong covalent bonding between two graphene layers
---
Carbon shows one of the richest chemistry in the periodic table and it is often found in allotropic forms. In molecules it is the basis for organic compounds, being central to different fields from biology to electronics in new materials. In solid state it shows very different properties drifting from a soft metal (graphite, the most stable phase at P=0 GPa, T= 0 K) to a hard wide gap semiconductor (diamond). New forms like fullerenes and nano-tubes have raised even more the interest in carbon for their potential applications. Recently, the realization of two-dimensional periodic systems made by the stacking of few graphene layers (FGL), going down to the single layer, has attracted much interest as the basis for new electronic devices[@novoselov04]. The peculiar linear dispersion found in the electronic band structure near the charge neutrality point (Dirac Point), where the carriers behave like mass-less chiral relativistic particles, translates in all sort of new phenomena related to transport properties on these systems[@heersche07]. Moreover, a variety of preparation techniques have been used giving rise to samples showing important differences[@rokuta99; @horiuchi03; @novoselov04; @berger04; @meyer07]; most notably: charge accumulation regions associated with physical corrugation found in free standing graphene[@meyer07], new properties induced in the graphene layers by the epitaxial growth on a SiC substrate[@berger04], or a modification of the stacking, from Bernal AB phase to AA, found in carbon nanofilms grown from graphite oxide[@horiuchi03]. Accurate and detailed information on these samples obtained from structural techniques is sometimes difficult to interpret; state-of-the-art theoretical total-energy methods are necessary to understand the precise atomic and electronic structure of these films. In this paper, we report on the formation of strong covalent bonds between graphene layers stacked directly on top of each other (AA) at a distance that is much smaller than $\sim 0.335$ nm, that is, the typical distance for an alternating (AB) stacking based on weak van der Waals forces (Fig. 1). On this meta-stable polymorphic form of a graphene bi-layer each carbon is bonded to the four nearest neighbours, at 0.154 and 0.156 nm for in-plane and out-of-plane bonds respectively. Under these conditions, the bi-layer is a wide gap semiconductor (indirect gap of 0.91 eV). As a function of the separation between layers, transport properties of the un-doped AA stacking are rich: at large distances between planes (e.g., as found in graphite) the system is very close to a semi-metal, mostly dominated by the single graphene layer properties. As the distance between layers decreases it is possible to find interlayer distances and/or different 2D unit cell sizes where the bi-layer becomes metallic.
![(Color online) Meta-stable extended 2D carbon allotrope formed by two graphene layers at covalent C-C bond distance and direct on-top stacking (AA). The 2D unit cell is shown ($a=b=0.267$ nm, $\gamma=120^{o}$). []{data-label="fig1"}](fig1.eps){width="0.9\columnwidth"}
Our ab-initio calculations are based in Density Functional Theory (DFT)[@hohenberg65] and a local approximation to describe exchange and correlation (LDA)[@kohn65; @gga]. LDA calculations performed with CASTEP[@payne02; @accelrys; @precision] reproduce very well distances and angles for the strong sp$^{2}$ bonds inside the graphene layer, predicting as the most stable configuration a honeycomb lattice with a C-C distance of 0.141 nm, and a bond population of 1.53. Experimental value is 0.142 nm (fractional error less than 1[%]{}), in between a carbon double bond (typical length 0.133 nm) and a single one (0.154 nm). A negligible charge transfer (0.3[%]{}) takes place from 2s to 2p orbitals. These geometrical results, together with those obtained for electronic and vibrational properties, demonstrate the ability of DFT to describe the C-C bond at typical distances allowing the formation of covalent bonds. A rough electron-counting picture for the graphene layer would be each C atom sharing one electron with each of the in-plane three nearest C neighbours, while the fourth electron is delocalized among them, making three stronger C-C bonds with a character somewhere in between a single and a double bond. In bulk graphite this fourth electron would be responsible for the appearance of pockets near the Fermi energy and the in-plane conductivity. This scenario makes plausible to use this extra electron to establish single bonds between carbons across the layers. While van der Waals interaction is weak and not accurately described by a local DFT, the formation of the new allotropic form of graphene bi-layer rather involves interactions between carbons at shorter covalent bonding distances where the DFT formalism is accurate and realistic. This is independent of the basis choosen to solve the equations; we have checked that quantum chemistry calculations made with localized basis sets and mixed functional methods in small clusters concur with the ones derived from plane-waves basis for extended 2D systems.
![(Color online) First-principles total energy landscape for the graphene bi-layer as a function of the distance between layers ($d$) and the 2D lattice parameter ($a$). Label A shows the local minimum reported in this work at $d=0.156$ nm and $a=0.267$ nm corresponding to the formation of covalent bonds across layers. Contour lines start at -618 eV and go up in steps of 0.05 eV (the minimum in A is at -617.208 eV). []{data-label="fig2"}](fig2a.eps "fig:"){width="0.85\columnwidth"} ![(Color online) First-principles total energy landscape for the graphene bi-layer as a function of the distance between layers ($d$) and the 2D lattice parameter ($a$). Label A shows the local minimum reported in this work at $d=0.156$ nm and $a=0.267$ nm corresponding to the formation of covalent bonds across layers. Contour lines start at -618 eV and go up in steps of 0.05 eV (the minimum in A is at -617.208 eV). []{data-label="fig2"}](fig2b.eps "fig:"){width="0.85\columnwidth"}
Recent papers have investigated the electronic structure of the standard alternating AB stacking since it is energetically favoured over the AA stacking[@latil06]. However, the expected energy difference is necessarily small due to the weak interaction between layers, about 0.02 eV/cell in our calculations. The barrier to transform one stacking into the other might be higher, but of the same order. Therefore, we have investigated the AA bi-layer, searching for new structural configurations; we find a meta-stable energy minimum at about half the usual distance between layers in graphite. This new structure implies an important lateral relaxation of the 2D unit cell, and displays electronic properties quite different from the global van der Waals-like minimum. Fig. 2 shows a 2D total energy map for the system near the new minimum: the meta-stable configuration appears around an interlayer distance $d=0.156$ nm and lattice parameter $a=0.267$ nm (label A in Fig. 2). The alternating Bernal stacking (AB) does not show a similar meta-stable local minimum in our calculations. The reason for the different behaviour lies on the different coordination of the C atoms in the bi-layer stacking AA and AB. While [*all*]{} C atoms can form an interlayer covalent bond in the AA stacking, only half of the atoms have this possibility for the AB case. Consequently, when a small separation is forced in the AB bi-layer, buckling of both planes can release stress efficiently, and sp$^{3}$ coordination with nearly tetrahedral angles appears (the resulting structure is a 2D diamond precursor). In the AA case, the formation of a meta-stable configuration is favoured because symmetry does not allow the relaxation of structural strain by buckling. A similar idea has been put forward to explain the meta-stability of n-prismanes[@jenkins00]. Stacking of carbon layers with covalent bonds accross layers seems unnoticed; because this configuration is meta-stable it should require contributing some external energy to the system. A natural way of doing this is to grow the layers epitaxially on a substrate imposing a stretched length for the 2D unit cell. However, we should mention that the AA stacking has been reported in the literature for some related system[@horiuchi03].
![Phonon spectrum calculated at the local minimum A in Fig. 2. The x-axis samples the boundary of the irreducible 2D Brillouin zone. Lines between points are only meant to guide the eye. []{data-label="fig3"}](fig3.eps){width="0.8\columnwidth"}
We have obtained the barrier to escape from minimum A to the global one (G) by applying first a Linear Synchronous Transit (LST) transition state search, followed by a Quadratic Synchronous Transit (QST) method. We find a barrier of $0.16 \pm 0.04$ eV/cell, allowing us to predict that the structure A is stable at room temperature. The energy barrier for the formation of the meta-stable state (from G to A) amounts to 4.80 eV/cell, the A configuration being 4.64 eV/cell higher in energy than the G one. The path from A to the transition state involves a simultaneous modification of parameters, $d$ and $a$ (Fig. 2) due to the correlation between bonds formed in and out the planes. Boundary conditions keeping the parameter $a$ fixed to a given value make a different scenario with interesting consequences. If $a$ is kept at a constant value of $0.279$ nm, the local minimum A is established at $d=0.155$ nm and the barrier grows to 0.8 eV/cell. For a constant value of $a= 0.291$ nm, A becomes the global minimum, and the barrier from A to G goes to 1.7 eV/cell, A being lower in energy than G by 1.03 eV/cell.
![ Evolution of the electronic band structure for the bi-layer as separation changes from van der Waals-like distance (a) to the small separation allowing the chemical bonding of graphene sheets (d). (a) $d=0.358$ nm, $a=0.243$ nm (global minimum G, semi-metallic); (b) $d=0.300$ nm, $a=0.250$ nm (2D metal); (c) $d= 0.1625$ nm, $a= 0.2645$ nm (near the transition state, 2D metal); and (d) $d= 0.156$ nm, $a=0.267$ nm (local minimum A, insulator). Fermi energy is used as the origin for energies. []{data-label="fig4"}](fig4.eps){width="1.0\columnwidth"}
Let us further characterize the new bonding configuration after the formation of chemical bonds between C atoms located in different layers. The building of these bonds produces a weakening of the sp$^{2}$-like in-plane bonds, that elongate from 0.141 nm to 0.154 nm. In A, we observe a 0.1 electron charge transfer from the 2p to the 2s orbital, and the formation of a single bond between carbons across the two graphene layers at 0.156 nm with a calculated bond order of 0.92. This distance is typical of single C-C bonding for substances like diamond, propane, etc.[@pauling], supporting the formation of a chemical bond in place of the previous weak van der Waals interaction. We notice that similar strained carbon structures have been observed in molecular systems known as n-prismanes[@allinger83]. Quantum chemical calculations performed with the program GAMESS[@gamess] confirm the building of single C-C bonds across parallel carbon hexagonal rings saturated with H to form the hexaprismane. C-C bond distances and angles are similar to those found in the graphene bi-layer (A). In agreement with our periodic solid-state calculations, this is a meta-stable molecular configuration w.r.t. van der Waals-like separation between two benzene molecules. Our calculations give a barrier between the meta-stable structure and the global minimum (C$_6$H$_6$-C$_6$H$_6$, one hexagonal ring) of about 0.83 eV per C-C bond. This value decreases consistently as more rings are added; already for three hexagonal rings (C$_{13}$H$_{8}$-C$_{13}$H$_{8}$) it goes down to several tenths of eV per C-C bond. As the number of hexagonal rings increases this barrier converges to our result for the graphene bi-layer.
Fig. 3 gives the phonon spectra at the minimum A calculated with a linear response formalism[@refson06]. The phonon spectra has no dispersion in the direction perpendicular to C layers and shows that the new minimum is stable with respect to small displacements that preserve the unit cell area[@nota]. The optical branches around 1600 cm$^{-1}$ at $\Gamma$ can be compared with those measured for graphite[@maultzsch04], although bonding in the layer is now weaker than for graphite. Near 1100 and 1250 cm$^{-1}$ we observe a couple of optical modes related to vibrations perpendicular to the layers that are similar in energy to that found for two C$_{6}$H$_{6}$ rings (hexaprismane) vibrating against each other at C-C covalent distances. These may be used to experimentally identify the bilayer.
Transport properties on FGL-based devices are determined by the band-structure of the material. Therefore, we study the electronic structure of the bi-layer for different structural parameters (size of the 2D unit cell, $a$, and separation between layers, $d$). A single graphene layer displays a semi-metallic character with valence and conduction bands touching in the corners of the Brillouin zone, {[**K**]{}}, and the dispersion relation being linear. At the van der Waals-like separation between layers (0.358 nm), the interaction is weak, but already a marginal 2D metal starts to form. The 2D Fermi circle is centred at the corners of the Brillouin zone, [**K**]{}, with a very small radius and the density of states at the Fermi energy is nearly zero (Fig. 4a). We notice that in the AA stacking the bands near [**K**]{} are still linear, unlike the AB stacking where the bands approach [**K**]{} quadratically[@latil06]. A new situation emerges if the two layers are forced to get closer to each other. Fig. 4b shows the band structure for such a non-equilibrium configuration ($a=0.250$ nm, $d=0.300$ nm). For this geometry, repulsive forces on atoms on each layer are 0.024 eV/nm. A comparison between panels a and b in Fig. 4 shows how the radius of the Fermi circle increases, yielding a distinctively non-zero density of states and making the bi-layer a 2D metal. This picture is still valid near the transition state, where the Fermi line is approaching the symmetry point $\Gamma$ in the Brillouin zone (Fig. 4c). Further down the distance between the two layers, the system develops strong single covalent bonds, and the bi-layer becomes a wide gap semiconductor. Finally, we have explored the role of external stresses on the bi-layer by applying in-plane tensile stresses of $\sigma_{xx}$=$\sigma_{yy}$= 3, 6 and 9 GPa. As expected, by forcing the 2D unit cell to extend, the minimum at A is stabilized and the barrier grows to 0.43, 0.88 and 1.4 eV/cell respectively. The local minimum A changes so the 2D unit cell size grows from 0.267 nm to 0.273, 0.280 and 0.289 nm respectively, while the two layers come closer together by a small distance (0.0008 nm for 6 GPa). We notice that around G the strain is approximately half the value around A (from 0.243 to 0.249 nm for the 6 GPa stress), a consequence of the existence of stronger sp$^{2}$ bonds.
In conclusion, we have found a new polymorphic form for two extended flat 2D graphene layers stacked with AA sequence where carbon atoms located in atop positions establish new covalent bonds. This meta-stable configuration is not subject to thermodynamic instability and shows a barrier large enough to make it feasible at room temperature. As a function of the separation between the two layers, their electronic properties range from a semi-metal (layers far away apart) to a weak 2D metal (van der Waals distances, low density of states at the Fermi energy) to a stronger 2D metal (intermediate distances, higher density of states at the Fermi energy), and finally to a wide gap semiconductor (covalent bonding distance). External stresses can help to further stabilize these configurations, as well as to control the separation between layers. The new predicted semiconductor should allow traditional doping with impurities (B, N) opening a well-defined way towards strict 2D electronics.
This work has been financed by the CICYT (Spain) under contracts MAT-2005-3866, MAT-2006-03741, FIS-2006-12117-C04-03, and NAN-2004-09183-C10-08. We acknowledge the use of the Spanish Supercomputing Network and the CTI (CSIC).
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---
abstract: 'We study the application of machine learning methods based on a geometrical and time-series character of data in the application to quantum control. We demonstrate that recurrent neural networks posses the ability to generalize the correction pulses with respect to the level of noise present in the system. We also show that the utilisation of the geometrical structure of control pulses is sufficient for achieving high-fidelity in quantum control using machine learning procedures.'
author:
- 'M. Ostaszewski'
- 'J.A. Miszczak'
- 'P. Sadowski'
bibliography:
- 'geometry\_vs\_time.bib'
date: 'v. 1.0 (14/03/2018)'
title: 'Geometrical versus time-series representation of data in learning quantum control'
---
=1
Introduction
============
The presented work aims to study two approaches to representing the information included in control pulses. In the first approach, the pulses are treated as time-series. In the second approach, we utilise only Euclidean geometry of the space of pulses. We focus on the correction scheme, which can be used to utilise *normal control pulses* (NCP), [*ie.*]{}control pulses corresponding to the system without the drift Hamiltonian, for obtaining *denoising control pulses* (DCP) for the system with the drift Hamiltonian, taking into account the undesired dynamics in the system.
The optimization of the control parameters of quantum systems is carried out mainly using greedy gradient methods [@khaneja2005optimal; @PhysRevLett.106.190501]. This is a different situation than in the field of control in classical systems, where machine learning methods are often used. In contrast to gradient methods, machine learning methods allow for the representation of the dynamics of the quantum system. The artificial neural network processing control impulses for the quantum system is an unambiguous representation of such dynamics. Therefore, it can be used to study the dynamics of the system, such as sensitivity to the disorder. Thus, the approach based on the use of neural networks in the theory of quantum control is important not only from the point of view of obtaining optimal control impulses [@august2018taking; @swaddle2017generating]. It allows to build and analyze the dynamics of the quantum system [@lloyd2014information].
We present the analysis of different methods for representing the knowledge about the correction scheme. We focus on two categories of such methods. The first one relies on the geometrical dependencies of data. The second one takes into account the time-series characteristics of data. To be more specific, we limit our considerations to two methods: construction of correction scheme by k-means and kNN algorithms, and an approximation of this scheme by recurrent neural networks. The methods based on k-means and kNN algorithms are machine learning algorithms utilizing the geometrical features of data. They are used in many applications where fast classification is required. On the other hand, recurrent neural networks were developed for the purpose of learning time correlations of sequences and are applied in the situations with time series with long-range correlations, such as in natural language processing.
Our analysis of the methods for representing the knowledge about the correction scheme is focused on two criteria.
- Efficiency in reproduction of the correction scheme.
- Ability of generalization with respect to the strength of the noise parameter.
The first criterion tells us how good the analysed methods are in terms of reproducing the correction scheme, [*ie.*]{}what is the quality of the approximation provided by them. The second criterion enables us the examination how much information about the reparation of control pulses for a fixed strength of noise can be used to generate the control pulses for different strengths of the noise. Such analysis can be used to investigate to what degree the used methods can encode the form of the undesired interaction present in the system.
We will present our results in the following order. We start by introducing the notation and technical details required for the description of the experiments. In particular, we introduce the architecture of artificial neural network suitable for processing quantum control pulses. Next, we focus on the efficiency of artificial neural networks and geometrical methods in the reconstruction of the quantum control pulses.
Preliminaries
=============
Single-qubit dynamics
---------------------
Let us consider a two-dimensional system, described by $\mathcal{H}={\mathbb{C}}^2$, with evolution described by Schrödinger equation of the form $$\frac{{\mathrm{d}}{\ensuremath{|\psi\rangle}}}{{\mathrm{d}}t} =-{\mathrm{i}}H(t){\ensuremath{|\psi\rangle}}.
\label{eq:schrod}$$ The Hamiltonian of the system is a sum of two terms corresponding to the control field and drift interaction respectively $$H(t) = H_c(t) + H_{\gamma},$$ with control Hamiltonian $$\begin{aligned}
H_c(t) &= h_x(t){\sigma_x}+ h_z(t){\sigma_z}, \label{eq:control_hamiltonian} \end{aligned}$$ and drift Hamiltonian $H_{\gamma}$, where $\gamma$ is the real parameter. To steer the system, one needs to choose the coefficients in Eq. , [*ie.*]{}$h(t)
= (h_x(t),h_z(t))$. In simulations we assume that function $h(t)$ is constant in time intervals/time slots $\Delta t_i =
[t_i,t_{i+1}]$, which are the equal partition of evolution time interval $T
=\bigcup_{i=0}^{n-1}\Delta t_i$. Moreover, we assume that $h(t)$ has values from the interval $[ -1,1]$.
Control parameters $h(t)$ will be denoted as a vector of values for time intervals. For $\gamma = 0$, control parameters are called NCP (normal control pulses), while for $\gamma>0$ – DCP (denoising control pulses). Thus NCP and DCP are vectors of pairs, with the first dimension corresponding to corresponding to time slots $\{0,\ldots , n-1\}$, and the second dimension corresponding to controls $\{x,z\}$, $$\begin{split}
NCP_{i,j} &= h_j(t),\ \mathrm{for}\ \gamma=0,\\
DCP_{i,j} &= h_j(t),\ \mathrm{for}\ \gamma\neq0,
\end{split}$$ with $t\in [t_i, t_{i+1}]$, $i \in \{0,\ldots ,n-1\}$, $j\in\{x,z\}$.
The figure of merit in our problem is the fidelity distance between superoperators, defined as [@floether12robust] $$F=1-F_{err},$$ with $$F_{err} = \frac{1}{2N^2} ({{\mathrm{Tr}}}(Y-X(T))^\dagger(Y-X(T))),\label{eq:fidelity-error}$$ where $N$ is the dimension of the system in question, $Y$ is superoperator of the fixed target operator, and $X(T)$ is evolution superoperator of operator resulting from the numerical integration of Eq. with given controls. One should note that superoperator $S$ of operator $U$ is given by formula $$S=U\otimes \bar{U}.$$
Machine learning methods
------------------------
In this section we introduce two machine learning algorithms, which will be compared in the next sections. It should be noted that the algorithms described below directly work on (NCP,DCP) pairs. However, the evaluation of their efficiency demands the comparison on the level of operators corresponding to the considered control pulses. Therefore, (NCP, $U_{target}$) pairs are used in the testing data. By the efficiency we understand the mean fidelity between operators generated from DCP using the considered approximation and target operators. We can distinguish two reference values for which the efficiency of considered approximation is compared
1. mean fidelity between operators obtained from NCP applied on system with drift, and target operators,
2. one [*ie.*]{}maximal value which can be obtained from fidelity function.
The necessary condition is to obtain the efficiency higher than the value from point 1). The desired condition is to obtain the efficiency close to the value from point 2).
### LSTM as an approximation of the correction scheme
The control pulses used to drive the quantum system with Hamiltonian given by Eq. \[eq:control\_hamiltonian\] are formally time series. This suggests that one may study their properties using the methods from pattern recognition and machine learning [@bishop1995neural; @goodfellow2016deep] that have been successfully applied to process data with similar characteristics. The mapping from NCP to DCP shares similar mathematical properties with that of statistical machine translation [@koehn2009statistical], a problem which is successfully modelled with artificial neural networks (ANN) [@bahdanau2014neural]. Because of this analogy, we use ANN as the approximation function to learn the correction scheme for control pulses. A trained artificial neural network will be used as a map from NCP to DCP $$\textrm{ANN}(\textrm{NCP})=\textrm{nnDCP},$$ where nnDCP, neural network DCP, is an approximation of DCP obtained using the neural network.
Our approach is based on bidirectional LSTM network, where the input will be the batch of NCP. After the last unit of bidirectional LSTM, we apply one dense layer, which processes the output of LSTMs into the final output. Because of time series character of control sequences, bidirectional long short-term memory (LSTM) units are the core of our network [@hochreiter1997long]. Long short-term memory (LSTM) block is a special kind of recurrent neural network (RNN), a type of neural network which has directed cycles between units. Similarly to other RNN, LSTM networks can take into account hidden states from their history. In other words, the output in given time depends not only on current input but also on earlier inputs. Therefore, this kind of neural network is applicable in situations with time series with long-range correlations, such as in natural language processing where the next word depends not only on the previous sentence but also on the context. Basic architectures of RNN are not suitable for maintaining long time dependences, due to the so called *vanishing/exploding gradient problem* – the gradient of the cost function may either exponentially decay or explode as a function of the hidden units. The LSTM unit has a structure specifically built to solve the vanishing/exploding gradient problem of other RNNs, and is adjusted to maintain memory over long periods of time. The bidirectional version of LSTM is characterized by the fact that it analyses input sequence/time series forwards and backwards, so it uses not only information from the past but also from the future [@schuster1997bidirectional; @graves2005bidirectional].
It should be noted that LSTM units are not the only ones in our network. Because a bidirectional LSTM returns forward and backward layers, the resulting signals have to be merged. The obtained results sugest that the fully connected layer at the end of network increases the efficiency.
For two qubit systems we found that three stacked bidirectional LSTM layers are sufficient. Moreover, at the end of the network we use one dense layer which precesses the output of stacked LSTMs to obtain our nnDCP. Experiments are performed using the `TensorFlow` library [@abadi2016tensorflow; @tensorflow].
The details of the proposed architecture are as follows:
- input representing a batch of NCP, with shape equal to \[batch\_size, time\_slots, 2\], where the last dimension is 2 because we have 2 controls;
- three layers of bidirectional LSTM with the number of hidden units respectively 200, 250, 300, resulting in two outputs with shapes \[batch\_size, time\_slots, 300\];
- merging of the outputs of the last LSTM unit [*ie.*]{}element-wise sum of forward LSTM output and backward LSTM product, resulting in the output shape \[batch\_size, time\_slots, 300\];
- joining batch\_size and time\_slots dimensions [*ie.*]{}reshape, resulting in the output shape \[batch\_size \* time\_slots, 300\];
- processing the data by dense layer [*ie.*]{}fully connected layer (MLP), with $\tanh$ as an activation function, resulting in the output shape \[batch\_size\*time\_slots, 2\]),
- reshape, resulting in the output shape \[batch\_size, time\_slots, 2\]),
- output with shape \[batch\_size, time\_slots, 2\].
As the cost function we choose mean squared error between nnDCP and target DCP. The scheme of generating data and utilizing networks is as follows
1. **Generating training set** Generate (NCP, DCP) pairs for random target unitary operators by [QuTIP]{} [@qutip; @qutip1; @qutip2].
2. **Training of the network** Train neural network using NCP as an input and treating the corresponding DCP as an output reference.
3. **Testing**
1. use [QuTIP]{}to generate a testing set consisting of NCP vectors for random $U_{target}$ operators,
2. use the trained neural network we generate a nnDCP vector for each NCP in the testing set,
3. construct operator $F_{nnDCP}$ for the system with the drift using resulting nnDCP as a control sequence,
4. construct operator $F_{NCP}$ for system with the drift from testing NCP,
5. calculate fidelity distance between $F_{nnDCP}$ and $U_{target}$,
6. calculate fidelity between $F_{NCP}$ and $U_{target}$ which corresponds to NCP (from the testing set),
7. compare the results from 5) and 6).
Clustering/classification algorithm
-----------------------------------
The considered geometric correction scheme approximation method is based on two steps. The first one is clustering of the training set and generating the set of representative corrections for the resulting clusters. In this step we generate the corrections which will be assigned in the next step during the classification procedure. The second step is building a classifier, that decides which correction cluster should be taken into account given the input NCP. This is motivated by the question whether is it possible to divide control pulses into clusters such that a common correction method would be effective.
### Generation of representative corrections {#sec:generation-kmeans}
In the first step of our procedure we need to extract the key information on the set of the training control pulses. To do this, we perform $k$-means algorithm on training data and calculate the means over each cluster.
In the simplest approach, the input data for the $k$-means algorithm are provided as raw pulses obtained from [QuTIP]{}. However, it is possible to reduce the dimensionality of the space by using various approximations. Here, we have checked three types of approximation of correction control pulses, namely
a) sinusoid $$a\sin(bx + c) + d\sin(ex + f) + g,
\label{eq:sine}$$ with $a>d$,
b) polynomial of the third degree (poly3) $$ax^3 + bx^2 + cx + d,
\label{eq:poly3}$$
c) polynomial of the forth degree (poly4) $$ax^4 + bx^3 + cx^2 + dx + e.
\label{eq:poly4}$$
Input
: A set of
${CCP_i} = {NCP_i} - {DCP_i}$ vectors, number of clusters $k$.
For each unitary matrix we calculate DCP and NCP using QuTIP. Next, we calculate correction control pulses (CCP) as the differences between DCP and NCP. The number of clusters $k$ should be chosen to maximize the mean fidelity, but it should be significantly smaller than the number of samples.
Output
: Cluster labels for each training point, set of $k$ corrections $\bar{C_1} ,\dots,\bar{C_k}$ representative for the clusters, mean efficiency of the corrections.
Step 0
: (Optional) Approximation of CCP vectors.
[We choose the type of approximation (one from Eqs. ,, or ) and change the space of considered data [*ie.*]{}we express every $\textrm{CCP}_i$ by a vector of suitable approximation coefficients – $\textrm{coeffs}_i$.]{}
Step 1
: Clustering of the set of CCP vectors.
[Apply $k$-means algorithm on the set of correction pulses $C = \{{CCP_i}\}_i$ or on $\textrm{coeffs}_i$. As every $\textrm{coeffs}_i$ corresponds to $\textrm{CCP}_i$, the result is the set of $k$ clusters $C_1,C_2,\ldots,C_k$. Note that this set of CCP vectors is disjoint from the initial training set $C$.]{}
Step 2
: Calculate output corrections.
For each of the clusters we calculate the output correction as the mean correction vector $$\bar{C_i}=\frac{\sum_{{CCP}\in C_i}{CCP}}{|C_i|}.$$
Step 3
: Calculate the efficiency of the output corrections.
1. For each training sample $i$ we calculate fidelity of the operator obtained from $NCP_i+CCP_i$ and the operator resulting from applying $NCP_i+\bar{C_j}$, for $j$ being the cluster label for the $i$th training sample.
2. The mean of obtained results is the score of the parameter $k$, $\mathrm{SoP}(k)$.
Step 4
: [Return calculated labels, corrections and the score of parameter $k$.]{}
It should be stressed that if Step 0 of the above algorithm is executed then the data from the new space are used only in the first step of the algorithm for calculating distances. The remaining part of the algorithm is based on the raw data and the resulting corrections are calculated and benchmarked against the raw data.
If SoP($k$) is lower than the expected efficiency, the above algorithm should be repeated for another $k$. In other words, if we do not know a suitable $k$, then the above algorithm can be used as a subprocedure for finding the parameter $k$ with assumed efficiency.
### Application of corrections
In the next step of our procedure we need to develop a method for deciding which correction should be used for given NCP. For this purpose we utilize kNN algorithm, which will be fitted on the set of training NCP with labels obtained during the clustering of CCP. Prediction of kNN on test NCP gives us the most probable cluster $j$, and in result a correction $\bar{C_j}$ which should be used to generate suitable DCP.
The choice of kNN algorithm is motivated by geometrical dependencies of data it utilizes. This is in line with our assumption about the examined approaches for representing control pulses.
Efficiency of learning
======================
In the experiments described below, we examine the two introduced methods [*ie.*]{}LSTM and $k$-means/kNN. We compare their efficiency as the approximations of the correction scheme and we analyse their ability to generalize the scheme additional information about noise strength in the input.
The training and testing data used in the experiments described below were generated using standards methods [@mezzadri2006generate; @miszczak12generating] and [QuTIP]{} [@qutip; @qutip1; @qutip2] package for Python programming language. First, we construct the target matrix $U$ which is a $2\times
2$ random unitary matrix. For the purpose of generating random unitary matrices we utilize the method proposed in [@mezzadri2006generate] (see also [@miszczak12generating]). Next, we generate NCP corresponding to the target operators. All control pulses were generated by [QuTIP]{}, with initialization ’ZERO’. This initialization is used to minimize the randomness of the optimization.
The parameters used in the numerical experiments are fixed as follows:
- time of evolution $T=2.1$,
- number of intervals $n=16$,
- control pulses in $[-1, 1]$.
Efficiency of LSTM
------------------
In this experiment we generate 5000 random unitary matrices and train the neural network to predict DCP based on NCP. The network is given thr reference (NCP, DCP) pairs as a training set. The following results were obtained with LSTM trained on 3000 control pulses. In this experiment we aim at analysing the limiting efficiency, [*ie.*]{}the maximal obtainable efficiency with unlimited number of training samples and we do not observe any significant improvement with the increasing number of training samples.
-------------------- -------- -------- -------- --------
Method $0.2$ $0.4$ $0.6$ $0.8$
nnDCP from LSTM $.994$ $.991$ $.982$ $.951$
DCP from [QuTIP]{} $.999$ $.999$ $.998$ $.989$
NCP from [QuTIP]{} $.903$ $.657$ $.372$ $.165$
-------------------- -------- -------- -------- --------
: Comparison of efficiency (mean fidelity) for $H_{\gamma}={\sigma_y}$ for different methods of obtaining control pulses.[]{data-label="tab:results-lstm"}
The test has been performed on the set of 2000 control pulses and the range of values of the parameter $\gamma$ is adjusted to the restrictions for the parameters of the model. We restrict the values of this parameter to $\gamma\in[0,1]$. One should note that we were not able to find satisfactory control pulses for higher values of $\gamma$ using gradient-based methods implemented in [QuTIP]{}.
The results presented in Table \[tab:results-lstm\] demonstrate that LSTM network can achieve efficiency with errors of the same order of magnitude as the reference data. However, one should note that for higher values of gamma parameter the data obtained from [QuTIP]{}contain many outliers, [*ie.*]{}there is a significant number of matrices for which the resulting fidelity is below the acceptable level of 0.90.
Efficiency of corrections from clustering
-----------------------------------------
In the second experiment we use $k$-means algorithm to utilize mean control from the cluster to correct NCP. As the input data we take 3000 unitary matrices and generate DCP, NCP and CCP vectors for them. It should be noted that at this level of algorithm, we test how the clusters are efficient. For this purpose we calculate the mean fidelity form operators obtained from NCP corrected by representative corrections and compare it to the efficiency of DCP and to the efficiency of NCP applied on system with noise. Also, on all CCP we perform three approximations.
As one can see in Fig. \[fig:comparison\_approx\_3000\], for each kind of approximation method, there exists $k$ such that applying the correction scheme, being mean CCP within corresponding cluster, is better than the mean fidelity of application of NCP with $\gamma\neq 0$. Moreover, the clustering on the raw data, without any kind of approximation, yields significantly better results. This demonstrates that the methods used for approximation are not suitable for reducing the dimensionality in this case.
In Fig. \[fig:comparison\_1000\_vs\_5000\_samples\] one can see the important characteristic of the algorithm, namely its invariance with respect to the number of samples. One can see that the clustering with a fixed parameter $k$ gives similar results for training sets consisting of 1000 and 5000 samples.
![Invariance of the clustering with respect to the number of samples. The graph plotted with a blue dashed line corresponds to the sample of size 1000, and the graph plotted with a yellow dot-dashed line corresponds to the sample of size 5000. The experiment was performed for $\gamma = 0.8$.[]{data-label="fig:comparison_1000_vs_5000_samples"}](clustering_comparison_1000_vs_5000_samples.pdf){width="50.00000%"}
The last part of the assessment procedure consists of the efficiency test of the whole correction scheme, [*ie.*]{}we aim at answering what is the efficiency when we use kNN classifier. In this case the efficiency of kNN should be limited by the efficiency resulting from clustering. As we can see in Fig. \[fig:kNN\_efficiency\], kNN classifier has similar results on a test set as the clustering on the training set. Moreover, one can see that for the number of clusters close to 300, we obtain the efficiency similar to LSTM.
From the above results one can see, that the data have some geometrical structure which can be captured by $k$-means and kNN algorithms. Moreover, the results obtained using this approach are very similar to the results obtained using significantly more complicated approach represented by LSTM.
Generalization in learning quantum control
==========================================
As we are interested not only in the efficiency of the reproduction of the correction scheme, the goal of the next experiment is to investigate the ability of the considered methods to provide a generalization of the correction scheme.
To achieve this goal, we modify inputs in such way that we add $\gamma$ as an additional parameter to NCP. Next we train the models for many choices of gamma, and check whether the analysed algorithms are able to infer the correction pulse for the new choices of gamma.
This allows the examination of the analysed algorithms ability to interpolate and extrapolate the correction scheme according to parameter $\gamma$. The analysis will be based on the mean efficiency of the considered algorithms. Moreover, we will compare their results to the reference point [*ie.*]{}to the efficiency of DCP generated for some gamma, but integrated within the system with different gammas.
We perform experiments on two sets of $gamma$ parameters, $\{0.1, 0.3, 0.5\}$ and $\{0.5,0.7,0.9\}$. We choose training gammas from different halves of $[0,1]$ interval. The training on different gammas has different efficiency, that is if gamma is bigger, then the efficiency of approximation is lower. Moreover, tests in all experiments are performed on the set of gammas $\{0.1,0.2,\ldots ,0.9\}$.
Generalization using LSTM
-------------------------
In previous experiments, the vectors of pairs [*ie.*]{}NCP were applied as the input of artificial neural network. Now, as the input we will take vectors of triples, where each triple will be of the form $(h_x(t), h_z(t),\gamma)$. We denote this inputs as (NCP, $\bar{\gamma}$). It can be interpreted as the addition of the third dimension to the time series, where at each time slot there is the same value of $\gamma$. Because of that, the architecture of our network remains unchanged. The elements of the training set
- were generated from 3000 random unitary matrices,
- consist of $9 000$ NCP with gammas $\gamma_1 ,\gamma_2 ,\gamma_3$, where we have 3000 vectors for each $\gamma$,
- each pair (NCP, $\bar{\gamma}$) corresponds to a different DCP.
The elements of the test set
- were generated from different 2000 random unitary matrices
- consist of $18 000$ NCP with gammas $0.1,0.2,\ldots ,0.9$, where we have 2000 vectors for each $\gamma$,
- each pair (NCP, $\gamma$) corresponds to a different DCP.
Generalization using $k$–means and kNN
--------------------------------------
Training of $k$–means
: We train the clustering algorithm with number of clusters equal to 500, on CCP obtained from DCP for $\gamma = 0.1,0.3,0.5$ and $0.5,0.7,0.9$.
Training of kNN
: We train the classification algorithm on the flattened NCP corresponding to CCP from the clustering. However, we include additional information about $\gamma$ by adding $\gamma$ multiplied by some large number to the flattened NCP vector. This separates the vectors on the subspace spanned by the added element. In our experiments, this number is equal to 1000. In kNN we choose $k=4$.
Testing of kNN
: This situation is analogical to training, [*ie.*]{}we take NCP from the test set and concatenate to it some label about gamma.
### Reference point
For the purpose of analysing the ability to generalize the correction scheme we are using a *reference point*, defined as the values of mean fidelity obtained by the algorithm trained on data with fixed parameter $\gamma$, applied to other values of $\gamma$. Reference point provided the minimal efficiency which should be obtained by the tested algorithm in order to consider it acceptable.
The reference points are constructed as follows. Let us suppose that we have an already trained artificial neural network and $k$-means/kNN algorithms. This trained approximations reproduce corrections schemes for a system with a fixed parameter $\gamma$ and can be applied on a test set of NCP. The generated approximations of DCP can be integrated with different values of parameter $\gamma$. The fidelity of the resulting operator with the target operator provides the efficiency of the approximation. In the other words, we test how efficient is DCP generated for some $\gamma$ when we apply it to system with different $\gamma$.
The reference points for kNN and LSTM are presented in Fig. \[fig:reference\_points\_gam03\_05\_07\]. One can observe that LSTM near the value of $\gamma$ for the reference point has efficiency higher than kNN.
### Results
As one can see in Fig. \[fig:generalization\_kMeanskNN\_gam01\_03\_05\], the $k$–means/kNN has three local extrema, which correspond to gamma on which the algorithm was trained. Therefore, this algorithm has noticeable drops in the interpolation. Comparing exact values, the $k$–means/kNN trained on data with $\gamma = 0.1,0.3,0.5$, has efficiency 0.962 and 0.945 for gammas 0.4 and 0.6 respectively. Reference values for kNN trained with $\gamma = 0.5$ are equal to 0.960 and 0.954 for gammas 0.4 and 0.6 respectively. Similar effect can be observed for reference points for kNN trained with $\gamma=0.3$. In this case the values of mean fidelity for $\gamma=0.2$ and $\gamma=0.4$ are almost identical for reference points and for kNN trained with three input values of $\gamma$. Moreover, for kNN trained with $\gamma=0.1,0.3,0.5$ the extrapolation is worse than the reference point for $\gamma=0.5$. For kNN trained with $\gamma=0.5,0.7,0.9$ the ability to generalize for other $\gamma$ is also limited. In this case this might be caused by the presence of outliers in the training data (see Fig \[fig:generalization\_kMeanskNN\_gam05\_07\_09\]). This suggests that this method does not utilize the information about the $\gamma$ parameter. The additional $\gamma$ in the input is not utilized and the algorithm obtains similar results as the algorithm without $\gamma$ in the input.
The situation is different in the case of utilizing of LSTM network, which displays the ability to generalize the correction scheme using information about the $\gamma$ parameter. This effect can be observed in Fig. \[fig:generalization\_LSTM\_gam01\_03\_05\]. As one can see, LSTM has high efficiency in the neighbourhood of the training points. The reference point, is the result obtained from LSTM trained on pairs (NCP, DCP) for the system with $\gamma=0.5$. For the tested cases $\gamma=0.4, 0.6$ reference point has efficiencies equal to 0.968 and 0.964, respectively. In the same cases the LSTM with $\gamma$ as additional parameter in input has efficiencies equal to 0.992 and 0.984, respectively. Thus the LSTM provides better results in the case of the interpolation. Similar effect can be observed for reference points for LSTM trained with $\gamma=0.3$. In this case, the values of mean fidelity for $\gamma=0.2$ and $\gamma=0.4$ are lower for the reference point than for the LSTM trained for three input gammas. Moreover, for LSTM trained with $\gamma=0.1,0.3,0.5$ the extrapolation is higher than the reference point for $\gamma=0.5$. Thus, one can conclude that LSTM has the ability to generalize for other $\gamma$.
One can observe the decrease of the efficiency for the generalization for LSTM trained on $\gamma=0.5,0.7,0.9$, applied for $\gamma=0.8$ and $\gamma=0.9$. This might be caused by the presence of outliers in the training data. However, one should note that this effect does not influence the ability to the provided correction scheme for lower values of $\gamma$, which is more efficient than the reference point. This is in contrast with the lack of such ability observed in the case of using kNN.
One should note that the ability of generalization does not depend on the absolute values of $\gamma$. This can be observed by the decrease in the efficiency of extrapolation which can be observed for the cases when we train the algorithms on small gammas and extrapolate for larger values (see Figs \[fig:generalization\_kMeanskNN\_gam01\_03\_05\] and \[fig:generalization\_LSTM\_gam01\_03\_05\]) or when we train the algorithms on large values and extrapolate to small vales (see Figs \[fig:generalization\_kMeanskNN\_gam05\_07\_09\] and \[fig:generalization\_LSTM\_gam05\_07\_09\]).
Concluding remarks
==================
The presented work demonstrates that the techniques used in machine learning can be applied for the purpose of generating quantum control pulses. The conducted experiments demonstrate that both neural networks and geometrical methods provide good approximations of correction schemes and enable counteracting the undesired interaction present in the system.
However, one should note that both methods have their specific advantages and disadvantages. Artificial neural networks are useful in the sense of approximation function as the trained network is a unique map from NCP to DCP. Because of this, one can examine a variation, continuity, and other mathematical features of this correction scheme [@ostaszewski18approximation]. Moreover, we demonstrated that recurrent neural networks have the ability to generalise their predictions. This can be seen in the presented experiments where the network that was trained on few gammas, has also good results for gammas which were absent in the training process.
On the other hand, the application of clustering shows that this repair scheme can be compressed to a relatively small number of corrections. This demonstrates that the continuous process of quantum control can be represented be a relatively small number of representative control pulses. Such method provides the efficiency of approximation similar as in the case of the recurrent neural networks. Unfortunately, the obtained results suggest that such purely geometrical approach is significantly less reliable in the process of generalization. This is especially visible in the situation where the extrapolation of the correction scheme is required. One should also note that the correction scheme based on the geometrical features of control pulses cannot be easily simplified by using standard approximation methods.
MO acknowledges support from Polish National Science Center grant 2011/03/D/ST6/00413. JAM acknowledges support from Polish National Science Center grant 2014/15/B/ST6/05204. Authors would like to thank Daniel Burgarth and Leonardo Banchi for discussions about quantum control, Bartosz Grabowski and Wojciech Masarczyk for discussions concerning the details of LSTM architecture, and Izabela Miszczak for reviewing the manuscript.
|
---
abstract: 'In this paper, we investigate the holographic dark energy model with interaction between dark energy and dark matter, from the statefinder viewpoint. We plot the trajectories of the interacting holographic dark energy model for different interaction cases as well as for different values of the parameter $c$ in the statefinder-plane. The statefinder diagrams characterize the properties of the holographic dark energy and show the discrimination between the two cases with and without interaction. As a result, we show the influence of the interaction on the evolution of the universe in the statefinder diagrams. Moreover, as a complement to the statefinder diagnosis, we study the interacting holographic dark energy model in the $w-w''$ plane, which can provide us with a dynamical diagnosis.'
---
[**Statefinder diagnosis for the interacting model of holographic dark energy**]{}
[Jingfei Zhang,$^{1}$ Xin Zhang,$^{2}$ and Hongya Liu$^{1}$]{}
Today it has been confirmed that our universe is undergoing an accelerating expansion through numerous cosmological observations, such as type Ia supernovae (SNIa) [@SN], large scale structure (LSS) [@LSS] and cosmic microwave background (CMB) [@CMB]. This cosmic acceleration is attributed to a mysterious dominant component, dark energy, with negative pressure. The combined analysis of cosmological observations suggests that the universe is spatially flat, and consists of about $70\%$ dark energy, $30\%$ dust matter, and negligible radiation. Many candidates have been proposed to interpret or describe the properties of dark energy, though its nature still remains enigmatic. The most obvious theoretical candidate of dark energy is the cosmological constant $\lambda$ [@Einstein:1917; @cc] which has the equation of state $w=-1$. However, as is well known, there are two difficulties arise from the cosmological constant scenario, namely the two famous cosmological constant problems — the “fine-tuning” problem and the “cosmic coincidence” problem [@coincidence]. Theorists have made lots of efforts to try to resolve the cosmological constant problem but all these efforts were turned out to be unsuccessful.
Also, there is an alternative proposal to dark energy — the dynamical dark energy scenario. The dynamical dark energy scenario is often realized by some scalar field mechanism which suggests that the energy form with negative pressure is provided by a scalar field evolving down a proper potential. A lot of scalar-field dark energy models have been studied, including quintessence [@quintessence], K-essence [@kessence], tachyon [@tachyon], phantom [@phantom], ghost condensate [@ghost] and quintom [@quintom] etc.. In addition, other proposals on dark energy include scenarios of interacting dark energy [@intde], braneworld [@brane], Chaplygin gas [@cg], and so forth. By far, obviously, it is not yet clear if dark energy is a cosmological constant or a dynamical field. Generally, theorists believe that we can not entirely understand the nature of dark energy before a complete theory of quantum gravity is established [@Witten:2000zk].
However, in this circumstance, we still can make some efforts to probe the properties of dark energy according to some principle of quantum gravity. The holographic dark energy model is an example of such effort, which stems from the holographic principle and can provide us with an intriguing way to interpret the dynamics of dark energy. The holographic principle is an important result of the recent researches of exploring the quantum gravity and is enlightened by investigations of the quantum property of black holes [@holoprin]. According to the holographic principle, the number of degrees of freedom for a system within a finite region should be finite and should be bounded roughly by the area of its boundary. In the cosmological context, the holographic principle will set an upper bound on the entropy of the universe. Motivated by the Bekenstein entropy bound, it seems plausible that one may require that for an effective quantum field theory in a box of size $L$ with UV cutoff $\Lambda$, the total entropy should satisfy $S=L^3\Lambda^3\leq S_{BH}\equiv\pi M_{\rm P}^2L^2$, where $S_{BH}$ is the entropy of a black hole with the same size $L$. However, Cohen et al. [@Cohen:1998zx] pointed out that to saturate this inequality some states with Schwartzschild radius much larger than the box size have to be counted in. As a result, a more restrictive bound, the energy bound, has been proposed to constrain the degrees of freedom of the system, requiring the total energy of a system with size $L$ not to exceed the mass of a black hole with the same size, namely, $L^3\Lambda^4=L^3\rho_{\rm de}\leq L M_{\rm P}^2$. This means that the maximum entropy is in the order of $S_{BH}^{3/4}$. When we take the whole universe into account, the vacuum energy related to this holographic principle is viewed as dark energy, usually dubbed holographic dark energy. The largest IR cut-off $L$ is chosen by saturating the inequality, so that we get the holographic dark energy density $$\rho_{\rm de}=3c^2M_{\rm P}^2L^{-2}~,\label{de}$$ where $c$ is a numerical constant[^1] (note that $c>0$ is assumed), and as usual $M_{\rm
P}$ is the reduced Planck mass. If we take $L$ as the size of the current universe, for instance the Hubble scale $H^{-1}$, then the dark energy density will be close to the observed value. However, Hsu [@Hsu:2004ri] pointed out that this yields a wrong equation of state for dark energy. Li [@Li:2004rb] subsequently proposed that the IR cutoff $L$ should be given by the future event horizon of the universe, $$R_{\rm eh}(a)=a\int\limits_t^\infty{dt'\over
a(t')}=a\int\limits_a^\infty{da'\over Ha'^2}~.\label{eh}$$ Such a holographic dark energy looks reasonable, since it may provide simultaneously natural solutions to both dark energy problems, as demonstrated in Ref. [@Li:2004rb]. Meanwhile, other applications of the holographic principle in cosmology [@otherholo] show that holography is an effective way to investigate cosmology. For other extensive studies, see e.g. [@holoext]–[@Wang:2005jx].
Besides, some interacting models are discussed in many works because these models can help to understand or alleviate the coincidence problem by considering the possible interaction between dark energy and cold dark matter due to the unknown nature of dark energy and dark matter. In addition, the proposal of interacting dark energy is compatible with the current observations such as the SNIa and CMB data [@Guo:2007zk]. For the interacting model of holographic dark energy see [@Wang:2005jx].
On the other hand, since more and more dark energy models have been constructed for interpreting or describing the cosmic acceleration, the problem of discriminating between the various contenders is becoming emergent. In order to be capable of differentiating between those competing cosmological scenarios involving dark energy, a sensitive and robust diagnosis for dark energy models is a must. In addition, for some geometrical models arising from modifications to the gravitational sector of the theory, the equation of state no longer plays the role of a fundamental physical quantity, so it would be very useful if we could supplement it with a diagnosis which could unambiguously probe the properties of all classes of dark energy models. For this purpose a diagnostic proposal that makes use of parameter pair $\{r,s\}$, the so-called “statefinder", was introduced by Sahni et al. [@sahni]. The statefinder probes the expansion dynamics of the universe through higher derivatives of the scale factor $\stackrel{...}{a}$ and is a “geometrical” diagnosis in the sense that it depends on the scale factor and hence on the metric describing space-time. Since different cosmological models involving dark energy exhibit different evolution trajectories in the $s-r$ plane, the statefinder can be used to diagnose different dark energy models [@Alam:2003sc].
In this paper, we focus on a model of holographic dark energy with interaction between dark energy and dark matter and study the influence of the interaction to the cosmic evolution. Moreover, we use the statefinder to diagnose various cases with different interaction strength and different parameter $c$ in the holographic model.
Let us start with a spatially flat Friedmann-Robertson-Walker (FRW) universe with dust matter and holographic dark energy. The Friedmann equation reads $$3M_{\rm P}^{2}H^{2}\label{f1}=\rho_{\rm de}+\rho_{\rm m},$$ where $\rho_{\rm m}$ is the energy density of matter and $\rho_{\rm
de}=3c^2M_{\rm P} ^2R_{\rm h}^{-2}$ is the dark energy density. The total energy density satisfies a conservation law, $$\dot{\rho}_{\rm de}+\dot{\rho}_{\rm m}=-3H(\rho+P),\label{f3}$$ where $\rho=\rho_{\rm m}+\rho_{\rm de}$ is the total energy density of the universe, and $P=P_{\rm de}=w\rho_{\rm de}$ is the total pressure ($w$ denotes the equation of state of dark energy). Note that since the matter component is mainly contributed by the cold dark matter, we ignore the contribution of the baryon matter here for simplicity. By introducing $\Omega_{\rm de}=\rho_{\rm
de}/(3M_{\rm P} ^{2}H^{2})$ and $\Omega_{\rm m}=\rho_{\rm
m}/(3M_{\rm P}^{2}H^{2})$, the Friedmann equation can also be written as $\Omega_{\rm de}+\Omega_{\rm m}=1$. Furthermore, if we proceed to consider a scenario of interacting dark energy, $\rho_{\rm m}$ and $\rho_{\rm de}$ do not satisfy independent conservation laws, they instead satisfy $$\dot{\rho}_{\rm m}+3H\rho_{\rm m}=Q, \label{a1}$$ and $$\dot{\rho}_{\rm de}+3H(1+w)\rho_{\rm de}=-Q,\label{a2}$$ where $Q$ describes the interaction between dark energy and dark matter. It is obvious that the interaction term $Q$ could not be introduced by considering some micro-process currently, so a phenomenological way is the must. One possible choice for the interaction term is setting $$Q=3b^2 H\rho, \label{Q}$$ where $b$ is a constant describing the coupling strength. This expression for the interaction term was first introduced in the study of the suitable coupling between a quintessence scalar field and a pressureless cold dark matter component, in order to get a scaling solution to the coincidence problem [@Pavon:2005yx].
Taking the ratio of energy densities as $\mu=\rho_{\rm m}/\rho_{\rm
de}$ and using the Friedmann equation $\Omega_{\rm de}+\Omega_{\rm
m}=1$, we have $\mu=(1-\Omega_{\rm de})/\Omega_{\rm de}$ and $\dot{\mu}=-\dot{\Omega}_{\rm de}/\Omega_{\rm de}^2$. Furthermore, from (\[a1\]), (\[a2\]) and (\[Q\]), we obtain $$\dot{\mu}=3b^2
H(1+\mu)^2+3H\mu w.\label{3}$$ Combining these results, we easily get the equation of state of dark energy $$\begin{aligned}
w & = &\frac{-\dot{\Omega}_{\rm de}/\Omega_{\rm de}^2-3b^2H(1+\mu)^2}{3H\mu} \nonumber \\
& = &-\frac{\Omega_{\rm de}'}{3\Omega_{de}(1-\Omega_{\rm
de})}-\frac{b^2}{\Omega_{\rm de}(1-\Omega_{\rm de})}, \label{4}\end{aligned}$$ where prime denotes the derivative with respect to $x=\ln a$.
Using the definition of the holographic dark energy (\[de\]) and the Friedmann equation, the future event horizon (\[eh\]) can be expressed as $R_{\rm h}=c\sqrt{1+\mu}/H$. Then, for this expression, taking the derivative with respect to $t$ and reducing the result, we get $$\frac{\Omega_{\rm de}'}{\Omega_{\rm de}^2}=(1-\Omega_{\rm
de})\left[\frac{1}{\Omega_{\rm de}}+ \frac{2}{c\sqrt{\Omega_{\rm
de}}}-\frac{3b^2}{\Omega_{\rm de}(1-\Omega_{\rm de})}\right].
\label{5}$$ It is notable that this differential equation governs the whole dynamics of the interacting model of holographic dark energy. Substituting (\[5\]) to (\[4\]) yields $$w= -\frac{1}{3}-\frac{2\sqrt{\Omega_{\rm
de}}}{3c}-\frac{b^2}{\Omega_{\rm de}} .\label{w}$$ Then we can compute the deceleration parameter $$\begin{aligned}
q& = & -\frac{\ddot{a}}{aH^2}=\frac{1}{2}+\frac{3}{2}w\Omega_{\rm
de} =\frac{1}{2}\left(1-3b^2-\Omega_{\rm de}-\frac{2}{c}\Omega_{\rm
de}^{\frac{3}{2}}\right).\label{6}\end{aligned}$$
{width="10cm"}
{width="10cm"}
In order to show the influence of interaction to the cosmic evolution, the cases with dependence of the parameter $b^2$ for the deceleration parameter $q$ are shown in Fig. 1. In Fig. 1, we fix $c=1$ and take the coupling constant $b^2$ as 0, 0.02, 0.06, and 0.10, respectively. Besides, the cases with a fixed $b^2$ and various values of $c$ are also interesting. In Fig. 2, fixing the coupling constant $b^2=0.10$, we plot the evolution diagram of the deceleration parameter $q$ with different values of parameter $c$ (here we take the values of $c$ as 0.9, 1.0, and 1.1, respectively). From Figs. 1 and 2 we learn that the universe experienced an early deceleration and a late time acceleration. Fig. 1 shows that, for a fixed parameter $c$, the cosmic acceleration starts earlier for the cases with interaction than the ones without coupling (for this point see also, e.g., [@Amendola:2002kd]). Moreover, the stronger the coupling between dark energy and dark matter is the earlier the acceleration of universe began. However, the cases with smaller coupling will get bigger acceleration finally in the far future. In addition, Fig. 2 shows that the acceleration starts earlier when $c$ is larger for the same coupling $b^2$, but finally a smaller $c$ will lead to a bigger acceleration. It should be pointed out that, in the interacting holographic dark energy model, the interaction strength has an upper limit because of the evolutionary behavior of the holographic dark energy. For detailed discussions about correlation of the coupling $b^2$ and the parameter $c$, see [@Wang:2005jx]. It is remarkable that, with the interaction between dark energy and dark matter, the case of $c=1$ could not enter a de Sitter phase in the infinite future. In short, the influence of the interaction between dark energy and dark matter to the cosmic evolution is obvious, as manifested by Figs. 1 and 2. On the other hand, nevertheless, as Eq. (\[6\]) shows, though the deceleration parameter $q$ carries the information of the equation of state of dark energy $w$, the property of dynamical evolution for $w$ can not be read out from $q$. For diagnosing properties and evolutionary behaviors of dark energy models exquisitely, more powerful diagnostic tool is a must.
Now we turn to the statefinder diagnosis. For characterizing the expansion history of the universe, one defines the geometric parameters $H=\dot{a}/a$ and $q=-\ddot{a}/aH^2$, namely the Hubble parameter and the deceleration parameter. It is clear that $\dot{a}>0$ means the universe is undergoing an expansion and $\ddot{a}>0$ means the universe is experiencing an accelerated expansion. From the cosmic acceleration, $q<0$, one infers that there may exist dark energy with negative equation of state, $w<-1/3$ and likely $w\sim -1$, but it is hard to deduce the information of the dynamical property of $w$ (namely the time evolution of $w$) from the value of $q$. In order to extract the information on the dynamical evolution of $w$, it seems that we need the higher time derivative of the scale factor, ${\stackrel{...}a}$. Another motivation for proposing the statefinder parameters stems from the merit that they can provide us with a diagnosis which could unambiguously probe the properties of all classes of dark energy models including the cosmological models without dark energy describing the cosmic acceleration. Though at present we can not extract sufficiently accurate information of $\ddot{a}$ and ${\stackrel{...}a}$ from the observational data, we can expect, however, the high-precision observations of next decade may be capable of doing this. Since different cosmological models exhibit different evolution trajectories in the $s-r$ plane, the statefinder parameters can thus be used to diagnose the evolutionary behaviors of various dark energy models and discriminate them from each other. In this paper, we apply the statefinder diagnosis to the interacting holographic dark energy model.
The expansion rate of the universe is described by the Hubble parameter $H$, and the rate of acceleration/deceleration of the expanding universe is characterized by the deceleration parameter $q$. Furthermore, in order to find a more sensitive discriminator of the expansion rate, let us consider the general expansion form for the scale factor of the universe $$a(t)=a(t_0)+\dot{a}|_0(t-t_0)+{\ddot{a}|_0\over
2}(t-t_0)^2+{{\stackrel{...}a}|_0\over
6}(t-t_0)^3+\dots.\label{aexp}$$ Generically, various dark energy models give rise to families of curves $a(t)$ having vastly different properties. In principle, we can confine our attention to small value of $|t-t_0|$ in (\[aexp\]) because the acceleration of the universe is a fairly recent phenomenon. Then, we see, following [@sahni], that a new diagnostic of dark energy dubbed statefinder can be constructed using both second and third derivatives of the scale factor. The second derivative is encoded in the deceleration parameter $q$, and the third derivative is contained in the statefinder parameters $\{r,s\}$. The statefinder parameters $\{r,s\}$ are defined as $$\label{rs1}
r\equiv\frac{\stackrel{...}a}{aH^3},~~~~~
s\equiv\frac{r-1}{3(q-\frac{1}{2})}.$$ Note that the parameter $r$ is also called cosmic jerk. Thus the set of quantities describing the geometry is extended to include $\{H,
q, r, s\}$. Trajectories in the $s-r$ plane corresponding to different cosmological models exhibit qualitatively different behaviors, so the statefinder can be used to discriminate different cosmological models. The spatially flat LCDM (cosmological constant $\lambda$ with cold dark matter) scenario corresponds to a fixed point in the diagram $$\{s,r\}\bigg\vert_{\rm LCDM} = \{ 0,1\} ~.\label{lcdm}$$ Departure of a given dark energy model from this fixed point provides a good way of establishing the “distance” of this model from spatially flat LCDM [@sahni]. As demonstrated in Refs. [@Alam:2003sc]–[@Chang:2007jr], the statefinder can successfully differentiate between a wide variety of dark energy models including the cosmological constant, quintessence, phantom, quintom, the Chaplygin gas, braneworld models and interacting dark energy models, etc.. We can clearly identify the “distance” from a given dark energy model to the LCDM scenario by using the $r(s)$ evolution diagram. The current location of the parameters $s$ and $r$ in these diagrams can be calculated in models. The current values of $s$ and $r$ are evidently valuable since we expect that they can be extracted from data coming from SNAP (SuperNovae Acceleration Probe) type experiments. Therefore, the statefinder diagnosis combined with future SNAP observations may possibly be used to discriminate between different dark energy models.[^2]. The statefinder parameter-pair also can be expressed as $$r=1+\frac{9(\rho+P)\dot{P}}{2\rho\dot{\rho}},~~~~~
s=\frac{(\rho+P)\dot{P}}{P\dot{\rho}},$$ where $\rho$ is the total density and $P$ is the total pressure. Then, by using the Friedmann equation, we can obtain the following concrete expressions $$\begin{aligned}
r&=&1-\frac{3}{2}\Omega_{\rm de}w'+
3\Omega_{\rm de}w\left(1-\frac{1}{c}\sqrt{\Omega_{\rm de}}\right),\label{r}\\
s&=&1+w-\frac{w'}{3w}+\frac{b^2}{\Omega_{\rm de}}.\label{s}\end{aligned}$$ Directly, from Eq. (\[w\]), we have $$\begin{aligned}
w'&=&\frac{\Omega_{\rm de}'}{\Omega_{\rm de}^2}
\left(b^2-\frac{1}{3c}\Omega_{\rm de}^{3/2}\right)\nonumber\\
&=&(1-\Omega_{\rm de})\left(b^2-\frac{\Omega_{\rm
de}^{3/2}}{3c}\right) \left[\frac{1}{\Omega_{\rm de}}-\frac{3b^2}
{\Omega_{\rm de}(1-\Omega_{\rm de})}+\frac{2}{c\sqrt{\Omega_{\rm
de}}}\right],\label{w1}\end{aligned}$$ where the prime denotes the derivative with respect to $x=\ln a$. Note that the whole dynamics of the universe in the interacting holographic dark energy model is governed by the differential equation (\[5\]). So by solving Eq. (\[5\]) we can get the evolution solution of $\Omega_{\rm de}$ and then hold all the cosmological quantities of interest and the whole dynamics of the universe.
{width="10cm"}
$\begin{array}{c@{\hspace{0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\
\includegraphics[scale=0.8]{rsb0.eps} &\includegraphics[scale=0.8]{rsb01.eps} \\
\end{array}$
In what follows we shall diagnose the interacting holographic dark energy model employing the statefinder method. We shall analyze the cases with fixed coupling constant $b^2$ and with fixed parameter $c$, respectively. As demonstrated above, the information of this model can be acquired by solving the differential equation (\[5\]). Making the redshift $z$ vary in a large enough range involving far future and far past, one can solve the differential equation (\[5\]) numerically and then get the evolution trajectories in the statefinder $s-r$ planes for this model. For instance, we plot the statefinder diagram in Fig. 3 for the cases of $c=1$ with various values of coupling such as $b^2=0$, $0.02$, $0.06$ and $0.10$, meanwhile the present density parameter of dark energy is taken to be $\Omega_{\rm de0}=0.73$. The case $b^2=0$ corresponds to the holographic dark energy model without interaction between dark energy and dark matter. The arrows in the diagram denote the evolution directions of the statefinder trajectories and the star corresponds to $\{r=1,s=0\}$ representing the LCDM model. This diagram shows that the evolution trajectories with different interaction strengths exhibit different features in the statefinder plane. When the interaction is absent, the $r(s)$ curve for holographic dark energy ends at the LCDM fixed point, i.e., the universe of this case will evolve to the de Sitter phase in the far future. However, taking the interaction into account, the endpoints of the $r(s)$ curves could not arrive at the LCDM fixed point $(0,
1)$, though all of the evolution trajectories tend to approach this point. It should be mentioned that the statefinder diagnosis for holographic dark energy model without interaction has been investigated in detail in [@Zhang:2005holo], where the focus is put on the diagnosis of the different values of parameter $c$. The statefinder analysis on the holographic dark energy in a non-flat universe see [@Setare:2006xu]. In [@Zhang:2005holo], it has been demonstrated that from the statefinder viewpoint $c$ plays a significant role in this model and it leads to the values of $\{r,
s\}$ in today and future tremendously different. In this paper, by far, we have clearly seen that the interaction between holographic dark energy and dark matter makes the statefinder evolutionary trajectories with the same value of $c$ tremendously different also. If the accurate information of $\{r_0, s_0\}$ can be extracted from the future high-precision observational data in a model-independent manner, these different features in this model can be discriminated explicitly by experiments, one thus can use this method to test the holographic dark energy model as well as other dark energy models. Hence, today’s values of $\{r, s\}$ play a significant role in the statefinder diagnosis. We thus calculate the present values of the statefinder parameters for different cases in the interacting holographic dark energy model and mark them on evolution curves with dots. It can be seen that stronger interaction results in longer distance to the LCDM fixed point. The interaction between holographic dark energy and dark matter prevents the holographic dark energy from behaving as a cosmological constant $\lambda$ ultimately in the far future.
We also plotted the statefinder diagram in the $s-r$ plane for different values of parameter $c$ with $b^2=0$ and $0.10$ in Fig. 4. The left panel is for the holographic dark energy without interaction while the right one is for the case involving the interaction. The star in the figure also corresponds to the LCDM fixed point and the dots marked on the curves represent the present values of the statefinder parameters. Note that the true values of $(s_0, r_0)$ of the universe should be determined in a model-independent way, we can only pin our hope on the future experiments to achieve this. We strongly expect that the future high-precision experiments (e.g. SNAP) may provide sufficiently large amount of precise data to release the information of statefinders $\{H, q, r, s\}$ in a model-independent manner so as to supply a way of discriminating different cosmological models with or without dark energy. From Fig. 4, we can learn that the $r(s)$ evolutions have the similar behavior, i.e. the curves almost start from a fixed point for both cases in the $s-r$ plane. Evidently, the interaction between dark components makes the value of $r$ smaller and the value of $s$ bigger. Also, obviously, the parameter $c$ plays a crucial role in the holographic model.
$\begin{array}{c@{\hspace{0.2in}}c}
\multicolumn{1}{l}{\mbox{}} &
\multicolumn{1}{l}{\mbox{}} \\
\includegraphics[scale=0.8]{ww1b0.eps} &\includegraphics[scale=0.8]{ww1b01.eps}
\end{array}$
As a complement to statefinder diagnosis, we investigate the dynamical property of the interacting holographic dark energy in the $w-w'$ phase plane, where $w'$ represents the derivative of $w$ with respect to $\ln a$. Recently, this method became somewhat popular for analyzing dark energy models. Caldwell and Linder [@Caldwell:2005tm] proposed to explore the evolving behavior of quintessence dark energy models and test the limits of quintessence in the $w-w'$ plane, and they showed that the area occupied by quintessence models in the phase plane can be divided into thawing and freezing regions. Then, the method was used to analyze the dynamical property of other dark energy models including more general quintessence models [@Scherrer:2005je], phantom models [@Chiba:2005tj] and quintom models [@Guo:2006pc], etc.. The $w-w'$ analysis undoubtedly provides us with an alternative way of classifying dark energy models using the quantities describing the dynamical property of dark energy. But, it is obviously that the $(w, w')$ pair is related to statefinder pair $(s, r)$ in a definite way, see Eqs. (\[r\]) and (\[s\]). The merit of the statefinder diagnosis method is that the statefinder parameters are constructed from the scale factor $a$ and its derivatives, and they are expected to be extracted in a model-independent way from observational data, although it seems hard to achieve this at present. While the advantage of the $w-w'$ analysis is that it is a direct dynamical diagnosis for dark energy. Hence, the statefinder $s-r$ geometrical diagnosis and the $w-w'$ dynamical diagnosis can be viewed as complementarity in some sense.
Now let us investigate the interacting model of holographic dark energy in the $w-w'$ plane. In Fig. 5, we plot the evolutionary trajectories of the holographic dark energy in the $w-w'$ plane where the selected curves correspond to $c=0.8$, $1.0$ and $1.2$, respectively. The left graph is an illustrative example without interaction to which we can compare the evolution of the interacting holographic dark energy in the right diagram. Fig. 5 shows clearly that the parameter $c$ and the interaction $b^2$ both play important roles in the evolution history of the universe. The left graph tells us: $c\geq1$ makes the holographic dark energy behave as quintessence-type dark energy with $w\geq-1$ and $c<1$ makes the holographic dark energy behave as quintom-type dark energy with $w$ crossing $-1$ during the evolution history. However, when the interaction between dark components is present, the situation becomes somewhat ambiguous because that the equation of state $w$ loses the ability of classifing dark energies definitely, due to the fact that the interaction makes dark energy and dark matter be entangled in each other. In this circumstance, the conceptions such as quintessence, phantom and quintom are not so clear as usual. But, anyway, we can still use these conceptions in an undemanding sense. It should be noted that when we refer to these conceptions the only thing of interest is the equation of state $w$. The right panel of Fig. 5 tells us: with the interaction (a case of strong coupling, $b^2=0.10$), $c\leq1$ makes the holographic dark energy behave as phantom-type dark energy with $w\leq -1$ and $c>1$ makes the holographic dark energy behave as quintom-type dark energy with $w$ crossing $-1$ during the evolution history.[^3] In this diagram, the effect of the interaction is shown again. When the coupling between the two components is absent, the value of $w'$ first decreases from zero to a minimum then increases again to zero meanwhile the value of $w$ decreases monotonically. Nevertheless, for the case involving the interaction, $w$ increases first to a maximum and then decreases meanwhile $w'$ decreases from a maximum to a negative minimum first and then increases to zero again. Therefore, we see that the $w-w'$ dynamical diagnosis can provide us with a useful complement to the statefinder geometrical diagnosis.
In summary, we have studied the interacting holographic dark energy model from the statefinder viewpoint in this paper. Since the accelerated expansion of the universe was found by astronomical observations, many cosmological models involving dark energy component or modifying gravity have been proposed to interpret this cosmic acceleration. This leads to a problem of how to discriminate between these various contenders. The statefinder diagnosis provides a useful tool to break the possible degeneracy of different cosmological models by constructing the parameters $\{r, s\}$ using the higher derivative of the scale factor. Thus the method of plotting the evolutionary trajectories of dark energy models in the statefinder plane can be used to as a diagnostic tool to discriminate between different models. Furthermore, the values of $\{r, s\}$ of today, if can be extracted from precise observational data in a model-independent way, can be viewed as a discriminator for testing various cosmological models. On the other hand, though we are lacking an underlying theory of the dark energy, this theory is presumed to possess some features of a quantum gravity theory, which can be explored speculatively by taking the holographic principle of quantum gravity theory into account. So the holographic dark energy model provides us with an attempt to explore the essence of dark energy within a framework of fundamental theory. In addition, some physicists believe that the involving of interaction between dark energy and dark matter leads to some alleviation and more understanding to the coincidence problem. It is thus worthwhile to investigate the interacting model of holographic dark energy. We analyzed the interacting holographic dark energy model employing the statefinder parameters as a diagnostic tool. The statefinder diagrams show that the interaction between dark sectors can significantly affect the evolution of the universe and the contributions of the interaction can be diagnosed out explicitly in this method. At last, as the complement to the statefinder geometrical diagnosis, a dynamical diagnosis was also studied, which diagnoses the dynamical property of the interacting holographic dark energy in the $w-w'$ phase plane. We hope that the future high-precision observations can offer more and more accurate data to determine these parameters precisely and consequently shed light on the essence of dark energy.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the grants from the China Postdoctoral Science Foundation (20060400104), the K. C. Wong Education Foundation (Hong Kong), the National Natural Science Foundation of China (10573003,10705041), and the National Basic Research Program of China (2003CB716300).
[99]{}
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[^1]: The parameter $c$ is introduced to parameterize some uncertainties, such as the species of quantum fields in the universe, the effect of curved spacetime, and so forth.
[^2]: It should be noted that the opinion of other authors may not be so optimistic, see, e.g. [@Cattoen:2007id]
[^3]: It should be noted that the old version of holographic cosmology is not compatible with the phantom energy, see e.g., [@Bak:1999hd; @Flanagan:1999jp]
|
---
abstract: 'Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-$k$ version of this problem, a naïve solution is to iteratively find the densest subgraph and remove it in each iteration. However, such a solution is impractical due to high processing cost. The problem is further complicated when dealing with dynamic graphs, since adding or removing an edge requires re-running the algorithm. In this paper, we study the top-$k$ densest-subgraph problem in the sliding-window model and propose an efficient fully-dynamic algorithm. The input of our algorithm consists of an edge stream, and the goal is to find the node-disjoint subgraphs that maximize the sum of their densities. In contrast to existing state-of-the-art solutions that require iterating over the entire graph upon any update, our algorithm profits from the observation that updates only affect a limited region of the graph. Therefore, the top-$k$ densest subgraphs are maintained by only applying local updates. We provide a theoretical analysis of the proposed algorithm and show empirically that the algorithm often generates denser subgraphs than state-of-the-art competitors. Experiments show an improvement in efficiency of up to five orders of magnitude compared to state-of-the-art solutions.'
author:
- |
Muhammad Anis Uddin Nasir[$^{\sharp 1}$]{}, Aristides Gionis[$^{\ddagger 2}$]{}, Gianmarco De Francisci Morales[$^{\diamond 3}$]{}\
Sarunas Girdzijauskas[$^{\sharp 4}$]{}
bibliography:
- 'biblio.bib'
title: 'Fully Dynamic Algorithm for Top-Densest Subgraphs'
---
Conclusion
==========
We studied the top-$k$ densest subgraphs problem for graph streams, and proposed an efficient one-pass fully-dynamic algorithm. In contrast to the existing state-of-the-art solutions that require iterating over the entire graph upon update, our algorithm maintains the solution in one-pass. Additionally, the memory requirement of the algorithm is independent of $k$. The algorithm is designed by leveraging the observation that graph updates only affect a limited region. Therefore, the top-$k$ densest subgraphs are maintained by simply applying local updates to small subgraphs, rather than the complete graph. We provided a theoretical analysis of the proposed algorithm and showed empirically that the algorithm often generates denser subgraphs than the state-of-the-art solutions. Further, we observed an improvement in performance of up to five orders of magnitude when compared to the baselines.
This work gives rise to further interesting research questions: Is it necessary to leverage k-core decomposition algorithm as a backbone? Is it possible to achieve stronger bounds on the threshold for high-degree vertices? Can we design an algorithm with a space bound on the size of the bag? Is it possible to achieve stronger approximation guarantees for the problem? We believe that solving these questions will further enhance the proposed algorithm, making it a useful tool for numerous practical applications.
|
---
abstract: 'Aggregating different pieces of similar information is necessary to generate concise and easy to understand reports in technical domains. This paper presents a general algorithm that combines similar messages in order to generate one or more coherent sentences for them. The process is not as trivial as might be expected. Problems encountered are briefly described.'
author:
- |
James Shaw\
Dept. of Computer Science\
Columbia University\
New York, NY 10027, USA\
[shaw@cs.columbia.edu]{}
title: Conciseness through Aggregation in Text Generation
---
Motivation
==========
Aggregation is any syntactic process that allows the expression of concise and tightly constructed text such as coordination or subordination. By using the parallelism of syntactic structure to express similar information, writers can convey the same amount of information in a shorter space. Coordination has been the object of considerable research (for an overview, see [@van; @Oirsouw87]). In contrast to linguistic approaches, which are generally analytic, the treatment of coordination in this paper is from a synthetic point of view — text generation. It raises issues such as deciding when and how to coordinate. An algorithm for generating coordinated sentences is implemented in PLANDoc [@Kukich; @et; @al.; @93; @McKeown; @et; @al.; @94], an automated documentation system.
PLANDoc generates natural language reports based on the interaction between telephone planning engineers and LEIS-PLAN[^1], a knowledge based system. Input to PLANDoc is a series of messages, or semantic functional descriptions (FD, Fig. 1). Each FD is an atomic decision about telephone equipment installation chosen by a planning engineer. The domain of discourse is currently limited to 31 message types, but user interactions include many variations and combinations of these messages. Instead of generating four separate messages as in Fig. 2, PLANDoc combines them and generates the following two sentences: “[*This refinement activated DLC for CSAs 3122 and 3130 in the first quarter of 1994 and ALL-DLC for CSA 3134 in 1994 Q3. It also activated DSS-DLC for CSA 3208 in 1994 Q3.*]{}”
------------------------------------------------------------------------
((cat message)
(admin ((PLANDoc-message-name RDA)
(runid r-reg1)))
(class refinement)
(action activation)
(equipment-type all-dlc)
(csa-site 3134)
(date ((year 1994) (quarter 3))))
------------------------------------------------------------------------
------------------------------------------------------------------------
This refinement activated ALL-DLC for CSA 3134 in 1994 Q3. (E1 S3 D2)
This refinement activated DLC for CSA 3130 in 1994 Q1. (E2 S2 D1)
This refinement activated DSS-DLC for CSA 3208 in 1994 Q3. (E3 S4 D2)
This refinement activated DLC for CSA 3122 in 1994 Q1. (E2 S1 D1)
Equipment: E1= ALL-DLC, E2= DLC, E3= DSS-DLC
Site: S1= CSA 3122, S2= CSA 3130, S3= CSA 3134, S4= CSA 3208
Date: D1= l994 Q1, D2= 1994 Q3
------------------------------------------------------------------------
System Architecture
===================
Fig. 3 is an overview of PLANDoc’s architecture. Input to the message generator comes from LEIS-PLAN tracking files which record user’s actions during a planning session. The ontologizer adds hierarchical structure to messages to facilitate further processing. The content planner organizes the overall narrative and determines the linear order of the messages. This includes combining atomic messages into aggregated messages, choosing cue words, and determining paraphrases that maintain focus and ensure coherence. Finally the FUF/SURGE package [@Elhadad91; @Robin-PhD] lexicalizes the messages and maps case roles into syntactic roles, builds the constituent structure of the sentence, ensures agreement, and generates the surface sentences.
------------------------------------------------------------------------
(500,40) (-20,0)[(55,35)]{} (45,0)[(55,35)]{} (120,0)[(55,35)]{} (195,0)[(55,35)]{} (270,0)[(55,35)]{} (345,0)[(55,35)]{} (420,0)[(55,40)]{}
(25,15)[(1,0)[20]{}]{} (100,15)[(1,0)[20]{}]{} (175,15)[(1,0)[20]{}]{} (250,15)[(1,0)[20]{}]{} (325,15)[(1,0)[20]{}]{} (400,15)[(1,0)[20]{}]{}
------------------------------------------------------------------------
Combining Strategy
==================
Because PLANDoc can produce many paraphrases for a single message, aggregation during the syntactic phase of generation would be difficult; semantically similar messages would already have different surface forms. As a result, aggregation in PLANDoc is carried out at the content planning level using semantic FDs. Three main criteria were used to design the combining strategy:
1. [**domain independence**]{}: the algorithm should be applicable in other domains.
2. [**generating the most concise text**]{}: it should avoid repetition of phrases to generate shortest text.
3. [**avoidance of overly-complex sentences**]{}: it should not generate sentences that are too complex or ambiguous for readers.
The first aggregation step is to identify semantically related messages. This is done by grouping messages with the same action attribute. Then the system attempts to generate concise and unambiguous text for each action group separately. This reduces the problem size from tens of messages into much smaller sizes. Though this heuristic disallows the combination of messages with different actions, the messages in each action group already contain enough information to produce quite complex sentences.
The system combines the maximum number of related messages to meet the second design criterion–generating the most concise text. But such combination is blocked when a sentence becomes too complex. A bottom-up 4-step algorithm was developed:
1. [**Sorting**]{}: putting similar messages right next to each other.
2. [**Merging Same Attribute**]{}: combining adjacent messages that only have one distinct attribute.
3. [**Identity Deletion**]{}: deletion of identical components across messages.
4. [**Sentence Breaking**]{}: determining sentence breaks.
Step 1: Sorting
---------------
The system first ranks the attributes to determine which are most similar across messages with the same action. For each potential distinct attribute, the system calculates its rank using the formula $m - d$, where $m$ is the number of messages and $d$ is the number of distinct attributes for that particular attribute. The rank is an indicator of how similar an attribute is across the messages. Combining messages according to the highest ranking attribute ensures that minimum text will be generated for these messages. Based on the ranking, the system reorders the messages by sorting, which puts the messages that have the same attribute right next to each other. In Fig. 2, [*equipment*]{} has rank 1 because it has 3 distinct equipment values – ALL-DLC, DLC, and DSS-DLC; [*date*]{} has rank 2 because it has two distinct date values – 1994 Q1 and 1994 Q3; [*site*]{} has rank 0. Attribute [*class*]{} and [*action*]{} (Fig. 1) are ignored because they are always the same at this stage. When two attributes have the same rank, the system breaks the tie based on a priority hierarchy determined by the domain experts. Because the final sorting operation dominates the order of the resulting messages, PLANDoc sorts the message list from the lowest rank attribute to the highest. In this case, the ordering for sorting is [*site*]{}, [*equipment*]{}, and then [*date*]{}. The resulting message list after sorting each attribute is shown in Fig. 4.
------------------------------------------------------------------------
(E2 S1 D1) (E1 S3 D2) (E2 S1 D1)
(E2 S2 D1) (E2 S1 D1) (E2 S2 D1)
(E1 S3 D2) --> (E2 S2 D1) --> (E1 S3 D2)
(E3 S4 D2) (E3 S4 D2) (E3 S4 D2)
by Site by Equipment by Date
------------------------------------------------------------------------
Step 2: Merging Same Attribute
------------------------------
The list of sorted messages is traversed. Whenever there is only one distinct attribute between two adjacent messages, they are merged into one message with a conjoined attribute, which is a list of the distinct attributes from both messages. What about messages with two or more distinct attributes? Merging two messages with two or more distinct attributes will result in a syntactically valid sentence but with an undesirable meaning: “[*This refinement activated ALL-DLC and DSS-DLC for CSAs 3122 and 3130 in the third quarter of 1993.*]{}”
By tracking which attribute is compound, a third message can be merged into the aggregate message if it also has the same distinct attribute. Continue from Step 1, (E2 S1 D1) and (E2 S2 D1) are merged because they have only one distinct attribute, [*site*]{}. A new FD, (E2 (S1 S2) D1), is assembled to replace those two messages. Note that although (E1 S3 D2) and (E3 S4 D2) have the date in common, they are not combined because they have more than one distinct attribute, [*site*]{} and [*equipment*]{}.
Step 2 is applied to the message list recursively to generate possible crossing conjunction, as in the following output which merges [*four*]{} messages: “[*This refinement activated ALL-DLC and DSS-DLC for CSAs 3122 and 3130 in the third quarter of 1993.*]{}” Though on the outset this phenomenon seems unlikely, it does happen in our domain.
Step 3: Identity Deletion
-------------------------
After merging at step 2, the message list left in an action group either has only one message, or it has more than one message with at least two distinct attributes between them. Instead of generating two separate sentences for (E2 (S1 S2) D1) and (E1 S3 D2), the system realizes that both the subject and verb are the same, thus it uses deletion on identity to generate “[*This refinement activated DLC for CSAs 3122 and 3130 in 1994 Q1 and \[this refinement activated\] ALL-DLC for CSA 3134 in 1994 Q3.*]{}” For identical attributes across two messages (as shown in the bracketed phrase), a “deletion” feature is inserted into the semantic FD, so that SURGE will suppress the output.
Step 4: Sentence Break
----------------------
Applying deletion on identity blindly to the whole message list might make the generated text incomprehensible because readers might have to recover too much implicit information from the sentence. As a result, the combining algorithm must have a way to determine when to break the messages into separate sentences that are easy to understand and unambiguous.
How much information to pack into a sentence does not depend on grammaticality, but on coherence, comprehensibility, and aesthetics which are hard to formalize. PLANDoc uses a heuristic that always joins the first and second messages, and continues to do so for third and more if the distinct attributes between the messages are the same. This heuristics results in parallel syntactic structure and the underlying semantics can be easily recovered. Once the distinct attributes are different from the combined messages, the system starts a new sentence. Using the same example, (E2 (S1 S2) D1) and (E1 S3 D2) have three distinct attributes. They are combined because they are the first two messages. Comparing the third message (E3 S4 D2) to (E1 S3 D2), they have different [*equipment*]{} and [*site*]{}, but not [*date*]{}, so a sentence break will take place between them. Aggregating all three messages together will results in questionable output. Because of the parallel structure created between the first 2 messages, readers are expecting a different [*date*]{} when reading the third clause. The second occurrence of “1994 Q3” in the same sentence does not agree with readers’ expectation thus potentially confusing.
Future Directions
=================
In this paper, I have described a general algorithm which not only reduces the amount of the text produced, but also increases the fluency of the text. While other systems do generate conjunctions, they deal with restricted cases such as conjunction of subjects and predicates[@Dalianis]. There are other interesting problems in aggregations. Generating marker words to indicate relationships in conjoined structures, such as “respectively”, is another short term goal. Extending the current aggregation algorithm to be more general is currently being investigated, such as combining related messages with different actions.
Acknowledgements
================
The author thanks Prof. Kathleen McKeown, and Dr. Karen Kukich at Bellcore for their advice and support. This research was conducted while supported by Bellcore project \#CU01403301A1, and under the auspices of the Columbia University CAT in High Performance Computing and Communications in Healthcare, a New York State Center for Advanced Technology supported by the New York State Science and Technology Foundation.
[AAAAAAAAAAAAAAAAAA]{}
Dalianis, Hercules, and Hovy, Edward. 1993. Aggregation in Natural Language Generation. In [*Proceedings of the Fourth European Workshop on Natural Language Generation*]{}, Pisa, Italy.
Elhadad, Michael. 1991. FUF: The universal unifier - user manual, version 5.0. , Columbia Univ.
Robin, Jacques. 1994. Ph.D. thesis, Computer Science Department, Columbia Univ.
Kukich, K., McKeown, K., Morgan, N., Phillips, J., Robin, J., Shaw, J., and Lim, J. 1993. User-Needs Analysis and Design Methodology for an Automated Documentation Generator. In [*Proceedings of the Fourth Bellcore/BCC Symposium on User-Centered Design*]{}, Piscataway, NJ.
McKeown, Kathleen, Kukich, Karen, and Shaw, James. 1994. Practical Issues in Automatic Documentation Generation. In [*Proceedings of the 4th Conference on Applied Natural Language Processing*]{}, Stuttgart, p.7-14.
van Oirsouw, Robert. 1987. Beckenham: Croom Helm.
[^1]: LEIS is a registered trademark of Bell Communications Research, Piscataway, NJ.
|
---
abstract: 'In this paper, we deal with the asymmetric Orlicz zonotopes by using the method of shadow system. We establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.'
address:
- '1. Department of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, People’s Republic of China'
- '2, 3. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang, Guizhou 550001, People’s Republic of China.'
author:
- 'Fangwei Chen$^1$, Congli Yang$^2$, Miao Luo$^{2}$,'
title: Volume inequalities for asymmetric Orlicz zonotopes
---
[^1]
introduction
============
A classical problem in convex geometry is to find the maximizer or minimizer of the volume product among convex bodies. The celebrated Blaschke-Santaló inequality characterizes ellipsoids are the maximizers of this function on convex bodies. However, finding the minimizer of this function is a trouble in convex geometry. Only in two dimensional case, this problem is solved by Mahler (see, e.g., [@mah-ein-min1939; @mah-ein-ube1939]). Moreover, it is conjectured by him that simplices are the solution of this function for all dimensional $n$, which is called the Mahler’s conjecture. Although it is extremely difficult to attack, but it attracts lots of author’s interests, many substantial inroads have been made. One can refer to e.g., [@bar-fra-the2013; @bou-mil-new1987; @cam-gro-on2006; @cam-gro-vol2006; @fra-mey-zva-an2012; @gor-mey-rei-zon1988; @hug-sch-rev2011; @kim-rei-loc2011; @kup-fro2008; @mey-rei-sha2006; @rei-zon1986; @rog-she-som1958] for more about this conjecture.
One aspect of the researches for the Mahler’s conjecture is to make studying the volume product of zonotopes or zonoids, that is the Minkowski sums of origin-symmetric line segments in $\mathbb R^n$, and their limits with respect to the Hausdorff distance (see, e.g., [@gor-mey-rei-zon1988; @rei-zon1986; @sch-wei-zon1983]). Although the restriction to zonotopes and zonoids is a regrettable drawback, but there seems no approach for general convex bodies for this problem. On the other hand, inequalities for zonoids can be applied to stochastic geometry (see [@sch-wei-sto2008]).
In the last century, the volume product inequalities in Euclidean space, $\mathbb R^n$, are widely been generalized with the development of the $L_p$-Minkowski theory. See, for example, [@lut-the1993; @lut-the1996; @lut-yan-zha-lp2005; @lut-yan-zha-vol2004; @lut-yan-zha-vol2007; @lut-yan-zha-lp2000; @wer-ye-new2008; @sta-the2002; @sta-on2003; @cam-gro-on2002; @cam-gro-the2002; @cam-gro-on2006; @fir-p1962] for more details about the volume product inequalities with $L_p$-Minkowski theory. The $L_p$-volume product inequalities for zonotopes, together with its dual volume ratio inequality, were established by Campi and Cronchi [@cam-gro-vol2006]. These results extend the results of Reisner [@rei-zon1986]. However, all of these results are restricted to the origin-symmetric setting. The asymmetric extension of the $L_p$-volume product inequality and $L_p$-volume ratio inequality, along with the characterization of its extremals are established by Weberndorfer in [@web-sha2013], and the Campi and Cronchi’s results as a special case. The seminal work in studying the asymmetric geometric inequalities are very important in convex geometry. For example, in the paper of Ludwig [@lud-min2005], she’s characterization of the asymmetric $L_p$-centroid body and asymmetric $L_p$-projection body, which establishes the classification of the $SL(n)$ invariant Minkowski valuation on convex set. After that, the asymmetric geometric inequalities involving the volume and other geometric invariant are emerged. For instance, the asymmetric $L_p$-centroid body operator turned out to be an extension of $L_p$ version of the Blaschke-Santaló inequality for all convex bodies, whereas established by Lutwak and Zhang [@lut-zha-bla1997] for origin-symmetric setting. One can refer to [@hab-sch-gen2009; @hab-sch-asy2009; @hab-sch-xia-an2012; @sch-web-vol2012] for more details.
Beginning with the articles [@hab-lut-yan-zha-the2010; @lut-yan-zha-orl-cen2010; @lut-yan-zha-orl-pro2010] of Haberl, Lutwak, Yang and Zhang, a more wide extension of the $L_p$-Brunn-Minkowski theory emerged, called the Orlicz Brunn-Minkowski theory. In these papers, the Orlicz Busemann projection inequality and Orlicz Busemann centroid inequality were established. Recently, in a paper of Gardner, Hug and Weil [@gar-hug-wei-the2014], a systematic studies are made on the Orlicz Minkowski addition, the Orlicz Brunn-Minkowski inequality and Orlicz Minkowski inequality are obtained. See, e.g., [@bor-str2013; @bor-lut-yan-zha-the2012; @bor-lut-yan-zha-the2013; @che-zho-yan-on2011; @che-yan-zho-the2014; @gar-hug-wei-the2014; @hab-lut-yan-zha-the2010; @hua-he-on2012; @zhu-zho-xu-dua2014; @zou-xio-orl2014] about the Orlicz Brunn-Minkowski theory.
In view of the importance of the volume product inequality in convex geometry, we tempted to consider the naturally posed problem in the wide interest of the Orlicz Brunn-Minkowski theory. What is like the volume product inequality or volume ratio inequality for asymmetric Orlicz zonotopes? In this context, the main goal of this paper is to establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes.
Throughout this paper, let $ \mathcal C$ be the class of convex, strictly increasing functions $\varphi:[0,\infty)\rightarrow [0,\infty)$ satisfying $\varphi(0)=0$ and $\varphi(1)=1$.
Suppose that $\Lambda$ is a finite set of vectors from $\mathbb R^n\setminus \{o\}$, the asymmetric Orlicz zonotope $Z^+_\varphi\Lambda$ is the unique compact convex set with support function $$\begin{aligned}
h_{Z^+_\varphi\Lambda}(u)=\inf\left\{\lambda>0:\sum_{w\in\Lambda}\varphi\left(\frac{\langle w, u\rangle_+}{\lambda}\right)\leq 1\right\}.\end{aligned}$$ Where $u\in \mathbb R^n$ and $\langle w,u\rangle_+=\max\{0, \langle w,u\rangle\}$ denotes the positive part of the Euclidean scalar product.
Specially, if take $\varphi(t)=t^p$, $p\geq 1$, then $Z^+_\varphi\Lambda$ is precisely the $L_p$-asymmetric zonotope $Z^+_p\Lambda$ defined in [@web-sha2013].
In this paper, our main results are the volume product inequality and volume ratio inequality for the asymmetric Orlicz zonotopes.
Let $\Lambda_\bot=\{e_1,\cdots,e_n\}$ denote the canonical basis of $\mathbb R^n$, $Z^{+,*}_\varphi\Lambda$ denotes the polar body of $Z^{+}_\varphi\Lambda$ with respect to the Santaló point. For the asymmetric Orlicz zonotopes, we establish the following volume product inequality.
Suppose $\varphi\in \mathcal C$ and $\Lambda$ is a finite and spanning multiset. Then $$\begin{aligned}
V(Z^{+,*}_\varphi\Lambda)V(Z^+_1\Lambda)\geq V(Z^{+,*}_\varphi\Lambda_\bot)V(Z^+_1\Lambda_\bot).
\end{aligned}$$ Equality holds with $\varphi\neq Id$ if and only if $\Lambda$ is a $GL(n)$ image of the canonical basis $\Lambda_\bot$. If $\varphi=Id$, the identity function, the equality holds if and only if $Z^+_1\Lambda$ is a parallelepiped.
We follow the notations of paper [@web-sha2013]. A set $\Lambda$ of vectors from $\mathbb R^n$ is called [*obtuse*]{} if every pair of distinct vectors $u,\,\, v$ from $\Lambda$ satisfies $$\begin{aligned}
\langle u,v\rangle_+=0.\end{aligned}$$
Another result regards to the volume ratio for asymmetric Orlicz zonotopes associate with the obtuse sets $\Lambda$ says that it attains its maximum if $\Lambda$ is a canonical basis of $\mathbb R^n$.
Suppose $\varphi\in\mathcal C$ and $\Lambda$ is a finite and spanning set. Then $$\begin{aligned}
\frac{V(Z^+_\varphi\Lambda)}{V(Z^+_1\Lambda)}\leq\frac{V(Z^+_\varphi \Lambda_\bot)}{V(Z^+_1\Lambda_\bot)}.
\end{aligned}$$ With equality if and only if $\Lambda$ is a $GL(n)$ image of an obtuse set.
The paper is organized as follows. In section 2, we introduce the asymmetric Orlicz zonotopes and show some of their properties. The shadow system and some results of them are given in Section 3. Section 4 deals with the equality case of the volume product inequality and volume ratio inequality for Orlicz zonotopes. The final proofs of the main theorems are presented in section 5.
preliminaries
=============
For quick reference we recall some basic definition and notations in convex geometry that is required for our results. Good references see Gardner [@gar-geo2006], Gruber [@gru-con2007], Schneider [@sch-con1993].
Let $\mathcal K^n$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean $n$-space, $\mathbb R^n$. If $K$ is a convex body, denote by $V(K)$ its $n$-dimensional volume, and by $h_K(\cdot):S^{n-1}\rightarrow \mathbb R$ the support function of $K$; i.e., for $u\in \mathbb S^{n-1}$, $$h_K(u)=max\big\{\langle u, x\rangle :x\in K\big\},$$ where $\langle u, x\rangle$ denotes the standard inner product in $\mathbb R^n$. It is shown that the sublinear support function characterizes a convex body and, conversely, every sublinear function on $\mathbb R^n$ is the support function of a nonempty compact convex set.
Two convex body $K,\,\,L$ satisfy $K\subseteq L$ if and only if $h_K(\cdot)\leq h_L(\cdot)$. By the definition of the support function, it follows immediately that the support function of the image $\phi K:=\{\phi y:y\in K\}$ is given by $$\begin{aligned}
h_{\phi K}(x)=h_K(\phi ^Tx)\end{aligned}$$ for $\phi\in GL(n)$. Here $\phi^T$ denotes the transpose of $\phi$.
Let $K$ be a convex body, for every interior point $s$ of $K$, $$\begin{aligned}
K^s=\{y\in \mathbb R^n:\langle y, x-s\rangle\leq 1 \,\,for \,\,\,all\,\, x\in K\}\end{aligned}$$ defines a convex body that is called the polar body of $K$ with respect to $s$. A well-known result of Santaló states that, in every convex body $K\in \mathbb R^n$, there exists a unique point $s(K)\in K$, the Santaló point, such that $$\begin{aligned}
V(K^{s(K)})=\min_{s\in K}V(K^s).\end{aligned}$$ To shorten the notation, we shall denote $K^{s(K)}$ by $K^*$. It is well known that the polarization with respect to the Santaló point is translation invariant and $GL(n)$ contravariant, that is, $$\begin{aligned}
(K+y)^*=K^*\,\,\,\,\,\,\,\,\,and \,\,\,\,\,\,\,\,\,\,(\phi K)^*=\phi^{-T}K^*,\end{aligned}$$ for $y\in \mathbb R^n$ and $\phi\in GL(n)$.
Suppose $\Lambda$ is a set in $\mathbb R^n$, it is called multiset if its members are allowed to appear more than once. More precisely, a multiset $\Lambda$ is identified with its multiplicity function $1_{\Lambda}:\mathbb R^n\rightarrow N\cup\{0\}$, that generalizes the characteristic function of sets. We say that a vector is an element of a multiset if the corresponding multiplicity function evaluated at the vector is greater than zero, and call a multiset finite if it contains only a finite number of vectors. If these vectors span $\mathbb R^n$, then we say that the multiset is spanning.
The operation between multisets can be defined using the multiset function. For instance, the union $\Lambda_1\uplus\Lambda_2$ of $\Lambda_1$ and $\Lambda_2$ is defined as $$\begin{aligned}
1_{\Lambda_1\uplus \Lambda_2}(x)=1_{\Lambda_1}(x)+1_{\Lambda_2}(x),\end{aligned}$$ and $\Lambda_1-\Lambda_2$ is defined as $$\begin{aligned}
1_{\Lambda_1- \Lambda_2}(x)=\max\{0, 1_{\Lambda_1}(x)-1_{\Lambda_2}(x)\}.\end{aligned}$$ We write multisets in usual set notation, that is, $\Lambda=\{v_1,\cdots,v_m\}$.
The asymmetric $L_p$-zonotopes associated with finite and spanning multisets $\Lambda=\{v_1,\cdots,v_m\}$ are defined by Weberdorfer [@web-sha2013]. Here we extend the notations to asymmetric Orlicz zonotopes.
Let $ \mathcal C$ be the class of convex, strictly increasing functions $\varphi:[0,\infty)\rightarrow [0,\infty)$ satisfying $\varphi(0)=0$ and $\varphi(1)=1$. Here the normalization is a matter of convenience and other choices are possible. It is not hard to conclude that $\varphi\in \mathcal C$ is continuous on $[0,\infty)$.
Asymmetric Orlicz zonotopes associated with finite and spanning multisets $\Lambda=\{v_1,\cdots,v_m\}$ are defined by $$\begin{aligned}
\label{asy-orl-zon}
h_{Z^+_\varphi\Lambda}(u)=\inf\Bigg\{\lambda>0:\sum_{i=1}^m\varphi\bigg(\frac{\langle v_i,u\rangle_+}{\lambda}\bigg)\leq 1\Bigg\},\end{aligned}$$ for all $u\in S^{n-1}$. Moreover, if $\langle v_i,u\rangle_+=0$ for all $i=1,\cdots,m$, we define $h_{Z^+_\varphi\Lambda}(u)=0$.
In fact, by the convexity of $\varphi$ and the sub-additive of $\langle v, \cdot\rangle_+$, we have $$\begin{aligned}
\varphi\bigg(\frac{\langle v,u_1+u_2\rangle_+}{\lambda_1+\lambda_2}\bigg)\leq\frac{\lambda_1}{\lambda_1+\lambda_2}\varphi\bigg(\frac{\langle v,u_1\rangle_+}{\lambda_1}\bigg)+\frac{\lambda_2}{\lambda_1+\lambda_2}\varphi\bigg(\frac{\langle v,u_2\rangle_+}{\lambda_2}\bigg),
\end{aligned}$$ it follows that the support function defined in (\[asy-orl-zon\]) is sublinear, which grants the existence of convex body $Z^+_\varphi\Lambda$.
Specially, if $p\geq 1$ and take $\varphi(t)=t^p$, then it turns out that $Z^+_\varphi\Lambda=Z^+_p\Lambda$.
Note that $\varphi\in\mathcal C$ is strictly convex and increasing on $[0,\infty)$, it follows that the function $$\begin{aligned}
\lambda\mapsto \sum_{i=1}^m\varphi\bigg(\frac{\langle v_i,u\rangle_+}{\lambda}\bigg)\end{aligned}$$ is strictly decreasing on $[0,\infty)$. The next lemma easily follows.
\[orl-nor-lem\] Suppose $\varphi\in \mathcal C$ and $\Lambda=\{v_1,\cdots,v_m\}$ spans $\mathbb R^n$. For $u_0\in S^{n-1}$, then
(1),$\sum_{i=1}^m\varphi\Big(\frac{\langle v_i,u_0\rangle_+}{\lambda_0}\Big)=1$ if and only if $\lambda_0 =h_{Z^+_\varphi\Lambda}(u_0)$;
(2),$\sum_{i=1}^m\varphi\Big(\frac{\langle v_i,u_0\rangle_+}{\lambda_0}\Big)>1$ if and only if $\lambda_0 <h_{Z^+_\varphi\Lambda}(u_0)$;
(3),$\sum_{i=1}^m\varphi\Big(\frac{\langle v_i,u_0\rangle_+}{\lambda_0}\Big)<1$ if and only if $\lambda_0 >h_{Z^+_\varphi\Lambda}(u_0)$.
A simple observe of definition (\[asy-orl-zon\]) is that the operator $Z^+_\varphi$ on finite and spanning multisets is $GL(n)$ equivariant, that is, $Z^+_\varphi \phi\Lambda=\phi Z^+_\varphi\Lambda$ holds for all $\phi\in GL(n)$. In fact, $$\begin{aligned}
h_{Z^+_\varphi\phi\Lambda}(u)&=\inf\Bigg\{\lambda>0:\sum_{i=1}^m\varphi\bigg(\frac{\langle \phi v_i, u\rangle_+}{\lambda}\bigg)\leq 1\Bigg\}\\
&=\inf\Bigg\{\lambda>0:\sum_{i=1}^m\varphi\bigg(\frac{\langle v_i,\phi^T u\rangle_+}{\lambda}\bigg)\leq 1\Bigg\}\\
&=h_{Z^+_\varphi\Lambda}(\phi^Tu)=h_{\phi Z^+_\varphi\Lambda}(u),\end{aligned}$$ holds for all $u\in S^{n-1}$. Moreover, the asymmetric Orlicz zonotopes defined by (\[asy-orl-zon\]) are closely related to the origin symmetric Orlicz zonotopes $Z_\varphi \Lambda$ defined in [@wan-len-hua-vol2012]. Specially, $$\begin{aligned}
\varphi\bigg(\frac{\rvert\langle v_i, u\rangle\lvert}{\lambda}\bigg)&=\varphi\bigg(\frac{\langle v_i, u\rangle_++\langle -v_i, u\rangle_+}{\lambda}\bigg)\\
&=\varphi\bigg(\frac{\langle v_i, u\rangle_+}{\lambda}\bigg)+\varphi\bigg(\frac{\langle -v_i, u\rangle_+}{\lambda}\bigg).\end{aligned}$$ Which implies that $Z_\varphi\Lambda=Z^+_\varphi(\Lambda\uplus-\Lambda)$.
In the following, the volume product and volume ratio for Orlicz zonotopes associate with the multisets $\Lambda$ always refer to $V(Z^{+,*}_\varphi\Lambda)V({Z^+_1\Lambda})$ and $\frac{V(Z^+_\varphi \Lambda)}{V(Z^+_1\Lambda)}$, respectively.
shadow system of multiset
=========================
The notation of shadow system, introduced by Rogers and Shephard (see [@rog-she-som1958; @she-sha1964]), play an important role in proving geometric inequalities in convex geometry. For example, this method was used by Campi, Gronchi, Meyer and Reisner (see, e.g., [@cam-col-gro-a1999; @cam-gro-the2002; @cam-gro-on2002; @cam-gro-on2006; @cam-gro-vol2006; @mey-rei-sha2006]). A shadow system $X_t$ of points from $\mathbb R^n$ is a family of sets which can be defined as follows: $$\begin{aligned}
X_t=\{x_i+t\beta_i v: x_i \in\mathbb R^n\}\end{aligned}$$ where $t\in[t_1,t_2]$, $\beta_i\in \mathbb R$. Here $t$ can be seen as a time-like parameter and $\beta_i$ as the speed of the point $x_i$ along the direction $v$.
Let $\Lambda=\{v_1,\cdots, v_m\}$ be a finite multiset such that $\Lambda\setminus v_1$ is spanning. Following the ideas of Campi and Gronchi [@cam-gro-vol2006], define $\Lambda^a_t=\{w_1(t),\cdots,w_m(t)\}$, where $$\begin{aligned}
\label{ort-sha-sys}
w_i(t)=\left\{
\begin{array}{ll}
(1+t a) v_1, & \hbox{i=1;} \\
v_i-t\frac{\langle v_1,v_i\rangle}{\lVert v_1\rVert^2}v_1, & \hbox{otherwise.}
\end{array}
\right.
\end{aligned}$$ Where $t$ varies in $[-a^{-1},1]$, and $$\begin{aligned}
\label{sha-sys-con}
a=\frac{\sum\limits_{2\leq i_1<\cdots<i_n\leq m}\big|[v_{i_1},\cdots ,v_{i_n}]\big|}{\sum\limits_{2\leq i_2<\cdots<i_n\leq m}\big|[v_{1},v_{i_2}\cdots ,v_{n}]\big|},
\end{aligned}$$ here $[v_{i_1},\cdots ,v_{i_n}]$ denotes the determinant of the matrix whose rows are $v_{i_1},\cdots ,v_{i_n}$. From the definition we have $\Lambda^a_0=\Lambda$, and $w_1(1)$ is orthogonal to the remaining vectors in $\Lambda_1^a$, while $w_1(-a^{-1})=o$. Moreover, by the construction (\[ort-sha-sys\]), $\Lambda^a_t$, $t\in[-a^{-1},1]$, is a shadow system of multisets along the direction $v=\frac{v_1}{\lVert v_1\rVert}\in S^{n-1}$.
The important result of Campi and Gronchi is that for $t\in [-a^{-1},1]$, the asymmetric $L_1$-zonotopes associate $\Lambda^a_t$ preserve the volume, and then be extended to asymmetric $L_p$-zonotopes by Weberndorfer. From now on, we use $\Lambda_t$ to denote $\Lambda^a_t$, the orthogonalization of $\Lambda$ with respect to $v_1$ if $a$ is determined by (\[sha-sys-con\]).
In the following, we will show that the asymmetric Orlicz zonotopes associate with the shadow system of a multiset along direction $v$ is independent of $t$.
\[pro-ind-lem\] Suppose that $\Lambda_t$, $t\in[-a^{-1}, 1]$ is a shadow system of multisets along the direction $v\in S^{n-1}$. Then the orthogonal projection of $Z_{\varphi}^{+}\Lambda_t$ onto $v^\bot$ is independent of $t$.
By the definition of $\Lambda_t$, for $x\in{v^\bot}$ we have $$\begin{aligned}
h_{Z^+_\varphi \Lambda_t}(x)&=\inf\left\{\lambda>0: \sum^m_{i=1}\varphi\left(\frac{\langle v_i+t\beta_i v,x\rangle_+ }{\lambda}\right)\leq 1\right\}\\
&=\inf\left\{\lambda>0: \sum^m_{i=1}\varphi\left(\frac{\langle v_i,x\rangle_+ }{\lambda}\right)\leq 1\right\}\\
&=h_{Z^+_\varphi \Lambda}(x).
\end{aligned}$$ Which shows the result.
Before characterizing the shadow system of convex bodies along with direction $v$ distinguish with others, we introduce the uppergraph function $\overline g_v(K, \cdot)$ and the lowergraph function $\underline g_v(K, \cdot)$ of a convex body $K$. $$\begin{aligned}
\begin{split}
\overline g_v(K, x):=\sup\{\lambda\in \mathbb R: x+\lambda v\in K\};\\
\underline g_v(K, x):=\inf\{\lambda\in \mathbb R: x+\lambda v\in K\}.
\end{split}\end{aligned}$$ An alternative representation of the above formulas are obtain by Weberndorfer [@web-sha2013]. Let $w\in v^\bot$, then $$\begin{aligned}
\label{up-low-gra}
\begin{split}
\overline g_v(K,x)=\inf_{w\in v^\bot}\{h_K(v+w)-\langle x, w\rangle\};\\
\underline g_v(K, x)=-\inf_{w\in v^\bot}\{h_K(-v-w)+\langle x, w\rangle\}.
\end{split}\end{aligned}$$ for all $x\in v^\bot$.
Now we present the characterization of a shadow system obtain by Campi and Gronchi.
[@cam-gro-on2002]\[cam-gro-thm\] Let $K_t$, $t\in[-a^{-1},1]$, be one parameter family of convex bodies such that $K_t|_{v^\bot}$ is independent of $t$. Then $K_t$, $t\in[-a^{-1},1]$, be a shadow system of convex bodies along the direction $v$ if and only if for every $x\in K_0|_{v^\bot}$, the functions $t\rightarrow \overline g_v(K_t,x)$ and $t\rightarrow -\underline{g}_v(K_t,x)$ are convex and $$\begin{aligned}
\label{upp-low-ine}
\underline g_v(K_{\lambda s+\mu t}, x)\leq \lambda\overline g_v(K_s,x)+\mu \underline g_v(K_t,x)\leq\overline g_v(K_{\lambda s+\mu t}, x)
\end{aligned}$$ for every $s, t\in[-a^{-1},1]$ and $\lambda,\mu\in (0,1)$ such that $\lambda+\mu=1$.
A remarkable result about the volume of a shadow system is due to Shephard.
[@she-sha1964] \[vol-con-lem\] Every mixed volume involving $n$ shadow systems along the same direction is a convex function of the parameter. In particular, the volume $V(K_t)$ and all quermassintegrals $W_i(K_t),\,\,i=1,\,2,\,\cdots, n$, of a shadow system are convex functions of $t$.
This result was largely used by Campi and Gronchi [@cam-gro-the2002; @cam-gro-on2002; @cam-gro-on2006; @cam-gro-vol2006], Li [@li-len-a2011] and Chen [@che-zho-yan-on2011]. In the following, we show that the support function of asymmetric Orlicz zonotopes associate with a shadow system of multisets $\Lambda_t$, $t\in[-a^{-1},1]$, is a Lipschitz function of t.
Let $f=(f_1,\cdots,f_n):\mathbb R^n\rightarrow \mathbb R$ be a real value function, $\overline v=(v_1,\cdots,v_n)$, $\beta=(\beta_1,\cdots,\beta_n)$ and $v(t)=(v_1(t),\cdots, v_n(t))$ are vectors in $\mathbb R^n$, where $v_i(t)=v_i+t\beta_i v$ for $i=1,\cdots,n$, as defined before. For notational convenience we define $$\begin{aligned}
\label{orl-nor-def}
\lVert f(t)\rVert_\varphi:=\inf\left\{\lambda>0; \sum^m_{i=1}\varphi\Big(\frac{\lvert f_i(t)\rvert}{\lambda}\Big)\leq 1\right\},
\end{aligned}$$ for real-valued functions $f$ on $\mathbb R^n$, and $[\cdot]_+:=\max\{\cdot, 0\}$. By Lemma \[orl-nor-lem\], if $c>0$, we have $\lVert c f\rVert=\lvert c\rvert \lVert f\rVert$. Moreover, if $f\leq g$ for all $t\in \mathbb R^n$, we have $$\begin{aligned}
\lVert f\rVert_\varphi\leq \lVert g\rVert_\varphi.\end{aligned}$$
\[lip-sup-lem\] Suppose $\varphi\in\mathcal C$ and $\Lambda_t$, $t\in [-a^{-1}, 1]$, is a shadow system of multiset $\Lambda=\{v_1,\cdots,v_m\}$ along the direction $v$ and speed function $\beta$. If $t_1, t_2\in [-a^{-1}, 1]$ and $x\in \mathbb R^n$, then $$\begin{aligned}
\big| h_{Z_{\varphi}^{+}\Lambda_{t_1}}(x)- h_{Z_{\varphi}^{+}\Lambda_{t_2}}(x)\big|\leq \lVert \beta\langle v, x\rangle \rVert_\varphi \lvert t_1-t_2\rvert.
\end{aligned}$$
From definition (\[orl-nor-def\]) and Lemma \[orl-nor-lem\], we have $$\begin{aligned}
\begin{split}
\rVert f(t)\lVert_\varphi=\lambda_1 \Leftrightarrow \sum^{m}_{i=1}\varphi\Big(\frac{|f_i(t)|}{\lambda_1}\Big)=1; \\
\rVert g(t)\lVert_\varphi=\lambda_2 \Leftrightarrow \sum^{m}_{i=1}\varphi\Big(\frac{|g_i(t)|}{\lambda_2}\Big)=1.
\end{split}\end{aligned}$$ The convexity of $\varphi$ shows $$\begin{aligned}
\label{con-lam}
\varphi\left(\frac{|f_i(t)+g_i(t)|}{\lambda_1+\lambda_2}\right)\leq\frac{\lambda_1}{\lambda_1+\lambda_2}
\varphi\left(\frac{|f_i(t)|}{\lambda_1}\right)+\frac{\lambda_2}{\lambda_1+\lambda_2}
\varphi\left(\frac{|g_i(t)|}{\lambda_2}\right).
\end{aligned}$$ Summing both sides of (\[con-lam\]) with respect to $i=1,\cdots, m$, gives $$\begin{aligned}
\sum^m_{i=1}\varphi\left(\frac{|f_i(t)+g_i(t)|}{\lambda_1+\lambda_2}\right) \leq 1.
\end{aligned}$$ By Lemma \[orl-nor-lem\] again we have $$\begin{aligned}
\label{orl-tri-ine}
\lVert f(t)+g(t)\lVert_\varphi\leq \lVert f(t)\lVert_\varphi+\lVert g(t)\lVert_\varphi.
\end{aligned}$$ If we take $f=f-g+g$, we have $$\begin{aligned}
\lVert f(t)\lVert_\varphi-\lVert g(t)\lVert_\varphi\leq \lVert f(t)-g(t)\lVert_\varphi.\end{aligned}$$ Which means $$\begin{aligned}
\label{tri-orl}
\big| \lVert f(t)\lVert_\varphi-\lVert g(t)\lVert_\varphi\big|\leq \lVert f(t)-g(t)\lVert_\varphi.\end{aligned}$$ Moreover, together with the definition of the support function $h_{Z^+_\varphi \Lambda_t}(x)$ and (\[orl-nor-def\]), we have $$h_{Z_{\varphi}^{+}\Lambda_t}(x)=\lVert \langle v(t), x\rangle_+\rVert_\varphi.$$ Then, by (\[tri-orl\]) we have $$\begin{aligned}
\begin{split}
\big|h_{Z_{\varphi}^{+}\Lambda_{t_1}}(x)-h_{Z_{\varphi}^{+}\Lambda_{t_2}}(x)\big|&=
\big|\lVert \langle v(t_1), x\rangle_+\rVert_\varphi-\lVert \langle v(t_2), x\rangle_+\rVert_\varphi\big|\\
&\leq\lVert \langle v(t_1), x\rangle_+-\langle v(t_2), x\rangle_+\rVert_\varphi\\
&\leq\lVert \beta\langle x,v\rangle_+(t_1-t_2)\rVert_\varphi=\lVert \beta\langle x,v\rangle_+\rVert_\varphi|t_1-t_2|.
\end{split}\end{aligned}$$ We complete the proof.
\[sha-orl-thm\] Suppose $\varphi\in \mathcal C$, $\Lambda_t$, $t\in [-a^{-1},1]$, is a shadow system of multisets along the direction $v\in S^{n-1}$. Then $Z^+_\varphi \Lambda_t$, $t\in [-a^{-1},1]$, is a shadow system of convex bodies along the direction $v$.
Let $x$ be a point in $Z^+_\varphi \Lambda_0|_{v^\bot}$, and $\nu, \mu\in (0,1)$ satisfy $\nu+\mu=1$. By Lemma \[pro-ind-lem\], it is remains to show that the hypotheses of Proposition \[cam-gro-thm\] on properties of the graph functions are satisfied.
By assumption, the shadow system $\Lambda_t$ is equal to, say, $\{v_1(t),\cdots,v_m(t)\}$ where $v_i(t)=v_i+t\beta_iv$. With these definitions, the support function of $Z_\varphi^+\Lambda_t$ can be written as $$\begin{aligned}
h_{Z_\varphi^+\Lambda_t}(u)=\inf\bigg\{\lambda>0: \sum^{m}_{i=1}\varphi\Big(\frac{\langle v_i(t), u\rangle_+}{\lambda}\Big)\leq 1\bigg\}=\lVert \langle v(t), u\rangle_+\rVert_\varphi.\end{aligned}$$ To establish the convexity of the uppergraph and lowergraph function as functions of $t$, we firstly prove the uppergraph function is a convex function of $t$. Notice that $\Lambda_t$ is also a shadow system in direction $-v$. Then the vector $v$ can be replaced by $-v$ and by application of the identity $\overline g_{-v}(\cdot,x)=-\underline g_v(\cdot, x)$, we can obtain that the lowergraph function is a convex function of $t$.
By the definition of $\overline g_v(Z_\varphi^+\Lambda_{t}, x)$, we have $$\begin{aligned}
\nonumber\overline g_v(Z_\varphi^+\Lambda_{\nu s+\mu t}, x)&=\inf_{w_1,w_2\in v^\bot}\left\{h_{Z_\varphi^+\Lambda_{\nu s+\mu t}}(v+\nu w_1+\mu w_2)-\langle x, \nu w_1+\mu w_2\rangle\right\}\\
&=\inf_{w_1,w_2\in v^\bot}\Big\{\lVert \langle v(
\nu s+\mu t), v+\nu w_1+\mu w_2\rangle_+\rVert_\varphi-\langle x, \nu w_1+\mu w_2\rangle\Big\}.\end{aligned}$$ By the inequality $\max\{u+v,0\}\leq\max\{u,0\}+\max\{v,0\}$ and definition (\[orl-nor-def\]), we have $$\begin{aligned}
& \lVert \langle v(\nu s+\mu t),v+\nu w_1+\mu w_2\rangle_+\lVert_\varphi\\
&=\inf\left\{\lambda>0: \sum^{m}_{i=1}\varphi\bigg(\frac{\langle v_i+(\nu s +\mu t)\beta_iv, v+\nu w_1+\mu w_2\rangle_+}{\lambda}\bigg)\leq 1\right\}.
\end{aligned}$$ The convexity of $\varphi$ and (\[orl-tri-ine\]) imply that $$\begin{aligned}
\label{orl-nor-ine}
\lVert \langle v(\nu s+\mu t),v+\nu w_1+\mu w_2\rangle_+\lVert_\varphi\leq\nu\lVert\langle v(s),v+w_1\rangle_+\lVert_\varphi+\mu\lVert\langle v(t),v+w_2\rangle_+\lVert_\varphi.
\end{aligned}$$ Thus, together with (\[orl-nor-ine\]) and the expression of $\overline g_v(K,x)$ we have $$\begin{aligned}
\begin{split}
\inf_{w_1,w_2\in v^\bot}&\Big\{\lVert \langle v(
\nu s+\mu t), v+\nu w_1+\mu w_2\rangle_+\rVert_\varphi-\langle x, \nu w_1+\mu w_2\rangle\Big\}\\
&\leq \inf_{w_1\in v^\bot}\Big\{\nu\lVert \langle v(s), v+ w_1\rangle_+\rVert_\varphi-\nu\langle x, w_1\rangle\Big\}\\
&+\inf_{w_2\in v^\bot}\Big\{\mu\lVert \langle v(t), v+ w_2\rangle_+\rVert_\varphi-\mu\langle x, w_2\rangle\Big\}.
\end{split} \end{aligned}$$ Which means $$\begin{aligned}
\label{upp-orl-fun}
\overline g_v(Z_\varphi^+\Lambda_{\nu s+\mu t}, x)\leq \nu \overline g_v(Z_\varphi^+\Lambda_{ s}, x)+\mu \overline g_v(Z_\varphi^+\Lambda_{t}, x).\end{aligned}$$ Hence $t\rightarrow \overline g_v(Z_\varphi^+\Lambda_{ t}, x)$ is convex.
Next we verify the inequality (\[upp-low-ine\]) of Proposition \[cam-gro-thm\]. First we show $$\begin{aligned}
\label{rig-ineq}
\nu \overline g_v(Z^+_\varphi \Lambda_s,x)+ \mu \underline g_v(Z^+_\varphi \Lambda_t,x)\leq \overline g_v(Z^+_\varphi \Lambda_{\nu s+\mu t},x).\end{aligned}$$ To see this, let $w\in v^\bot$, $$\begin{aligned}
\label{low-ine}
\nu\overline g_v(Z^+_\varphi \Lambda_s,x)=\inf_{w\in v^\bot}\Big\{\lVert \nu\langle v(s),v+w\rangle_+\rVert_\varphi-\nu\langle x, w\rangle\Big\}.\end{aligned}$$ Let $w=\nu^{-1}(w_1-\mu w_2)$, $w_1, w_2\in v^\bot$, in (\[low-ine\]), we have $$\begin{aligned}
\nu\overline g_v(Z^+_\varphi \Lambda_s,x)=\inf_{w_1,w_2\in v^\bot}\Big\{\lVert \langle v(s),\nu v+w_1-\mu w_2\rangle_+\rVert_\varphi-\langle x, w_1-\mu w_2\rangle\Big\},\end{aligned}$$ where $$\begin{aligned}
\langle v(s),\nu v&+w_1-\mu w_2\rangle_+= \langle v_i+s\beta_iv,(1-\mu) v+w_1-\mu w_2\rangle_+\\
&=\mu\langle v_i+ t\beta_iv, -v-w_2\rangle_++\langle v_i+(\nu s+\mu t)\beta_iv, v+w_1\rangle_+.\end{aligned}$$ By (\[orl-tri-ine\]) we have, $$\begin{aligned}
\nu\overline g_v&(Z^+_\varphi \Lambda_s,x)=\inf_{w_1,w_2\in v^\bot}\Big\{\lVert \langle v(s),\nu v+w_1-\mu w_2\rangle_+\rVert_\varphi-\langle x, w_1-\mu w_2\rangle\Big\}\\
&=\inf_{w_1,w_2\in v^\bot}\Big\{\lVert \mu\langle v_i+ t\beta_iv, -v-w_2\rangle_++\langle v_i+(\nu s+\mu t)\beta_iv, v+w_1\rangle_+\rVert_\varphi-\langle x, w_1-\mu w_2\rangle\Big\}\\
&\leq\mu\inf_{w\in v^\bot}\Big\{\lVert\langle v(t), -v-w_2\rangle_+\rVert_\varphi-\langle x, - w_2\rangle\Big\}+\inf_{w\in v^\bot}\Big\{\lVert\langle v(\nu s+\mu t), v+w_1\rangle_+\rVert_\varphi-\langle x, w_1\rangle\Big\}\\
&=-\mu\underline g_v(Z^+_\varphi\Lambda_t,x)+\overline g_v(Z^+_\varphi \Lambda_{\nu s+\mu t},x).\end{aligned}$$ Which implies the inequality (\[rig-ineq\]).
The left hand of inequality (\[upp-low-ine\]) can be derived from inequality (\[rig-ineq\]) by replacing $v$ by $-v$, and using the following facts $$\begin{aligned}
\underline g_{-v}(\cdot,x)=-\overline g_v(\cdot, x) \,\,\,\,\,\,\, and \,\,\,\,\,\,\,\,\overline g_{-v}(\cdot,x)=-\underline g_v(\cdot, x).\end{aligned}$$ Now we complete the proof.
Now Theorem \[sha-orl-thm\] together with Lemma \[pro-ind-lem\] imply the following Theorem.
\[orl-vol-rat-thm\] Suppose $\varphi\in\mathcal C$ and $\Lambda$ is a finite and spanning multiset. If $\Lambda_t$, $t\in [-a^{-1},1]$, is an orthogonalization of $\Lambda$ defined by (\[ort-sha-sys\]), then:
\(1) The volume $V(Z^{+,*}_\varphi \Lambda_t)^{-1}$ is a convex function of $t$. In particular, the inverse volume product for asymmetric Orlicz zonotopes associated with $\Lambda_t$ is a convex function of $t$.
\(2) The volume $V(Z^{+}_\varphi \Lambda_t)^{-1}$ is a convex function of $t$. In particular, the volume ratio for asymmetric Orlicz zonotopes associate with $\Lambda_t$ is a convex function of $t$.
Theorem \[orl-vol-rat-thm\] shows that the inverse volume product and the volume ratio for asymmetric Orlicz zonotopes are nondecreasing if $\Lambda$ is replaced by either $\Lambda_{-a^{-1}}$ or $\Lambda_1$, because convex functions attain global maxima at the boundary of compact intervals.
the equality condition
======================
The following lemma is crucial for our proof of main results.
\[spa-obt-lem\] Suppose $\varphi\in \mathcal C $, and $\Lambda$ is a finite and spanning multiset. Replace all vectors in $\Lambda$ that point in the same direction by their sum, and denote this new multiset by $\overline \Lambda$. Then the following inequalities $$\begin{aligned}
\begin{split}
\label{orl-vol-pro-eq}
\frac{V(Z^+_\varphi\Lambda)}{V(Z^+_1 \Lambda)}\leq \frac{V(Z^+_\varphi\overline\Lambda)}{V(Z^+_1 \overline\Lambda)},\\
V(Z^{+,*}_\varphi\Lambda)V(Z^+_1 \Lambda)\geq(V^{+,*}_\varphi\overline \Lambda)(V(Z^+_1 \overline\Lambda),
\end{split}\end{aligned}$$ hold. With equalities, when $\varphi\neq Id$, if and only if $\Lambda=\overline \Lambda$.
Let multiset $\Lambda=\{v_1,\cdots,v_m\}$, and $\overline \Lambda=\{w_1,\cdots,w_k\}$. By the construction of $\overline \Lambda$ we have $$\begin{aligned}
w_j=\sum_{i\in I_j}v_i,\end{aligned}$$ where $I_j$, $j=1,\cdots k$, is a partition of $\{1,\cdots m\}$, and the vectors in every $\{v_i:i\in I_j\}$ point in the same direction. If $\varphi=Id$, that means $\varphi(t)=t$, here we write $Z^+_{Id}\Lambda=Z^+_1\Lambda$. It is easy to know that $$\begin{aligned}
\label{eur-zon-pro}Z_1^+\Lambda=Z_1^+\overline\Lambda.\end{aligned}$$
Now assume that $\varphi\neq Id$. Let $u\in S^{n-1}$, and set $$\begin{aligned}
h_{Z^+_\varphi\Lambda}(u)=\lambda \,\,\,\,\,\,and \,\,\,\,\,\,\, h_{Z^+_\varphi\overline\Lambda}(u)=\overline\lambda.\end{aligned}$$ By the definition of $Z^+_\varphi\overline\Lambda$, and note that all $v_i$ point in the same direction for $i\in I_j$, then we have $$\langle \sum_{i\in I_j}v_i,u\rangle_+=\sum_{i\in I_j}\langle v_i,u\rangle_+.$$ It follows that $$\begin{aligned}
\sum^{k}_{j=1}\varphi\left(\frac{\langle\sum_{i\in I_j} v_i,u\rangle_+}{\overline\lambda}\right)=\sum^{k}_{j=1}\varphi\left(\frac{\sum_{i\in I_j}\langle v_i,u\rangle_+}{\overline\lambda}\right)=1.\end{aligned}$$ Since the fact that $\varphi$ is convex and increasing, we have that if $x_1,\cdots, x_l\in[0,\infty)$, then $$\begin{aligned}
\label{con-fun}
\varphi(x_1+\cdots+x_l)\geq \varphi(x_1)+\cdots+\varphi(x_l),\end{aligned}$$ with equality if and only if $\varphi$ is a linear function. So we obtain $$\begin{aligned}
\label{orl-var-equ}
1=\sum_{j=1}^k\varphi\left(\frac{\sum_{i\in I_j}\langle v_i,u\rangle_+}{\overline\lambda}\right)\geq\sum_{i=1}^m\varphi\left(\frac{\langle v_i,u\rangle_+}{\overline\lambda}\right).\end{aligned}$$ Since $ h_{Z^+_\varphi\Lambda}(u)=\lambda$, by Lemma \[orl-nor-lem\], we obtain $\lambda\leq\overline \lambda$. Since $\varphi\neq Id$, with equality holds only if all sum over $i\in I_j$ contain at most one positive summand, that means $\Lambda=\overline\Lambda$. In fact, if $\Lambda\neq\overline \Lambda$, say $v_1$ and $v_2$ point in the same direction, by the convexity of $\varphi$, then $h_{Z^+_\varphi \overline\Lambda}(v_1)> h_{Z^+_\varphi \Lambda}(v_1)$. Hence we obtain $Z^+_\varphi \Lambda\subset Z^+_\varphi \overline\Lambda.$
On the other hand, if $\Lambda\neq \overline\Lambda$, with equality in (\[orl-var-equ\]) if and only if $$\begin{aligned}
\sum_{j=1}^k\varphi\left(x_j\right)=\varphi\left(\sum_{i=1}^k x_i\right),\end{aligned}$$ holds for arbitrary $k$ and $x_i\in \mathbb R$. Combine with the convexity and the normalization of $\varphi$, and solve this functional equation we know that $\varphi(t)=t$. Then $$Z^{+}_\varphi \Lambda=Z^+_1\Lambda=Z^+_1\overline\Lambda.$$ The first inequality of (\[orl-vol-pro-eq\]) now follows immediately. To the second inequality of (\[orl-vol-pro-eq\]), if $\varphi\neq Id$, note that $$\begin{aligned}
Z^{+,*}_\varphi\Lambda=(Z^{+}_\varphi\Lambda-s(Z^{+}_\varphi\Lambda))^o\supseteq(Z^{+}_\varphi\overline\Lambda-s(Z^{+}_\varphi\Lambda))^o,\end{aligned}$$ with equality if and only if $\Lambda=\overline\Lambda$. Thus $$\begin{aligned}
V(Z^{+,*}_\varphi\Lambda)\geq V((Z^{+}_\varphi\overline\Lambda-s(Z^{+}_\varphi\Lambda))^o)\geq V(Z^{+,*}_\varphi\overline\Lambda).\end{aligned}$$ Together with (\[eur-zon-pro\]) show the second inequality of (\[orl-vol-pro-eq\]).
In the following, we observe that a set that can be written as a disjoint union $\Lambda_\bot\cup \{v_1,\cdots,v_l\}$ is obtuse if and only if there are disjoint nonempty subset $I_1,\cdots, I_l$ of $\{1,\cdots,n\}$ and positive numbers $\mu_i$ such that, for every $j\in\{1,\cdots,l\}$, $$\begin{aligned}
v_j=\sum_{i\in I_j}-\mu_ie_i.\end{aligned}$$
The following Lemma shows that every spanning obtuse set has a linear image of above type.
[@web-sha2013]\[spa-mul-lem\] Suppose $\Lambda$ is a spanning obtuse set, then the following three statements holds:
\(1) If $B\subset A$ is a basis, then the vectors in $A\setminus B$ are pairwise orthogonal and have nonpositive components with respect to the basis $B$.
\(2) Every $GL(n)$ image of $\Lambda$ that contains the canonical basis $\Lambda_\bot$ is obtuse.
\(3) Suppose in addition that $\Lambda$ contains the canonical basis. For every $y\in Z^+_\varphi\Lambda$ there is a $\phi\in GL(n)$ such that $\phi y$ has nonnegative coordinates with respect to the canonical basis and $\Lambda_\bot \subset \phi \Lambda$.
This Lemma is established by Webermdorfer in [@web-sha2013], here we omit the proof this lemma.
One of the immediate implications of the above Lemma is that a spanning obtuse set contains at least $n$ and not more than $2n$ vectors. Now we give the equality condition of our main results.
\[orl-vol-lem\] Suppose $\varphi\in\mathcal C$ and $\Lambda$ is a spanning obtuse set. Then $$\begin{aligned}
\frac{V(Z^+_\varphi\Lambda)}{V(Z^+_1\Lambda)}=\frac{V(Z^+_\varphi\Lambda_\bot)}{V(Z^+_1\Lambda_\bot)}.
\end{aligned}$$
Let $\Lambda$ be a spanning obtuse set. The $GL(n)$ invariance of the volume ratio for asymmetric Orlicz zonotopes and Lemma \[spa-mul-lem\], we may assume that $\Lambda=\{w_1,\cdots,w_{m}\}$, where $n\leq m\leq 2n$, contains the canonical basis $\Lambda_\bot=\{e_1,\cdots,e_n\}$. In the following, if we can establish the dissection formula $$\begin{aligned}
\label{dis-orl-asy}
Z^+_\varphi\Lambda=\bigcup_{1\leq i_1<\cdots <i_n\leq m}Z^+_\varphi\big\{w_{i_1},\cdots,w_{i_n}\}.
\end{aligned}$$ Then, we have $$\begin{aligned}
\label{vol-dis}
V(Z^+_\varphi \Lambda)=\sum_{1\leq i_1<\cdots<i_n\leq m}V\Big(Z^+_\varphi\{v_{i_1},\cdots,v_{i_n}\}\Big).\end{aligned}$$ The $GL(n)$ equivariance of $Z^+_\varphi $ together with (\[vol-dis\]) for $\varphi(t)=t$, we have $$\begin{aligned}
\frac{V(Z^+_\varphi \Lambda)}{V(Z^+_1 \Lambda)}=\frac{\sum\limits_{1\leq i_1<\cdots<i_n\leq m}V\Big(Z^+_1\{v_{i_1},\cdots,v_{i_n}\}\Big)}{\sum\limits_{1\leq i_1<\cdots<i_n\leq m}V\Big(Z^+_1\{v_{i_1},\cdots,v_{i_n}\} \Big)}=\frac{V(Z^+_\varphi \Lambda_\bot)}{V(Z^+_1 \Lambda_\bot)}.\end{aligned}$$ Here we used the $GL(n)$ equivariance of $Z^+_\varphi$ and the fact $V\big(Z^+_\varphi \{v_{i_1},\cdots v_{i_n}\}\big)=0$, if $\{v_{i_1},\cdots v_{i_n}\}$ is not a $GL(n)$ image of conical basis $\Lambda_\bot$. Hence we have $$\begin{aligned}
\frac{V(Z^+_\varphi \Lambda)}{V(Z^+_1\Lambda)}=\frac{V(Z^+_\varphi \Lambda_\bot)}{V(Z^+_1 \Lambda_\bot)}.\end{aligned}$$
In the following, we will show the dissection formula (\[dis-orl-asy\]) holds. Let $y\in\bigcup_{1\leq i_1<\cdots <i_n\leq m}Z^+_\varphi\big\{w_{i_1},\cdots,w_{i_n}\}$, it must belong to, we say, $Z^+_\varphi\{w_1,\cdots,w_n\}$. In order to prove $y\in Z^+_\varphi\Lambda$. Let $$\begin{aligned}
h_{Z^+_\varphi\{w_1,\cdots,w_n\}}(u)=\lambda_0\,\,\,\,\,\,\,and \,\,\,\,\,\,\,h_{Z^+_\varphi\Lambda}(u)=\lambda_1.\end{aligned}$$ By the definition of the support function, we have $$\begin{aligned}
\sum_{i=1}^{n}\varphi\left(\frac{\langle w_i,u\rangle_+}{\lambda_0}\right)= 1 \,\,\,\,\,\,\,\,\, and\,\,\,\,\,\,\,\,\,\,\,
\sum_{j=1}^{m}\varphi\left(\frac{\langle w_j,u\rangle_+}{\lambda_1}\right)= 1.\end{aligned}$$ Since $\varphi$ is increasing, which implies $$\begin{aligned}
\sum_{i=1}^{n}\varphi\left(\frac{\langle w_i,u\rangle_+}{\lambda_1}\right)\leq\sum_{j=1}^{m}\varphi\left(\frac{\langle w_j,u\rangle_+}{\lambda_1}\right).\end{aligned}$$ By Lemma \[orl-nor-lem\] we have $\lambda_1\geq \lambda_0$. That means $$\begin{aligned}
\label{asy-con-equ}
Z^+_\varphi\{w_1,\cdots,w_n\}\subseteq Z^+_\varphi\Lambda.\end{aligned}$$ We prove $Z^+_\varphi\Lambda$ contains the right hand side of (\[dis-orl-asy\]). Now it remains to prove that $Z^+_\varphi\Lambda$ is a subset of the right hand side of (\[dis-orl-asy\]). Let $y\in Z^+_\varphi\Lambda$, it is sufficient to show that there is a $\phi\in GL(n)$ such that $y\in Z^+_\varphi \phi^{-1}\Lambda_\bot$ and $\phi^{-1}\Lambda_\bot\subseteq \Lambda$. By Lemma \[spa-mul-lem\], there is a $\phi\in GL(n)$ such that $\phi y$ has nonnegative coordinates with respect to the canonical basis and $\Lambda_\bot\subseteq \phi\Lambda$. Moreover, $\phi \Lambda$ is obtuse, then we can write $$\begin{aligned}
\phi \Lambda=\Lambda_\bot\cup\{w_1,\cdots, w_{m-n}\},\end{aligned}$$ and there are disjoint subsets $I_1,\cdots, I_{m-n}$, and positive number $\mu_i$ such that, for $1\leq j\leq m-n$, $$\begin{aligned}
\label{neg-com}
w_j=\sum_{i\in I_j}-\mu'_ie_{i}.\end{aligned}$$ Let $h_{Z^+_\varphi \phi\Lambda}=\lambda_0$. Then we have $$\begin{aligned}
\sum_{i=1}^{n}\varphi\left(\frac{\langle e_i,u\rangle_+}{\lambda_0}\right)+\sum_{i=j}^{m-n}\varphi\left(\frac{\langle w_j,u\rangle_+}{\lambda_0}\right)= 1.\end{aligned}$$ Note that the convexity and strictly increasing of $\varphi$ imply that, there exists a constant $\nu>0$ such that $$\begin{aligned}
\sum_{j=1}^{n}\varphi\left(\frac{\nu\langle -\mu'_je_j,u\rangle_+}{\lambda_0}\right)\geq\sum_{j=1}^{m-n}\varphi\left(\frac{\langle w_j,u\rangle_+}{\lambda_0}\right).\end{aligned}$$ We write $ \mu_j=\nu\mu'_j$, $j=1,\cdots ,n$, and define the set $\widetilde\Lambda=\{e_1,\cdots,e_n,-\mu_1e_1,\cdots,-\mu_ne_n\}$. Obviously, $\widetilde \Lambda$ is an obtuse set. Moreover, We have $$\begin{aligned}
\sum_{i=1}^{n}\varphi\left(\frac{\langle e_i,u\rangle_+}{\lambda_0}\right)+\sum_{j=1}^{n}\varphi\left(\frac{\langle -\mu_je_j,u\rangle_+}{\lambda_0}\right)\geq 1.\end{aligned}$$ Then by Lemma \[orl-nor-lem\] we have $\lambda_0\leq h_{Z^{+}_\varphi \widetilde\Lambda}(u)$. Then we have $$\begin{aligned}
\label{asy-con-equ}
Z^+_\varphi\phi\Lambda\subseteq Z^+_\varphi\widetilde\Lambda=Z^+_\varphi\big\{e_1,\cdots,e_n,-\mu_1e_1,\cdots,-\mu_ne_n\big\}.\end{aligned}$$
It remains to show that $Z^+_{\varphi} \phi\Lambda\subseteq Z^+_\varphi \Lambda_\bot$. First note that $\Lambda$ is an obtuse, $w_i$ has negative coordinates with respect to the canonical basis $\Lambda_\bot$. In order to simply the computation, we assume that $\Lambda'(\mu)=\{e_1,\cdots,e_n,-\mu e_1\}$, where $ \mu\geq 0$. For $x\in e^\bot_1\cap e_2^\bot$, by (\[up-low-gra\]) we have $$\begin{aligned}
\label{inf-upp-fun}\overline g_{e_2}(Z^+_\varphi\Lambda'(\mu),x)&=\inf_{w\in e^\bot_2}\left\{h_{Z^+_\varphi\Lambda'(\mu)}(e_2+w)- \langle x, w\rangle\right\}.
\end{aligned}$$ Here $$\begin{aligned}
\begin{split}\label{inf-orl-upp}
h_{Z^+_\varphi\Lambda'(\mu)}(e_2+w)&=\inf\bigg\{\lambda>0:\varphi\Big(\frac{\langle e_1,w\rangle_+}{\lambda}\Big)+\varphi\Big(\frac{\langle -\mu e_1,w\rangle_+}{\lambda}\Big)\\
&+\sum_{i=2}^n\varphi \Big(\frac{\langle e_i, e_2+w\rangle_+}{\lambda}\Big)\leq 1\bigg\},
\end{split}\end{aligned}$$ Note that, for all $w\in e^\bot_2$, the scalar product $\langle x, w\rangle$ does not dependent on the first component of $w$. The increasing of $\varphi$ together with the expression of (\[inf-orl-upp\]) show that it suffices to compute the infimum of (\[inf-upp-fun\]) over all $w\in e_1^\bot\cap e_2^\bot$. It is now obvious that the uppergraph function $\overline g_{e_2}(Z^+_\varphi\Lambda'(\mu),x)$ is independent of $\mu$ for every $x\in e^\bot\cap e^\bot_2$. The same argument applied to the lowergraph function leads to the same conclusion, so we infer that $$\begin{aligned}
Z^+_\varphi\Lambda'(\mu)\cap e^\bot_1
\end{aligned}$$ is independent of $\mu$. Moreover, the support function of $Z^+_\varphi \Lambda'(\mu)$ evaluated at vectors $w\in e^\bot_1$, $$\begin{aligned}
h_{Z^+_\varphi\Lambda'(\mu)}(w)=\inf\bigg\{\lambda>0:\sum_{i=2}^n\varphi\Big(\frac{\langle e_2,w\rangle_+}{\lambda}\Big)\leq 1\bigg\},
\end{aligned}$$ is a constant function of $\mu$. Equivalently, $$\begin{aligned}
Z^+_\varphi\Lambda'(\mu)|_{e^\bot_1},
\end{aligned}$$ is independent of $\mu$.
If $\mu=1$, the convex body $Z^+_\varphi\Lambda'(1)$ is symmetric with respect to reflections in the hyperplane $e^\bot_1$. Then for $y\in Z^+_\varphi \Lambda'(\mu)$, we have $$\begin{aligned}
y|_{e^\bot_1}\in Z^+_\varphi\Lambda'(\mu)|_{e^\bot_1}= Z^+_\varphi\Lambda'(1)|_{e^\bot_1}= Z^+_\varphi\Lambda'(1)\cap{e^\bot_1}= Z^+_\varphi\Lambda'(\mu)\cap{e^\bot_1}
\end{aligned}$$ for all $\mu$. In particular, $\underline g_{e_1}(Z^+_\varphi\Lambda'(\mu), y|_{e^\bot_1})$ is negative for all $\mu$. Moreover, the uppergraph function $\overline g_{e_1}(Z^+_\varphi\Lambda'(\mu),y|_{e_1^\bot})$ is independent of $\mu$. Because, $h_{Z^+_\varphi \Lambda'(\mu)}(e_1+w)$ is independent of $\mu$, for $w\in e^\bot_1$. Hence $$\begin{aligned}
\begin{split}
y&\in\Big\{y|_{e^\bot_1}+re_1:0\leq r\leq \overline g_{e_1}(Z^+_\varphi \Lambda'(\mu), y|_{e^\bot_1})\Big\}\\
&=\Big\{y|_{e^\bot_1}+re_1:0\leq r\leq \overline g_{e_1}(Z^+_\varphi \Lambda'(0), y|_{e^\bot_1})\Big\}\subset Z^+_\varphi\Lambda' (0).
\end{split}\end{aligned}$$ Then we have $Z^+_\varphi\Lambda' (\mu)\subseteq Z^+_\varphi \Lambda_\bot.$ This together with (\[asy-con-equ\]) we have $Z^+_\varphi \Lambda'(\mu)= Z^+_\varphi\Lambda_\bot.$ Repeating this argument for $\mu_2,\cdots,\mu_n$, if them are not zero. We have that $\phi y$ is contained in $Z^+_\varphi \Lambda_\bot$, which shows the equality of (\[dis-orl-asy\]).
Moreover, if we can show the intersection of any two distinct parts in the dissection (\[dis-orl-asy\]) has volume zero, we can complete the proof. To see this, let $\Lambda^1,\Lambda^2\in \Lambda$, each contain $n$ vectors and assume that $\Lambda^1\neq\Lambda^2 .$ If one of these sets is not spanning, then the intersection $Z^+_\varphi\Lambda^1\cap Z^+_\varphi\Lambda^2$ is a set of volume zero contained in a hyperplane. Otherwise, without loss of generality, $\Lambda^1=\Lambda_\bot$ and $\Lambda^2$ does not contain $e_1$. Then by the definition of support function (\[asy-orl-zon\]), we have $h_{Z^+_\varphi\Lambda^1}(-e_1)=0,$ and $h_{Z^+_\varphi\Lambda^2}(e_1)=0$. Then we obtain that $Z^+_\varphi\Lambda^1\cap Z^+_\varphi\Lambda^2$ is a set of volume zero contained in the hyperplane $e^\bot_1$. So we complete the proof.
If take $\varphi(t)=t^p$, $p\geq 1$, this result reduces to $L_p$ case.
Suppose $p\geq 1$ and $\Lambda$ is a spanning obtuse set. Then $$\begin{aligned}
\frac{V(Z^+_p\Lambda)}{V(Z^+_1\Lambda)}=\frac{V(Z^+_p\Lambda_\bot)}{V(Z^+_1\Lambda_\bot)}.
\end{aligned}$$
In paper [@cam-gro-on2006], Campi and Gronchi proved that if $K_t$ is a shadow system of origin symmetric convex bodies in $\mathbb R^n$, then $V(K^*_t)^{-1}$ is a convex function of t. This result is developed by Meyer and Reisner [@mey-rei-sha2006] to more general setting.
[@mey-rei-sha2006] \[pol-sha-pro\] Suppose $K_t$, $t\in[-a^{-1},1]$, is a shadow system of convex bodies along the direction $v=e_1$ and $V(K_t)$ is independent of $t$. Then the volume of $K_t^*$ is independent of $t$ if and only if there are a real number $\alpha$ and a vector $z\in \mathbb R^{1\times n-1}$ such that $$\begin{aligned}
K_t=t\alpha e_1+\bigg(\begin{array}{cc}
1 &tz \\
0 & I_{n-1}
\end{array}\bigg)K_0.\end{aligned}$$
Unfortunately, there is no analogue result for volume product of asymmetric Orlicz zonotopes with equality holds.
\[vol-pro-lem\] Let $\varphi\in\mathcal C$, and $\Lambda=\Lambda_\bot\cup\{-\mu e_1\}$. Then $$\begin{aligned}
\label{vol-orl-rat-lem}
V(Z^{+,*}_\varphi\Lambda)V(Z^{+}_1\Lambda)\geq V(Z^{+,*}_\varphi\Lambda_\bot)V(Z^{+}_1\Lambda_\bot),
\end{aligned}$$ with equality if and only if $\varphi=Id$.
If $\varphi=Id $, that means $Z^{+}_\varphi\Lambda=Z^{+}_1\Lambda=Z_1\Lambda$. It is an immediate consequence of the fact that all parallelepipeds have the same volume product.
Now assume that $\varphi\neq Id$, let $\Lambda_t$, $t\in[-a^{-1},1]$, denote the orthogonalization of $\Lambda$ with respect to $e_1$ defined by (\[ort-sha-sys\]). Theorem (\[orl-vol-rat-thm\]) shows that the inverse volume product of asymmetric Orlicz zonotopes associate with $\Lambda_t$ is a convex function of $t$, together with the convex function attains it’s maxima at the boundary of compact intervals, we obtain $$\begin{aligned}
\frac{1}{V(Z^{+,*}_\varphi\Lambda)V(Z^{+}_1\Lambda)}\leq \max_{t\in\{-a^{-1},1\}}\bigg\{\frac{1}{V(Z^{+,*}_\varphi\Lambda_t)V(Z^{+}_1\Lambda_t)}\bigg\}.\end{aligned}$$ By the $GL(n)$ invariance of the volume product of asymmetric Orlicz zonotopes and the definition of $\Lambda_t$, the right hand side of this inequality is just $$\begin{aligned}
\frac{1}{V(Z^{+,*}_\varphi\Lambda_\bot)V(Z^{+}_1\Lambda_\bot)}.\end{aligned}$$ Thus the equality condition of inequality (\[vol-orl-rat-lem\]) means that the $V(Z_\varphi^{+,*}\Lambda_t)$ is a constant function of $t$. On the other hand, by the definition of (\[ort-sha-sys\]), we have $$\Lambda_t=\big\{(1+ta)e_1,\mu(t-1)e_1,e_2,\cdots, e_n\big\},$$ here $\Lambda_t,$ $t\in[-a^{-1},1]$, is a spanning obtuse set. Together with Lemma \[orl-vol-lem\] and the fact $Z^+_1\Lambda_t$ is independent of $t$, we obtain that $V(Z^+_\varphi\Lambda_t)$ is independent of $t$. Proposition (\[pol-sha-pro\]) implies that $Z^+_\varphi \Lambda_t$ are affine images of each other, which means there is a number $\alpha$ and a vector $z\in \mathbb R^{1\times(n-1)}$ such that $$Z^+_\varphi\Lambda_t=t\alpha e_1+\phi_t Z^+_\varphi\Lambda,$$ where $\phi_t=\bigg(\begin{array}{cc}
1 & tz \\
0 & I_{n-1}
\end{array}\bigg).$ Note that $Z^+_\varphi$ is $GL(n)$ equivariant, we can rewrite it as $$\begin{aligned}
\label{orl-aff}
Z^+_\varphi\Lambda_t=t\alpha e_1+ Z^+_\varphi\phi_t\Lambda.\end{aligned}$$ Equivalent, for all $u\in\mathbb R^n$, $$\begin{aligned}
\label{orl-aff-equ}
h_{Z^+_\varphi\Lambda_t}(u)=t\alpha\langle e_1,u\rangle+h_{Z^+_\varphi\phi_t\Lambda}(u).\end{aligned}$$ Now we determined the constant $\alpha$. Note that $t\in[a^{-1},1]$, the zonotope $Z^+_\varphi\Lambda_t$ is symmetric with respect to permutations of all coordinates except the first. Due to (\[orl-aff\]), this implies that $z$ has $n-1$ equal components, say $\xi$. Note that the coefficient $a$ is nothing to do with the $\xi$, without loss of generality, we may assume $\xi\leq 0$, let $u=e_1$ and $t=1$ in (\[orl-aff-equ\]), after a simple computation we obtain $$\begin{aligned}
\alpha=\frac{a}{\varphi^{-1}(1)}=a.\end{aligned}$$ In order to determine $\xi$, firstly, by the normalization of $\varphi$ and together with Lemma \[orl-nor-lem\], we have $$\begin{aligned}
\label{e2-int}
h_{Z^+_\varphi \Lambda_1}(e_i)=1, \,\,\,\,\,\,\,\,\,\,\,\,i=2,\,\,\cdots, n.\end{aligned}$$ Note that $Z^+_\varphi\Lambda_1$ is convex, specially, let $|e_2|=1=h_{Z^+_\varphi \Lambda_1}(e_2)$, which means that $e_2$ is contained in a plan intersect with $Z^+_\varphi\Lambda_1$, we say, $$\begin{aligned}
\label{e2-poi}
\{e_2\}=Z^+_\varphi\Lambda_1\cap({e_2}+span\{e_1\}).\end{aligned}$$ On other hand, by the definition of convex hull $conv$ and the support function of $Z^+_\varphi\Lambda$, together with Lemma (\[orl-nor-lem\]), we have $Z^+_\varphi\Lambda$ contains the convex hull of $\Lambda$, that is $$\begin{aligned}
conv\{\Lambda\}\subseteq Z^+_\varphi\Lambda.\end{aligned}$$ Then we have $Z^+_\varphi \phi_1 \Lambda$ contains the convex hull of $\phi_1\Lambda$, $conv\{\phi_1\Lambda\}$. In particular, it contains $\phi_1 e_2=\xi e_1+e_2$. Combine this observation with (\[orl-aff\]) and (\[e2-poi\]) for $t=1$, we obtain $$\begin{aligned}
e_2=(a+\xi)e_1+e_2,\end{aligned}$$ which means $\xi=-a$.
Now putting $u= e_1 +e_2$ and $t=-a^{-1}$ in equation (\[orl-aff-equ\]). Let $h_{Z^+_\varphi \Lambda_{-a^{-1}}}(e_1+e_2)=\lambda,\,\,
h_{Z^+_\varphi\phi_{-a^{-1}}\Lambda}(e_1+e_2)=\lambda'$. Note that $\Lambda_{-a^{-1}}=\{\mu(-a^{-1}-1)e_1,\,\,e_2,\cdots, e_n\}$ and $\phi_{-a^{-1}}\Lambda=\{e_1,\,\,e_1+e_2, \cdots, e_1+e_n, -\mu e_1\}$, then we have $$\begin{aligned}
\label{lam}
\lambda=1,\,\,\,\,\,\,\,\,\,and\,\,\,\,\,\,\,\,\,\, (n-1)\varphi\Big(\frac{1}{\lambda'}\Big)+\varphi\Big(\frac{2}{\lambda'}\Big)=1.\end{aligned}$$ Note that by equation (\[orl-aff-equ\]), $\lambda$ and $\lambda'$ should satisfy $\lambda={-1}+\lambda'$. On other hand, by (\[lam\]), they contradict with equality (\[orl-aff-equ\]). Then $Z^+_\varphi\Lambda_t$ are not the affine images of each other. So the equality (\[vol-orl-rat-lem\]) does not hold when $\varphi\neq Id.$ We complete the proof.
If take $\varphi(t)=t^p$, $p\geq 1$, it is established in [@web-sha2013].
Let $p\geq 1$, and $\Lambda=\Lambda_\bot\cup\{-\mu e_1\}$. Then $$\begin{aligned}
V(Z^{+,*}_p\Lambda)V(Z^{+}_1\Lambda)\geq V(Z^{+,*}_p\Lambda_\bot)V(Z^{+}_1\Lambda_\bot),
\end{aligned}$$ with equality if and only if $p=1$.
proofs of the main results
==========================
Now we are in a position of proving the main results. Before giving the main results, let us present the following lemma established by Weberndorfer, which we will use in the proof of our results.
[@web-sha2013] \[gen-vol-rat-lem\] Suppose $\Phi$ is a real-valued $GL(n)$ invariant function on finite and spanning multisets. Moreover, assume that $\Phi(\Lambda_t)$ is a convex function of $t$ whenever $\Lambda_t$, $t\in[-a^{-1},1]$ is an orthogonalization of a multiset $\Lambda$ defined by (\[ort-sha-sys\]). Then for ever finite and spanning multiset $\Lambda$, there exists a multiset $\Lambda_{e_1}$ of multiples of $e_1$ such that $$\begin{aligned}
\label{gen-vol-ine}
\Phi(\Lambda)\leq \Phi(A_\bot \uplus \Lambda_{e_1}).
\end{aligned}$$ Moreover,
\(1) If $\Lambda$ is not a $GL(n)$ image of $\Lambda_\bot$ and equality holds in (\[gen-vol-ine\]), then $\Lambda_{e_1}$ is not the empty set.
\(2) If $\Lambda$ is not a $GL(n)$ image of an obtuse set and equality holds in (\[gen-vol-ine\]), then $\Lambda_{e_1}$ contains a positive multiple of $e_1$.
Let $\varphi\in\mathcal C$, and $\Lambda$ is a finite and spanning multiset. Then $$\begin{aligned}
\label{the-one}
V(Z^{+,*}_\varphi\Lambda)V(Z^+_1\Lambda)\geq V(Z^{+,*}_\varphi\Lambda_\bot)V(Z^+_1\Lambda_\bot).
\end{aligned}$$ If $\varphi\neq Id$, with equality if and only if $\Lambda$ is a $GL(n)$ image of the canonical basis $\Lambda_\bot$. If $\varphi=Id$, the equality holds if and only if $Z^+_1\Lambda$ is a parallelepiped.
First, if $\varphi=Id$, it is established in [@web-sha2013], so we only need to show the case $\varphi\neq Id$.
For $\varphi\neq Id$, let $\mathcal P(\Lambda)=\frac{1}{V(Z^{+,*}_\varphi\Lambda)V(Z^+_1\Lambda)}$ denote the inverse volume product of asymmetric Orlicz zonotopes, $\Lambda_t$, $t\in[-a^{-1},1]$, denotes the orthogonalization of $\Lambda$. Firstly, by the $GL(n)$ invariance of $\mathcal P(\Lambda)$, there is nothing to show if $\Lambda$ is a $GL(n)$ image of the canonical basis $\Lambda_\bot$. Otherwise, by Theorem \[orl-vol-rat-thm\], we know that $\mathcal P(\Lambda_t)$, $t\in[-a^{-1},1]$, satisfies the hypotheses of Lemma \[gen-vol-rat-lem\]. Then there exists a multiset $\Lambda_{e_1}$ of $e_1$ such that $$\begin{aligned}
\label{vol-pro-ort}
\mathcal P(\Lambda)\leq \mathcal (\Lambda_\bot\uplus\Lambda_{e_1}).
\end{aligned}$$ If $\Lambda_{e_1}$ is empty then the inequality (\[the-one\]) holds. If $\Lambda_{e_1}$ contains the only positive multiples of $e_1$, then $\overline{\Lambda_\bot\uplus \Lambda_{e_1} }$ is a $GL(n)$ image of $\Lambda_\bot$. Then we have $$\begin{aligned}
\mathcal P(\Lambda)\leq \mathcal P (\Lambda_\bot\uplus\Lambda_{e_1})=\mathcal P (\Lambda_\bot).
\end{aligned}$$ It remains to show that if $\Lambda_{e_1}$ contains negative multiples, we say, $\Lambda_{e_1}=\{-\mu e_1\}$, where $\mu>0$. By Lemma \[vol-pro-lem\] we have $$\begin{aligned}
\mathcal P(\Lambda)\leq \mathcal P (\Lambda_\bot\uplus\Lambda_{e_1})<\mathcal P (\Lambda_\bot).
\end{aligned}$$ Then the inequality of (\[the-one\]) holds. Now we deal with the equality condition. Since the equality holds in (\[vol-pro-ort\]) only if $\Lambda_{e_1}$ is not empty. By Lemma \[spa-obt-lem\], we have $$\begin{aligned}
\mathcal P(\Lambda\uplus\Lambda_{e_1})\leq\mathcal P(\overline{\Lambda_\bot\uplus \Lambda_{e_1}}),
\end{aligned}$$ with equality if and only if $\Lambda_{e_1}=\{-\mu{e_1}\}$, where $\mu\geq 0$. Note that $\Lambda_{e_1}$ is not the empty set, then $\mu>0$. So we have $$\begin{aligned}
\label{equ-lam}
\mathcal P(\Lambda)=\mathcal P(\Lambda_\bot\uplus \Lambda_{e_1})=\mathcal P(\overline{\Lambda_\bot\cup\{-\mu e_1\}})=\mathcal P(\Lambda_\bot).\end{aligned}$$ Note that $\varphi\neq Id$, by Lemma \[vol-pro-lem\], we have the equalities of (\[equ-lam\]) hold if and only if $\Lambda$, $\Lambda_\bot\uplus \Lambda_{e_1}$, $\overline{\Lambda_\bot\cup\{-\mu e_1\}}$, and $\Lambda_\bot$ are $GL(n)$ images of each other. So we obtain the desired inequality together with its equality conditions.
If $\varphi(t)=t^p$, $p\geq1$, this result reduces to the asymmetric $L_p$-volume ratio inequality in [@web-sha2013].
Let $p\geq 1$, and $\Lambda$ is a finite and spanning multiset. Then $$\begin{aligned}
V(Z^{+,*}_p\Lambda)V(Z^+_1\Lambda)\geq V(Z^{+,*}_p\Lambda_\bot)V(Z^+_1\Lambda_\bot).
\end{aligned}$$ If $p> 1$, with equality if and only if $\Lambda$ is a $GL(n)$ image of the canonical basis $\Lambda_\bot$. If $p=1$, the equality holds if and only if $Z^+_1\Lambda$ is a parallelepiped.
Suppose $\varphi\in\mathcal C$ and $\Lambda$ is a finite and spanning multiset. Then $$\begin{aligned}
\label{vol-rat-ine}
\frac{V(Z^+_\varphi\Lambda)}{V(Z^+_1\Lambda)}\leq\frac{V(Z^+_\varphi\Lambda_\bot)}{V(Z^+_1\Lambda_\bot)},
\end{aligned}$$ with equality if and only if $\Lambda$ is a $GL(n)$ image of an obtuse set.
Let $\mathcal R(\Lambda)=\frac{V(Z^+_\varphi\Lambda)}{V(Z^+_1\Lambda)}$, Theorem \[orl-vol-rat-thm\] implies that $\mathcal R(\Lambda)$ satisfies the hypotheses of Lemma \[gen-vol-rat-lem\]. Then $$\begin{aligned}
\label{rad-orl}
\mathcal R(\Lambda)\leq \mathcal R(\Lambda_\bot\uplus\Lambda_{e_1}),\end{aligned}$$ where $\Lambda_{e_1}$ contains multiples of $e_1$. If $\Lambda_{e_1}$ is empty, then (\[vol-rat-ine\]) holds. If $\Lambda_{e_1}$ contains the only positive multiples of $e_1$, then $\overline{\Lambda_\bot\uplus \Lambda_{e_1} }$ is a $GL(n)$ image of $\Lambda_\bot$. Then we have $$\begin{aligned}
\label{b1}
\mathcal R (\overline{\Lambda_\bot\uplus\Lambda_{e_1}})=\mathcal R (\Lambda_\bot).
\end{aligned}$$ Moreover, by Lemma \[spa-obt-lem\], $$\begin{aligned}
\label{b2}
\mathcal R ({\Lambda_\bot\uplus\Lambda_{e_1}})\leq\mathcal R (\overline{\Lambda_\bot\uplus\Lambda_{e_1}}).
\end{aligned}$$ Together with (\[rad-orl\]), (\[b1\]) and (\[b2\]) we obtain $$\begin{aligned}
\mathcal R(\Lambda)\leq\mathcal R (\Lambda_\bot).\end{aligned}$$ Note that if $\Lambda_{e_1}=\{-\mu e_1\}$, where $\mu>0$, then $\Lambda_\bot\uplus\Lambda_{e_1}=\Lambda_\bot\cup\{-\mu e_1\}$ is an obtuse set. Lemma \[orl-vol-lem\] implies that the inequality in (\[vol-rat-ine\]) holds.
Now we deal with the equality case of (\[vol-rat-ine\]). We assume that $\Lambda$ is not a $GL(n)$ image of an obtuse set. Note that (\[rad-orl\]) with equality holds only if $\Lambda_{e_1}$ contains a positive multiple of $e_1$. In this case, then $\overline{\Lambda_\bot\uplus \Lambda_{e_1} }$ is a $GL(n)$ image of $\Lambda_\bot$. Then we have $$\begin{aligned}
\mathcal R (\overline{\Lambda_\bot\uplus\Lambda_{e_1}})=\mathcal R (\Lambda_\bot).
\end{aligned}$$ By Lemma \[spa-obt-lem\], we have $$\begin{aligned}
\mathcal R ({\Lambda_\bot\uplus\Lambda_{e_1}})\leq\mathcal R (\overline{\Lambda_\bot\uplus\Lambda_{e_1}}).
\end{aligned}$$ With equality if and only if ${\Lambda_\bot\uplus\Lambda_{e_1}}=\overline{\Lambda_\bot\uplus\Lambda_{e_1}}$, which means $\Lambda_{e_1}$ must be an negative multiples of $e_1$, then it contradicts with $\Lambda_{e_1}$ contains positive multiples of $e_1$. We prove that if the equality hold in (\[vol-rat-ine\]), holds then $\Lambda$ is a $GL(n)$ image of an obtuse set.
On the other hand, if $\Lambda$ is a $GL(n)$ image of an obtuse set, by the $GL(n)$ invariance of the volume ratio for the Orlicz zonotopes and Lemma \[orl-vol-lem\], the equality of (\[vol-rat-ine\]) holds.
Together with the above we have the equality of (\[vol-rat-ine\]) hold if and only if $\Lambda$ is a $GL(n)$ image of an obtuse set. We complete the proof.
If we take $\varphi(t)=t^p$, $p>1$, then it reduces to the following.
Suppose $p>1$ and $\Lambda$ is a finite and spanning multiset. Then $$\begin{aligned}
\frac{V(Z^+_p\Lambda)}{V(Z^+_1\Lambda)}\leq\frac{V(Z^+_p\Lambda_\bot)}{V(Z^+_1\Lambda_\bot)},
\end{aligned}$$ with equality if and only if $\Lambda$ is a $GL(n)$ image of an obtuse set.
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[^1]: The work is supported in part by CNSF (Grant No. 11161007, Grant No. 11101099), Guizhou (Unite) Foundation for Science and Technology (Grant No. \[2014\] 2044, No. \[2012\] 2273, No. \[2011\] 16), Guizhou Technology Foundation for Selected Overseas Chinese Scholar and Doctor foundation of Guizhou Normal University.
|
---
abstract: 'We report experimental evidence of a remarkable spontaneous time reversal symmetry breaking in two dimensional electron systems formed by atomically confined doping of phosphorus (P) atoms inside bulk crystalline silicon (Si) and germanium (Ge). Weak localization corrections to the conductivity and the universal conductance fluctuations were both found to decrease rapidly with decreasing doping in the Si:P and Ge:P $\delta-$layers, suggesting an effect driven by Coulomb interactions. In-plane magnetotransport measurements indicate the presence of intrinsic local spin fluctuations at low doping, providing a microscopic mechanism for spontaneous lifting of the time reversal symmetry. Our experiments suggest the emergence of a new many-body quantum state when two dimensional electrons are confined to narrow half-filled impurity bands.'
address:
- '$^1$ Department of Physics, Indian Institute of Science, Bangalore 560 012, India'
- '$^2$ Centre for Quantum Computation and Communication Technology, University of New South Wales, Sydney NSW 2052, Australia'
author:
- 'S. Shamim,$^1$ S. Mahapatra,$^2$ G. Scappucci,$^2$ W.M.Klesse,$^2$ M.Y.Simmons,$^2$ and A. Ghosh,$^1$'
title: Spontaneous breaking of time reversal symmetry in strongly interacting two dimensional electron layers in silicon and germanium
---
Invariance to time reversal is among the most fundamental and robust symmetries of nonmagnetic quantum systems. Its violation often leads to new and exotic phenomena, particularly in two dimensions (2D), such as the quantized Hall conductance in semiconductor heterostructures [@Klitzing_PRL1980], the quantum anomalous Hall effect in topological insulators [@Chang_TI_Science2013] or the predicted chiral superconductivity in graphene [@Levitov_NatPhys2012]. The breaking of time reversal invariance is experimentally achieved either by an external magnetic field or intentional magnetic doping. Here we show that strong Coulomb interactions can also lift the time reversal symmetry in nonmagnetic 2D systems at zero magnetic field.
While bulk P-doped Si and Ge have been extensively studied in the context of electron localization in three dimensions [@Rosenbaum_PRB1983; @Sarachik_PRB1992; @Loh_PRL1989; @Sachdev_Paalanen_PRL1988; @Sachdev_Localmoment_PRB1989; @Sachdev_Bhatt_LocalMoment_PRL1989], confining the dopants to one or few atomic planes ($\delta-$layers) of the host semiconductor has recently led to a new class of 2D electron system [@Goh2006; @Giordano_Nanolett2012; @BentScience; @Fuechsle2012]. Electron transport in these atomically confined 2D layers occurs within a 2D impurity band where the effective Coulomb interaction is parameterized in terms of $U/\gamma$, with $U$ being the Coulomb energy required to add an additional electron to a dopant site, and $\gamma$, the hopping integral between adjacent dopants. Since each dopant P atom contributes one valence electron, the impurity band is intrinsically ’half filled’ (schematic in Fig. 1a), which reinforces the interaction effects due to the in-built electron-hole symmetry, and forms an ideal platform to explore the rich phenomenology of the 2D Mott-Hubbard model, ranging from Mott metal-insulator transition (MIT) to novel spin excitations and magnetic ordering [@Vollhardt_PRL2005; @NandiniTrivedi_PRL1999; @Kohno_PRL2012; @BhatPRB2007].
In this Letter we show evidence of spontaneously broken time reversal symmetry in 2D Si:P and Ge:P $\delta$-layers as the on-site effective Coulomb interaction is increased by decreasing the doping density of P atoms. Quantum transport and noise experiments indicate a strong suppression of quantum interference effects at low doping densities. We could attribute this to a spontaneous breaking of time reversal symmetry which manifest in an unambiguous suppression of universal conductance fluctuations (UCF) at zero magnetic field.
The preparation of the P $\delta$-layers in Si and Ge have been detailed in earlier publications [@Goh2006; @Johnson_PhDThesis; @Giordano_Nanolett2012], and parameters relevant to the present work is supplied in the Supplementary Information (SI). The Drude conductivity ($\sigma_D$) of the $\delta$-layers decreases with decreasing doping as $\sigma_D \propto n^{3/2}$ (Fig. 1b), where $n$ is the electron density measured from Hall effect, implying significant scattering from charged dopants [@SDSarma_SiP_PRB2013]. We find $\sigma_D \gg e^2/h$ in all devices, ensuring a nominally weakly localized regime. All electrical transport measurements were carried out in a dilution refrigerator with an electron temperature of $0.15$ K using low frequency ac lock-in technique. The electron transport was strictly diffusive with $k_BT\tau_0/\hbar \ll 10^{-2}$, because of short momentum relaxation times $\tau_0 \sim 10 - 100$ fs, and displays negative logarithmic correction to conductivity in the quantum coherent regime (Fig. 1c) [@Giordano_Nanolett2012].
{width="1\linewidth"}
The key advantage of using both Si and Ge as host semiconductors is the factor of three difference in the Bohr radius, $a_B^*$, which allows us to achieve a wide range of average effective dopant separation ($r_P/a_B^*$) within the similar range of doping density ($r_P \approx 2/\sqrt{\pi n}$). As shown in the scale bar of Fig. 1b, $r_P/a_B^*$ has an overall range from $\approx 0.6$ to 3. This corresponds to a range of $\gamma \sim 10-20$ meV and $\sim 20-50$ meV for the Ge:P and Si:P devices respectively, assuming hydrogenic orbitals [@Efros_Shklovskii_Book]. Since $U \sim 200$ meV and $\sim 50$ meV for single P donor in Si and Ge, respectively, the effective on-cite Coulomb interaction $U/\gamma$ can be $\gg 1$, particularly in lightly doped Si devices.
{width="1\linewidth"}
In Fig. 1d, we show the transverse magnetic field ($B_\perp$) dependence of the quantum correction to conductivity, $\sigma_{QI}(B_\perp)=\sigma(B_\perp)-\sigma(0)-\sigma_{cl}$, where $\sigma_{cl} = -(\sigma_D^3/n^2e^2)B_\perp^2$, is the classical correction to the Drude conductivity. Due to diffusive nature of our devices the quantum correction from the electron-electron interaction is only perturbative ($\sim (\omega_c\tau_0)^2$ $\lesssim 10^{-4}$, where $\omega_c$ is the cyclotron frequency) [@Interaction_MR_PRL] and $\sigma_{QI}(B_\perp)$ represents the contribution primarily from the quantum interference effect. $\sigma_{QI}$ for three 2D Si:P $\delta$-layers at $0.28$ K is shown in Fig. 1d. For comparison, $\sigma_{QI}$ is scaled by $\sigma_{WL}$, where $\sigma_{WL} = (e^2/\pi h)\ln{(\tau_\phi/\tau_0)}$ is the universal weak localization correction to conductivity for a diffusive 2D conductor with free electrons. For each device, both $\sigma_{WL}$ and the phase breaking field $B_\phi = \hbar/4eD\tau_\phi$ (shown by vertical lines in Fig. 1d) were experimentally estimated from the low-$B_\bot$ magnetoconductivity data (see SI, section S3), where $\tau_\phi$ and $D$ are the phase coherence time and electron diffusivity, respectively. Since the magnitude of $\sigma_{QI}$ at $B \gg B_\phi$ represents the net correction to conductivity due to quantum interference, it is evident from Fig. 1d that the contribution of weak localization effect on transport decreases with decreasing doping density (see SI, section S1). It is important to note that a major shift in the dominant dephasing mechanism in the lightly doped samples is ruled out because we find $\tau_\phi$ to be similar in magnitude in all three devices, and $\propto T$ down to $T = 0.2$ K (Fig. S2 in SI). This confirms the predominance of the electron-electron scattering mediated dephasing which has been reported earlier in such $\delta$-layers [@Giordano_Nanolett2012].
The reduced quantum correction cannot be due to finite experimental range ($\approx 0-14$ T) of $B_\perp$, which exceeds both $B_\phi$ and $B_0$ ($=\hbar/4eD\tau_0$, the upper cutoff field due to momentum relaxation) by factors of $1000$ and $2$ respectively even for the least doped devices at $0.28$ K (Table I in SI). Spin-orbit interaction is also known to be small for P-doped (bulk) Si and Ge [@AGPRL2000; @Critical_exponent_SiP_PRB], and independent of the density of the dopants. Any long range magnetic order is also unlikely because the Hall resistance was found to vary linearly with $B_\bot$ at all $T$ (see SI, section S7) in all our devices [@BhatPRB2007].
The suppression of quantum correction to conductivity has been observed in low density electron gases in Si MOSFETs near the apparent MIT [@Kravcehnko_PRL2003] although its microscopic origin remains unclear with both temperature dependant screening of disorder and interaction driven spin fluctuations suggested as competing mechanisms. However, the formation of local magnetic moments in the presence of strong Coulomb interactions, is known to occur in three dimensional P-doped Si close to the MIT [@Sachdev_Paalanen_PRL1988; @Sachdev_Localmoment_PRB1989; @Sachdev_Bhatt_LocalMoment_PRL1989]. These moments serve to remove the time reversal symmetry, suppressing the coherent back-scattering of electrons. In 2D, the possibility of localized spin excitations at the Mott transition has been suggested theoretically [@Kohno_PRL2012; @Imada_RMP1998], but without any experimental evidence so far.
{width="1\linewidth"}
To probe whether the observed suppression of localization correction indeed manifests a breaking of the time reversal symmetry, we have measured the UCF as a function of $T$ and $B_\perp$ from slow time-dependent fluctuations in the conductance ($G$) of the $\delta$-layers which represents the ensemble fluctuations via the ergodic hypothesis [@Stone_PRB; @AGPRL2000; @BirgePRB1990; @BirgePRB1993; @FengPRL1986]. The time dependant conductance fluctuations (inset of Fig. 2a) are analyzed to obtain the power spectral density, $S_G$, which on integration over the experimental bandwidth gives the normalized variance, $N_G=\int{S_G/G^2}df=\langle\delta{G}^2\rangle/\langle{G}\rangle^2$ as shown in Fig. 2a (see Ref [@SaquibPRB2011] and SI, section S3 for details). Fig. 2b shows $N_G$ as a function of $T$ for Si\_HD. For $T \lesssim 15$ K, $N_G$ increases with decreasing $T$, which is a hallmark of UCF. In this regime, one expects $N_G \propto L_\phi^4n_T \propto 1/T$, where $L_\phi (\propto T^{-0.5})$ and $n_T (\propto T)$ are the phase coherence length and density of active two level fluctuators [@BirgePRB1990] (Fig. 2b). The absolute magnitude of $N_G$ in all devices correspond to the change in conductance by $\sim O[e^2/h]$ due to a single fluctuator within a phase coherent box (see SI, section S5), establishing the observed noise to be indeed from mesoscopic fluctuations.
As a function of $B_\perp$, the magnitude of UCF is expected to decrease by an exact factor of two at two field scales, first at $B_\perp \sim B_\phi$ when the time reversal symmetry, and hence the Cooperon (self-intersecting diffusion trajectories) contribution, is removed [@Altshuler_Spivak_1985; @Stone_PRB; @Beenakker_Houten_Review] and second at $B_\perp \sim B_Z = k_BT/g\mu_B$ due to removal of spin degeneracy [@Stone_PRB; @BirgePRB1996_Zeeman; @Beenakker_Houten_Review], where $g$ and $\mu_B$ are the $g$-factor and $\mu_B$ respectively. The inset of Fig. 2d shows schematically the two reductions in UCF magnitude as a function of $B_\perp$. Fig. 2d shows that the UCF magnitude in heavily doped Ge\_HD (violet symbols) consists of both factors of two reduction at $B_\bot \approx B_\phi$ and $B_\bot \approx B_Z$, corresponding to the removal of time reversal symmetry and spin degeneracy, respectively, whereas the lightly doped devices, such as Si\_MD, shows almost no variation in the UCF magnitude on the scale of $B_\phi$ but decreases by a factor of two at $B_\bot \approx B_Z$. To confirm this scenario, we have also recorded the variation of $N_G$ in Si\_MD as a function of parallel magnetic field, $B_\|$, which couples only to spin degree of freedom (Fig. 2c). The factor of two reduction at $B_\| \sim B_Z$ (shown by vertical arrows in Fig. 2c) for $T=0.5$ K and $4.2$ K establishes that the $1/f$ noise in our devices indeed arises from the UCF mechanism.
Since the reduction in UCF at $B_\bot \sim B_\phi$ is associated only to removal of the fundamental time reversal symmetry of the underlying Hamiltonian [@Altshuler_Spivak_1985], its absence in the lightly doped $\delta-$layers is unique, and has not been previously observed in interacting 2D systems in semiconductors [@MYS_NatPhys2008; @Kravchenko_RMP2001]. To elaborate, we have compiled the $B_\bot$-dependence of $N_G$ normalized by $N_{G\phi}$, where $N_{G\phi}$ is the value of $N_G$ at $B_\perp \gg B_\phi$ but $< B_Z$ , for all devices in Fig. 3. $N_{G\phi}$ was chosen at $B_\perp \sim 20B_\phi$ which was $< B_Z$ for all the devices at all temperatures. The peak in $N_G$ around $B_\bot = 0$ is progressively suppressed with decreasing doping density, and eventually for $r_P/a_B^* \gtrsim 1.5$, the Cooperon contribution to UCF noise at low $B_\perp$ becomes immeasurably small, implying a spontaneous breaking of time reversal symmetry even at $B_\perp = 0$ (Inset of Fig. 3).
{width="1\linewidth"}
To explore the origin of lifting of the time reversal symmetry in the $\delta$-layers, we subjected the devices to [*in-plane*]{} magnetic field, $B_\|$, that resulted in a nonmonotonic magnetoconductivity in the lightly doped $\delta$-layers. The logarithmic increase in the magnetoconductivity at large $B_\|$, as shown in Fig. 4a, was observed in all devices irrespective of doping level, and known to represent suppression of weak localization due to the finite width of the $\delta$-layers [@InplaneMR_Meyer_arxiv]. However, the negative magnetoconductivity around $B_\| = 0$ often indicates the presence of local moments, because localization strengthens as phase coherence increases with the freezing of spin-flip scattering [@MR_CeRhIn_PRB2002; @InplaneMR_Meyer_arxiv]. In such a case, the activated spin-flip processes across the Zeeman gap, leads to magnetoconductivity decreasing linearly with $B_\|$ as $\Delta\sigma(B_\|) = -\eta B_\|/T$, where $\eta\sim e^2g_{imp}\mu_B/hk_B$, and $g_{imp}$ is the $g$-factor of the magnetic impurity [@InplaneMR_Meyer_arxiv]. As shown in Fig. 4c, we indeed find the $\Delta\sigma(B_\|,T) \propto B_\|/T$ in Si\_LD. The negative magnetoconductivity in $B_\|$ is entirely absent in the heavily doped devices (Fig. 4b). This establishes that the spin fluctuations are entirely due to strong Coulomb interactions, and hence observable only in the lightly doped $\delta$-layers. Importantly, the experimental value of $\eta$ was found to be a factor of $\sim 50$ smaller than that expected theoretically (assuming $g_{imp}=2$), suggesting that the impact of local moments on the dephasing process is anomalously small.
The compelling analogy with the bulk P-doped Si close to MIT provides a “two-fluid” framework to address transport in our $\delta$-layers. This consists of itinerant electrons in disordered Hamiltonian and local magnetic moments [@Sachdev_Paalanen_PRL1988; @Sachdev_Localmoment_PRB1989; @Sachdev_Bhatt_LocalMoment_PRL1989]. The interaction between the local moments and itinerant electrons suppresses localization, although the spin-scattering process is quasi-elastic (energy exchange $\ll k_BT$), causing only minor modification to the dephasing mechanism (as confirmed by the linear $T$ dependence of $\tau_\phi^{-1}$ in Fig. S2 of SI and small $\Delta\sigma(B_\|)$). In addition, the two-fluid model allows a phenomenological generalized Hikami-Larkin-Nagaoka expression for the total quantum interference correction that includes the quasi-elastic spin scattering rate ($\tau_s^{-1}$) as,
$$\begin{aligned}
\label{deltasigma_total}
\Delta\sigma(B_\bot,T)=\frac{\alpha e^2}{\pi h}\left[F\left(\frac{B_\bot}{B_\phi}\right) - F\left(\frac{B_\bot}{B_0}\right)\right] - \frac{\beta e^2}{\pi h} F\left(\frac{B_\bot}{B_s}\right)\end{aligned}$$
where $\alpha$ and $\beta$ are positive constants close to unity, and $F(x) = \ln(x)+\psi(0.5+1/x)$, with $\psi(x)$ being the digamma function. As shown by the solid lines in Fig. 1d, Eq. \[deltasigma\_total\] describes the magnetoconductivity very well over the entire range of $B_\bot$. The fit parameter $B_s = \hbar/4eD\tau_s$, provides an estimate of the spin scattering time $\tau_s$. We note the following: (i) As evident in Fig. 4d, $\tau_s^{-1}$ is more than ten times larger than experimentally measured $\tau_\phi^{-1}$ (see SI), confirming that the spin-scattering is mostly elastic. (ii) Second, $\tau_s^{-1}$ varies nonmonotonically with $n$. The filled squares represent $\tau_s^{-1}$ analyzed from data of Ref [@Johnson_PhDThesis]. At low $n$, $\tau_s^{-1} \sim n^{0.5}$ irrespective of host material, disorder or carrier mobility, indicating that the number of local spins are only related with the number of P dopant sites. However $\tau_s^{-1}$ drops abruptly around $n \sim 1.5\times10^{14}$ cm$^{-2}$, suggesting a quenching of the spins and commencement of free-electron weakly localized quantum transport. The T-dependence of $\tau_s^{-1}$ (Fig. 4e), in accordance with the two-fluid model, shows a power law variation as $\tau_s^{-1} \propto T^p$, with $p \approx 0.7$. This sets the exponent for susceptibilty and specific heat divergence in the $\delta$-layers to be $\approx 0.3$, which is about half of that observed in the bulk Si:P close to MIT [@Sachdev_Paalanen_PRL1988; @Lohneysen_SpecificHeat_PRL1989].
Finally, to estimate the fraction of P-dopants that host a local moment, we compare the estimated $\tau_s^{-1}$ in lightly doped Si\_LD ($n = 5\times10^{13}$ cm$^{-2}$) with (1) the total momentum relaxation rate $\tau_0^{-1} \approx 10^{14}$ s$^{-1}$ from the experimental Drude conductivity, although this involves scattering from neutral defects as well, and (2) calculated momentum relaxation rate ($\approx 2\times10^{13}$ s$^{-1}$) expected purely from the P-dopants (charged impurities) (see calculation details in Ref [@SDSarma_SiP_PRB2013] and SI, section S6). This gives a bound between $2\% - 10\%$ of the P-dopants to host local moments which is consistent with the fraction expected for half-filled impurity bands in bulk Si:P [@Sachdev_Bhatt_LocalMoment_PRL1989]. Importantly, while the weak localization correction is reduced only partially ( 30% in Si\_LD), the UCF noise due to the Cooperons is completely suppressed for the weakly doped devices. It is possible that because the UCF noise involves interference between two Feynman propagators, it is more likely to be affected by the localized spins than the WL correction which is determined by a single self intersecting propagator. Note that we have not discussed spatial inhomogeneity or clustering in the distribution of dopants which can lead to coexistence of localized and delocalized phases [@Vollhardt_PRL2005], impact of multiple valleys [@Punnoose_Science2005; @Medini_NatPhys2006], or the inter-site Coulomb interaction [@Shepelyansky_PRL1994; @MYS_NatPhys2008; @Kravchenko_RMP2001] which are unlikely to affect the time reversal symmetry.
In summary, magnetoconductivity and noise measurements reveal an unexpected spontaneous breaking of time reversal symmetry in 2D electron systems hosted in atomically confined Si:P and Ge:P crystals. The universal conductance fluctuations and in-plane magnetoconductivity suggest that local spin fluctuations in the presence of strong Coulomb interaction play an important role in the lifting the time reversal symmetry. Whether this indeed leads to a true interaction-induced metallic ground state in two dimensions needs further experimental and theoretical exploration.
Acknowledgement
===============
We acknowledge Sankar Das Sarma, Ravin N.Bhatt, Vijay Shenoy, Sanjoy Sarker and Jainendra Jain for discussions. We thank Department of Science and Technology (DST), Government of India and Australian-Indian Strategic Research Fund (AISRF) for funding the project. The research was undertaken in collaboration with the Australian Research Council, Centre of excellence for Quantum Computation and Communication Technology (Project number CE110001027) and the US Army Research Office under contract number W911NF-08-1-0527. SS thanks CSIR for financial support. GS acknowledges support from UNSW under the GOLDSTAR Award 2012 scheme. MYS acknowledges a Federation Fellowship.
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---
abstract: 'We tackle the problem of automatically reconstructing a complete 3D model of a scene from a single RGB image. This challenging task requires inferring the shape of both visible and occluded surfaces. Our approach utilizes viewer-centered, multi-layer representation of scene geometry adapted from recent methods for single object shape completion. To improve the accuracy of view-centered representations for complex scenes, we introduce a novel “Epipolar Feature Transformer” that transfers convolutional network features from an input view to other virtual camera viewpoints, and thus better covers the 3D scene geometry. Unlike existing approaches that first detect and localize objects in 3D, and then infer object shape using category-specific models, our approach is fully convolutional, end-to-end differentiable, and avoids the resolution and memory limitations of voxel representations. We demonstrate the advantages of multi-layer depth representations and epipolar feature transformers on the reconstruction of a large database of indoor scenes.'
author:
- Daeyun Shin
- Zhile Ren
- 'Erik B. Sudderth'
- 'Charless C. Fowlkes'
bibliography:
- 'scene3d.bib'
title: '3D Scene Reconstruction with Multi-layer Depth and Epipolar Transformers'
---
@cmdkey[subfigpos]{}\[sfp@\][vsep]{}\[0.6\]@cmdkey[subfigpos]{}\[sfp@\][hsep]{}\[2.5pt\]
Introduction
============
When we examine a photograph of a scene, we not only perceive the 3D shape of visible surfaces, but effortlessly infer the existence of many invisible surfaces. We can make strong predictions about the complete shapes of familiar objects despite viewing only a single, partially occluded aspect, and can infer information about the overall volumetric occupancy with sufficient accuracy to plan navigation and interactions with complex scenes. This remains a daunting visual task for machines despite much recent progress in detecting individual objects and making predictions about their shape. *Convolutional neural networks* (CNNs) have proven incredibly successful as tools for learning rich representations of object identity which are invariant to intra-category variations in appearance. Predicting 3D shape rather than object category has proven more challenging since the output space is higher dimensional and carries more structure than simple regression or classification tasks.
![Given a single input view of a scene (top left), we would like to predict a complete geometric model. Depth maps (top right) provide an efficient representation of scene geometry but are incomplete, leaving large holes (e.g., the wardrobe). We propose multi-layer depth predictions (bottom left) that provide complete view-based representations of shape, and introduce an epipolar transformer network that allows view-based inference and prediction from virtual viewpoints (like overhead views, bottom right).[]{data-label="fig:splash"}](figures/figure1-lowres.png){width="\columnwidth"}
Early successes at using CNNs for shape prediction leveraged direct correspondences between the input and output domain, regressing depth and surface normals at every input pixel [@eigen2014depth]. However, these so-called 2.5D representations are incomplete: they don’t make predictions about the back side of objects or other occluded surfaces. Several recent methods instead manipulate voxel-based representations [@song2016ssc] and use convolutions to perform translation-covariant computations in 3D. This provides a more complete representation than 2.5D models, but suffers from substantial storage and computation expense that scales cubically with resolution of the volume being modeled (without specialized representations like octrees [@riegler2017octnet]). Other approaches represent shape as an unstructured point cloud [@qi2017pointnet; @su18splatnet], but require development of suitable convolutional operators [@Gadelha2018multiresolution; @wang2018pixel2mesh] and fail to capture surface topology.
In this paper, we tackle the problem of automatically reconstructing a [*complete*]{} 3D model of a scene from a single RGB image. As depicted in Figures \[fig:splash\] and \[fig:etn\_overview\], our approach uses an alternative shape representation that extends view-based 2.5D representations to a complete 3D representation. We combine [*multi-layer*]{} depth maps that store the depth to multiple surface intersections along each camera ray from a given viewpoint, with [*multi-view*]{} depth maps that record surface depths from different camera viewpoints.
While multi-view and multi-layer shape representations have been explored for single object shape completion, for example by [@shin2018pixels], we argue that multi-layer depth maps are particularly well suited for representing full 3D scenes. [*First*]{}, they compactly capture high-resolution details about the shapes of surfaces in a large scene. Voxel-based representations allocate a huge amount of resources to simply modeling empty space, ultimately limiting shape fidelity to much lower resolution than is provided by cues like occluding contours in the input image [@song2016ssc]. A multi-layer depth map can be viewed as a run-length encoding of dense representations that stores only transitions between empty and occupied space. [*Second*]{}, view-based depths maintain explicit correspondence between input image data and scene geometry. Much of the work on voxel and point cloud representations for single object shape prediction has focused on predicting a 3D representation in an object-centered coordinate system. Utilizing such an approach for scenes requires additional steps of detecting individual objects and estimating their pose in order to place them back into some global scene coordinate system [@tulsiani2018factoring]. In contrast, view-based multi-depth predictions provide a single, globally coherent scene representation that can be computed in a “fully convolutional” manner from the input image.
One limitation of predicting a multi-layer depth representation from the input image viewpoint is that the representation cannot accurately encode the geometry of surfaces which are nearly tangent to the viewing direction. In addition, complicated scenes may contain many partially occluded objects that require a large number of layers to represent completely. We address this challenge by predicting additional (multi-layer) depth maps computed from virtual viewpoints elsewhere in the scene. To link these predictions from virtual viewpoints with the input viewpoint, we introduce a novel *Epipolar Feature Transformer* (EFT) network module. Given the relative poses of the input and virtual cameras, we transfer features from a given location in the input view feature map to the corresponding epipolar line in the virtual camera feature map. This transfer process is modulated by predictions of surface depths from the input view in order to effectively re-project features to the correct locations in the overhead view.
To summarize our contributions, we propose a view-based, multi-layer depth representation that enables fully convolutional inference of 3D scene geometry and shape completion. We also introduce EFT networks that provide geometrically consistent transfer of CNN features between cameras with different poses, allowing end-to-end training for multi-view inference. We experimentally characterize the completeness of these representations for describing the 3D geometry of indoor scenes, and show that models trained to predict these representations can provide better recall and precision of scene geometry than existing approaches based on object detection.
Related Work
============
The task of recovering 3D geometry from 2D images has a rich history dating to the visionary work of Roberts [@roberts1963machine].
[****]{} Single-view 3D shape reconstruction is challenging because the output space is under-constrained. Large-scale datasets like ShapeNet [@chang2015shapenet; @wu20153d] facilitate progress in this area, and recent methods have learned geometric priors for object categories [@kar2015category; @wu2018shapehd], disentangled primitive shapes from objects [@girdhar2016learning; @zou20173d], or modeled surfaces [@hane2017hierarchical; @shin2018pixels; @zhang2018learning]. Other work aims to complete the occluded geometric structure of objects from a 2.5D image or partial 3D scan [@rock2015completing; @dai2017shape; @wu2017marrnet; @yang20183d]. While the quality of such 3D object reconstructions continues to grow [@katoneural; @wang2018pixel2mesh], applications are limited by the assumption that input images depict a single, centered object.
[****]{} We seek to predict the geometry of full scenes containing an unknown number of objects; this task is significantly more challenging than object reconstruction. Tulsiani [@tulsiani2018factoring] factorize 3D scenes into detected objects and room layout by integrating separate methods for 2D object detection, pose estimation, and object-centered shape prediction. Given a depth image as input, Song [@song2016ssc] propose a volumetric reconstruction algorithm that predicts semantically labeled 3D voxels. Another general approach is to retrieve exemplar CAD models from a large database and reconstruct parts of scenes [@izadinia2017im2cad; @zou2017complete; @gupta2015aligning], but the complexity of CAD models may not match real-world environments. While our goals are similar to Tulsiani , our multi-layered depth estimates provide a denser representation of complex scenes.
[****]{} Most recent methods use voxel representations to reconstruct 3D geometry [@choy20163dr2n2; @song2016ssc; @Edward173D; @drcTulsiani17; @smith2018multi], in part because they easily integrate with 3D CNNs [@wu20153d] for high-level recognition tasks [@maturana2015voxnet]. Other methods [@fan2017point; @lin2018learning] use dense point clouds representations. Classic 2.5D depth maps [@eigen2014depth; @chen2016single] recover the geometry of visible scene features, but do not capture occluded regions. Shin [@shin2018pixels] empirically compared these representations for object reconstruction. We extend these ideas to whole scenes via a multi-view, multi-layer depth representation that encodes the shape of multiple objects.
[****]{} Layered representations [@wang94] have proven useful for many computer vision tasks including segmentation [@ghosh2012nonparametric] and optical flow prediction [@sun2012layered]. For 3D reconstruction, decomposing scenes into layers enables algorithms to reason about object occlusions and depth orderings [@isola2013scene; @smith2004layered; @wang2019geometric]. Layered 2.5D representations such as the two-layer decompositions of [@tulsiani2018layer; @dhamo2018peeking] infer the depth of occluded surfaces facing the camera. Our multi-layer depth representation extends this idea by including the depth of back surfaces (equiv. object thickness). We also infer depths from virtual viewpoints far from the input view for more complete coverage of 3D scene geometry. Our use of layers generalizes [@richter2018matryoshka], who used multiple intersection depths to model non-convexities for constrained scenes containing a single, centered object. Concurrently to our work, [@nicastro2019x] predicts object-level thicknesses for volumetric RGB-D fusion and [@gabeur2019moulding] estimates 3D human shape.
[****]{} Many classic 3D reconstruction methods utilize multi-view inputs to synthesize 3D shapes [@izadi2011kinectfusion; @snavely2008modeling; @dai2017scannet]. Given monocular inputs, several recent methods explore ways of synthesizing object appearance or image features from novel viewpoints [@zhou2018stereo; @yan2016perspective; @ji2017deep; @choy20163dr2n2; @tvsn_cvpr2017; @su20143d]. Other work uses unsupervised learning from stereo or video sequences to reason about depths [@zhou2017unsupervised; @jiang2017self]. Instead of simply transferring the pixel colors associated with surface points to novel views, we transfer whole CNN feature maps over corresponding object volumes, and thereby produce more accurate and complete 3D reconstructions.
Reconstruction with Multi-Layer Depth
=====================================
Traditional depth maps record the depth at which a ray through a given pixel first intersects a surface in the scene. Such 2.5D representations of scene geometry accurately describe visible surfaces, but cannot encode the shape of partially occluded objects, and may fail to capture the complete 3D shape of unoccluded objects (due to self-occlusion). We instead represent 3D scene geometry by recording multiple surface intersections for each camera ray. As illustrated in Figure \[fig:volumeinference\](a), some rays may intersect many object surfaces and require several layers to capture all details. But as the number of layers grows, multi-layer depths completely represent 3D scenes with multiple non-convex objects.
[X|X|X|X|X|Y]{} $\bar{D}_{1}$ & $\bar{D}_{1,2}$ & $\bar{D}_{1,2,3}$ & $\bar{D}_{1..4}$ & $\bar{D}_{1..5}$ & $\bar{D}_{1..5}\text{ +Ovh.}$\
0.237 & 0.427 & 0.450 & 0.480 & 0.924 & 0.932\
We use experiments to empirically determine a fixed number of layers that provides good coverage of typical natural scenes, while remaining compact enough for efficient learning and prediction. Another challenge is that surfaces that are nearly tangent to input camera rays are not well represented by a depth map of fixed resolution. To address this, we introduce an additional virtual view where tangent surfaces are sampled more densely (see Section \[sec:eft\]).
![ Epipolar transfer of features from the input image to a virtual overhead view. Given multi-layer depth predictions of surface entrances and exits, each pixel in the input view is mapped to zero, one, or two segments of the corresponding epipolar line in the virtual view.[]{data-label="fig:volumeinference"}](figures/volume-inference4.png){width="0.9\columnwidth"}
![ A volumetric visualization of our predicted multi-layer surfaces and semantic labels on SUNCG. We project the center of each voxel into the input camera, and the voxel is marked occupied if the depth value falls in the first object interval $({D}_1, {D}_2)$ or the occluded object interval $({D}_3, {D}_4)$. []{data-label="fig:voxelization"}](figures/voxelization4.jpg){width="0.925\columnwidth"}
Multi-Layer Depth Maps from 3D Geometry
---------------------------------------
In our experiments, we focus on a five-layer model designed to represent key features of 3D scene geometry for typical indoor scenes. To capture the overall room layout, we model the room envelope (floors, walls, ceiling, windows) that defines the extent of the space. We define the depth $D_5$ of these surfaces to be the *last* layer of the scene.
To model the shapes of observed objects, we trace rays from the input view and record the first intersection with a visible surface in depth map $D_1$. This resembles a standard depth map, but excludes the room envelope. If we continue along the same ray, it will eventually exit the object at a depth we denote by $D_2$. For non-convex objects the ray may intersect the same object multiple times, but we only record the *last* exit in $D_2$. As many indoor objects have large convex parts, the $D_1$ and $D_2$ layers are often sufficient to accurately reconstruct a large proportion of foreground objects in real scenes. While room envelopes typically have a very simple shape, the prediction of occluded structure behind foreground objects is more challenging. We define layer $D_3 > D_2$ as the depth of the next object intersection, and $D_4$ as the depth of the exit from that second object instance.
We let $(\bar{D}_1, \bar{D}_2, \bar{D}_3, \bar{D}_4,\bar{D}_5)$ denote the ground truth multi-layer depth maps derived from a complete 3D model. Since not all viewing rays intersect the same number of objects (e.g., when the room envelope is directly visible), we define a binary mask $\bar{M}_\ell$ which indicates the pixels where layer $\ell$ has support. Note that $\bar{M}_1=\bar{M}_2$, and $\bar{M}_3=\bar{M}_4$, since $D_2$ (first instance exit) has the same support as $D_1$. Experiments in Section \[sec:experiments\] evaluate the relative importance of different layers in modeling realistic 3D scenes.
Predicting Multi-Layer Depth Maps {#sec:predicing}
---------------------------------
To learn to predict five-channel multi-layer depths $\mathcal{D}
= (D_1, D_2, D_3, D_4, D_5)$ from images, we train a standard encoder-decoder network with skip connections, and use the Huber loss $\rho_h(., .)$ to measure prediction errors: $$L_d (\mathcal{D}) = \sum_{\ell=1}^5
\left( \frac{\bar{M}_\ell}{||\bar{M}_\ell||_1} \right)
\cdot \rho_h(D_\ell, \bar{D}_\ell).
\label{eq:loss_func}$$ We also predict semantic segmentation masks for the first and third layers. The structure of the semantic segmentation network is similar to the multi-layer depth prediction network, except that the output has 80 channels (40 object categories in each of two layers), and errors are measured via the cross-entropy loss. To reconstruct complete 3D geometry from multi-layer depth predictions, we use predicted masks and depths to generate meshes corresponding to the front and back surfaces of visible and partially occluded objects, as well as the room envelope. Without the back surfaces [@shade1998layered], ground truth depth layers $\bar{D}_{1,3,5}$ cover only 82% of the scene geometry inside the viewing frustum (vs. 92% including back surfaces of objects, see Table \[fig:ub\_table\]).
Epipolar Feature Transformer Networks {#sec:eft}
=====================================
To allow for richer view-based scene understanding, we would like to relate features visible in the input view to feature representations in other views. To achieve this, we transfer features computed in input image coordinates to the coordinate system of a “virtual camera” placed elsewhere in the scene. This approach more efficiently covers some parts of 3D scenes than single-view, multi-layer depths.
Figure \[fig:etn\_overview\] shows a block diagram of our *Epipolar Feature Transformer* (EFT) network. Given features $F$ extracted from the image, we choose a virtual camera location with transformation mapping $T$ and transfer weights $W$, and use these to warp $F$ to create a new “virtual view” feature map $G$. Like *spatial transformer networks* (STNs) [@jaderberg2015spatial] we perform a parametric, differentiable “warping” of a feature map. However, EFTs incorporate a weighted pooling operation informed by multi-view geometry.
[****]{} Image features at spatial location $(s,t)$ in an input view correspond to information about the scene which lies somewhere along the ray $$\begin{bmatrix}
x \\ y \\ z
\end{bmatrix}
= z
\mathbf{K_I}^{-1}
\begin{bmatrix}
s \\ t \\ 1
\end{bmatrix},
\quad \quad z \geq 0,$$ where $\mathbf{K}_I \in \mathbb{R}^{3 \times 3}$ encodes the input camera intrinsic parameters, as well as the spatial resolution and offset of the feature map. $z$ is the depth along the viewing ray, whose image in a virtual orthographic camera is given by $$\begin{bmatrix}
u(s,t,z) \\ v(s,t,z)
\end{bmatrix}
=
\mathbf{K_V}
\left(
z
\mathbf{R}
\mathbf{K_I}^{-1}
\begin{bmatrix}
s \\ t \\ 1
\end{bmatrix}
+ \mathbf{t}
\right),
\quad \quad z \geq 0.$$ Here $\mathbf{K}_V \in \mathbb{R}^{2 \times 3}$ encodes the virtual view resolution and offset, and $\mathbf{R}$ and $\mathbf{t}$ the relative pose.[^1] Let $T(s,t,z) = (u(s,t,z),v(s,t,z))$ denote the forward mapping from points along the ray into the virtual camera, and $\Omega(u,v) = \{ (s,t,z) : T(s,t,z) =
(u,v) \}$ be the pre-image of $(u,v)$.
Given a feature map computed from the input view $F(s,t,f)$, where $f$ indexes the feature dimension, we synthesize a new feature map $G$ corresponding to the virtual view. We consider general mappings of the form $$G(u,v,f) = \frac{\sum_{(s,t,z) \in \Omega(u,v)} F(s,t,f) W(s,t,z)}{\sum_{(s,t,z) \in \Omega(u,v)} W(s,t,z)},$$ where $W(s,t,z) \geq 0$ is a gating function that may depend on features of the input image.[^2] When $\Omega(u,v)$ is empty, we set $G(u,v,f)=0$ for points $(u,v)$ outside the viewing frustum of the input camera, and otherwise interpolate feature values from those of neighboring virtual-view pixels.
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[****]{} By design, the transformed features are differentiable with respect to $F$ and $W$. Thus in general we could assign a loss to predictions from the virtual camera, and learn an arbitrary gating function $W$ from training data. However, we instead propose to leverage additional geometric structure based on predictions about the scene geometry produced by the frontal view.
Suppose we have a scene depth estimate $D_1(s,t)$ at every location in the input view. To simplify occlusion reasoning we assume that relative to the input camera view, the virtual camera is rotated around the $x$-axis by $\theta < 90^{\circ}$ and translated in $y$ and $z$ to sit above the scene so that points which project to larger $t$ in the input view have larger depth in the virtual view. Setting the gating function as
$$W^1_{\text{surf}}(s,t,z) = \delta[D_1(s,t) = z] \prod_{\hat t=0}^{t-1} \delta[D_1( s,\hat t) + (t - \hat t) \cos \theta \not= z]$$
yields an epipolar feature transform that [*re-projects*]{} each feature at input location $(s,t)$ into the overhead view via the depth estimate $D_1$, but only in cases where it is not occluded by a patch of surface higher up in the scene. In our experiments we compute $W^\ell_{\text{surf}}$ for each $D_\ell$, $\ell \in \{1,2,3,4\}$, and use $W_{\text{surf}} = \max_\ell W^\ell_{\text{surf}}$ to transfer input view features to both visible and occluded surfaces in the overhead feature map. We implement this transformation using a z-buffering approach by traversing the input feature map from bottom to top, and overwriting cells in the overhead feature map.
Figure \[fig:volumeinference\](b) illustrates this feature mapping applied to color features using the ground-truth depth map for a scene. In some sense, this surface-based reprojection is quite conservative because it leaves holes in the interior of objects (e.g., the interior of the orange wood cabinet). If the frontal view network features at a given spatial location encode the presence, shape, and pose of some object, then those features really describe a whole volume of the scene behind the object surface. It may thus be preferable to instead transfer the input view features to the entire expected volume in the overhead representation.
To achieve this, we use the multi-layer depth representation predicted by the frontal view to define a range of scene depths to which the input view feature should be mapped. If $D_1(s,t)$ is the depth of the front surface and $D_2(s,t)$ is the depth at which the ray exits the back surface of an object instance, we define a volume-based gating function: $$W_{\text{vol}}(s,t,z) = \delta[z \in (D_1(s,t),D_2(s,t))].$$ As illustrated in Figure \[fig:volumeinference\](a), volume-based gating copies features from the input view to entire segments of the epipolar line in the virtual view. In our experiments we use this gating to generate features for $(D_1,D_2)$ and concatenate them with a feature map generated using $(D_3,D_4)$.
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---------- ---------- ----------
[****]{} For cluttered indoor scenes, there may be many overlapping objects in the input view. Overhead views of such scenes typically have much less occlusion and should be simpler to reason about geometrically. We thus select a virtual camera that is roughly overhead and covers the scene content visible from the reference view. We assume the input view is always taken with the gravity direction in the $y,z$ plane. We parameterize the overhead camera relative to the reference view by a translation $(t_x,t_y,t_z)$ which centers it over the scene at fixed height above the floor, a rotation $\theta$ which aligns the overhead camera to the gravity direction, and a scale $\sigma$ that captures the radius of the orthographic camera frustum.
{width="0.92\columnwidth"}
[@XYY@]{} & Precision & Recall\
$D_{1,2,3,4,5}$ & Overhead & **0.221** & **0.358**\
Tulsiani [@tulsiani2018factoring] & 0.132 & 0.191\
Experiments {#sec:experiments}
===========
Because we model complete descriptions of the ground-truth 3D geometry corresponding to RGB images, which is not readily available for natural images, we learn to predict multi-layer and multi-view depths from physical renderings of indoor scenes [@zhang2016physically] provided by the SUNCG dataset [@song2016ssc].
Generation of Training Data {#sec:trainingData}
---------------------------
The SUNCG dataset [@song2016ssc] contains complete 3D meshes for 41,490 houses that we render to generate our training data. For each rendered training image, we extract the subset of the house model that is relevant to the image content, without making assumptions about the room size. We transform the house mesh to the camera’s coordinate system and truncate polygons that are outside the left, top, right, bottom, and near planes of the perspective viewing frustum. Objects that are projected behind the depth image of the room envelope are also removed. The final ground truth mesh that we evaluate against contains all polygons from the remaining objects, as well as the true room envelope.
For each rendered training image, we generate target multi-depth maps and segmentation masks by performing multi-hit ray tracing on the ground-truth geometry. We similarly compute ground-truth height maps for a virtual orthographic camera centered over each scene. To select an overhead camera viewpoint for training that covers the relevant scene content, we consider three heuristics: (i) Convert the true depth map to a point cloud, center the overhead camera over the mean of these points, and set the camera radius to 1.5 times their standard deviation; (ii) Center the overhead camera so that its principal axis lies in the same plane as the input camera, and offset in front of the input view by the mean of the room envelope depths; (iii) Select a square bounding box in the overhead view that encloses all points belonging to objects visible from the input view. None of these heuristics worked perfectly for all training examples, so we compute our final overhead camera view via a weighted average of these three candidates.
Model Architecture and Training
-------------------------------
As illustrated in Figure \[fig:etn\_overview\], given an RGB image, we first predict a multi-layer depth map as well as a 2D semantic segmentation map. Features used to predict multi-layer depths are then mapped through our EFT network to synthesize features for a virtual camera view, and predict an orthographic height map. We then use the multi-layer depth map, semantic segmentation map, and overhead height map to predict a dense 3D reconstruction of the imaged scene.
![ Qualitative comparison of our viewer-centered, end-to-end scene surface prediction (left) and the object-based detection and voxel shape prediction of [@tulsiani2018factoring] (right). Object-based reconstruction is sensitive to detection and pose estimation errors, while our method is more robust. []{data-label="fig:soa"}](figures/soa2.png){width="1\columnwidth"}
We predict multi-layer depth maps and semantic segmentations via a standard convolutional encoder-decoder with skip connections. The network uses dilated convolution and has separate output branches for predicting each depth layer using the Huber loss specified in Section \[sec:predicing\]. For segmentation, we train a single branch network using a softmax loss to predict 40 semantic categories derived from the SUNCG mesh labels (see supplement for details).
Our overhead height map prediction network takes as input the transformed features of our input view multi-layer depth map. The overhead model integrates 232 channels (see Figure \[fig:etn\_overview\]) including epipolar transformations of a $48$-channel feature map from the depth prediction network, a $64$-channel feature map from the semantic segmentation network, and the RGB input image. These feature maps are extracted from the frontal networks just prior to the predictive branches. Other inputs include a “best guess” overhead height map derived from frontal depth predictions, and a mask indicating the support of the input camera frustum. The frustum mask can be computed by applying the epipolar transform with $F=1$, $W=1$. The best-guess overhead depth map can be computed by applying an unnormalized gating function $W(s,t,z) = z \cdot
\delta[D_1(s,t)=z]$ to the $y$-coordinate feature $F(s,t)=t$.
We also train a model to predict the virtual camera parameters which takes as input feature maps from our multi-depth prediction network, and attempts to predict the target overhead viewpoint (orthographic translation $(t_x,t_y)$ and frustum radius $\sigma$) chosen as in Section \[sec:trainingData\]. While the EFT network is differentiable and our final model can in principle be trained end-to-end, in our experiments we simply train the frontal model to convergence, freeze it, and then train the overhead model on transformed features without backpropagating overhead loss back into the frontal-view model parameters. We use the Adam optimizer to train all of our models with batch size 24 and learning rate 0.0005 for 40 epochs. The Physically-based Rendering [@zhang2016physically] dataset uses a fixed downward tilt camera angle of 11 degrees, so we do not need to predict the gravity angle. At test time, the height of the virtual camera is the same as the input frontal camera and assumed to be known. We show qualitative 3D reconstruction results on the SUNCG test set in Figure \[fig:exp\_recon\].
Evaluation
----------
\[sec:evaluation\]
To reconstruct 3D surfaces from predicted multi-layer depth images as well as the overhead height map, we first convert the depth images and height maps into a point cloud and triangulate vertices that correspond to a $2\times2$ neighborhood in image space. If the depth values of two adjacent pixels is greater than a threshold $\delta \cdot a$, where $\delta$ is the footprint of the pixel in camera coordinates and $a=7$, we do not create an edge between those vertices. We do not predict the room envelope from the virtual overhead view, so only pixels with height values higher than 5 cm above the floor are considered for reconstruction and evaluation.
[****]{} We use precision and recall of surface area as the metric to evaluate how closely the predicted meshes align with the ground truth meshes, which is the native format for SUNCG and ScanNet. Coverage is determined as follows: We uniformly sample points on surface of the ground truth mesh then compute the distance to the closest point on the predicted mesh. We use sampling density $\rho = 10000/\text{meter}^2$ throughout our experiments. Then we measure the percentage of inlier distances for given a threshold. *Recall* is the coverage of the ground truth mesh by the predicted mesh. Conversely, *precision* is the coverage of the predicted mesh by the ground truth mesh.
[@XYY@]{} & Precision & Recall\
$D_{1}$ & 0.525 & 0.212\
$D_{1}$ & Overhead & **0.553** & **0.275**\
$D_{1,2,3,4}$ & 0.499 & 0.417\
$D_{1,2,3,4}$ & Overhead & **0.519** & **0.457**\
[****]{} To provide an upper-bound on the performance of our multi-layer depth representation, we evaluate how well surfaces reconstructed from ground-truth depths cover the full 3D mesh. This allows us to characterize the benefits of adding additional layers to the representation. Table \[fig:ub\_table\] reports the coverage (recall) of the ground-truth at a threshold of 5cm. The left panels of Figure \[fig:pr\_objects\] show a breakdown of the precision and recall for the individual layers of our model predictions along with the upper bounds achievable across a range of inlier thresholds.
Since the room envelope is a large component of many scenes, we also analyze performance for objects (excluding the envelope). Results summarized in Table \[fig:pr\_table\] show that the addition of multiple depth layers significantly increases recall with only a small drop in precision, and the addition of overhead EFT predictions boosts both precision and recall.
[****]{} To further demonstrate the value of the EFT module, we evaluate the accuracy of the overhead height map prediction while incrementally excluding features. We first exclude channels that correspond to the semantic segmentation network features and compare the relative pixel-level L1 error. We then exclude features from the depth prediction network, using only RGB, frustum mask and best guess depth image. This baseline corresponds to taking the prediction of the input view model as an RGB-D image and re-rendering it from the virtual camera viewpoint. The L1 error increases respectively from $0.132$ to $0.141$ and $0.144$, which show that applying the EFT to the whole CNN feature map outperforms simple geometric transfer.
[****]{} Finally, we compare the scene reconstruction performance of our end-to-end approach with the object-based Factored3D [@tulsiani2018factoring] method using their pre-trained weights, and converting voxel outputs to surface meshes using marching cubes. We evaluated on 3960 examples from the SUNCG test set and compute precision and recall on objects surfaces (excluding envelope). As Figure \[fig:pr\_objects\] shows, our method yields roughly 3x improvement in recall and 2x increase in precision, providing estimates which are both more complete and more accurate. Figure \[fig:soa\] highlights some qualitative differences. To evaluate with an alternative metric, we voxelized scenes at 2.5cm$^3$ resolution (shown in Figure \[fig:voxelization\]). Using the voxel intersection-over-union metric, we see significant performance improvements over Tulsiani [@drcTulsiani17] (0.102 to 0.310) on objects (see supplement for details).
[****]{} Our network model is trained entirely on synthetically generated images [@zhang2016physically]. We test the ability of the model to generalize to the NYUv2 dataset [@SilbermanECCV12] via the promising comparison to Tulsiani [@tulsiani2018factoring] in Figure \[fig:nyu\].
We additionally test our model on images from the ScanNetv2 dataset [@dai2017scannet]. The dataset contains RGB-D image sequences taken in indoor scenes, as well as 3D reconstructions produced by BundleFusion [@dai2017bundlefusion]. For each video sequence from the 100 test scenes, we randomly sample $5\%$ of frames, and manually select 1000 RGB images to compare our algorithm to Tulsiani [@tulsiani2018factoring]. We select images where the pose of the camera is almost perpendicular to the gravity orientation, the amount of motion blur is small, and the image does not depict a close-up view of a single object. We treat the provided 3D reconstructions within each viewing frustum as ground truth annotations. As summarized in Table \[fig:scannet\_table\], our approach has significantly improved precision and recall to Tulsiani [@tulsiani2018factoring].
Conclusion
==========
Our novel integration of deep learning and perspective geometry enables complete 3D scene reconstruction from a single RGB image. We estimate multi-layer depth maps which model the front and back surfaces of multiple objects as seen from the input camera, as well as the room envelope. Our epipolar feature transformer network geometrically transfers input CNN features to estimate scene geometry from virtual viewpoints, providing more complete coverage of real-world environments. Experimental results on the SUNCG dataset [@song2016ssc] demonstrate the effectiveness of our model. We also compare with prior work that predicts voxel representations of scenes, and demonstrate the significant promise of our multi-view and multi-layer depth representations for complete 3D scene reconstruction.
[****]{} This research was supported by NSF grants IIS-1618806, IIS-1253538, CNS-1730158, and a hardware donation from NVIDIA.
[**Appendix**]{}
System Overview
===============
We provide an overview of our 3D reconstruction system and additional qualitative examples in our [**supplementary video**]{} (see project website).
Training Data Generation
========================
As we describe in Section \[sec:trainingData\] of our paper, we generate the target multi-layer depth maps by performing multi-hit ray tracing on the ground-truth 3D mesh models. If an object instance is completely occluded (i.e. not visible at all from the first-layer depth map), it is ignored in the subsequent layers. The Physically-based Rendering [@zhang2016physically] dataset ignores objects in “person” and “plant” categories, so those categories are also ignored when we generate our depth maps. The complete list of room envelope categories (according to NYUv2 mapping) are as follows: wall, floor, ceiling, door, floor\_mat, window, curtain, blinds, picture, mirror, fireplace, roof, and whiteboard. In our experiments, all room envelope categories are merged into a single “background” category. In Figure \[fig:meshlayers\], we provide a layer-wise 3D visualization of our multi-layer depth representation. Figure \[fig:evaluation\] illustrates our surface precision-recall metrics.
![ Layer-wise illustration of our multi-layer depth representation in 3D. Table \[fig:ub\_table\] in our paper reports an empirical analysis which shows that the five-layer representation ($\bar{D}_{1,2,3,4,5}$) covers 92% of the scene geometry inside the viewing frustum.[]{data-label="fig:meshlayers"}](figures/layers/all.png){width="1\columnwidth"}
![Illustration of ground-truth depth layers. (a, b): 2.5D depth representation cannot accurately encode the geometry of surfaces which are nearly tangent to the viewing direction. (b): We model both the front and back surfaces of objects as seen from the input camera. (c): The tangent surfaces are sampled more densely in the additional virtual view (dark green). Table \[fig:pr\_table\] in our paper shows the effect of augmenting the frontal predictions with the virtual view predictions. []{data-label="fig:gtlayers"}](figures/gtlayers.jpg){width="1\columnwidth"}
Representing the Back Surfaces of Objects
=========================================
Without the back surfaces, ground truth depth layers ($\bar{D}_{1,3,5}$) cover only 82% of the scene geometry inside the viewing frustum (vs. 92% including frontal surfaces — refer to Table \[fig:ub\_table\] in our paper for full comparison). Figure \[fig:gtlayers\](a) visualizes $\bar{D}_{1,3,5}$, without the back surfaces. This representation, *layered depth image* (LDI) [@shade1998layered], was originally developed in the computer graphics community [@shade1998layered] as an algorithm for rendering textured depth images using parallax transformation. Works based on prediction of LDI or its variants [@tulsiani2018layer; @zhou2018stereo] therefore do not represent the invisible back surfaces of objects. Prediction of back surfaces enables volumetric inference in our epipolar transformation.
![Illustration of our 3D precision-recall metrics. *Top*: We perform a bidirectional surface coverage evaluation on the reconstructed triangle meshes. *Bottom*: The ground truth mesh consists of all 3D surfaces within the viewing frustum and in front of the room envelope. We take the union of the predicted meshes from different views in world coordinates. This allows us to perform a layer-wise evaluation (e.g. Figure \[fig:pr\_objects\] in our paper). []{data-label="fig:evaluation"}](figures/evaluation.png){width="1\columnwidth"}
Multi-layer Depth Prediction
============================
See Figure \[fig:network1\] for network parameters of our multi-layer depth prediction model. All batch normalization layers have momentum 0.005, and all activation layers are Leaky ReLUs layers with $\alpha=0.01$. We use In-place Activated BatchNorm [@rotabulo2017place] for all of our batch normalization layers. We trained the network for 40 epochs.
Multi-layer Semantic Segmentation {#sec:mls}
=================================
See Figure \[fig:network2\] for network parameters of multi-layer semantic segmentation. We construct a binary mask for all foreground objects, and define segmentation mask $M_l$ as all non-background pixels at layer $l$. As mentioned in section 3.1, $D_1$ and $D_2$ have the same segmentation due to symmetry, so we only segment layers $1$ and $3$. The purpose of the foreground object labels is to be used as a supervisory signal for feature extraction $F_{\text{seg}}$, which is used as input to our Epipolar Feature Transformer Networks.
Virtual-view Prediction
=======================
See Figure \[fig:network4\] and \[fig:network5\] for network parameters of our virtual-view height map prediction and segmentation models. The height map prediction network is trained to minimize foreground pixel losses. At test time, the background mask predicted by the segmentation network is used to zero out the floor pixels. The floor height is assumed to be zero in world coordinates. An alternate approach is minimizing both foreground and background losses and thus allowing the height map predictor to implicitly segment the floor by predicting zeros. We experimented with both architectures and found the explicit segmentation approach to perform better.
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![Volumetric evaluation of our predicted multi-layer depth maps on the SUNCG [@song2016ssc] dataset.[]{data-label="fig:extravoxels"}](figures/extra-voxelization.jpg){width="1\columnwidth"}
![Distribution of voxel intersection-over-union on SUNCG (excluding room layouts). We observe that object-based reconstruction is sensitive to detection failure and misalignment on thin structures.[]{data-label="fig:voxelhist"}](figures/voxel-iou-hist.pdf){width="1\columnwidth"}
Voxelization of Multi-layer Depth Prediction
============================================
Given a 10m$^3$ voxel grid of resolution 400 (equivalently, 2.5cm$^3$) with a bounding box ranging from (-5,-5,-10) to (5,5,0) in camera space, we project the center of each voxel into the predicted depth maps. If the depth value for the projected voxel falls in the first object interval $({D}_1, {D}_2)$ or the occluded object interval $({D}_3, {D}_4)$, the voxel is marked occupied. We evaluate our voxelization against the SUNCG ground truth object meshes inside the viewing frustum, voxelized using the Binvox software which implements z-buffer based carving and parity voting methods. We also voxelize the predicted Factored3D [@tulsiani2018factoring] objects (same meshes evaluated in Figure \[fig:pr\_objects\] of our paper) using Binvox under the same setting as the ground truth. We randomly select 1800 examples from the test set and compute the intersection-over-union of all objects in the scene. In addition to Figure \[fig:voxelization\] of our paper, Figure \[fig:extravoxels\] shows a visualization of our voxels, colored according to the predicted semantic labeling. Figure \[fig:voxelhist\] shows a histogram of voxel intersection-over-union values.
Predictions on NYU and SUNCG
============================
Figures \[fig:exp\_recon2\] and \[fig:nyu2\] show additional 3D scene reconstruction results. We provide more visualizations of our network outputs and error maps on the SUNCG dataset in the last few pages of the supplementary material.
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[^1]: For a perspective model the righthand side is scaled by $z'(s,t,z)$, the depth from the virtual camera of the point at location $z$ along the ray.
[^2]: For notational simplicity, we have written $G$ as a sum over a discrete set of samples $\Omega$. To make $G$ differentiable with respect to the virtual camera parameters, we perform bilinear interpolation.
|
---
abstract: 'If an active Janus particle is trapped at the interface between a liquid and a fluid, its self-propelled motion along the interface is affected by a net torque on the particle due to the viscosity contrast between the two adjacent fluid phases. For a simple model of an active, spherical Janus colloid we analyze the conditions under which translation occurs along the interface and we provide estimates of the corresponding persistence length. We show that under certain conditions the persistence length of such a particle is significantly larger than the corresponding one in the bulk liquid, which is in line with the trends observed in recent experimental studies.'
author:
- 'P. Malgaretti'
- 'M.N. Popescu'
- 'S. Dietrich'
bibliography:
- 'swimmers.bib'
title: Active colloids at fluid interfaces
---
Introduction\[Intro\]
=====================
Micro- and nanometer scale particles capable of self-induced motility within liquid environments [@ebbens; @LaugaRev; @SenRev; @Sanchez2015; @Poon] are promising candidates for the development of novel lab-on-a-chip cell-sorting devices, chemical sensors [@RevLabChip], or targeted-drug-delivery systems [@RevDrugDelivery], to cite just a few potential applications. One proposal, which has generated significant experimental and theoretical attention within the last decade (see, e.g., the recent reviews in Refs. [@SenRev; @ebbens; @Sanchez2015]), is to achieve self-motility by designing “active” particles capable to induce chemical reactions within the surrounding liquid. One such system, which will be of particular interest for the present study, is represented by spherical beads partially covered over a spherical cap region by a catalyst which promotes, in the suspending solution, a chemical conversion of reactant (“fuel”) into product molecules[^1]. Due to the partial coverage by catalyst, the spherical symmetry of the system is broken in two ways. First, the material properties of the particle vary across the surface and an axis of symmetry, which passes through the center of the particle and the pole inside the catalyst covered cap, can be defined. Second, the chemical reaction takes place only on the catalytic part of the surface, and therefore the chemical composition of the surrounding solution is varying along the surface of the particle. The out-of-equilibrium chemical composition gradients along the surface, due to the chemical reaction, couple to the particle via the interactions of the molecules in solution with the surface of the particle. This interplay eventually leads to hydrodynamic flows and to the motion of the particle relative to the solution, analogous to classic phoresis [@Anderson1989; @Golestanian2005]. The active motion of such particles in the homogeneous bulk of the fluids has been the subject of numerous experimental (see, e.g., Refs. [@SenRev; @ebbens; @Golestanian2012; @Fisher2014; @Sanchez2015]) and theoretical (see, e.g., Refs. [@Golestanian2005; @Julicher; @Popescu2009; @Kapral2013; @Lowen2011]) studies.
However, in many cases such active Janus particles are suspended in a solution bounded by a liquid-fluid interface, which raises several new issues. It is known that in thermal equilibrium, i.e., in the *absence* of such motility-promoting chemical reactions, owing to their amphiphilic nature Janus particles tend to accumulate at liquid-fluid interfaces. (This effect can be exploited, e.g., for the stabilization of binary emulsions [@Aveyard2012; @Park2012; @Rezvantalab2013].) If the Janus particles are trapped at and confined to liquid-fluid interfaces their collective behavior, e.g., when externally driven or when relaxing towards equilibrium after a perturbation, can be strongly affected by this quasi two-dimensional ($2D$) confinement itself and by interface-promoted interactions, such as capillary interactions (see, e.g., Refs. [@alvaro2013; @alvaro2011; @alvaro2005]).
Turning to the case of an *active* Janus colloid, being trapped at the interface (i.e., being unable to move in the direction normal to the interface) can, on one hand, affect the particle dynamics due to the effects discussed above. On the other hand, this trapping may induce novel self-propulsion mechanisms. For example, it has been recently predicted that if one of the reaction products exhibits a preference for the surface and thus tends to accumulate at the interface, the trapped Janus sphere will be set into motion along the interface by Marangoni flow, induced by the spatially non-uniform distribution of the reaction products [@Lauga2011; @Stone2014]. (A similar motility mechanism can originate from thermally induced Marangoni flows, e.g., if the Janus particle contains a metal cap which is heated by a laser beam [@Wurger2014]; furthermore, as reported recently, induced Maragoni flows can drive the motion of active particles even if they are *not trapped at* but *located nearby* the interface [@Dominguez2015].)
If none of the reaction products exhibits a preference for the interface, the Marangoni type of propulsion is no longer in action. The question arises if for an active Janus particle, trapped at the interface, sustained motion along the interface can still occur due to the self-induced phoresis mechanism, which works in the bulk solution. For reasons of simplicity, in the following we focus solely on the case of planar interfaces. In thermal equilibrium (i.e., in the absence of diffusiophoresis), a Janus particle trapped at the interface typically exhibits a configuration in which the particle axis is not aligned with the normal of the interface (see Fig. \[fig:stab-janus\]). Therefore, at first it seems that, upon “turning on” the chemical reaction, motion along the interface may be achieved[^2]. However, it is known that the motion at the interface between two fluids generally involves a coupling between translation and rotation [@LaugaRev; @BrennerBook; @Pozrikidis2007]. Thus the possibility arises that the translation along the interface may lead to a rotation of the axis of the particle towards alignment with the interface normal. Since self-phoresis of active particles is, in general, characterized by very small Reynolds (Re) numbers (i.e., inertia does not play a role) [@Anderson1989; @ebbens; @SenRev; @Julicher], such a rotation of the axis of the particle leads to a motionless state once the axis is aligned with the normal, i.e., the motile state is just a transient. Therefore, predicting whether or not for a particular system sustained motion along the interface may occur via self-diffusiophoresis requires an understanding of the interplay between the equilibrium configuration of the Janus particle at the interface, the distribution of reactant and product molecules upon turning on the chemical reaction, and the induced hydrodynamic flows in the liquid and the fluid.
Here we study theoretically the issue of sustained self-diffusiophoresis along a liquid-fluid interface for a simple model of a spherical, chemically active Janus colloid trapped at a liquid-fluid interface. Nonetheless we expect this simple model to qualitatively capture some of the main physical features of the phenomenon. The chemical activity of the particle is modeled via the production of one species of solute molecules, with a uniform rate across the catalytic region. We determine the conditions under which this model system exhibits sustained motility. These conditions involve the interplay between the equilibrium configuration, the difference in viscosity between the two adjacent fluids, and the interactions between the particle and the product (solute) molecules. Finally, for particles trapped at the interface we analyze the persistence length and the stability of the motile state against thermal fluctuations and compare it with the corresponding motion in unbounded fluids.
Model\[model\]
==============
In this section we discuss our model for a catalytically active, spherical Janus colloid [@Golestanian2005] trapped at a liquid-fluid interface (see Fig. \[fig:stab-janus\](a)). The colloid (red disk in Fig. \[fig:stab-janus\](a)) of radius $R$ has a spherical cap region (the green patch in Fig. \[fig:stab-janus\](a)) decorated by a catalyst which promotes the conversion $A \longrightarrow B$ of “fuel” molecules $A$ into product (solute) molecules $B$. The particle is trapped at the interface (the horizontal black line in Fig. \[fig:stab-janus\](a)) between the two fluids “1” and “2” (denoted also as fluid and liquid, respectively) with bulk viscosities $\eta_1$ and $\eta_2$, respectively. For simplicity we assume that the fuel ($A$) and the product ($B$) molecules diffuse freely in both fluids, that neither of the two species $A$ and $B$ exhibits a preference for either the interface or one of the two fluid phases, and that the concentration of $A$ molecules in the two fluid phases is at a steady state. Furthermore, it is assumed that the fuel molecules $A$ are present in abundance such that their number density is not affected by the reaction. (Under the latter assumption, the dynamics of the fuel molecules is irrelevant and their sole role is, similarly to that of the catalyst, that of a “spectator” enabling the reaction due to which active motion emerges. Thus within this model the diffusion constants of the $A$ particles in the two fluids do not enter the description.) Accordingly, we assume that the effects of the chemical reaction can be approximated by representing the catalyst area as an effective source of solute which releases molecules $B$ (at a time independent rate per area of catalyst). We denote the diffusion constants of the solute molecules in the two fluids by $D_1$ and $D_2$, respectively.
For a Janus particle trapped at the interface several scenarios can emerge if the parameters controlling the effective particle-interface interactions, such as the coverage by catalyst or the three-phase contact angle of the “bare” particle (i.e., without catalyst) are changed. Here we restrict the discussion to the case that the particle and the two fluid phases have densities and surface tensions for which the deformations of the interface due to buoyancy are negligible, and we assume that the catalytic cap is completely immersed in one of the two fluid phases. Thus the Janus particle forms the three-phase contact angle of the bare particle [@Aveyard2012]. Furthermore, we assume that the effective interaction between the particle and the interface is such that the catalytic cap cannot jut into the other fluid, i.e., the circular boundary between the catalytic cap and the bare regions gets pinned at the three-phase contact line (between fluid 1, fluid 2, and the particle surface) upon touching it. (Accordingly, if the symmetry axis of the particle, i.e., the axis passing through the center $C$ of the particle and the center of the catalytic spherical cap, would rotate beyond this touching point, the interface would no longer be planar.) For simplicity, we further restrict our discussion to the case in which the three-phase contact angle of the bare particle is $\pi/2$, which implies that the catalytic cap has to be smaller than a hemisphere. Actually, the size of the catalytic cap should be sufficiently small so that thermal fluctuations of the orientation around the equilibrium one do not lead to the aforementioned touching, which causes pinning of the liquid-fluid interface. With these assumptions, the center of the Janus particle lies in the plane of the interface. (As it will be discussed in Sec. \[sec:phoretic\_velocities\], this particular configuration significantly simplifies the technical details, and thus provides transparent and physically intuitive results.) The orientation of the symmetry axis is determined by the effective interactions between the catalytic cap and the interface. For net repulsive interactions [@Aveyard2012] we expect the equilibrium distribution of the symmetry axis of the particle to be peaked at the direction normal to the interface. If, on the other hand, the effective interaction between the catalytic cap and the interface includes a long-ranged attractive part and a dominant short-ranged repulsive part, the equilibrium distribution is expected to be peaked at a direction which is close to, but distinct from, an orientation parallel to the interface. (Since the catalytic material is assumed to be completely immersed in one of the fluids, a net attractive interaction between the catalytic cap and the interface would be incompatible with our model.)
Since the surface tension compensates any action of the active Janus particle in the direction of the interface normal (which implies that the particle is trapped at the interface), translation of the particle upon turning on the catalytic reaction is possible only within the planar interface. Thus a motile state can be reached only if the axis of the Janus particle is not oriented perpendicular to the interface. Due to the symmetry of the problem, all lateral directions of the particle translation are equivalent; in other words, at a given tilting angle of the symmetry axis with respect to the normal, upon rotating the symmetry axis around the normal a state of motion emerges which is identical to the one in the original configuration. This allows us to consider the particle motion in the plane spanned by the axis of the particle (in its orientation at the moment when the catalytic reaction is turned on) and the normal of the planar interface. Thus we neglect the effects of thermal fluctuations leading to a rotation of the axis of symmetry out of this plane. Under this assumption, and in accordance with Fig. \[fig:stab-janus\](b), we choose the coordinate system with the $y$-axis along the interface normal, pointing towards the upper fluid, the $x$-axis as the intersection of the interface with the plane of motion, and the $z$-axis as the normal of the latter (see Fig. \[fig:stab-janus\](b)). For future reference, we also introduce a system of coordinates – with the origin $O'$ at the center $C$ of the particle and co-moving with the particle –, which is denoted by primed quantities. As shown in Fig. \[fig:stab-janus\](b), in the plane spanned by the axis of symmetry and the normal to the interface passing through the center of the particle we choose the $z'$-axis to point through the center of the catalytic cap and the $x'$-axis to lie in the plane of the interface and to be parallel to the $z$-axis. (These choices for the primed and unprimed coordinates are taken as to facilitate more convenient calculations in Sects. \[sec:phoretic\_velocities\] and \[sec:results\] below.)
A viscosity contrast between the two fluids forming the interface leads to the onset of net torques on particles translating along the interface. Therefore the orientation of the symmetry axis of the trapped Janus particle relative to the $y$-axis will change once the chemical reaction is turned on and the particle is set into translation. Depending on the viscosity contrast between the two fluids, a small fluctuation of the orientation of the symmetry axis can be either amplified by the induced torque, leading to a different, yet motile, state, or suppressed. In the former case the steady state orientation of the axis of the Janus particle is ultimately determined by the geometry of the particle including the shape of the catalytic patch and the details of the effective interactions between the catalytic cap and the interface, which is a complex problem. Here we shall assume that the motion is quasi-adiabatic, in the sense that the rotation of the particle is much slower than the time it takes for the distribution of solute molecules $B$ and for the flow of the solution to reach a quasi-steady-state corresponding to the instantaneous orientation of the particle. Thus we focus solely on sustained motile states. The determination of the steady-state orientation of the symmetry axis of the particle (which, as noted above, ultimately involves the details of the effective interaction between the catalytic cap and the interface) is left to future research.
As discussed above, the rotations (spinning) of the particle around the symmetry axis are neither contributing to, nor being induced by, the motility along the interface, whereas the rotations of the symmetry axis around the interface normal have the sole effect of changing the direction of the in-plane translation. Therefore describing the motion of the particle at the interface requires only to account for one translational velocity component $U_x$ and one angular velocity component $\Omega_z$ corresponding to translation along the $x$-axis and rotation around the $z$-axis, respectively. These will be determined by assuming the motion of the fluids to be described by the Stokes equations and the translation and rotation of the particle to be quasi-adiabatic in the sense that the hydrodynamics obeys the (steady state) Stokes equations at the instantaneous state, i.e., the orientation and velocity, of the particle. Finally, we assume that the interface exerts no force on the particle when it is translating along the interface, that the torques exerted by the interface with respect to rotations of the particle around the $z$-axis are vanishingly small, and that the eventual, small deformations of the interface at the contact line region accompanying such a motion also have negligibly small effects.
The assumption of negligible torques deserves further consideration. In fact, if the catalytic cap leaves its equilibrium orientation, a torque will arise due to the effective interaction between the catalytic cap and the interface. Here we focus our attention on the case in which such contributions are negligible[^3]. Furthermore, the rotation around the $x'$-axis involves a moving contact line, which is a well-known conceptual issue in classical hydrodynamics [@Huh1971; @Dusan1974]. Here we do not attempt to provide a mechanism through which the associated contact line singularity (i.e., translation of the contact line while at the same time the fluids fulfill the no-slip boundary condition) is removed and motion occurs (see, e.g., Refs. [@Dusan1976; @Hocking1977; @Brenner1986]). Instead we assume that the region, where a microscopic description is necessary, is very small compared to the typical length scales in the system and that there the expected macroscopic velocity values provide a smooth interpolation.
Translational and angular velocities of Janus particles at liquid-fluid interfaces \[sec:phoretic\_velocities\]
===============================================================================================================
In order to calculate the angular and translational velocity of Janus particles trapped at a liquid-fluid interface, we model the two immiscible fluids separated by a planar interface as having a continuous, but steeply varying, viscosity profile $\eta(y)$ interpolating between $\eta_2$ at $y=-\infty$ and $\eta_1$ at $y = +\infty$ across the plane $y = 0$ (compare Fig. \[fig:stab-janus\]): $$\eta(y) = \eta_{0} + \dfrac{\Delta\eta}{2} \tanh\left(\frac{y}{\xi}\right)\,,
\label{eq:def-visc-1}$$ where $\eta_0 = (\eta_1+\eta_2)/2$ and $\Delta\eta = \eta_1 - \eta_2$ denote the mean viscosity $\eta_0 = \eta(0)$ and the viscosity contrast $\Delta \eta$, respectively, while $\xi$, which is of molecular size, characterizes the width of the interface. In the following we consider the case of a vanishingly thin interface, i.e., we take $\xi \to 0$.
Reciprocal theorem for a particle at liquid-fluid interfaces
------------------------------------------------------------
We exploit the reciprocal theorem [@Stone1996; @BrennerBook; @Stone2014], derived first by Lorentz [@Lorentz_original] (the English translation of the original paper is provided in Ref. [@Lorentz_transl]) for the case of a homogeneous fluid and later extended by Brenner [@BrennerBook] to the case in which the viscosity of the fluid varies spatially.[^4] The reciprocal theorem states that in the absence of volume forces any two incompressible flow fields $\mathbf{u}(\mathbf{r})$ and $\hat{\mathbf{u}}(\mathbf{r})$, which are distinct solutions of the Stokes equations *within the same domain $\cal D$*, i.e., solutions subject to different boundary conditions but *on the very same boundaries $\partial \cal D$*, obey the relation $$\int\limits_{\partial \cal D}^{} \mathbf{u} \cdot \hat{\boldsymbol{\sigma}} \cdot
\mathbf{n} \,dS
= \int\limits_{\partial \cal D}^{} \hat{\mathbf{u}} \cdot \boldsymbol{\sigma} \cdot
\mathbf{n} \,dS\,,
\label{rec-theo-1}$$ where $\boldsymbol{\sigma}$ and $\hat{\boldsymbol{\sigma}}$ denote the stress tensors corresponding to the two flow fields.
For our system, the assumption of immiscibility of the two fluids translates into the kinematic boundary condition that the velocity components of the flow fields above and below the interface along the direction of the normal to the interface must vanish at the interface. This effectively enforces the interface as a physical boundary across which, concerning the hydrodynamics, there is momentum transfer but no mass transfer. Therefore it is necessary that the hydrodynamic flow is obtained by solving the Stokes equations in the domains above (${\cal D}_1$) and below (${\cal D}_2$) the interface and by subsequently working out the problem by connecting the solutions corresponding to the upper and lower fluids via appropriate boundary conditions at the interface (see below). ${\cal D}_1$ is delimited by that part ${\Sigma_p}_1$ of the particle surface ${\Sigma_p}$ exposed to the upper fluid, the surface of the fluid at infinity in the half-plane $y > 0$, and the upper part, $y = 0^+$ of the fluid interface $\Gamma$ (note that this is the plane $y = 0$ less the area occupied by the particle); ${\cal D}_2$ is defined similarly[^5]. We note that the inner normals of the upper ($\mathbf{n}_1$) and lower ($\mathbf{n}_2$) parts of the interface $\Gamma$ are $\mathbf{n}_1~= \mathbf{e}_y = -\mathbf{n}_2$.
We restrict the discussion to the case in which for both flow fields (i.e., the un-hatted and the hatted one, which are to be considered in the reciprocal theorem), there are no (e.g., externally imposed, or due to surface tension gradients) tangential stresses at the interface. For both the un-hatted and the hatted flow fields, the corresponding flow velocities and stress tensors within the upper and lower domains, which are connected via the boundary conditions they have to obey at the interface, are denoted by the indices “1” and “2”, respectively. By i) applying the reciprocal theorem (Eq. (\[rec-theo-1\])) in each of the domains ${\cal D}_1$ and ${\cal D}_2$ (which can be done because $\Gamma$ is a physical boundary at which the boundary conditions are formally prescribed by imposing the tangential velocity and stress tensor to take the values given by the (yet unknown) velocity and stress tensor on the other side of the interface, respectively); ii) adding the left and right hand sides of the two results of applying the reciprocal theorem in ${\cal D}_1$ and ${\cal D}_2$; iii) noting that for flow fields, which decay sufficiently fast with the distance from the particle (which typically is the case), the contribution from the integrals over the surfaces at infinity are vanishingly small; iv) using the relation (see above) $\mathbf{n}_1 = -\mathbf{n}_2 = \mathbf{e}_y$ between the interface normals; v) noting that the boundary conditions for the flow velocity at the interface $\Gamma$ impose zero normal components, i.e., $(\mathbf{u}_1 \cdot \mathbf{e}_y)|_{y = 0} = 0, (\mathbf{u}_2
\cdot \mathbf{e}_y|)_{y = 0} = 0$ and continuous tangential components [@BrennerBook] so that $\mathbf{u}_1|_{y = 0_+} = \mathbf{u}_2|_{y = 0_-} =:
u_{||} \mathbf{e}_{||}$ (with similar relations for the hatted velocity field), where $\mathbf{e}_{||} \cdot \mathbf{e}_y = 0$, we arrive at $$\label{rec-theo-interf}
\int\limits_{\Gamma}^{} (u_{||} \mathbf{e}_{||}) \cdot ({\hat{\boldsymbol{\sigma}}}_1 -
{\hat{\boldsymbol{\sigma}}}_2)|_{y = 0} \cdot \mathbf{e}_y \,dS +
\int\limits_{\Sigma_p}^{} \mathbf{u} \cdot \hat{\boldsymbol{\sigma}} \cdot
\mathbf{n} \,dS
= \int\limits_{\Gamma}^{} ({\hat{u}}_{||} \mathbf{e}_{||}) \cdot
({\boldsymbol{\sigma}}_1 -
{\boldsymbol{\sigma}}_2)|_{y = 0} \cdot \mathbf{e}_y \,dS +
\int\limits_{\Sigma_p}^{} \hat{\mathbf{u}} \cdot \boldsymbol{\sigma} \cdot
\mathbf{n} \,dS
\,.$$ In the absence of tangential stresses at the interface the difference of normal stresses at the interface $({\boldsymbol{\sigma}}_1 - {\boldsymbol{\sigma}}_2)|_{y
= 0} \cdot \mathbf{e}_y$ is the force corresponding to the Laplace pressure [@BrennerBook], and thus is a vector oriented along the normal $\mathbf{e}_y$ of the interface. Therefore the first integral on the left hand side of Eq. (\[rec-theo-interf\]) vanishes due to $\mathbf{e}_{||}~\perp~\mathbf{e}_y$. By a similar argument, the first integral on the right hand side of Eq. (\[rec-theo-interf\]) vanishes, too. Thus for the present system the reciprocal theorem takes the simple form given in Eq. (\[rec-theo-1\]) but with $\cal D$ replaced by $\Sigma_p$.
In order to determine the translational and the angular velocity of the Janus particle, we shall select a proper set of “dual problems” (the “hatted” quantities), typically associated with known solutions for spatially uniform translations or rotations (under the action of external forces or torques) of solid spheres with prescribed boundary conditions at their surfaces. To this end we consider a solid sphere of radius $R$ with *no-slip* boundary conditions translating with velocity $\hat{\mathbf{U}}$ and rotating with angular velocity $\hat{\mathbf{\Omega}}$ under the action of the external force $\hat{\mathbf{F}}$ and the external torque $\hat{\mathbf{L}}$. At the surface of the particle, the flow $\hat{\mathbf{u}}(\mathbf{r}_p)$, where $\mathbf{r}_p$ denotes a point at the particle surface $\Sigma_p$, is given by $$\hat{\mathbf{u}}(\mathbf{r}_p) = \hat{\mathbf{U}} + \hat{\mathbf{\Omega}} \times
(\mathbf{r}_p - \mathbf{r}_{\mathrm{c}}) \,,
\label{eq:surf_flow_reciproc}$$ where $\mathbf{r}_\mathrm{C}$ is the position of the center $C$ of the sphere. (Both $\mathbf{r}_p$ and $\mathbf{r}_\mathrm{C}$ are measured from a common, arbitrary origin, the location of which drops out from $\mathbf{r}_p - \mathbf{r}_\mathrm{C}$.) Similarly, we consider a Janus particle, which translates with velocity $\mathbf{U} =
U_x \mathbf{e}_x$ and rotates with an angular velocity $\mathbf{\Omega}=\Omega_z\mathbf{e}_z$ around the axis, which is parallel to $Oz$ and passes through the moving center $C$ of the particle at its instantaneous position. If the particle exhibits boundary conditions given by a *phoretic slip* velocity $\mathbf{v}(\mathbf{r}_p)$, the flow $\mathbf{u}(\mathbf{r}_p)$ at its surface is given by $$\mathbf{u}(\mathbf{r}_p) = \mathbf{U} + \mathbf{\Omega}\times(\mathbf{r}_p -
\mathbf{r}_\mathrm{C}) + \mathbf{v}(\mathbf{r}_p) \,.
\label{eq:surf_flow_Janus}$$ By using Eq. (\[eq:surf\_flow\_reciproc\]) and noting that $\hat{\mathbf{U}}$ and $\hat{\mathbf{\Omega}}$ are spatially constant vectors, the right hand side (rhs) of Eq. (\[rec-theo-1\]) can be re-written as $$\int\limits_{\Sigma_p}^{} \hat{\mathbf{u}} \cdot \boldsymbol{\sigma} \cdot \mathbf{n} \,dS
= \hat{\mathbf{U}} \cdot \mathbf{F} + \hat{\mathbf{\Omega}} \cdot \mathbf{L}\,,
\label{rhs_rec_theo}$$ where $\mathbf{F} = \int_{\Sigma_p} \boldsymbol{\sigma} \cdot \mathbf{n}\, dS$ and $\mathbf{L} = \int_{\Sigma_p} (\mathbf{r}_p-\mathbf{r}_\mathrm{C}) \times
\boldsymbol{\sigma} \cdot \mathbf{n}\, dS$ denote the force and the torque, respectively, experienced by the Janus particle. (In Eq. (\[rhs\_rec\_theo\]), $\mathbf{n}$ denotes the normal of the surface $\Sigma_p$ of the particle, oriented into the fluid.) We note that while the active motion of Janus particles in the bulk is force- and torque-free, this is, in general, not the case if the motion occurs at the interface because the interface can exert forces and torques on the particle. However, because we have assumed that the interface does not exert a force in the case of translations of the Janus particle along the interface or a torque in the case of rotations around the $z$-axis (in the sense of spinning around an axis parallel to the $z$-axis, as discussed above), the components $F_x$ and $L_z$ vanish. Therefore, if the reciprocal problem involves only translations along the $x$-axis and/or such rotations around the $z$-axis, the rhs of Eq. (\[rhs\_rec\_theo\]), and, consequently, of Eq. (\[rec-theo-1\]) is zero. Restricting now the dual problem as discussed above to such a choice, and using Eq. (\[eq:surf\_flow\_Janus\]) for the left hand side of Eq. (\[rec-theo-1\]), we arrive at $$U_x \hat{F}_x + \Omega_z \hat{L}_z = -\int\limits_{\Sigma_p}^{} \mathbf{v}(\mathbf{r}_p)
\cdot \hat{\boldsymbol{\sigma}} \cdot \mathbf{n} \,dS \\.
\label{eq:rec-theo-4}$$
Calculation of the translational and angular velocities
-------------------------------------------------------
We proceed by selecting two so-called dual problems, each involving only one of the two types of motion (translation or rotation only) for both of which Eq. (\[eq:rec-theo-4\]) holds. These will provide two relations allowing one to determine $U_x$ and $\Omega_z$. The first one, denoted by the index “1”, is that of a sphere of radius $R$, the center of which lies in the plane of the flat, sharp ($\xi\rightarrow 0$) liquid-fluid interface (see Eq. (\[eq:def-visc-1\])), translating *without rotation* with velocity $\hat{\mathbf{U}} = \hat U_x
\mathbf{e}_x$ along the interface. This problem has been solved analytically [@Pozrikidis2007], and the result of interest here is $$\left.(\mathbf{n}\cdot\hat{\boldsymbol{\sigma}}_1)\right|_{\Sigma_p}=
-\frac{3}{2R}\eta(\mathbf{r}_p)\hat{\mathbf{U}}\,.
\label{trasl-norot-noslip}$$ This leads to (see Appendix A for the details of the calculation)
\[eq:recipr-prob-1\] $$\label{F1x}
\hat{F}_{1x} = -6 \pi R \,\eta_{0} \,\hat{U}_x := \alpha_1 \hat{U}_x$$ and $$\label{L1z}
\hat{L}_{1z} = +\frac{3 \pi}{2} R^{2} \Delta\eta \, \hat{U}_x := \beta_1 \hat{U}_x\,.$$
After inserting Eqs. (\[trasl-norot-noslip\]) and (\[eq:recipr-prob-1\]) into Eq. (\[eq:rec-theo-4\]) and canceling the common factor $\hat {U}_x$, we obtain $$\alpha_1 U_x + \beta_1 \Omega_z = C_1:= \frac{3}{2R}
\int\limits_{\Sigma_p}^{} \eta(\mathbf{r}_p) \, v_{x}(\mathbf{r}_p) \,dS\,,
\label{eq:rec-ther-1}$$ where $v_x(\mathbf{r}_p)$ is the $x$-component of the phoretic slip velocity at the point $\mathbf{r}_p$ on the surface of the particle.
The second problem which we consider, denoted by the index “2”, is that of the driven rotation, *without translation*, with angular velocity $\hat{\mathbf{\Omega}} = \hat{\Omega}_z \, \mathbf{e}_z$ of a spherical particle of radius $R$, the center of which lies in the plane of the flat, sharp ($\xi\rightarrow 0$) liquid-fluid interface. As noted before, this is a more involved problem due to the concomitant issue of contact line motion. For the case in which one of the two fluids has a vanishingly small viscosity, an exact solution was constructed in Ref. [@Brenner1986] under the assumption that a slip boundary condition, with a spatially uniform slip length along the surface, applies across that surface region which is immersed in the fluid of non-vanishing viscosity, called liquid. The result of this calculation shows that for typical slip lengths $l_0$, which are much smaller than the size of the particle,[^6] at distances $|y|/l_0\gg 1$ from the interface the hydrodynamic flow within the liquid is *de facto* identical with the one which would have occurred if the rotating sphere would have been completely immersed in the liquid and a no-slip boundary condition would have been applied. Thus in this context the only role played by the slip is to remove the contact line singularity, as discussed in Sect. \[model\]. This can be interpreted in the sense that the same solution would emerge if one assumes that the fluid slips only in a narrow region localized close to the three-phase contact line, while the no-slip condition holds for the rest of the sphere. This view is confirmed by an alternative solution presented in Ref. [@Sterr2009] for the same problem of the rotation of a sphere at the interface between a liquid and a fluid of vanishing viscosity.
In the following we shall adopt the latter interpretation and make the *ansatz* that for our above problem “2” (in which the viscosities of both fluids are, in general, certain non-zero quantities) with a sharp interface ($\xi\rightarrow 0$) the expression for the stress tensor at the surface of the particle is given by the one in Ref. [@Brenner1986], i.e., $$\left.(\mathbf{n}\cdot\hat{\boldsymbol{\sigma}}_2)\right|_{\Sigma_p} = -3 \,
\eta(\mathbf{r}_p) \,\hat{\Omega}_z \,
\left.(\mathbf{e}_z\times \mathbf{n})\right|_{\Sigma_p}\,,
\label{notrasl-rot-noslip}$$ except for a small region localized close to the three-phase contact line. As noted, the expression above is expected to provide a reliable approximation if the viscosity contrast between the two fluids is large [@Brenner1986]. In the limiting case of the two fluids becoming identical, by construction Eq. (\[notrasl-rot-noslip\]) reduces to the exact result corresponding to a sphere rotating without slip in a spatially homogeneous fluid.
With the assumption that the small region near the three-phase contact line contributes negligibly to the integrals over the surface of the particle, Eq. (\[notrasl-rot-noslip\]) implies that the corresponding components of the forces and torques required for the reciprocal theorem (Eq. (\[eq:rec-theo-4\])) are given by (see Appendix A)
\[eq:recipr-prob-2\] $$\label{F2x}
\hat{F}_{2x} \simeq +3 \pi R^{2} \Delta\eta \,\hat{\Omega}_z
:= \alpha_2 R \,\hat{\Omega}_z$$ and $$\label{L2z}
\hat{L}_{2z} \simeq -8 \pi \eta_{0} R^{3} \hat{\Omega}_z
:= \beta_2 R \,\hat{\Omega}_z\,.$$
After inserting Eqs. (\[notrasl-rot-noslip\]) and (\[eq:recipr-prob-2\]) into Eq. (\[eq:rec-theo-4\]) and canceling the common factor $\hat {\Omega}_z$, we obtain $$\alpha_2 U_x + \beta_2 \Omega_z = C_2 := \dfrac{3}{R} \int\limits_{\Sigma_p}^{}
\eta(\mathbf{r}_p) \left(\mathbf{n} \times \mathbf{v}(\mathbf{r}_p) \right)_{z}\,dS\,.
\label{eq:rec-ther-2}$$
Once a particular model is given for the mechanism through which the chemical activity determines the phoretic slip $\mathbf{v}(\mathbf{r}_p)$, the quantities $C_1$ and $C_2$ can be computed and $U_x$ and $\Omega_z$ follow from Eqs. (\[eq:rec-ther-1\]) and (\[eq:rec-ther-2\]). This concludes the calculation of the translational $(U_x)$ and angular $(\Omega_z)$ velocities of the Janus particle trapped at the interface. We note that in the limit $\Delta\eta \rightarrow 0$ Eqs. (\[eq:rec-ther-1\]) and (\[eq:rec-ther-2\]) reduce to the corresponding components for a Janus particle moving in a homogeneous fluid [@Stone1996], i.e., $U_x = - \frac{1}{4 \pi R^{2}} \int\limits_{\Sigma_p}^{} v_{x} dS$ and $\Omega_z = -\frac{3}{8\pi R^{3}} \int\limits_{\Sigma_p}^{} \left(\mathbf{n} \times
\mathbf{v}\right)_{z}dS$, respectively.
This latter result deserves further consideration. In the case of translation along the interface, in the limit $\Delta \eta\rightarrow 0$ the recovery of the result corresponding to a particle moving in a homogeneous fluid is to be expected in the case of a particle *having its center located at the interface*. This is so, because the flow around the Janus particle translating at the interface converges, as the viscosity contrast approaches zero, towards the solution corresponding to the motion in a homogeneous fluid [^7]. On the other hand, this expectation does not hold in the case of rotation: the immiscibility of the fluids requires that the interface modifies the flow by “forcing” the fluid to flow along the interface. This different structure of the flow survives even if the viscosity contrast is vanishing. Therefore, recovering nonetheless the result for rotation in a homogeneous fluid simply means that the *ansatz* for the stress tensor *at the surface of the particle* (Eq. (\[notrasl-rot-noslip\])) renders the correct limiting behavior for vanishing viscosity contrast, irrespective of the corrections provided by the presence of the interface.
Results and discussion \[sec:results\]
======================================
Solving Eqs. (\[eq:rec-ther-1\]) and (\[eq:rec-ther-2\]) for $U_x$ and $\Omega_z$ leads to
\[eq:solu-tot-1\] $$\label{Ux}
U_x = \dfrac{C_1 \beta_2 - \beta_1 C_2}{\alpha_1 \beta_2 - \alpha_2 \beta_1}$$ and $$\label{Omz}
\Omega_z = \dfrac{\alpha_1 C_2 - C_1 \alpha_2}{\alpha_1 \beta_2 - \alpha_2 \beta_1}\,.$$
For the discussion of these results we find it more convenient to employ the following alternative description of the orientation of the particle with respect to the interface. We define the director $\mathbf{p}$ of the Janus particle as being the unit vector corresponding to the axis of symmetry of the Janus particle oriented towards the catalytic cap. The acute angle between the director and the interface, in the plane $(xOy)$, is denoted by $\chi$ (Fig. \[fig:stability-condition\](a)). It is defined as a signed quantity, with the sign convention that $\chi$ is positive if $\mathbf{p}$ points towards the half-space $y > 0$ and negative otherwise; thus $-\pi/2 \leq \chi
\leq \pi/2$. (The angle $\chi$ is thus connected with $\delta$ (see Fig. \[fig:stab-janus\](b)) via $\chi = \mathrm{min}(\delta,\pi-\delta)$, if $0 < \delta
< \pi$, and $\chi = - \mathrm{min}(\delta-\pi,2 \pi-\delta)$, if $\pi < \delta < 2
\pi$.) The states with $\chi = \pm \,\pi/2$ correspond to the director being parallel and antiparallel, respectively, to the interface normal $\mathbf{e}_y$, for which $U_x$ vanishes. Therefore, in order for a state of motion along the surface, i.e., $U_x
\neq 0$, to be sustainable, any change in $\chi$ occurring as a result of motion along the interface should be such that $|\chi|$ decreases (i.e., the director rotates towards the interface).
Configurations of sustained motility
------------------------------------
Referring now to Fig. \[fig:stability-condition\](b) and considering as an example the situation shown in the upper part with the catalytic cap tilted slightly to the left of the normal $\mathbf{e}_y$, for which $\chi > 0$, one infers that, upon turning on the chemical reaction, for repulsive (attractive) interactions between the solute (i.e., the reaction products) and the particle the latter will tend to move towards the right (left), so that $U_x > 0$ ($U_x < 0$). If $\Delta \eta > 0$, i.e., the upper fluid is more viscous than the lower one, translation with $U_x > 0$ gives rise to a torque on the particle which induces a counterclockwise rotation, i.e., $\Omega_z > 0$. (The upper part of the particle experiences a stronger, retarding friction than the lower part of the particle.) This corresponds to a decrease of $\chi$ towards zero and thus promotes motility. This situation is shown in the right upper quadrant of Fig. \[fig:stability-condition\](b). On the other hand, translation with $U_x < 0$ (and still for $\chi >0$ as well as the cap tilted to the left of the interface normal; not shown in the right upper quadrant of Fig. \[fig:stability-condition\](b)) gives rise to a clockwise rotation (i.e., $\Omega_z <
0$, for the same reason as above) and therefore to an increase of $\chi$ towards $\pi/2$, i.e., rotation opposes motility. If $\Delta \eta < 0$ (and still $\chi > 0$ with the cap tilted to the left of the interface normal), the sign of those torques (which are described by the same color), and thus of the corresponding angular velocities, is reversed. In this case, the translation towards the left ($U_x < 0$) is accompanied by a rotation which decreases $\chi$ (i.e., $\Omega_z > 0$) and thus promotes motility (see the left upper quadrant of Fig. \[fig:stability-condition\](b)), while translation towards the right (i.e., $U_x > 0$ and still $\chi > 0$ with the cap tilted to the left of the interface normal; not shown in the left upper quadrant) is opposed by the rotation of the director. Following the above reasoning for the various possible configurations (i.e., catalytic cap above or below the interface, attractive or repulsive solute-particle interactions, viscosity contrast positive or negative), in the plane $(\Delta \eta, \chi)$ one can identify the cases in which sustained motion would occur, depending on the repulsive or attractive character of the interactions between the solute and the particle. These configurations are summarized in Fig. \[fig:stability-condition\](b), where the arrows indicate the corresponding directions of the translation and rotation. The colors blue and orange of the arrows refer to repulsive and attractive interactions, respectively.
From the discussion above (see also the schematic diagram in Fig. \[fig:stability-condition\] (b)) one infers that the states with sustained motion must satisfy $U_x/(R \Omega_z) > 0$ for $\Delta \eta > 0$ or $U_x/(R \Omega_z) < 0$ for $\Delta \eta < 0$. Therefore, for a given system these signs of $U_x/(R \Omega_z)$ as a function of the viscosity contrast $\Delta \eta$ provide necessary conditions for the occurrence of such motile states. However, in order to explicitly calculate the sign of the ratio $U_x/(\Omega_z R)$ one needs to provide an explicit form for $\mathbf{v_s}(\mathbf{r}_p)$ which determines $U_x$ and $\Omega_z$ (Eqs. (\[eq:rec-ther-1\]), (\[eq:rec-ther-2\]), and (\[eq:solu-tot-1\])). In order to determine $\mathbf{v_s}(\mathbf{r}_p)$ it is in principle necessary (i) to specify the geometrical properties of the catalytic cap responsible for the reaction within the fluids; (ii) to specify the reaction; (iii) to provide the diffusion constants of the reactant and product molecules in the two fluids (for example, they can either diffuse in both fluids but with different diffusion constants, or some of the reactants or products may effectively be confined to one of the two fluids), as well as any effective interaction between these molecules and the interface (e.g., whether or not they act as surfactants); (iv) to provide the interaction potentials of the various molecular species in the two solutions (i.e., the two fluids plus the reactants and the products) with the Janus particle as a whole as well as with its surface (for both the catalyst covered part and the inert part).
Motility of a model chemically active Janus colloid at fluid interfaces
-----------------------------------------------------------------------
Here we focus on the simple model of a chemically active Janus particle as introduced in Sec. \[model\], for which there is only a single reaction product (“solute”) diffusing in both fluids and for which the reactant molecules are present in abundance and diffusing very fast in both fluids, such that in both fluids the number density of reactant molecules is *de facto* time-independent and spatially uniform. The effective interaction of the solute with the colloidal particle is assumed to be of a range which is much smaller than the radius $R$ of the colloid and to be similar for the catalyst-covered part and the inert part of the colloid. Furthermore, we assume a sharp interface (i.e., $\xi \to 0$ so that $\eta(\mathbf{r}_p) = \eta(y)$ with $\eta(y) =
\eta_1$ for $y > 0$, while $\eta(y) = \eta_2$ for $y < 0$). The latter assumptions imply that, by adopting the classical theory of phoresis [@Anderson1989; @Golestanian2005; @Popescu2009] which has been developed for homogeneous (i.e., constant viscosity) fluids, one can express the phoretic slip as being proportional to the solute concentration gradient along the surface at all points of the surface of the particle except for a small region near the interface. Within the corresponding proportionality factor $\mathcal{L}/(\beta\,\eta)$ (the so-called “phoretic mobility”; see, c.f., Eq.(\[def:slip\_vel\])), where $\beta = 1/(k_B T)$, $k_B$ is the Boltzmann constant, and $T$ denotes the absolute temperature, it is possible to identify the contribution $\mathcal{U}(h)$ of the solute-particle interaction (relative to the solvent-particle interaction). $\mathcal{U}(h)$ is encoded in $\mathcal{L}$ (which has the units of an area) according to [@Anderson1989] $$\mathcal{L}=\int\limits_{0}^{\infty} dh \, h
\left(e^{-\beta \mathcal{U}(h)} - 1 \right)\,,
\label{eq:phor-mob}$$ where $h$ is the distance between the point-like solute and the particle surface. The potential $\mathcal{U}(h)$ is assumed to be such that $\mathcal{U}(h
\to 0) = + \infty$, i.e., right at the particle surface the solvent is strongly preferred. The potential can be either repulsive at all distances, or it can become attractive beyond a certain distance $h_0$ (and thus has to have an attractive minimum because at large distances it decays to zero); this latter case corresponds to *adsorption* of the solute. Note that $\mathcal{L} < 0$ for purely repulsive interactions $\mathcal{U}(h)$, while if $\mathcal{U}(h)$ has an attractive part and $h_0$ is sufficiently small, one has $\mathcal{L} > 0$. In the following, the notion of “attractive interactions” will refer strictly to the latter case, i.e., potentials $\mathcal{U}(h)$ which have attractive parts and satisfy $\mathcal{L} > 0$. At this stage we do not yet particularize the cap to more than the assumed spherical cap shape and to being completely immersed into one of the two fluids.
Under the above assumptions, the phoretic slip $\mathbf{v}(\mathbf{r}_p)$ follows from the solute distribution around the surface of the Janus particle. We further assume that the diffusivity $D(\mathbf{r})$ of the solute molecules is sufficiently high such that the number density distribution $\rho(\mathbf{r},t)$ of the solute is not affected by the convection of the fluids (i.e., we assume that the Péclet number $Pe$ is small) and that a steady state distribution $\rho(\mathbf{r})$ of solute is established at time scales which are much shorter than the characteristic translation time $R/U_x$ of the colloid. With this, $\rho(\mathbf{r})$ obeys the diffusion equation $$\nabla\cdot[D\nabla\rho]=0
\label{eq:diff-eq}$$ subject to the boundary conditions $$\lim_{r\rightarrow\infty}\rho=0,\,\,\,\,\, -D\nabla \rho\cdot\mathbf{n}|_{r=R} = Q \,
H(\theta_0 - \theta),
\label{eq:diff-eq-bond-cond}$$ where $\mathbf{n}$ is the outer normal of the particle, $Q$ denotes the number of solute molecules generated per area and per time at the location of the catalytic cap, and $H(x)$ is the Heaviside step function ($H(x > 0) = 1$, $H(x < 0) = 0$). (In accordance with the assumptions of the model (see Sec. \[model\]), in the above diffusion equation there are no terms to account for eventual interactions of the solute with the interface or with external fields.)
Since actually only the distribution of solute at the particle surface is required in order to calculate the phoretic slip, instead of seeking for the full solution $\rho(\mathbf{r})$ of Eq. (\[eq:diff-eq\]), which is a difficult problem, we only focus on the solute distribution at the particle surface. In the co-moving (primed) coordinate system (see Fig. \[fig:stab-janus\](b)), in which the phoretic slip velocity is most conveniently calculated, we introduce, in the usual manner, the common spherical coordinates $(r',\theta',\phi')$ defined via $$x' = r' \sin\left(\theta'\right)\cos\left(\phi'\right)\,,~
y' = r' \sin\left(\theta'\right)\sin\left(\phi'\right)\,,~
z' = r' \cos\left(\theta'\right)\,.
\label{spherical-coord}$$ Accordingly, the solute distribution at the particle surface $\rho(\mathbf{r'}_p) =
\rho(R,\theta',\phi')$ can be expressed as a series expansion in terms of the spherical harmonics $Y_{\ell\,m} (\theta',\phi')$ [@Jackson_book]: $$\rho(R,\theta',\phi')= \sum\limits_{\ell = 0}^\infty
\sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\, Y_{\ell\,m}(\theta',\phi') \,,
\label{def:rho_sph_harm}$$ where the coefficients $A_{\ell,m}$ are functions of the radius $R$ and of the other parameters (temperature, diffusion constants, viscosities, rate of solute production, etc.) characterizing the system. In the co-moving system, the phoretic slip can be expressed in terms of the gradients of $\rho(\mathbf{r}'_p)$ along the surface of the particle [@Anderson1989; @Golestanian2005; @Popescu2009; @Poon] (*except* at the three-phase contact line): $$\mathbf{v} (\mathbf{r}'_p) = - \dfrac{\mathcal{L}}{\beta\,\eta(\mathbf{r}_p)}
\nabla'_{||} \rho(\mathbf{r}'_p)\,:= v_{\theta'}\mathbf{e}_{\theta'}
+v_{\phi'}\mathbf{e}_{\phi'}
\label{def:slip_vel}$$ where $\nabla'_{||}=\frac{1}{R}\mathbf{e}_{\theta'}\partial_{\theta'}+
\frac{1}{R\sin\theta'}\mathbf{e}_{\phi'}\partial_{\phi'}$ denotes the projection of the gradient operator along the surface of the particle.
In order to determine the $x$-component $v_x$ of the slip-velocity in the spatially fixed coordinate system, we use Eqs. (\[def:rho\_sph\_harm\]) and (\[def:slip\_vel\]) and employ the relation between the unit vectors of the spatially fixed coordinate system and the co-moving one (see Fig. \[fig:stab-janus\](b)). Knowledge of $v_x$ allows one to determine the quantities $C_1$ and $C_2$ introduced in Eqs. (\[eq:rec-ther-1\]) and (\[eq:rec-ther-2\]) (see Appendix \[deriv-C1-C2\] for details): $$C_1 = -2 \sqrt{3\pi} \, \dfrac{\mathcal{L} \cos(\delta)}{\beta} A_{1,0} \,,
\label{eq:C1B}$$ and $$C_2 = 0 \,. \label{eq:C2B}$$ It is interesting to note that, apart from materials properties ($\cal L$) and temperature, $C_1$ depends solely on the projection ($\cos(\delta)$) of the particle director onto the plane of the interface and on the real amplitude $A_{1,0}$ (see Eq. (\[def:rho\_sph\_harm\])) of $Y_{1\,0}(\theta',\phi') =
\sqrt{3/(4 \pi)} \, \cos(\theta')$. This is that contribution to the angular dependence of $\rho$ along the particle surface which varies slowest between the poles at $\theta' = 0$ (center of the cap) and $\theta' = \pi$. This can be interpreted as an indication that for the model considered here the difference in the solute density between that at the catalytic pole and at the inert antipole is the dominant characteristics while the details of the variation of the density along the surface between these two values are basically irrelevant for the motion of the particle.
In the case that the two fluids have the same viscosity, i.e., $\Delta \eta = 0$, and the diffusion constant for the product molecules is the same in the two fluids (e.g., being related to the viscosity via the Stokes-Einstein relation), for the model considered here, according to which the reactant and product molecules can diffuse freely in both fluids and unhindered by the interface, the diffusion equation (Eqs. (\[eq:diff-eq\]) and (\[eq:diff-eq-bond-cond\])) becomes identical to the one in a homogeneous bulk fluid which can be solved analytically [@Golestanian2005; @Popescu2009]. (Thus, in this limit there is no signature of the interface left in the diffusion problem.) The corresponding expansion into spherical harmonics of the solute density at the surface of the particle (for a *b*ulk solvent without an interface) leads to the expression $$\label{eq:hom_A10}
A_{1,0}^{(b)} = \varkappa \dfrac{Q R}{D_0}\,,$$ where $\varkappa$ is a dimensionless factor determined by the geometry of the Janus sphere (i.e., the extent of the catalyst covered area) and $D_0$ is the diffusion constant of the product molecules in the fluids of viscosities $\eta_1 = \eta_2 =
\eta_0$. This leads to the ansatz $$\label{eq:def_varsig}
A_{1,0} = \varsigma(\Delta\eta/\eta_0) A_{1,0}^{(b)}\,,$$ for a system with an interface, where the dimensionless function $\varsigma$ is expected to depend on the viscosities solely via the dimensionless ratio $\epsilon = \Delta\eta/\eta_0$. Since in the limit $\epsilon \to 0$, which renders the homogeneous bulk fluid case, one has $A_{1,0} \to A_{1,0}^{(b)}$, the function $\varsigma$ must obey the constraint $\varsigma(\epsilon \to 0) = 1$.
By combining Eqs. (\[eq:recipr-prob-1\]), (\[eq:recipr-prob-2\]), (\[eq:solu-tot-1\]), (\[eq:C1B\]), (\[eq:C2B\]), (\[eq:hom\_A10\]), and (\[eq:def\_varsig\]), we obtain
\[eq:solu-tot-2-explicit\] $$\label{Ux_expl}
U_{x} = \dfrac{1}{\sqrt{3 \pi}}
\dfrac{\varsigma\left(\dfrac{\Delta\eta}{\eta_0}\right)}
{1-\dfrac{3}{32}\left(\dfrac{\Delta\eta}{\eta_0}\right)^2} \cos(\delta) V_0 + {\cal
O} \left(\left(\dfrac{\Delta \eta}{\eta_0}\right)^3\right)
\,,$$ $$\label{Omz_expl}
\Omega_{z} = \dfrac{3}{8} \dfrac{1}{R} \dfrac{\Delta\eta}{\eta_0} \, U_x + {\cal
O} \left(\left(\dfrac{\Delta \eta}{\eta_0}\right)^3\right)
\,,$$
where $$V_0 := \dfrac{\mathcal{L} \,A_{1,0}^{(b)}}{\beta \eta_0 R}
= \varkappa \dfrac{\mathcal{L} Q}{\beta \eta_0 D_0}\,
\label{eq:def_V0}$$ renders the characteristic translational and angular velocity scales $|V_0|$ and $\Omega_0 = |V_0|/R$, respectively. $V_0$ is independent of the particle radius $R$ as well as of the value of the viscosity $\eta_0$ because, under the assumption of the Stokes-Einstein relation, $\beta \eta_0 D_0$ depends only on the radius $R_m$ of the product *m*olecules. This implies that the translational velocity is independent of the radius $R$ of the particle while the angular velocity is proportional to $1/R$ (up to eventual additional dependences on $R$ arising from $\varsigma$).
Equation (\[eq:solu-tot-2-explicit\]) shows that both the translational and the angular velocity are proportional to $\cos\left(\delta\right)$. Therefore both vanish for $\delta=\pi/2$ which matches with the fact in this case the particle is in fully upright orientation and thus cannot propel laterally. Moreover, both $U_x$ and $\Omega_z$ change sign when the director $\mathbf{p}$ (Fig. \[fig:stability-condition\](a)) changes from pointing mainly to the right to pointing mainly to the left (Fig. \[fig:stability-condition\](b)). On the other hand, the sign of the ratio $U_x
/\Omega_z$, which, according to the discussion of Fig. \[fig:stability-condition\](b) in the main text, decides on the sustainability of the motile state, is independent of $\delta$ but is determined by the sign of $\Delta\eta$. This is in agreement with the symmetry exhibited by the diagram shown in Fig. \[fig:stability-condition\](b). In the limit of a vanishing viscosity contrast $\Delta\eta/\eta_0$, $U_{x}$ approaches the constant value $V_0
\cos(\delta)/\sqrt{3 \pi}$. (This corresponds to the motion in a homogeneous bulk fluid under the constraint of moving along a plane at an angle $\delta$ with respect to the orientation of the director position.) Thus, for small values of $\Delta \eta$, $U_x(\Delta \eta)$ does not vary much, while, as expected, the angular velocity vanishes linearly $\propto \Delta\eta/\eta_0$. In the limiting case $\Delta\eta \to 0$ translation and rotation are decoupled and the particle translates without any rotation because there is no viscosity contrast. In such a case the net effect of the interface is to keep the particle center bound to the plane of the interface.
While the diagram in Fig. \[fig:stability-condition\] is entirely determined by the ratio $U_x/\Omega_z$, which is independent of $\varsigma$, the magnitudes of both $U_x$ and $\Omega_z$ do depend on it via the amplitude $A_{1,0}$ (Eqs. (\[eq:solu-tot-2-explicit\]) and (\[eq:def\_V0\])). Since determining the exact form of $\varsigma(\epsilon)$ is clearly analytically intractable, one can try to analyze its behavior for $\epsilon \ll 1$. One option is to employ a perturbation series in terms of the small parameter $\epsilon$ in order to calculate the distribution of solute for $\epsilon \ll 1$, starting from the known solution $\rho_0(\mathbf{r})$ for $\epsilon
= 0$ (i.e., for a homogeneous bulk fluid without interface), from which one can estimate $A_{1,0}$ and implicitly $\varsigma(\epsilon)$. (Note that $\rho_0(\mathbf{r})$ varies spatially due to the solute sources located at the surface of the particle and the solute sink at infinity.) We denote by $\tilde{\rho}(\mathbf{r}) := \rho(\mathbf{r}) -
\rho_0(\mathbf{r})$ and $\tilde{D}(\mathbf{r}) := D(\mathbf{r})-D_0$ the deviations (first order in $\epsilon$) of the number density distribution and of the diffusion coefficient from their corresponding values $\rho_0(\mathbf{r})$ and $D_0$ (spatially constant) in a homogeneous medium. (Note that by assuming the Stokes-Einstein relation between the diffusion coefficient and the viscosity $\tilde{D}(\mathbf{r})$ is a known function determined by $D_0$, $\epsilon$, and the known variation of the viscosity across the interface (Eq. (\[eq:def-visc-1\])).) From Eqs. (\[eq:diff-eq\]) and (\[eq:diff-eq-bond-cond\]) one obtains that $\tilde{\rho}$ is the solution of the differential equation $$\nabla \cdot \left(\tilde{D} \nabla\rho_0 +
D_0 \nabla \tilde{\rho}\right)=0\,,
\label{eq:diff-1-order-a}$$ subject to the boundary conditions $$\lim_{r\rightarrow\infty}\tilde\rho=0,\,\,\,\,\,\, \left(\tilde D\nabla
\rho_0+D_0\nabla \tilde{\rho}\right)\cdot\mathbf{n}|_{r=R}=0\,.
\label{eq:diff-1-order-c}$$ We have been unable to find an analytical solution of Eqs. (\[eq:diff-1-order-a\]) and (\[eq:diff-1-order-c\]) for a general orientation of the (small) cap. Therefore we cannot make any further rigorous statements. Instead, we only formulate expectations concerning the behavior of $\varsigma(\epsilon)$. For example, considering the case in which the catalytic cap is in the upper fluid region ($y > 0$), $\epsilon >
0$ (i.e., enhanced \[reduced\] viscosity in the upper \[lower\] fluid) leads to a reduction \[increase\] of the diffusion coefficient in the upper \[lower\] fluid. Compared with the homogeneous fluid ($\epsilon=0$), intuitively this should lead to a relative accumulation of product molecules near the catalytic pole (located in the upper fluid) and to a relative depletion near the inert antipole (which is located in the lower fluid). For $\epsilon<0$ the behavior is reversed. Since (as discussed after Eq. (\[eq:C2B\])) the coefficient $A_{1,0}$ can be viewed as a measure of the difference between the densities at the catalytic pole and at the antipole, the reasoning above suggests that, upon deviating from the homogeneous state (with $A_{1,0}^{(b)}$), $A_{1,0}$ varies oppositely if the viscosity of fluid “1” relative to that of fluid “2” increases or decreases, respectively. Therefore, to first order in $\epsilon$ the function $\zeta(\epsilon)$ is expected to vary as $\zeta(\epsilon\rightarrow 0)=1+\text{const}\cdot\epsilon+\mathcal{O}(\epsilon^2)$.
Persistence length and effective diffusion coefficient for a chemically active Janus colloid at fluid interfaces {#sec:persitence length}
----------------------------------------------------------------------------------------------------------------
The motion of active particles is characterized by distinct regimes occurring at different time scales. At short time scales the active motion amounts to a ballistic trajectory whereas at larger time scales the behavior is diffusive. A key parameter, which characterizes the motion of active particles, is the persistence length (which can be defined as below irrespective of whether the active particle is trapped at an interface or moving in a bulk fluid) $$\lambda=\bar{v}\tau\,.
\label{def:lambda}$$ This is the typical distance a Janus particle, moving at an instantaneous velocity $\bar{v}$, covers before thermal fluctuations will eventually change its direction. The time $\tau$ is determined by the rotational diffusion of the particle (see Ref. [@GolestanianPRL2007]). In the present case of the active particle being trapped at the interface there are two types of rotations. First, there are rotations of the catalytic cap orientation, i.e., of $\mathbf{p}$ around the interface normal, with a characteristic time $\tau_\parallel$. These rotations lead $\mathbf{p}$ out of the initial plane of motion spanned by $\mathbf{p}$ and the interface normal. This clearly changes the direction of motion. Second, there are fluctuations of $\mathbf{p}$ within the plane of motion with the normal of the plane of motion acting as the rotation axis. Small fluctuations of this kind do not change the direction of motion because $\mathbf{p}$ stays within the initial plane of motion. However, large fluctuations can rotate $\mathbf{p}$, within the plane of motion, from a predominantly forward direction to a predominantly backward direction so that the particle runs backwards along the same straight line. This flipping of directions is associated with a time scale $\tau_\perp$. The minimum of these two time scales sets the rotational diffusion time $\tau_i = \min(\tau_\perp,\tau_\parallel)$ for an active particle trapped at an interface.
In the absence of thermal fluctuation the distribution function of the orientation of the axis $\mathbf{p}$ of the particle is peaked at the steady state value. Thermal fluctuations promote a broadening of the distribution. Both cases of rotations translate into fluctuations of the value of the instantaneous velocity of the particle. Accordingly, the typical velocity $\bar{v}_i$ of an active particle trapped at an *i*nterface is defined as the mean velocity of the Janus particle obtained as a weighted integral over all those possible configurations which give rise to a velocity with the same prescribed sign[^8]. Before entering into further technical details concerning the definition of $\bar{v}_i$, it is convenient to focus on one of the eight cases shown in Fig. \[fig:stability-condition\](b), namely the case of a Janus particle characterized by $V_0 < 0$ (see Eq. (\[eq:def\_V0\]) for repulsive solute-particle interactions so that $\mathcal{L} < 0$) and with the catalytic cap in the *upper* phase with $\delta < \pi/2$ ( so that $\chi = \delta > 0$). For $\Delta\eta > 0$, and within the linear regime $\epsilon \ll 1$, Eq. (\[eq:solu-tot-2-explicit\]) renders, in this case, $U_x < 0$ and $\Omega_z < 0$. This is the situation illustrated in the right part of the top right quadrant of Fig. \[fig:stability-condition\](b). The other cases can be discussed along the same line. Accordingly, we define $\bar{v}_i$ as $$\bar{v}_i = \left|\, \int\limits_{\delta_m}^{\pi/2} \mathcal{P}
(\delta) U_x(\delta)d\delta \right|,
\label{def:bar-v}$$ where $\mathcal{P}(\delta)$ is the steady state probability distribution to find a Janus particle with its axis forming an angle $\delta\in (\delta_m,\pi/2)$ with the plane of the interface[^9]; $\delta_m$ is the value of $\delta$ for which the catalytic cap would touch the interface (see Fig. \[fig:stab-janus\](b)).
In thermal equilibrium, $\mathcal{P}(\delta)=\mathcal{P}_{eq}(\delta)$ depends only on the effective interactions between the catalytic cap and the interface. For example, in the absence of such interactions one has $\mathcal{P}_{eq}(\delta) =
(\pi/2-\delta_m)^{-1}$. In contrast, in the presence of an effective attraction we expect that $\mathcal{P}_{eq}(\delta)$ exhibits a peak closer to the interface (i.e., close to $\delta = 0$) while the opposite holds in the case of an effective repulsion for which one expects $\mathcal{P}_{eq}(\delta)$ to be peaked at $\delta = \pi/2$. When particles are active, an additional torque arises due to the catalytic activity, hence modifying the shape of $\mathcal{P}(\delta)$. In order to estimate $\mathcal{P}(\delta)$ for active particles, we assume that $\mathcal{P}(\delta)$ factorizes as $\mathcal{P}(\delta)=\mathcal{P}_{eq}(\delta)\mathcal{P}_{neq}(\delta)$ into an equilibrium part $\mathcal{P}_{eq}(\delta)$ (as discussed above) and a modulation $\mathcal{P}_{neq}(\delta)$ due to the particle activity. As far as $\mathcal{P}_{neq}(\delta)$ is concerned, we assume that, although the system is out of thermal equilibrium, we can express it as a Boltzmann weight $\mathcal{P}_{neq}(\delta)\propto e^{-\beta\Phi(\delta)}$ of an effective potential $\Phi(\delta)$ accounting for the torque arising due to the particle activity. Since within the present model there are no effective interactions with the interface, $\mathcal{P}_{eq}(\delta)$ is a constant which can be absorbed into the normalization: $$\mathcal{P}(\delta):= \mathcal{P}_{eq}(\delta) \mathcal{P}_{neq}(\delta) =
\dfrac{e^{-\beta\Phi(\delta)}}{Z}
\label{def:distr-prob}$$ where $Z = \int_{\delta_m}^{\pi/2}e^{-\beta\Phi(\delta)}d\delta$ ensures that $\int_{\delta_m}^{\pi/2} \mathcal{P}(\delta)d\delta = 1$.
An estimate of the potential $\Phi$ can be obtained as follows. According to Eq. (\[eq:Lz\_expr\]), in our model a Janus particle, trapped at the interface and spinning with angular velocity $\Omega_z$ around an axis which is contained in the plane of the interface and passes through the center of the particle, experiences a torque $L_z = -8 \pi \eta_0 R^3 \Omega_z$. Therefore, in order to maintain the angular velocity $\Omega_z(\delta)$ of the particle an external torque equal to $-L_z$ must be applied to the particle. Accordingly, for the Janus particle translating with $U_x(\delta)$ while simultaneously rotating with $\Omega_z(\delta)$, we introduce an effective torque $$L(\delta) = 8 \pi \eta_0 \Omega_z(\delta) R^3$$ analogous to the external one which would have accounted for the same angular velocity, and define, via $L(\delta) = - d\Phi(\delta)/d\delta$, the effective potential $\Phi(\delta)$ as $$\beta\Phi(\delta) :=- \beta\int\limits^{\delta}_{\delta_{m}} L(\delta')d\delta'
~~\stackrel{Eqs. (\ref{eq:solu-tot-2-explicit}),\,(\ref{eq:def_V0})}{=}
\Pi \,[ \sin(\delta_m)- \sin(\delta) ]\,.
\label{def:pot-PHI}$$ In this equation one has $\Pi = \sqrt{3 \pi} \beta V_0 R^2 \Delta\eta$, the reference potential is set to $\Phi(\delta_m) = 0$, and we have accounted for the fact that here we discuss only the case in which $\delta_m \leq \delta \leq \pi/2$ (each of the other three quadrants can be analyzed following the same line of reasoning).
The sign of $\Pi$ is determined by the sign of $V_0$ and $\Delta\eta$. As we noted above, here we focus on the case in which $V_0<0$ (i.e., there is a repulsive interaction between the Janus particle and the product molecules of the catalysis) so that $\Pi = -\sqrt{3 \pi}\, \beta |V_0| R^2 \Delta\eta = - 1/(2 \sqrt{3 \pi})
(\Delta\eta/\eta_0) Pe_0$, where $Pe_0 = |V_0| R/D_P > 0$ is the Péclet number of a Janus particle in a homogeneous fluid of viscosity $\eta_0$ and $D_P = k_B T /(6 \pi \eta_0R)$ is the diffusion constant of the Janus particle defined via the Stokes-Einstein relation. The above expression for $\Pi$ shows that, for $\Delta\eta > 0$ and a Janus particle characterized by $V_0 < 0$, $\Pi$ is negative. In this case one has $\Phi(\delta > \delta_m) > 0$ (Eq. (\[def:pot-PHI\])) and thus $\Phi(\delta)$ attains its minimum at $\delta = \delta_m$. This means that the action of the effective torque is consistent with $\Omega_z < 0$ (see Eq. (\[Omz\_expl\])) which “drives” the particle towards its steady-state orientation $\delta_m$, as discussed in Fig. \[fig:stability-condition\](b) (right part of the top right quadrant). Accordingly, in the right part of the top right quadrant, corresponding to $\Delta \eta > 0$ and to the director pointing into the upper fluid, consistent with $\delta_m \leq \delta < \pi/2$ and thus $\chi = \mathrm{min}(\delta,\pi-\delta) > 0$, repulsive interactions (orange arrows) ensure a sustained motility state by providing a torque which tilts the director towards the interface, i.e., which in the present case leads to a decrease of $\delta$ towards $\delta_m$. As one can read off from Eq. (\[def:pot-PHI\]), the characteristics of the Janus particle (see Eq. (\[eq:def\_V0\])) and of the fluid phases are all encoded in $\Pi$. Therefore the above conclusions can be extended directly to the case of attractive interactions between the Janus particle and the product molecules of the catalytic reaction by changing the sign of $V_0$, and hence of $\Pi$. Therefore, in the case of attractive interactions (i.e., $V_0 > 0 $) and with the cap oriented such that $\delta_m \leq \delta <\pi/2$ and $\chi = \delta > 0$, $\Pi$ is negative for $\Delta \eta < 0$ and thus $\Phi(\delta > \delta_m) > 0$ (in agreement with the situation illustrated in the left part of the top left quadrant of Fig. \[fig:stability-condition\](b) for which $U_x > 0$ and $\Omega_z
< 0$). Therefore, if particles have the catalytic cap in the *upper* phase, for which $\sin(\delta) > 0$, the states with small values of $\sin(\delta)$, i.e., with the catalytic patch being closer to the interface and thus promoting the motile state, are favored if $\Pi < 0$. Similarly, if the catalytic cap of the particle is in the *lower* phase, where $\sin(\delta) < 0$, large values of $\sin(\delta)$, i.e., the catalytic patch being closer to the interface and thus promoting the motile state, are favored if $\Pi > 0$.
Concerning the persistence length $\lambda_i = \bar{v}_i \tau_i$ of an active particle trapped at an interface (Eqs. (\[def:lambda\]) and (\[def:bar-v\])) one would like to understand its relation to the persistence length $\lambda_b = \bar{v}_b
\tau_b$ of a similar active particle moving freely in a homogeneous bulk fluid. To this end, we proceed by assuming that the characteristic time $\tau_i$ for the loss of orientation of a Janus particle, trapped at and moving along an interface between two fluids characterized by a (not too large) viscosity contrast $\Delta \eta \neq 0$, is similar to the corresponding characteristic rotational diffusion time $\tau_b$ for the loss of orientation in a homogeneous bulk fluid[^10] of viscosity $\eta_0$. However, since one of the three possible independent rotations of a rigid body would affect the directionality for the particle trapped at the interface only if it is associated with a large fluctuation which would flip the director $\mathbf{p}$ with respect to the interface normal (see the discussion of $\tau_{\perp}$ after Eq. (\[def:lambda\])), it is a reasonable to expect that $\tau_i > \tau_b$. Furthermore, the argument concerning the weak influence of the rotations associated with $\tau_{\perp}$ suggests $\tau_i = \nu_0
\tau_b$ with $\nu_0 \simeq 3/2$ as a good *ansatz* for the relation between the two characteristic time scales. Furthermore, we note that Eq. (\[Ux\_expl\]) has the form of a projection onto the $x$-axis (due to the factor $\cos(\delta)$) of a velocity (the factor multiplying $\cos(\delta)$) oriented along the director $\mathbf{p}$. Thus in the limit $\Delta \eta \to 0$ this latter factor can be identified with the velocity ${\bar v}_b$ of the active particle moving in a homogeneous bulk fluid of viscosity $\eta_0$. With $\varsigma(\epsilon \to 0) = 1$ this renders ${\bar v}_b = \frac{1}{\sqrt{3\pi}} |V_0|$.
From Eqs. (\[def:bar-v\]) and (\[def:distr-prob\]), after disregarding corrections to $U_x$ of order $\epsilon = \frac{\Delta\eta}{\eta_0}$ (Eq.(\[Ux\_expl\])), one obtains $$\label{eq:Lamb_calc}
\Lambda := \dfrac{\lambda_i}{\lambda_b} \simeq \dfrac{\tau_i}{\tau_b}
\dfrac{\bar{v}_i}{\bar{v}_b} \simeq
\nu_0 \int \limits_{\delta_m}^{\pi/2}
\dfrac{e^{-\beta\Phi(\delta)}}{Z} \cos(\delta)d\delta$$ By inserting $Z = \int \limits_{\delta_m}^{\pi/2} e^{-\beta\Phi(\delta)} d\delta$ and Eq. (\[def:pot-PHI\]) into Eq. (\[eq:Lamb\_calc\]), the dependences of $\Lambda$ on $\delta_m$ and on $\Pi$ can be calculated. For the choice $\nu_0 = 3/2$ these are shown in Figs. \[fig:lambda\](a) and (b), respectively. (We recall that we have focused on the case of the catalytic cap exposed to the *upper* phase.) As shown in Fig. \[fig:lambda\], for sufficiently negative values of $\Pi$ the persistence length of a Janus particle moving at a liquid-fluid interface may be *larger* than the one in the corresponding bulk case, i.e., $\Lambda > 1$. According to the discussion in the previous paragraphs, the case of $\Pi < 0$ with the catalytic cap exposed to the upper phase corresponds to either $\Delta \eta > 0$ and repulsive interactions or $\Delta \eta < 0$ and attractive interactions, i.e., the cases for which sustained motility emerges (see Fig. \[fig:stability-condition\](b)). On the other hand, we have noted that for a given type of interactions (i.e., a given sign of $V_0$) and a given viscosity contrast $\Delta \eta$, the amplitude $\Pi$ of the potential $\Phi$ changes sign if the catalytic cap is exposed to the lower phase, i.e., for $\delta > \pi$, relative to the the case of the catalytic cap being exposed to the upper phase. Therefore, these corresponding dependences on $\delta_m$ and on $\Pi$ are given by the curves in Fig. \[fig:lambda\](a) and (b) but with the opposite sign of $\Pi$ and with $\delta_m \to -\delta_m$. Consequently, one infers that in this case one has $\Lambda > 1$ for sufficiently large positive values of $\Pi$. Thus also in the case that the cap is immersed in the lower phase the persistence length at the interface may be enhanced relative to the bulk one for those states in which sustained motility emerges. In summary, this implies that in all cases of sustained motility (i.e., the system corresponds to any of the cases shown in Fig. \[fig:stability-condition\](b)) the particle trapped at the interface exhibits an enhanced persistence length for sufficiently large values of $|\Pi|$.
Figure \[fig:lambda\](b) shows the dependence of $\Lambda$ on $\Pi$ for the case in which the catalytic cap is exposed to the upper phase. Interestingly, $\Lambda$ saturates at negative values of $\Pi$ with large $|\Pi|$. The saturation occurs at larger values of $|\Pi|$ upon increasing $\delta_m$. Concerning the magnitude of $\Pi$ at which $\Lambda$ starts to saturate, we recall that $|\Pi|=\frac{1}{2 \sqrt{3 \pi}}\frac{\Delta \eta}{\eta_0} Pe_0$. Therefore, for $\frac{\Delta\eta}{\eta_0}\rightarrow 0$ the onset of saturation at $|\Pi|\approx 5$ requires, even for very small caps, i.e., $\delta_m \to 0$, Péclet numbers $Pe_0 \simeq 30 \times (\Delta \eta/\eta_0)^{-1}$ much larger than the typical values $Pe_0 \simeq 10$ for Janus particles in a homogeneous bulk fluid. If, however, the viscosity contrast is high, the required corresponding $Pe_0$ numbers are significantly smaller. For example, for the water-air interface the viscosity of the air is negligible so that $\Delta\eta = - 2 \eta_0$ which implies $|\Pi|
\simeq \frac{1}{3} Pe_0$. In such a situation, as well as for other liquid-fluid interfaces characterized by high viscosity contrasts, large values of $|\Pi|$ are encountered already for typical values of $Pe_0$ and the persistence length at the interface may be enhanced relative to its bulk value, i.e., $\Lambda > 1$. As shown in Fig. \[fig:lambda\], this effect is particularly pronounced for small catalytic caps (i.e., $\delta_m$ small).
While the persistence length $\lambda$ characterizes the active motion of a particle at time scales shorter than the characteristic rotational diffusion time $\tau$, at time scales much larger than $\tau$ the motion of the particle crosses over to diffusion with an effective diffusion constant [@GolestanianPRL2007]: $$D_{eff} = D_{tr} + \dfrac{\lambda^2}{\tau} : = D_{tr} + \delta D\,,
\label{eq:D_def}$$ where $D_{tr}$ is the translational diffusion constant of the particle in the absence of activity and $\delta D = \lambda^2/\tau$ is the activity-induced enhancement of the diffusion constant. Equation (\[eq:D\_def\]) allows us to compare the enhancement $\delta D^{(i)}$ for the Janus particle trapped at, and moving along, the *interface* to the one, $\delta D^{(b)}$, which holds for the same active Janus particle moving in a *bulk* fluid: $$\dfrac{\delta D^{(i)}}{\delta D^{(b)}} = \left(\dfrac{\lambda_i}{\lambda_b}\right)^2
\dfrac{\tau_b}{\tau_i} \simeq \dfrac{1}{\nu_0} \Lambda^2 \,.
\label{eq:deltaD}$$ Therefore, according to the values of $\Lambda$ shown in Fig. \[fig:lambda\], for $\nu_0 = 3/2$ the enhancement of the diffusion constant due to the activity of a Janus particle trapped at a liquid-fluid interface can be up to $1.5$ times larger than the enhancement observed in a homogeneous bulk fluid. Finally, we note that for a particle trapped at the interface the activity induced contribution $\delta
D^{(i)}$ can become much larger than the passive translational diffusion constant $D_P$ in bulk fluid if $({\bar v}_i^2 \tau_i^2)/ (\tau_i D_P) \gg 1$ (see the definition of $\delta D^{(i)}$ above and Eq. (\[def:lambda\])). By using ${\bar v}_i = \Lambda {\bar v}_b/\nu_0$ (Eq. (\[eq:Lamb\_calc\])), $\bar{v}_b=V_0/\sqrt{3\pi}$, $Pe_0 = |V_0| R/D_P$, and $\tau_i/\tau_b = \nu_0$, and by taking $\tau_b = 1/D_P^{(rot)}$, where $D_P^{(rot)} = 4 D_P/(3 R^2)$ is the rotational diffusion constant of the particle in a homogeneous bulk fluid [@GolestanianPRL2007], for $\nu_0 = 3/2$ and by using Eqs. (\[eq:Lamb\_calc\])-(\[eq:deltaD\]) the condition $\delta D^{(i)}/D_P \gg 1$ translates into the condition $Pe_0 \gg 4.3/\Lambda$ for the P[éclet]{} number of the particle.
Summary and conclusions
=======================
We have studied the behavior of a chemically active Janus particle trapped at a liquid- fluid interface, under the assumptions that the activity of the particle does not affect the surface tension of the interface and that the interface can be assumed to be flat. If particles are moving in such a set-up (Fig. \[fig:stab-janus\]), a coupling between rotation and translation arises due to the viscosity contrast $\Delta \eta$ between the two adjacent fluids. Assuming that the particles are axisymmetric, and that both fluid phases are homogeneous and isotropic, the motile state of the particles is characterized by their linear velocity $U_x$ in the plane of the interface and their angular velocity $\Omega_z$ about an axis perpendicular to the plane of motion spanned by the interface normal and the velocity.
In Sec. \[sec:phoretic\_velocities\] we have determined the linear and angular velocity $U_x$ and $\Omega_z$, respectively, by using the Lorentz reciprocal theorem [@Lorentz_original]. Therein the stress-free interface is accounted for by imposing corresponding boundary conditions on the fluid flow in both phases and the fluids are taken to be quiescent far away from the particle. The result in Eq. (\[rhs\_rec\_theo\]) is valid for an arbitrary viscosity contrast $\Delta\eta$, including the limit of vanishing values of $\Delta\eta$ as well as the case that one of the two phases has a vanishing viscosity.
Determining $U_x$ and $\Omega_z$ via the reciprocal theorem requires to solve two independent auxiliary problems involving translation and rotation of a particle trapped at a liquid-fluid interface. In order to be able to obtain analytical solutions, we have considered neutrally buoyant particles exhibiting a contact angle of $\pi/2$ with the planar interface. Under these assumptions it is possible to exploit the available analytical solution for the stress exerted on the fluid by a particle which is translating without rotation [@Pozrikidis2007]. The case of a particle rotating at the interface is more challenging because it requires to determine the fluid flow close to the three-phase contact line formed as the intersection of the interface and the particle surface. In order to circumvent the issue of the motion of the three-phase contact line and in order to gain analytical insight into the problem, we have assumed that the fluid slips along the particle only in a small region close to the three-phase contact line. Accordingly, we can consider the fluid flow on each part of the surface of the particle to be *de facto* equal to the one which a particle experiences in a corresponding homogeneous bulk fluid under a no-slip condition on its surface (Eq. (\[notrasl-rot-noslip\])). The general expressions for $U_x$ and $\Omega_z$ (Eqs. (\[Ux\]) and (\[Omz\])) show that, for $\Delta\eta\neq 0$, $\Omega_z$ is nonzero. Therefore the motility of the particle along the interface is strongly affected by the change in the orientation of the axis of the particle relative to the interface normal. Accordingly, the velocity of the particle along the interface can be either enhanced or reduced.
In Sec. \[sec:results\] A we have established a diagram (Fig. \[fig:stability-condition\]) describing the situations for which $\Omega_z$ promotes orientations of the Janus particle axis to be parallel to the interface, hence enforcing the motile state of the particle. In particular, for repulsive interactions between the particle and the self-generated solute (e.g., for catalytic platinum caps on polystyrene particles suspended in water-peroxide solutions[^11]) we have found that the motile state is fostered if the catalytic cap is immersed into the more viscous phase, while the opposite conclusion holds for an attractive interaction. Therefore, by tuning the viscosity contrast $\Delta\eta$, one can control the motility of Janus particles trapped at liquid-fluid interfaces.
In Sec. \[sec:results\] B these general consideration have been extended further by specifying model particles which allow one to analyze the density profiles of the reaction product. The expansion of these profiles in terms of spherical harmonics shows that only the amplitude $A_{1,0}$ of the largest wavelength mode affects $U_x$ and $\Omega_z$ (see Appendix \[deriv-C1-C2\]). Accordingly, for the model considered here, different systems characterized by diverse physical properties (such as the viscosity contrast $\Delta\eta$, the catalytic reaction, or the interaction between the reaction product and the particle) but exhibiting the same value of $A_{1,0}$ lead to the same values of $U_x$ and $\Omega_z$ (see Eqs. (\[Ux\_expl\]) and (\[Omz\_expl\])). In particular we have found that both $U_x$ and $\Omega_z$ are proportional to the velocity scale $V_0$ (Eq. (\[eq:def\_V0\])) which depends linearly on the prefactor $\mathcal{L}$ (see Eq.\[eq:phor-mob\]), the reaction rate $Q$ per area, the inverse mean viscosity $\eta_0$ and the inverse diffusivity $D_0$ of the reaction product. The angular velocity experienced by the particle of radius $R$ is proportional to $V_0/R$ and to the viscosity contrast $\Delta\eta$.
If the angular velocity promotes the alignment of the axis of the particle with the interface, the persistence length of the particle increases. In order to quantify this effect, in Sec. \[sec:persitence length\] we have proposed a factorization of the probability distribution (Eq. (\[def:distr-prob\])) for the orientation of the axis of the particle into an equilibrium and into a non-equilibrium distribution induced by the angular velocity and we have constructed an effective potential $\Phi$ (Eq. (\[def:pot-PHI\])) describing the latter. The strength $|\Pi|$ (Eq. (\[def:pot-PHI\])) of this potential is proportional to the bulk Péclet number of the particle, which is of the order of $10$, and therefore may lead to an increase of the persistence length of a trapped active particle relative to its value in the bulk fluid. Figure \[fig:lambda\](b) shows that this enhancement increases with $|\Pi|$ as well as upon decreasing the size of the catalytic cap (which allows for smaller values of $\delta_m$ (see Figs. \[fig:stab-janus\](b) and \[fig:lambda\](a)). At long timescales the motion of active Janus particles is characterized by an effective diffusion coefficient (see Eq. (\[eq:D\_def\])). Concerning this regime our results predict that $\Delta\eta$ as well as $\Pi$ control the enhancement of the effective diffusion coefficient. In particular, by using Eq. (\[eq:deltaD\]) and the data in Fig. \[fig:lambda\], we have found that the presence of the interface can almost double the activity induced enhancement of the diffusion coefficient compared with the one in a homogeneous bulk fluid.
In sum we have obtained the following main results:
- [Within a minimalistic model of active Janus particles trapped at a liquid-fluid interface, we have characterized their dynamics and have shown that their motility is strongly affected by the angular velocity induced on the particle due to the viscosity contrast $\Delta \eta$ between the adjacent fluids.]{}
- [We have shown that the rotation-translation coupling induced by $\Delta\eta$ can affect experimentally observable quantities such as the persistence length and the effective diffusion coefficient of active Janus particles trapped at liquid-fluid interfaces. In particular, the behavior described by our model is in agreement with recently reported, corresponding experimental observations of increased persistence lengths for chemically active Janus particles at water-air interfaces [@Stocco], and it sheds light on the proposition of an alternative explanation for the observed phenomenon.]{}
- [Since the viscosity contrast $\Delta \eta$ can control the performance of active particles moving at liquid-fluid interfaces, we suggest that it can be relevant also for the onset of instabilities of thin films covered by active particles [@Stark2014].]{}
Finally, we mention a few interesting extensions of the present study. Relaxing some of the simplifying assumptions employed here might shed light on alternative means to control active particles motility at liquid-fluid interfaces. In this respect we recall that we have assumed that the contact angle of the particle with the interface is $\pi/2$, and that pinning of the three-phase contact line is absent. Concerning the contact angle, we expect particles with a contact angle unequal $\pi/2$ to experience extra torques due to the offset of their center of mass from the plane of the interface. A similar scenario has been reported for particles which are pulled, without rotating, under the action of suitably distributed external forces and torques [@Pozrikidis2007]. On the other hand, pinning of the three-phase contact line might affect the effective rotational diffusion and, possibly, suppress it, as shown recently for an equilibrium system [@Stocco2]. Therefore we expect that for active Janus particles trapped at liquid-fluid interfaces the pinning of the three-phase contact line can enhance the persistence length, and therefore the effective diffusion, as argued in Ref. [@Stocco], too.
Acknowledgments {#acknowledgments .unnumbered}
===============
P.M. thanks Dr. Antonio Stocco for useful discussions and for providing the preprint version of Ref. [@Stocco].
Forces and torques
==================
Here we present the steps of the derivations leading to Eqs. (\[eq:recipr-prob-1\]) and (\[eq:recipr-prob-2\]). In order to simplify the calculations, here it is convenient to translate the origin $O$ of the unprimed coordinate system (fixed in space, see Fig. \[fig:stab-janus\](b)) to the center $C$ of the moving particle and to use spherical coordinates $(r,\theta,\phi)$, which are defined as usual: $x = r \sin(\theta)
\cos(\phi)$, $y = r \sin(\theta) \sin(\phi)$, and $z = r \cos(\theta)$. (Note that these are defined in the unprimed coordinate system which, although exhibiting here the same origin as the primed (co-moving) one, has different orientations of the axes as compared with the primed one. The unprimed coordinate system offers a less cumbersome parametrization of the location of the interface as compared with the primed coordinate system.) We start with deriving Eq. (\[eq:recipr-prob-1\]).
By using that for a translation (only) with velocity $\hat{\mathbf{U}}=\hat{U}_x\mathbf{e}_x$ one has [@Pozrikidis2007] $$\left.\mathbf{n} \cdot \hat{\boldsymbol{\sigma}} \right|_{\Sigma_p}= -\frac{3}{2R}
\eta(\mathbf{{r}}_p) \hat{\mathbf{U}}\,$$ on the surface $\Sigma_p$ of the particle. By noting that $$\hat{F}_x=\int_{\Sigma_p}(\mathbf{n} \cdot \hat{\boldsymbol{\sigma}})_x \, dS$$ is the $x$ component of the integral over the surface of the normal pressure tensor as given by the above expression, using Eq. (\[eq:def-visc-1\]), and assuming a sharp interface ($\xi \to 0$ so that $\eta(\mathbf{r}_p) = \eta(\phi)$ with $\eta(\phi) = \eta_1$ for $0 < \phi < \pi$, i.e., $y > 0$, while $\eta(\phi) = \eta_2$ for $\pi < \phi < 2\pi$, i.e., $y < 0$) one obtains $$F_{x} =-\dfrac{3}{2R} \hat{U}_{x} R^{2} \int\limits_{0}^{\pi} d \theta \,
\sin\left(\theta \right)
\int\limits_{0}^{2\pi} d \phi \, \eta\left(\phi\right) =
-6 \pi \eta_{0} R \hat{U}_{x}\,,$$ which agrees with Eq. (\[F1x\]). The torque is defined as $$\mathbf{L} = \int_{\Sigma_p}(\mathbf{r}_p-\mathbf{r}_C)
\times \hat{\boldsymbol{\sigma}} \cdot \mathbf{n}\,dS = R \int\limits_{\Sigma_p}^{}
\mathbf{n} \times \hat{\boldsymbol{\sigma}} \cdot \mathbf{n} \, dS
= -\dfrac{3}{2} \int\limits_{\Sigma_p}^{} \eta(\mathbf{{r}}) \, \mathbf{n} \times
\hat{\mathbf{U}}\, dS\,,$$ where we have used $\mathbf{r}_p-\mathbf{r}_C = R\,\mathbf{n}$ (see Fig. \[fig:stab-janus\]). With the above choice of the coordinate system the torque is along the $z$-direction. Since $\left( \mathbf{n} \times \hat{\mathbf{U}} \right)_{z} = n_{x} \hat{U}_{y} -
n_{y}\hat{U}_{x}$ and $\hat{U}_{y} = 0$ (due to the choice of the dual problem “1”), with $n_y = \sin(\theta) \sin(\phi)$ one obtains $$L_{z}= \dfrac{3}{2} R^{2}\hat{U}_{x}
\int\limits_{0}^{\pi} d\theta \,\sin^{2} \left(\theta \right)
\int\limits_{0}^{2\pi} d \phi \sin\left(\phi \right) \eta\left(\phi \right) =
\dfrac{3}{2} \pi R^{2} \Delta\eta \hat{U}_{x}\,,$$ which agrees with Eq. (\[L1z\]).
Concerning the derivation of Eq. (\[eq:recipr-prob-2\]) we start from the relation (compare Eq. (\[notrasl-rot-noslip\]) with $\hat{\mathbf{\Omega}} = \Omega_z
\mathbf{e}_z$) $$\left.\mathbf{n} \cdot \hat{\boldsymbol{\sigma}} \right|_{\Sigma_p}=
- 3 \eta(\mathbf{r}_p) \,\hat{\mathbf{\Omega}} \times \mathbf{n}\,;$$ we note that $\left( \hat{\mathbf{\Omega}} \times \mathbf{n} \right)_{x} =
\hat{\Omega}_{y} n_{z}-\hat{\Omega}_{z} n_{y}$ and recall that by definition of the dual problem “2” only the component $\hat{\Omega}_{z}$ of the angular velocity is non-zero. Thus we obtain $$F_{x} = 3 \hat{\Omega}_{z} R^{2} \int\limits_{0}^{\pi} d\theta
\sin^{2}\left(\theta \right)
\int\limits_{0}^{2\pi} d\phi \, \sin\left(\phi \right) \eta\left(\phi \right) d\phi
= 3 \pi R^{2} \Delta\eta \hat{\Omega}_{z}\,,$$ which agrees with Eq. (\[F2x\]). For the torque one has, with $\mathbf{r}_p-\mathbf{r}_C = R \mathbf{n}$, $$\begin{aligned}
\label{eq:Lz_expr}
L_{z} = R \int\limits_{\Sigma_p}^{} \left[\mathbf{n} \times \hat{\boldsymbol{\sigma}}
\cdot \mathbf{n} \right]_{z} \, dS &=& - 3 R \int\limits_{\Sigma_p}^{}
\eta(\textbf{{r}}) \left[\mathbf{n} \times \left(\hat{\mathbf{\Omega}} \times
\mathbf{n} \right)\right]_{z}\, dS =
-3 R \int\limits_{\Sigma_p}^{} \eta(\textbf{{r}}) \hat{\Omega}_{z} \sin^2(\theta)
\, dS \nonumber\\
& = & -3 R^3 \hat{\Omega}_{z} \int\limits_{0}^{\pi} d \theta \sin^{3}
\left(\theta\right) \int\limits_{0}^{2 \pi} d\phi \,\eta(\phi) = -8 \pi \eta_{0} R^{3}
\hat{\Omega}_{z} \,, \end{aligned}$$ which agrees with Eq. (\[L2z\]) and where the following relations have been used: $$\hat{\mathbf{\Omega}} \times \mathbf{n} = -\left(
\begin{array}{c}
n_{y}\hat{\Omega}_{z}-n_{z}\hat{\Omega}_{y}\\
n_{z}\hat{\Omega}_{x}-n_{x}\hat{\Omega}_{z}\\
n_{x}\hat{\Omega}_{y}-n_{y}\hat{\Omega}_{x}
\end{array}
\right)$$ and $$\label{A10}
\left[\mathbf{n} \times \left( \hat{\mathbf{\Omega}} \times \mathbf{n} \right) \right]_{z}
= n_{x} n_{y} \hat{\Omega}_{x} + n_{x}^{2} \hat{\Omega}_{z} + n_{y}^{2} \hat{\Omega}_{z}
+ n_{y} n_{z} \hat{\Omega}_{y}
= \hat{\Omega}_{z} \left( n_{x}^{2} + n_{y}^{2} \right)
= \hat{\Omega}_{z} \sin^{2}\left(\theta\right).$$ In Eq. (\[A10\]) we have used, according to the definition of the dual problem “2”, $\hat{\Omega}_{x} = \hat{\Omega}_{y} = 0$.
Diffusiophoretic slip \[deriv-C1-C2\]
=====================================
We start the derivation of Eqs. (\[eq:C1B\]) and (\[eq:C2B\]) by employing spherical coordinates $(r',\theta',\phi')$: $$x' = r' \sin\left(\theta'\right)\cos\left(\phi'\right)\,,~
y' = r' \sin\left(\theta'\right)\sin\left(\phi'\right)\,,~
z' = r' \cos\left(\theta'\right)\,
\label{sph-coord}$$in the co-moving (primed) coordinate system (see Fig. \[fig:stab-janus\](b)) with $\rho(\mathbf{r'}_p) =
\rho(R,\theta',\phi')$; in the following, for reasons of shorter notations we shall not indicate explicitly the dependence on $R$. The density can be expressed as a series expansion in terms of the spherical harmonics $Y_{\ell\,m} (\theta',\phi')$: $$\rho(\theta',\phi') = \sum\limits_{\ell = 0}^\infty
\sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\, Y_{\ell\,m}
(\theta',\phi') \,,
\label{exp:rho_sph_harm}$$ where [@Jackson_book] $$\label{sph_harm}
Y_{\ell\,m} (\theta',\phi') = \alpha_{\ell,m} P_{\ell}^m (\cos\theta')
e^{i m \phi'}\,,~i = \sqrt{-1}\,,$$ and $$P_{\ell}^m (x) = \dfrac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}
\dfrac{d^{\ell+m}}{dx^{\ell+m}} (x^2-1)^\ell\,,~ \ell \geq 0\,,~|m| \leq \ell$$ is the associated Legendre polynomial of degree $\ell$ and order $m$ with $$\alpha_{\ell,m} = \sqrt{\dfrac{2 \ell + 1}{4 \pi}\dfrac{(\ell-m)!}{(\ell+m)!}}
\label{eq:def-alph}$$ as a normalization constant. Before proceeding, we list a few relations (obtained straightforwardly from the corresponding definitions) satisfied by $Y_{\ell\,m}$, $P_{\ell}^m$, and $\alpha_{\ell,m}$, which will be needed below[^12] : $$\begin{aligned}
&&Y_{0\,0} = \dfrac{1}{\sqrt{4 \pi}}\,, \label{Y_00} \\
&&P_{\ell}^{-m} = (-1)^m \,\dfrac{(\ell-m)!}{(\ell+m)!} \, P_{\ell}^m\,, \label{Pl_min}\\
&&Y_{\ell(-m)} = (-1)^m \, Y^*_{\ell\,m} \,, \label{Yl_min} \\
&&\alpha_{\ell,-m} = \dfrac{(\ell + m)!}{(\ell - m)!} \, \alpha_{\ell,m}\,,
\label{alfl_min} \\
&&\int\limits_0^{2 \pi} d\phi' \int\limits_0^{\pi} d\theta'\, \sin\theta \,
{Y^*_{\ell'\,m'}}(\theta',\phi') \, Y_{\ell \,m}(\theta',\phi') =
\delta_{\ell,\ell'}\delta_{m,m'}\,, \label{normal} \\
&& \dfrac{\partial Y_{\ell\,m}}{\partial \phi} = i \, m \,Y_{\ell\,m}\,,
\label{phi_der}\end{aligned}$$ and $$\begin{aligned}
&& \sin\theta' \dfrac{\partial Y_{\ell\,m}}{\partial\theta'} = \ell\,
\sqrt{\dfrac{(\ell+1)^2 - m^2}{(2 \ell+1) (2 \ell+3)}} \, Y_{(\ell+1) \,m} -
(\ell+1)\,
\sqrt{\dfrac{\ell^2 - m^2}{(2 \ell-1) (2 \ell+1)}} \, Y_{(\ell-1) \,m} \nonumber\\
&&\hspace*{.6in}:= a_{\ell,m} Y_{(\ell+1) \,m} + b_{\ell,m} Y_{(\ell-1)\,m}\,,
\label{theta_der}\end{aligned}$$ where $^*$ indicates the complex conjugate quantity.
From the definition of the phoretic slip $\mathbf{v}(\mathbf{r}'_p)$ (Eq. (\[def:slip\_vel\])) one obtains $$\eta(\mathbf{r'}_p) v_{\theta'} = - \dfrac{\mathcal{L}}{\beta R}
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\,
\dfrac{\partial Y_{\ell\,m}(\theta',\phi')} {\partial {\theta'}} \label{eq:B_vel_theta}$$ and $$\eta(\mathbf{r}'_p) v_{\phi'} = - \dfrac{\mathcal{L}}{\beta R}
\dfrac{1}{\sin\theta'} \sum\limits_{\ell = 0}^\infty
\sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\,
\dfrac{\partial Y_{\ell\,m}(\theta',\phi')} {\partial {\phi'}} \,.\label{eq:B_vel_phi}$$ Noting that the unit vectors $\mathbf{e}_{r'}$, $\mathbf{e}_{\theta'}$, and $\mathbf{e}_{\phi'}$ are given by $$\begin{aligned}
\label{unit_vec}
\mathbf{e}_{r'} &=& \sin\theta' \cos\phi' \mathbf{e}_{x'} +
\sin\theta' \sin\phi' \mathbf{e}_{y'} + \cos\theta' \mathbf{e}_{z'} \,,\nonumber\\
\mathbf{e}_{\theta'} &=& \cos\theta' \cos\phi' \mathbf{e}_{x'} +
\cos\theta' \sin\phi' \mathbf{e}_{y'} - \sin\theta' \mathbf{e}_{z'}\,,\\
\mathbf{e}_{\phi'} &=& -\sin\phi' \mathbf{e}_{x'} + \cos\phi' \mathbf{e}_{y'} \nonumber\end{aligned}$$ and using geometry (see Fig. \[fig:stab-janus\](b)) one obtains $$\label{slip_x}
v_x = v_{z'} \cos \delta = (\mathbf{v} \cdot \mathbf{e}_{z'}) \cos\delta =
(v_{\theta'} \mathbf{e}_{\theta'} \cdot \mathbf{e}_{z'} +
v_{\phi'} \mathbf{e}_{\phi'} \cdot \mathbf{e}_{z'}) \cos\delta
\stackrel{(\ref{unit_vec})}{=} - v_{\theta'} \sin\theta' \cos\delta\,;$$ according to Eq. (\[def:slip\_vel\]) $v_{r'} = 0$. Therefore $$\begin{aligned}
C_{1} &:=& \dfrac{3}{2 R} \int\limits_{\Sigma_p}^{} dS\, \eta(\mathbf{r}'_p) \, v_x
\nonumber\\
&\underset{(\ref{eq:B_vel_theta})}{\overset{(\ref{slip_x})}{=}}&
\dfrac{3}{2} \dfrac{\mathcal{L} \cos \delta}{\beta} \int\limits_0^{2 \pi} d \phi'
\int\limits_0^{\pi} d\theta'\, \sin\theta'
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\,
\sin\theta' \,
\dfrac{\partial Y_{\ell\,m}(\theta',\phi')} {\partial {\theta'}} \nonumber\\
&\underset{(\ref{Y_00})}{\overset{(\ref{theta_der})}{=}}&
\dfrac{\mathcal{L} \cos \delta}{\beta}
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\,
\left[a_{\ell,m}
\int\limits_0^{2 \pi} d \phi' \int\limits_0^{\pi} d\theta'\, \sin\theta'
Y^*_{0\,0}(\theta',\phi') Y_{(\ell+1)\,m}(\theta',\phi') \right. \nonumber\\
&&\hspace*{1.5in} \left. + ~ b_{\ell,m}
\int\limits_0^{2 \pi} d \phi' \int\limits_0^{\pi} d\theta'\, \sin\theta'
Y^*_{0\,0}(\theta',\phi') Y_{(\ell-1)\,m}(\theta',\phi')
\right]\nonumber\\
&\stackrel{(\ref{normal})}{=}&
3 \sqrt{\pi} \dfrac{\mathcal{L} \cos \delta}{\beta}
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell} A_{\ell,m}\,
\left(a_{\ell,m} \delta_{0,\ell+1}\delta_{0,m} + b_{\ell,m} \delta_{0,\ell-1}
\delta_{0,m}\right) = 3 \sqrt{\pi} \dfrac{\mathcal{L} \cos \delta}{\beta}
A_{1,0}\, b_{1,0}\nonumber\\
&\stackrel{(\ref{theta_der})}{=}& -2 \sqrt{3 \pi} \,\dfrac{\mathcal{L} \cos \delta}{\beta}
A_{1,0}\,,\end{aligned}$$ which agrees with Eq. (\[eq:C1B\]).
We now proceed with the calculation of $C_2$. First, we note that $\mathbf{e}_z$ = $\mathbf{e}_{x'}$ (see Fig. \[fig:stab-janus\](b)), and therefore $(\mathbf{n} \times \mathbf{v})_{z} := (\mathbf{n} \times \mathbf{v}) \cdot \mathbf{e}_z
= (\mathbf{n} \times \mathbf{v})_{x'}$, where $\mathbf{v} =
\mathbf{v}(\mathbf{r}'_p)$. The latter but one expression is calculated as follows (note that in the primed coordinate system $\mathbf{n} = \mathbf{e}_{r'}$): $$\begin{aligned}
\label{cross_prod}
(\mathbf{n} \times \mathbf{v})_{x'}
&=&
n_{y'} v_{z'} - n_{z'} v_{y'}
= (\mathbf{e}_{r'} \cdot \mathbf{e}_{y'}) v_{z'} - (\mathbf{e}_{r'} \cdot \mathbf{e}_{z'})
\left[v_{\theta'} (\mathbf{e}_{\theta'} \cdot \mathbf{e}_{y'}) +
v_{\phi'} (\mathbf{e}_{\phi'} \cdot \mathbf{e}_{y'})\right]\nonumber\\
&\underset{(\ref{unit_vec})}{\overset{(\ref{slip_x})}{=}}&
v_{\theta'} - \cos\theta' (\cos\theta'\sin\phi' v_{\theta'} + \cos\phi' v_{\phi'})
\nonumber\\
&=& - (\sin\phi' v_{\theta'} + \cos\theta' \cos\phi' v_{\phi'})\,.\end{aligned}$$ Therefore $C_2$ takes the form (Eq. (\[eq:rec-ther-2\])) $$\begin{aligned}
\label{def_I1_I2}
C_2 &:=& \dfrac{3}{R} \int\limits_{\Sigma_p}^{} d S' \,\eta(\mathbf{r}'_p)
(\mathbf{n} \times \mathbf{v})_{z} \nonumber\\
&=& - 3 R \left(
\int\limits_0^{2 \pi} d \phi' \sin\phi' \int\limits_0^{\pi} d\theta'\, \sin\theta'
\eta(\mathbf{r'}_p) v_{\theta'} +
\int\limits_0^{2 \pi} d \phi' \cos\phi' \int\limits_0^{\pi}
d\theta'\, \sin\theta' \cos\theta' \eta(\mathbf{r'}_p) v_{\phi'}\right) \nonumber\\
&=:& 3 \dfrac{\mathcal{L}}{\beta} \left(J_1 + J_2\right)\,.\end{aligned}$$ The integrals $J_1$ and $J_2$ are evaluated as follows. By introducing the notations $$\label{Z_def}
Z_\ell^m(\theta') = \dfrac{d P_\ell^m(\cos\theta')}{d \theta'}\,$$ and $$\begin{aligned}
\label{z_p_def}
z_{\ell,m} &=& \int\limits_0^{\pi} d\theta' \, \sin\theta' \,
Z_\ell^m(\theta')\,,\nonumber\\
p_{\ell,m} &=& \int\limits_0^{\pi} d\theta' \, \cos\theta' \,
P_\ell^m(\cos\theta')\,,\end{aligned}$$ after observing that $$\begin{aligned}
\label{z_min}
z_{\ell,-m} \stackrel{(\ref{Pl_min})}{=} (-1)^m \,\dfrac{(\ell-m)!}{(\ell+m)!}
z_{\ell,m} \,,\nonumber\\
p_{\ell,-m} \stackrel{(\ref{Pl_min})}{=} (-1)^m \,\dfrac{(\ell-m)!}{(\ell+m)!}
p_{\ell,m} \,,\end{aligned}$$ and with $$\label{eq:zm_pm}
z_{\ell,m} ~
\underset{(\ref{z_p_def})}{\overset{(\ref{Z_def})}{=}} ~
\left. \left(\sin\theta' \, P_\ell^m(\cos\theta')\right)\right|_0^\pi -
\int\limits_0^{\pi} d\theta' \,
[d(\sin\theta')/d\theta'] \, P_\ell^m(\cos\theta') = - p_{\ell,m}\,$$ one arrives at $$\begin{aligned}
J_1 &:=&
\int\limits_0^{2 \pi} d \phi' \sin\phi' \int\limits_0^{\pi} d\theta'\, \sin\theta'
\left(- \dfrac{\beta R}{\mathcal{L}} \, \eta(\mathbf{r'}_p) v_{\theta'} \right) \nonumber\\
&\underset{(\ref{Z_def})}{\overset{(\ref{eq:B_vel_theta})}{=}}&
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell}
A_{\ell,m}\, \alpha_{\ell,m} \int\limits_0^{\pi} d\theta'\, \sin\theta' Z_\ell^m(\theta')
\int\limits_0^{2 \pi} d \phi' \sin\phi' e^{i m \phi'} \nonumber\\
&=&
- i \pi
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell}
A_{\ell,m}\, \alpha_{\ell,m} z_{\ell,m} \left(\delta_{m,-1}-\delta_{m,1} \right)
\nonumber\\
&\underset{(\ref{alfl_min})}{\overset{(\ref{z_min})}{=}}&
i \pi \sum\limits_{\ell = 1}^\infty (A_{\ell,-1} + A_{\ell,1})
\alpha_{\ell,1} z_{\ell,1} \nonumber\end{aligned}$$ (note that $z_{0,1} = 0$ due to $|m| \leq \ell$) and $$\begin{aligned}
J_2 &:=& \int\limits_0^{2 \pi} d \phi' \cos\phi' \int\limits_0^{\pi}
d\theta'\, \sin\theta'
\cos \theta' \left(- \dfrac{\beta R}{\mathcal{L}} \,
\eta(\mathbf{r'}_p) v_{\phi'} \right)
\nonumber\\
&\underset{(\ref{eq:B_vel_phi})}{\overset{(\ref{phi_der})}{=}}&
\sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell}
i \,m \, A_{\ell,m}\, \alpha_{\ell,m}
\int\limits_0^{\pi} d\theta'\, \cos\theta' P_\ell^m(\cos\theta')
\int\limits_0^{2 \pi} d \phi' \cos\phi' e^{i m \phi'} \nonumber\\
&=&
i \,\pi\, \sum\limits_{\ell = 0}^\infty \sum\limits_{m = -\ell}^{m = \ell}
m A_{\ell,m}\, \alpha_{\ell,m} p_{\ell,m} \left(\delta_{m,-1} + \delta_{m,1} \right)
\nonumber\\
&\underset{(\ref{alfl_min})}{\overset{(\ref{z_min})}{=}}&
i \pi \sum\limits_{\ell = 1}^\infty (A_{\ell,-1} + A_{\ell,1}) \alpha_{\ell,1}
p_{\ell,1}\,.\nonumber\\
&\stackrel{(\ref{eq:zm_pm})}{=}&
- i \pi \sum\limits_{\ell = 1}^\infty (A_{\ell,-1} + A_{\ell,1}) \alpha_{\ell,1}
z_{\ell,1} = -J_1\,.\nonumber\end{aligned}$$ Therefore $C_2 = 3 \dfrac{\mathcal{L}}{\beta} (J_1 + J_2) = 0$, which verifies Eq. (\[eq:C2B\]).
[^1]: Such particles, which have distinct material properties across the two regions of their surface, are often called Janus particles; in the following we shall use this notion, too.
[^2]: Even if the equilibrium configuration would correspond to the symmetry axis being aligned with the interface normal, as, e.g., in the case of strongly repulsive, effective interactions between the interface and the catalytic patch, fluctuations will perturb this state of alignment and the previous scenario is recovered. These fluctuations can be thermal fluctuations of the orientation of the axis around the equilibrium position, or non-equilibrium fluctuations of the rate of the catalytic reaction along the surface.
[^3]: Since the typical effective forces between the interface and the catalytic cap decay rapidly with the distance from the interface, the assumption remains valid as long as the catalytic cap is not very close to the interface.
[^4]: A recent extension of this version of the reciprocal theorem to the case of a free interface, in which the viscosity exhibits an abrupt change across the liquid-air interface, can be found in Ref. [@Stone2014].
[^5]: Formally, the domains ${\cal D}_1$ and ${\cal D}_2$ as well as the interface $\Gamma$ could be closed along the $Ox$ direction by assuming periodic boundary conditions at $|x| \to \infty$; alternatively, one may choose for the surface at infinity, which is closing the domains ${\cal D}_1$ and ${\cal D}_2$, a spherical one, centered at $C$ and with a radius $R_\infty \to \infty$.
[^6]: For example, for water on PDMS or on glass surfaces the estimated slip length $l_0$ is well below 100 nm [@Joseph2005].
[^7]: In the case of a homogeneous fluid the flow around the particle which translates is symmetric with respect to any plane containing the translation direction. Thus the flow is characterized by a vanishing velocity normal to such a symmetry plane and a continuous velocity tangential to that symmetry plane (which thus can be regarded as an “imaginary planar interface” where the kinematic boundary conditions of an actual interface between immiscible liquids are obeyed).
[^8]: Since in the present case the system does not undergo any spontaneous symmetry breaking, the velocity obtained by averaging over *all* possible configurations, rather than only over those with a prescribed sign of the velocity, is zero.
[^9]: The mean velocity $\bar{v}_i$ for the same particle moving in the positive direction would be $\bar{v}_i = \int_{\pi/2}^{\pi-\delta_m}
\mathcal{P}(\delta) \,U_x(\delta) \,d\delta$; see the left part of the top right quadrant of Fig. \[fig:stability-condition\](b). Here $\mathcal{P}$ is the distribution of the angle $\delta\in (\pi/2,\pi-\delta_m)$.
[^10]: It is particularly difficult to determine $\tau_i$ because it depends on the details of the effective interaction between the Janus particle and the interface and it involves the dynamics of the moving three-phase contact line.
[^11]: In this case the repulsive character of interaction is inferred from the experimentally observed motion away from the platinum cap and under the assumption that the mechanism of motion is self-diffusiophoresis and that only the oxygen production and the corresponding surface gradients of oxygen are relevant.
[^12]: For completeness, we note that (i) the density must take real values, i.e., $\rho^*(\theta,\phi) =
\rho(\theta,\phi)$, and (ii) the system is symmetric with respect to the plane $y'z'$ (see Fig. \[fig:stab-janus\](b)). Thus the solute distribution must be invariant with respect to the transformation $\phi' = \pi/2 - \epsilon \to \phi' = \pi/2 +
\epsilon$, i.e., $\rho(\theta,\pi/2 - \epsilon) = \rho(\theta,\pi/2 + \epsilon)$. These properties require that the coefficients $A_{\ell,m}$ obey the relations $A^{^*}_{\ell,-m} = (-1)^m A_{\ell,m}$ and $A_{\ell,-m} = A_{\ell,m}$. This implies that all even coefficients, and in particular $A_{1,0}$, are real numbers.
|
---
abstract: |
The astrophysical factor $S_{\rm pp}(0)$ for the solar proton burning p + p $\to$ D + e$^+$ + $\nu_{\rm e}$ is recalculated in the relativistic field theory model of the deuteron (RFMD). We obtain $S_{\rm pp}(0) = 4.08 \times 10^{-25}\,{\rm MeV\,\rm b}$ which agrees good with the recommended value $S_{\rm pp}(0) = 4.00 \times 10^{-25}\,{\rm
MeV\,\rm b}$. The amplitude of low–energy elastic proton–proton (pp) scattering in the ${^1}{\rm S}_0$–state with the Coulomb repulsion contributing to the amplitude of the solar proton burning is described in terms of the S–wave scattering length and the effective range. This takes away the problem pointed out by Bahcall and Kamionkowski (Nucl. Phys. A625 (1997) 893) that in the RFMD one cannot describe low–energy elastic pp scattering with the Coulomb repulsion in agreement with low–energy nuclear phenomenology. The cross section for the neutrino disintegration of the deuteron $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p is calculated with respect to $S_{\rm pp}(0)$ for neutrino energies up to $E_{\nu_{\rm e}} \le 10\,{\rm MeV}$. The results can be used for the analysis of the data which will be obtained in the experiments planned by SNO. The astrophysical factor $S_{\rm pep}(0)$ for the process p + e$^-$ + p $\to$ $\nu_{\rm e}$ + D (or pep–process) is calculated relative to $S_{\rm pp}(0)$ in complete agreement with the result obtained by Bahcall and May (ApJ. 155 (1969) 501).
author:
- |
A. N. Ivanov [^1] ${
^\ddagger}$, H. Oberhummer [^2] , N. I. Troitskaya [^3] , M. Faber [^4]
title: 'Solar proton burning, neutrino disintegration of the deuteron and pep process in the relativistic field theory model of the deuteron'
---
[*Institut für Kernphysik, Technische Universität Wien,\
Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria*]{}
PACS: 11.10.Ef, 13.75.Cs, 14.20.Dh, 21.30.Fe, 26.65.+t\
Keywords: deuteron, proton burning, proton–proton scattering
Introduction
============
The weak nuclear process p + p $\to$ D + e$^+$ + $\nu_{\rm e}$, the solar proton burning or proton–proton (pp) fusion, plays an important role in Astrophysics \[1,2\]. It gives start for the p–p chain of nucleosynthesis in the Sun and the main–sequence stars \[1,2\]. In the Standard Solar Model (SSM) \[3\] the total (or bolometric) luminosity of the Sun $L_{\odot} = (3.846\pm
0.008)\times 10^{26}\,{\rm W}$ is normalized to the astrophysical factor $S_{\rm pp}(0)$ for pp fusion. The recommended value $S_{\rm
pp}(0) = 4.00\times 10^{-25}\,{\rm MeV b}$ \[4\] has been found by averaging over the results obtained in the Potential model approach (PMA) \[5,6\] and the Effective Field Theory (EFT) approach \[7,8\]. However, as has been shown recently in Ref.\[9\] [*the inverse and forward helioseismic approach indicate the higher values of $S_{\rm pp}(0)$ seem more favoured*]{}, for example, $S_{\rm pp}(0) =
4.20\times 10^{-25}\,{\rm MeV b}$ and higher \[9\]. Of course, accounting for the experimental errors the recommended value does not contradict the result obtained in Ref.\[9\].
In Refs.\[10-13\] we have developed a relativistic field theory model of the deuteron (RFMD). In turn, in Ref.\[14\] we have suggested a modified version of the RFMD which is not well defined due to a violation of Lorentz invariance of the effective four–nucleon interaction describing N + N $\to$ N + N transitions. This violation has turned out to be incompatible with a dominance of one–nucleon loop anomalies which are Lorentz covariant. Thereby, the astrophysical factor $S_{\rm
pp}(0)$ calculated in the modified version of the RFMD \[14\] and enhanced by a factor of 1.4 with respect to the recommended value \[4\] is not good established. This result demands the confirmation within the original RFMD \[10–13\] by using the technique expounded in Ref.\[13\].
As has been shown in Ref.\[12\] the RFMD is motivated by QCD. The deuteron appears in the nuclear phase of QCD as a neutron–proton collective excitation – a Cooper np–pair induced by a phenomenological local four–nucleon interaction. Strong low–energy interactions of the deuteron coupled to itself and other particles are described in terms of one–nucleon loop exchanges. The one–nucleon loop exchanges allow to transfer nuclear flavours from an initial to a final nuclear state by a minimal way and to take into account contributions of nucleon–loop anomalies determined completely by one–nucleon loop diagrams. The dominance of contributions of nucleon–loop anomalies has been justified in the large $N_C$ expansion, where $N_C$ is the number of quark colours \[13\]. Unlike the PMA and the EFT approach the RFMD takes into account non–perturbative contributions of high–energy (short–distance) fluctuations of virtual nucleon ($N$) and anti–nucleon ($\bar{N}$) fields, $N\bar{N}$ fluctuations, in the form of one–nucleon loop anomalies. In accord the analysis carried out in Refs.\[15\] nucleon–loop anomalies can be interpreted as non–perturbative contributions of the nucleon Dirac sea. The description of one–nucleon loop anomalies goes beyond the scope of both the PMA and the EFT approach due to the absence in these approaches anti–nucleon degrees of freedom related to the nucleon Dirac sea. However, one should notice that in low–energy nuclear physics the nucleon Dirac sea cannot be ignored fully \[16\]. For example, high–energy $N\bar{N}$ fluctuations of the nucleon Dirac sea polarized by the nuclear medium decrease the scalar nuclear density in the nuclear interior of finite nuclei by 15$\%$ \[16\]. This effect has been obtained within quantum field theoretic approaches in terms of one–nucleon loop exchanges.
In this paper we revise the value of $S_{\rm pp}(0)$ obtained in Ref.\[14\]. For this aim we apply the technique developed in the RFMD \[13\] for the description of contributions of low–energy elastic nucleon–nucleon scattering in the ${^1}{\rm S}_0$–state to amplitudes of electromagnetic and weak nuclear processes. This technique implies the summation of an infinite series of one–nucleon loop diagrams and the evaluation of the result of the summation in leading order in large $N_C$ expansion \[13\]. The application of this method to the evaluation of the cross sections for the anti–neutrino disintegration of the deuteron induced by charged $\bar{\nu}_{\rm e}$ + D $\to$ e$^+$ + n + n and neutral $\bar{\nu}_{\rm e}$ + D $\to$ $\bar{\nu}_{\rm e}$ + n + p weak currents gave the results agreeing good with the experimental data. The reaction $\bar{\nu}_{\rm e}$ + D $\to$ e$^+$ + n + n is, in the sense of charge independence of weak interaction strength, equivalent to the reaction p + p $\to$ D + e$^+$ + $\nu_{\rm e}$. Therefore, the application of the same technique to the description of the reaction p + p $\to$ D + e$^+$ + $\nu_{\rm e}$ should give a result of a good confidence level.
The paper is organized as follows. In Sect.2 we evaluate the amplitude of the solar proton burning. We show that the contribution of low–energy elastic pp scattering in the ${^1}{\rm S}_0$–state with the Coulomb repulsion is described in agreement with low–energy nuclear phenomenology in terms of the S–wave scattering length and the effective range. This takes away the problem pointed out by Bahcall and Kamionkowski \[17\] that in the RFMD one cannot describe low–energy elastic pp scattering with the Coulomb repulsion in agreement with low–energy nuclear phenomenology. In Sect.3 we evaluate the astrophysical factor for the solar proton burning and obtain the value $S_{\rm pp}(0) = 4.08\times 10^{-25}\,{\rm MeV\, b}$ agreeing good with the recommended one $S_{\rm pp}(0) = 4.00\times
10^{-25}\,{\rm MeV\, b}$. In Sect.4 we evaluate the cross section for the neutrino disintegration of the deuteron $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p caused by the charged weak current with respect to $S_{\rm pp}(0)$. In Sect.5 we adduce the evaluation of the astrophysical factor $S_{\rm pep}(0)$ of the reaction p + e$^-$ + p $\to$ D + $\nu_{\rm e}$ or pep–process relative to $S_{\rm
pp}(0)$. In the Conclusion we discuss the obtained results.
Amplitude of solar proton burning and low–energy elastic proton–proton scattering
=================================================================================
For the description of low–energy transitions N + N $\to$ N + N in the reactions n + p $\to$ D + $\gamma$, $\gamma$ + D $\to$ n + p, $\bar{\nu}_{\rm e}$ + D $\to$ e$^+$ + n + n and p + p $\to$ D + e$^+$ + $\nu_{\rm e}$, where nucleons are in the ${^1}{\rm S}_0$–state, we apply the effective local four–nucleon interactions \[11–13\]: $$\begin{aligned}
\label{label2.1}
&&{\cal L}^{\rm NN \to NN}_{\rm eff}(x)=G_{\rm \pi
NN}\,\{[\bar{n}(x)\gamma_{\mu}
\gamma^5 p^c(x)][\bar{p^c}(x)\gamma^{\mu}\gamma^5 n(x)]\nonumber\\
&&+\frac{1}{2}\,
[\bar{n}(x)\gamma_{\mu} \gamma^5 n^c(x)][\bar{n^c}(x)\gamma^{\mu}\gamma^5
n(x)] +
\frac{1}{2}\,[\bar{p}(x)\gamma_{\mu} \gamma^5 p^c(x)]
[\bar{p^c}(x)\gamma^{\mu}\gamma^5 p(x)]\nonumber\\
&&+ (\gamma_{\mu}\gamma^5 \otimes \gamma^{\mu}\gamma^5 \to \gamma^5 \otimes
\gamma^5)\},\end{aligned}$$ where $n(x)$ and $p(x)$ are the operators of the neutron and the proton interpolating fields, $n^c(x) = C \bar{n}^T(x)$ and so on, then $C$ is a charge conjugation matrix and $T$ is a transposition. The effective coupling constant $G_{\rm \pi NN}$ is defined by \[11–13\] $$\begin{aligned}
\label{label2.2}
G_{\rm \pi NN} = \frac{g^2_{\rm \pi NN}}{4M^2_{\pi}} - \frac{2\pi a_{\rm
np}}{M_{\rm N}} = 3.27\times 10^{-3}\,{\rm MeV}^{-2},\end{aligned}$$ where $g_{\rm \pi NN}= 13.4$ is the coupling constant of the ${\rm \pi NN}$ interaction, $M_{\pi}=135\,{\rm MeV}$ is the pion mass, $M_{\rm p} = M_{\rm n} = M_{\rm N} = 940\,{\rm MeV}$ is the mass of the proton and the neutron neglecting the electromagnetic mass difference, which is taken into account only for the calculation of the phase volumes of the final states of the reactions p + p $\to$ D + e$^+$ + $\nu_{\rm e}$, $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p and p + e$^-$ + p $\to$ D + $\nu_{\rm e}$, and $a_{\rm np} = (-23.75\pm
0.01)\,{\rm fm}$ is the S–wave scattering length of np scattering in the ${^1}{\rm S}_0$–state.
The effective Lagrangian for the low–energy nuclear transition p + p $\to$ D + e$^+$ + $\nu_{\rm e}$ has been calculated in Ref.\[12\] and reads $$\begin{aligned}
\label{label2.3}
{\cal L}_{\rm pp\to D e^+ \nu_{\rm e}}(x) = - i g_{\rm A}G_{\rm \pi
NN}M_{\rm N}\frac{G_{\rm V}}{\sqrt{2}}\frac{3g_{\rm
V}}{4\pi^2}\,D^{\dagger}_{\mu}(x)\,[\bar{p^c}(x)\gamma^5
p(x)]\,[\bar{\psi}_{\nu_{\rm e}}(x)\gamma^{\mu}(1 - \gamma^5) \psi_{\rm
e}(x)].\end{aligned}$$ where $G_{\rm V} = G_{\rm F}\,\cos \vartheta_C$ with $G_{\rm F} =
1.166\,\times\,10^{-11}\,{\rm MeV}^{-2}$ and $\vartheta_C$ are the Fermi weak coupling constant and the Cabibbo angle $\cos \vartheta_C =
0.975$, $g_{\rm A} = 1.2670 \pm 0.0035$ \[18\] and $g_{\rm V}$ is a phenomenological coupling constant of the RFMD related to the electric quadrupole moment of the deuteron $Q_{\rm D} = 0.286\,{\rm fm}^2$ \[11\]: $g^2_{\rm V} = 2\pi^2 Q_{\rm D}M^2_{\rm N}$. Then, $D_{\mu}(x)$, $\psi_{\nu_{\rm e}}(x)$ $\psi_{\rm e}(x)$ are the interpolating fields of the deuteron and leptonic pair, respectively.
The effective Lagrangian Eq.(\[label2.3\]) defines the effective vertex of the low–energy nuclear transition p + p $\to$ D + e$^+$ + $\nu_{\rm e}$ $$\begin{aligned}
\label{label2.4}
i{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+ + \nu_{e})&=&
\,G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi NN}\,\frac{3g_{\rm V}}{4\pi^2}\,
e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm e^+})]\nonumber\\
&&\times\,[\bar{u^c}(p_2) \gamma^5
u(p_1)],\end{aligned}$$ where $e^*_{\mu}(k_{\rm D})$ is a 4–vector of a polarization of the deuteron, $u(k_{\nu_{\rm e}})$, $v(k_{\rm e^+})$, $u(p_2)$ and $u(p_1)$ are the Dirac bispinors of neutrino, positron and two protons, respectively.
In order to evaluate the contribution of low–energy elastic pp scattering we have to determine the effective vertex of the p + p $\to$ p + p transition accounting for the Coulomb repulsion between the protons. For this aim we suggest to use the effective local four–nucleon interaction Eq.(\[label2.1\]) and take into account the Coulomb repulsion in terms of the explicit Coulomb wave function of the protons. This yields $$\begin{aligned}
\label{label2.5}
V_{\rm pp \to pp}(k',k) = G_{\rm \pi NN}\,\psi^*_{\rm pp}(k'\,)\,
[\bar{u}(p'_2) \gamma^5 u^c(p'_1)]\,[\bar{u^c}(p_2) \gamma^5
u(p_1)]\,\psi_{\rm pp}(k),\end{aligned}$$ where $\psi_{\rm pp}(k)$ and $\psi^*_{\rm pp}(k'\,)$ are the explicit Coulomb wave functions of the relative movement of the protons taken at zero relative radius, and $k$ and $k'$ are relative 3–momenta of the protons $\vec{k} = (\vec{p}_1 - \vec{p}_2)/2$ and $\vec{k}^{\,\prime} = (\vec{p}^{\;\prime}_1 - \vec{p}^{\;\prime}_2 )/2$ in the initial and final states. The explicit form of $\psi_{\rm pp}(k)$ we take following Kong and Ravndal \[8\] (see also \[19\]) $$\begin{aligned}
\label{label2.6}
\psi_{\rm pp}(k) = e^{\textstyle - \pi/4k r_C}\,\Gamma\Bigg(1 +
\frac{i}{2k r_C}\Bigg),\end{aligned}$$ where $r_C = 1/M_{\rm N}\alpha = 28.82\,{\rm fm}$ and $\alpha = 1/137$ are the Bohr radius of a proton and the fine structure constant. The squared value of the modulo of $\psi_{\rm pp}(k)$ is given by $$\begin{aligned}
\label{label2.7}
|\psi_{\rm pp}(k)|^2 = C^2_0(k) = \frac{\pi}{k r_C}\,
\frac{1}{\displaystyle e^{\textstyle \pi/k r_C} - 1},\end{aligned}$$ where $C_0(k)$ is the Gamow penetration factor \[1,2,19\]. We would like to emphasize that the wave function Eq.(\[label2.6\]) is defined only by a regular solution of the Schrödinger equation for the pure Coulomb potential \[19\].
By taking into account the contribution of the Coulomb wave function and summing up an infinite series of one–proton loop diagrams the amplitude of the solar proton burning can be written in the form $$\begin{aligned}
\label{label2.8}
\hspace{-0.5in}&&i{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+ +
\nu_{e}) = G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi
NN}\,\frac{3g_{\rm V}}{4\pi^2}\, e^*_{\mu}(k_{\rm
D})\,[\bar{u}(k_{\nu_{\rm e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm
e^+})]\,{\cal F}^{\rm e}_{\rm pp}\nonumber\\
\hspace{-0.5in}&&\times\,\frac{[\bar{u^c}(p_2) \gamma^5 u(p_1)]\,\psi_{\rm
pp}(k)}{\displaystyle 1 + \frac{G_{\rm \pi NN}}{16\pi^2}\int
\frac{d^4p}{\pi^2i}\,|\psi_{\rm pp}(|\vec{p} + \vec{Q}\,|)|^2 {\rm
tr}\Bigg\{\gamma^5 \frac{1}{M_{\rm N} - \hat{p} - \hat{P} -
\hat{Q}}\gamma^5 \frac{1}{M_{\rm N} - \hat{p} - \hat{Q}}\Bigg\}}.\end{aligned}$$ where $P = p_1 + p_2 = (2\sqrt{k^2 + M^2_{\rm N}}, \vec{0}\,)$ is the 4–momentum of the pp–pair in the center of mass frame; $Q =a\,P +
b\,K = a\,(p_1 + p_2) + b\,(p_1 - p_2)$ is an arbitrary shift of virtual momentum with arbitrary parameters $a$ and $b$, and in the center of mass frame $K = p_1 - p_2 = (0,2\,\vec{k}\,)$ \[14\]. The parameters $a$ and $b$ can be functions of $k$. The factor ${\cal
F}^{\rm e}_{\rm pp}$ describes the overlap of the Coulomb and strong interactions \[10\]. It is analogous the overlap integral in the PMA \[5\]. We calculate this factor below.
The evaluation of the momentum integral runs the way expounded in \[14\]. Keep only leading contributions in the large $N_C$ expansion \[13,14\] we obtain $$\begin{aligned}
\label{label2.9}
&&\int \frac{d^4p}{\pi^2i}\,|\psi_{\rm
pp}(|\vec{p} + \vec{Q}\,|)|^2 {\rm tr}\Bigg\{\gamma^5 \frac{1}{M_{\rm N}
- \hat{p} - \hat{P} - \hat{Q}}\gamma^5 \frac{1}{M_{\rm N} - \hat{p} -
\hat{Q}}\Bigg\} =\nonumber\\
&&= - 8\, a\,(a + 1)\,M^2_{\rm N} + 8\,(b^2 - a\,(a +
1))\,k^2 - i\,8\pi\,M_{\rm N}\,k\,|\psi_{\rm pp}(k)|^2 = \nonumber\\
&&= - 8\, a\,(a + 1)\,M^2_{\rm N} + 8\,(b^2 - a\,(a +
1))\,k^2 - i\,8\pi\,M_{\rm N}\,k\,C^2_0(k).\end{aligned}$$ Substituting Eq.(\[label2.9\]) in Eq.(\[label2.8\]) we get $$\begin{aligned}
\label{label2.10}
\hspace{-0.7in}&& i{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+
+ \nu_{e}) = G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi
NN}\,\frac{3g_{\rm V}}{4\pi^2}\,{\cal F}^{\rm e}_{\rm pp}\nonumber\\
\hspace{-0.7in}&&\times \, e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm e^+})]\,[\bar{u^c}(p_2)
\gamma^5 u(p_1)]\,e^{\textstyle - \pi/4k
r_C}\,\Gamma\Bigg(1 + \frac{i}{2k r_C}\Bigg)\nonumber\\
\hspace{-0.7in}&&\Bigg[ 1 - a(a+1) \frac{G_{\rm
\pi NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2 - i\,\frac{G_{\rm \pi NN}M_{\rm
N}}{2\pi}\,k\,C^2_0(k)\Bigg]^{-1}\!\!\!.\end{aligned}$$ In order to reconcile the contribution of low–energy elastic pp scattering with low–energy nuclear phenomenology \[19\] we should make a few changes. For this aim we should rewrite Eq.(\[label2.10\]) in more convenient form $$\begin{aligned}
\label{label2.11}
\hspace{-0.7in}&& i{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+
+ \nu_{e}) = G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi
NN}\,\frac{3g_{\rm V}}{4\pi^2}\,{\cal F}^{\rm e}_{\rm pp}\nonumber\\
\hspace{-0.7in}&&\times \, e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm e^+})]\,[\bar{u^c}(p_2)
\gamma^5 u(p_1)]\,e^{\textstyle i\sigma_0(k)}\,C_0(k)\nonumber\\
\hspace{-0.7in}&&\Bigg[ 1 - a(a+1) \frac{G_{\rm
\pi NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2 - i\,\frac{G_{\rm \pi NN}M_{\rm
N}}{2\pi}\,k\,C^2_0(k)\Bigg]^{-1}\!\!\!.\end{aligned}$$ We have denoted $$\begin{aligned}
\label{label2.12}
e^{\textstyle - \pi/4k r_C}\,\Gamma\Bigg(1 + \frac{i}{2k r_C}\Bigg)
= e^{\textstyle i\sigma_0(k)}\,C_0(k)\;,\;
\sigma_0(k)&=&{\rm arg}\,\Gamma\Bigg(1 + \frac{i}{2k r_C}\Bigg),\end{aligned}$$ where $\sigma_0(k)$ is a pure Coulomb phase shift.
Now, let us rewrite the denominator of the amplitude Eq.(\[label2.11\]) in the equivalent form $$\begin{aligned}
\label{label2.13}
&& \Bigg\{\cos\sigma_0(k)\Bigg[1 - a(a+1) \frac{G_{\rm \pi
NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2\Bigg]\nonumber\\
&& - \sin\sigma_0(k)\,\frac{G_{\rm \pi NN}M_{\rm
N}}{2\pi}\,k\,C^2_0(k)\Bigg\}- i\,\Bigg\{\cos\sigma_0(k)\,\frac{G_{\rm
\pi NN}M_{\rm N}}{2\pi}\,k\,C^2_0(k)\nonumber\\
&& + \sin\sigma_0(k)\Bigg[1 - a(a+1) \frac{G_{\rm \pi
NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2\Bigg]\Bigg\}=\nonumber\\
&&= \frac{1}{Z}\Bigg[1 -
\frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_C}\,h(2 k r_C) + i\,a^{\rm e}_{\rm pp}\,k\,C^2_0(k)\Bigg],\end{aligned}$$ where we have denoted $$\begin{aligned}
\label{label2.14}
\hspace{-0.3in}&& \frac{1}{Z}\Bigg[1 - \frac{1}{2}\,a^{\rm e}_{\rm pp}
r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm e}_{\rm pp}}{r_C}\,h(2 k r_C)
\Bigg]= - \sin\sigma_0(k)\,\frac{G_{\rm \pi NN}M_{\rm
N}}{2\pi}\,k\,C^2_0(k)\nonumber\\
\hspace{-0.3in}&& + \cos\sigma_0(k)\Bigg[1 - a(a+1) \frac{G_{\rm
\pi NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2\Bigg],\nonumber\\
\hspace{-0.3in}&& - \frac{1}{Z}\,a^{\rm e}_{\rm
pp}\,k\,C^2_0(k) =\cos\sigma_0(k)\frac{G_{\rm \pi NN}M_{\rm
N}}{2\pi}\,k\,C^2_0(k)\nonumber\\
\hspace{-0.3in}&& + \sin\sigma_0(k)\Bigg[1 - a(a+1) \frac{G_{\rm \pi
NN}}{2\pi^2}\,M^2_{\rm N} + \frac{G_{\rm \pi NN}}{2\pi^2}\,(b^2 -
a\,(a + 1))\,k^2\Bigg].\end{aligned}$$ Here $Z$ is a constant which will be removed the renormalization of the wave functions of the protons, $a^{\rm e}_{\rm pp} = ( - 7.8196\pm
0.0026)\,{\rm fm}$ and $r^{\rm e}_{\rm pp} = 2.790\pm 0.014\,{\rm
fm}$ \[20\] are the S–wave scattering length and the effective range of pp scattering in the ${^1}{\rm S}_0$–state with the Coulomb repulsion, and $h(2 k r_C)$ is defined by \[19\] $$\begin{aligned}
\label{label2.15}
h(2 k r_C) = - \gamma + {\ell n}(2 k r_C) +
\sum^{\infty}_{n=1}\frac{1}{n(1 + 4n^2k^2r^2_C)}.\end{aligned}$$ The validity of the relations Eq.(\[label2.14\]) assumes the dependence of parameters $a$ and $b$ on the relative momentum $k$.
After the changes Eq.(\[label2.11\])–Eq.(\[label2.14\]) the amplitude Eq.(\[label2.10\]) takes the form $$\begin{aligned}
\label{label2.16}
\hspace{-0.2in}&& i{\cal M}({\rm p} + {\rm p} \to {\rm D} +
{\rm e}^+ + \nu_{e}) =
G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi NN}\,\frac{3g_{\rm
V}}{4\pi^2}\,e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm e^+})]\,{\cal F}^{\rm e}_{\rm
pp}\nonumber\\ \hspace{-0.2in}&&\times \,\frac{C_0(k)}{\displaystyle1 -
\frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_C}\,h(2 k r_C) + i\,a^{\rm e}_{\rm pp}\,k\,C^2_0(k)
}\,Z\,[\bar{u^c}(p_2) \gamma^5 u(p_1)].\end{aligned}$$ Following \[14\] and renormalizing the wave functions of the protons $\sqrt{Z}u(p_2) \to u(p_2)$ and $\sqrt{Z}u(p_1) \to u(p_1)$ we obtain the amplitude of the solar proton burning $$\begin{aligned}
\label{label2.17}
\hspace{-0.2in}&& i{\cal M}({\rm p} + {\rm p} \to {\rm D}
+ {\rm e}^+ + \nu_{e}) =
G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi NN}\,\frac{3g_{\rm
V}}{4\pi^2}\,e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) v(k_{\rm e^+})]\,{\cal F}^{\rm e}_{\rm
pp}\nonumber\\ \hspace{-0.2in}&&\times \,
\frac{\displaystyle C_0(k)}{\displaystyle1 -
\frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_C}\,h(2 k r_C) + i\,a^{\rm e}_{\rm pp}\,k\,C^2_0(k)
}\,[\bar{u^c}(p_2) \gamma^5 u(p_1)]\,F_{\rm D}(k^2),\end{aligned}$$ where we have introduced too an universal form factor \[14\] $$\begin{aligned}
\label{label2.18}
F_{\rm D}(k^2) = \frac{1}{1 + r^2_{\rm D}k^2}\end{aligned}$$ describing a spatial smearing of the deuteron coupled to the NN system in the ${^1}{\rm S}_0$–state at low energies; $r_{\rm D} =
1/\sqrt{\varepsilon_{\rm D}M_{\rm N}} = 4.315\,{\rm fm}$ is the radius of the deuteron and $\varepsilon_{\rm D} = 2.225\,{\rm MeV}$ is the binding energy of the deuteron.
The real part of the denominator of the amplitude Eq.(\[label2.17\]) is in complete agreement with a phenomenological relation \[19\] $$\begin{aligned}
\label{label2.19}
{\rm ctg}\delta^{\rm e}_{\rm pp}(k) = \frac{1}{\displaystyle
C^2_0(k)\,k}\,\Bigg[ - \frac{1}{a^{\rm e}_{\rm pp}} + \frac{1}{2}\,r^{\rm
e}_{\rm pp}k^2 - \frac{1}{r_{\rm C}}\,h(2 k r_{\rm C})\Bigg],\end{aligned}$$ describing the phase shift $\delta^{\rm e}_{\rm pp}(k)$ of low–energy elastic pp scattering in terms of the S–wave scattering length $a^{\rm e}_{\rm pp}$ and the effective range $r^{\rm e}_{\rm pp}$. As has been pointed out \[19\] the expansion Eq.(\[label2.19\]) is valid up to $T_{\rm pp}\le 10\,{\rm MeV}$, where $T_{\rm pp} = k^2/M_{\rm
N}$ is a kinetic energy of the relative movement of the protons.
Thus, we argue that the contribution of low–energy elastic pp scattering to the amplitude of the solar proton burning is described in agreement with low–energy nuclear phenomenology in terms of the S–wave scattering length $a^{\rm e}_{\rm pp}$ and the effective range $r^{\rm e}_{\rm pp}$ taken from the experimental data \[20\]. This takes away the problem pointed out by Bahcall and Kamionkowski \[17\] that in the RFMD with the local four–nucleon interaction given by Eq.(\[label2.1\]) one cannot describe low–energy elastic pp scattering with the Coulomb repulsion in agreement with low–energy nuclear phenomenology.
Now let us proceed to the evaluation of ${\cal F}^{\rm e}_{\rm
pp}$. For this aim we should write down the matrix element of the transition p + p $\to$ D + e$^+$ + $\nu_{\rm e}$ with the Coulomb repulsion. The required matrix element has been derived in Refs.\[11,14\] and reads $$\begin{aligned}
\label{label2.20}
\hspace{-0.3in}&&i{\cal M}_C({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+ +
\nu_{e}) =\nonumber\\
\hspace{-0.3in}&&= G_{\rm V}\,g_{\rm A} M_{\rm
N}\,G_{\rm \pi NN}\,\frac{3g_{\rm
V}}{4\pi^2}\,C_0(k)\,[\bar{u}(k_{\nu_{\rm e}})\gamma^{\mu}
(1-\gamma^5) v(k_{\rm e^+})]\,e^*_{\mu}(k_{\rm
D})\,\nonumber\\
\hspace{-0.3in}&&\times\,\{- [\bar{u^c}(p_2)\gamma_{\alpha} \gamma^5
u(p_1)]{\cal J}^{\alpha\mu\nu}_C(k_{\rm D}, k_{\ell}) -
[\bar{u^c}(p_2)\gamma^5 u(p_1)]\,{\cal J}^{\mu\nu}_C(k_{\rm D},
k_{\ell})\},\end{aligned}$$ where $k_{\rm D}$ and $k_{\ell}$ are 4–momenta of the deuteron and the leptonic pair, respectively. The structure functions ${\cal
J}^{\alpha\mu\nu}(k_{\rm D}, k_{\ell})$ and ${\cal J}^{\mu\nu}(k_{\rm
D}, k_{\ell})$ are determined by \[11,14\] $$\begin{aligned}
\label{label2.21}
&&{\cal J}^{\alpha\mu\nu}_C(k_{\rm D}, k_{\ell}) =
\int\frac{d^4p}{\pi^2i}\,e^{\textstyle - \pi/4|\vec{q}\,|
r_C}\,\Gamma\Bigg(1 - \frac{i}{2 |\vec{q}\,| r_C}\Bigg)\nonumber\\
&&\times\,{\rm tr} \Bigg\{\gamma^{\alpha}\gamma^5\frac{1}{M_{\rm N} - \hat{p}
+ \hat{k}_{\rm D}}\gamma^{\mu}\frac{1}{M_{\rm N} -
\hat{p}}\gamma^{\nu}\gamma^5 \frac{1}{M_{\rm N} - \hat{p} -
\hat{k}_{\ell}}\Bigg\},\nonumber\\
&&{\cal J}^{\mu\nu}_C(k_{\rm D},k_{\ell}) =
\int\frac{d^4p}{\pi^2i}\,e^{\textstyle - \pi/4|\vec{q}\,|
r_C}\,\Gamma\Bigg(1 - \frac{i}{2 |\vec{q}\,| r_C}\Bigg)\nonumber\\
&&\times\,{\rm tr} \Bigg\{\gamma^5\frac{1}{M_{\rm N} - \hat{p}
+ \hat{k}_{\rm D}}\gamma^{\mu}\frac{1}{M_{\rm N} -
\hat{p}}\gamma^{\nu}\gamma^5 \frac{1}{M_{\rm N} - \hat{p} -
\hat{k}_{\ell}}\Bigg\},\end{aligned}$$ where $\vec{q} = \vec{p} + (\vec{k}_{\ell} - \vec{k}_{\rm D})/2$.
For the subsequent analysis it is convenient to represent the structure functions in the form of two terms $$\begin{aligned}
\label{label2.22}
{\cal J}^{\alpha\mu\nu}_C(k_{\rm D}, k_{\ell})&=&{\cal
J}^{\alpha\mu\nu}_{SS}(k_{\rm D}, k_{\ell}) + {\cal
J}^{\alpha\mu\nu}_{SC}(k_{\rm D}, k_{\ell}),\nonumber\\
{\cal J}^{\mu\nu}_C(k_{\rm D}, k_{\ell})&=&{\cal
J}^{\mu\nu}_{SS}(k_{\rm D}, k_{\ell}) + {\cal
J}^{\mu\nu}_{SC}(k_{\rm D}, k_{\ell}).\end{aligned}$$ The decomposition is caused by the change $$\begin{aligned}
\label{label2.23}
e^{\textstyle - \pi/4|\vec{q}\,| r_C}\,\Gamma\Bigg(1 - \frac{i}{2
|\vec{q}\,| r_C}\Bigg) = 1 + \Bigg[e^{\textstyle - \pi/4|\vec{q}\,|
r_C}\,\Gamma\Bigg(1 - \frac{i}{2 |\vec{q}\,| r_C}\Bigg) - 1\Bigg],\end{aligned}$$ where the first term gives the contribution to the $SS$ part of the structure functions defined by strong interactions only, while the second one vanishes at $r_C \to \infty$ ( or $\alpha \to 0$) and describes the contribution to the $SC$ part of the structure functions caused by both strong and Coulomb interactions.
The procedure of the evaluation of the structure functions Eq.(\[label2.21\]) and Eq.(\[label2.22\]) has been described in detail in Ref.\[11,14\]. Following this procedure we obtain ${\cal
F}^{\rm e}_{\rm pp}$ in the form $$\begin{aligned}
\label{label2.24}
\hspace{-0.3in}&&{\cal F}^{\rm e}_{\rm pp} =\nonumber\\
\hspace{-0.3in}&&= 1 +
\frac{32}{9}\int\limits^{\infty}_0 dp\,p^2\,\Bigg[e^{\textstyle -
\pi/4p r_C}\,\Gamma\Bigg(1 - \frac{i}{2pr_C}\Bigg)-
1\Bigg]\Bigg[\frac{M^2_{\rm N}}{(M^2_{\rm N} + p^2)^{5/2}} -
\frac{7}{16}\,\frac{1}{(M^2_{\rm N} + p^2)^{3/2}}\Bigg]\nonumber\\
\hspace{-0.3in}&&= 1 + \frac{32}{9}\int\limits^{\infty}_0
dv\,v^2\,\Bigg[e^{\textstyle - \alpha\pi/4v}\,\Gamma\Bigg(1 -
\frac{i\alpha}{2v}\Bigg)- 1\Bigg]\Bigg[\frac{1}{(1 + v^2)^{5/2}} -
\frac{7}{16}\,\frac{1}{(1 + v^2)^{3/2}}\Bigg].\end{aligned}$$ The integral can be estimated perturbatively. The result reads $$\begin{aligned}
\label{label2.25}
{\cal F}^{\rm e}_{\rm pp} = 1 +
\alpha\,\Bigg(\frac{5\pi}{54} - i\,\frac{5\gamma}{27}\Bigg) + O(\alpha^2).\end{aligned}$$ The numerical value of $|{\cal F}^{\rm e}_{\rm pp}|^2$ is $$\begin{aligned}
\label{label2.26}
|{\cal F}^{\rm e}_{\rm pp}|^2 = 1 + \alpha\,\frac{5\pi}{27} +
O(\alpha^2) = 1 +(4.25 \times 10^{-3}) \simeq 1.\end{aligned}$$ The contribution of the Coulomb field Eq.(\[label2.26\]) inside the one–nucleon loop diagrams is found small. This is because of the integrals are concentrated around virtual momenta of order of $M_{\rm
N}$ which is of order $M_{\rm N} \sim N_C$ in the large $N_C$ expansion \[12\]. For the calculation of the astrophysical factor $S_{\rm pp}(0)$ we can set ${\cal F}^{\rm e}_{\rm pp} = 1$.
Astrophysical factor for solar proton burning
=============================================
The amplitude Eq.(\[label2.17\]) squared, averaged over polarizations of protons and summed over polarizations of final particles reads $$\begin{aligned}
\label{label3.1}
\hspace{-0.5in}&&\overline{|{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm
e}^+ + \nu_{e})|^2}= G^2_{\rm V} g^2_{\rm A} M^4_{\rm
N}\,G^2_{\rm \pi NN}\,\frac{9Q_{\rm D}}{8\pi^2}\,F^2_{\rm D}(k^2)\nonumber\\
\hspace{-0.5in}&&\times\,\frac{\displaystyle C^2_0(k)}
{\displaystyle \Big[1 - \frac{1}{2}\,a^{\rm e}_{\rm pp}
r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm e}_{\rm pp}}{r_C}\,h(2 k r_C)\Big]^2 +
(a^{\rm e}_{\rm pp})^2 k^2 C^4_0(k)}\,\Bigg(-
g^{\alpha\beta}+\frac{k^{\alpha}_{\rm D}k^{\beta}_{\rm D}}{M^2_{\rm
D}}\Bigg)\nonumber\\
\hspace{-0.5in}&&\times {\rm tr}\{(- m_{\rm e} + \hat{k}_{\rm
e^+})\gamma_{\alpha}(1-\gamma^5) \hat{k}_{\nu_{\rm
e}}\gamma_{\beta}(1-\gamma^5)\}\times \frac{1}{4}\times {\rm tr}\{(M_{\rm
N} - \hat{p}_2) \gamma^5 (M_{\rm N} + \hat{p}_1) \gamma^5\},\end{aligned}$$ where $m_{\rm e}=0.511\,{\rm MeV}$ is the positron mass, and we have used the relation $g^2_{\rm V} = 2\,\pi^2\,Q_{\rm D}\,M^2_{\rm
N}$.
In the low–energy limit the computation of the traces yields $$\begin{aligned}
\label{label3.2}
\hspace{-0.5in}&&\Bigg(- g^{\alpha\beta}+\frac{k^{\alpha}_{\rm
D}k^{\beta}_{\rm D}}{M^2_{\rm D}}\Bigg)\,\times\,{\rm tr}\{( - m_{\rm e} +
\hat{k}_{\rm e^+})\gamma_{\alpha}(1-\gamma^5) \hat{k}_{\nu_{\rm
e}}\gamma_{\beta}(1-\gamma^5)\}= \nonumber\\
\hspace{-0.5in}&& = 24\,\Bigg( E_{\rm e^+} E_{\nu_{\rm e}} -
\frac{1}{3}\vec{k}_{\rm e^+}\cdot \vec{k}_{\nu_{\rm e}}\,\Bigg) ,\nonumber\\
\hspace{-0.5in}&&\frac{1}{4}\,\times\,{\rm tr}\{(M_{\rm N} - \hat{p}_2)
\gamma^5 (M_{\rm N} + \hat{p}_1) \gamma^5\} = 2\,M^2_{\rm N},\end{aligned}$$ where we have neglected the relative kinetic energy of the protons with respect to the mass of the proton.
Substituting Eq. (\[label3.2\]) in Eq. (\[label3.1\]) we get $$\begin{aligned}
\label{label3.3}
\hspace{-0.2in}&&\overline{|{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+ +
\nu_{e})|^2} = \,G^2_{\rm V}\, g^2_{\rm A} M^6_{\rm N}\,G^2_{\rm
\pi NN}\,\frac{54 Q_{\rm D}}{\pi^2}\,F^2_{\rm D}(k^2)\nonumber\\
\hspace{-0.2in}&&\times\,\frac{\displaystyle C^2_0(k)}
{\displaystyle \Big[1 - \frac{1}{2}\,a^{\rm e}_{\rm pp}
r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm e}_{\rm pp}}{r_C}\,h(2 k r_C)\Big]^2 +
(a^{\rm e}_{\rm pp})^2 k^2 C^4_0(k)}\,\Bigg( E_{\rm e^+} E_{\nu_{\rm e}} -
\frac{1}{3}\vec{k}_{\rm e^+}\cdot \vec{k}_{\nu_{\rm e}}\Bigg).\end{aligned}$$ The integration over the phase volume of the final ${\rm D}{\rm e}^+
\nu_{\rm e}$–state we perform in the non–relativistic limit $$\begin{aligned}
\label{label3.4}
\hspace{-0.5in}&&\int\frac{d^3k_{\rm D}}{(2\pi)^3 2E_{\rm
D}}\frac{d^3k_{\rm e^+}}{(2\pi)^3 2E_{\rm e^+}}\frac{d^3k_{\nu_{\rm
e}}}{(2\pi)^3 2 E_{\nu_{\rm e}}}\,(2\pi)^4\,\delta^{(4)}(k_{\rm D} +
k_{\ell} - p_1 - p_2)\,\Bigg( E_{\rm e^+} E_{\nu_{\rm e}} -
\frac{1}{3}\vec{k}_{\rm e^+}\cdot \vec{k}_{\nu_{\rm e}}\,\Bigg)\nonumber\\
\hspace{-0.5in}&&= \frac{1}{32\pi^3 M_{\rm N}}\,\int^{W + T_{\rm
pp}}_{m_{\rm e}}\sqrt{E^2_{\rm e^+}-m^2_{\rm e}}\,E_{\rm e^+}(W + T_{\rm
pp} - E_{\rm e^+})^2\,d E_{\rm e^+} = \frac{(W + T_{\rm pp})^5}{960\pi^3
M_{\rm N}}\,f(\xi),\end{aligned}$$ where $W = \varepsilon_{\rm D} - (M_{\rm n} - M_{\rm p}) = (2.225
-1.293)\,{\rm MeV} = 0.932\,{\rm MeV}$ and $\xi = m_{\rm e}/(W +
T_{\rm pp})$. The function $f(\xi)$ is defined by the integral $$\begin{aligned}
\label{label3.5}
\hspace{-0.5in}f(\xi)&=&30\,\int^1_{\xi}\sqrt{x^2 -\xi^2}\,x\,(1-x)^2 dx=(1
- \frac{9}{2}\,\xi^2 - 4\,\xi^4)\,\sqrt{1-\xi^2}\nonumber\\
\hspace{-0.5in}&&+ \frac{15}{2}\,\xi^4\,{\ell
n}\Bigg(\frac{1+\sqrt{1-\xi^2}}{\xi}\Bigg)\Bigg|_{T_{\rm pp} = 0} = 0.222\end{aligned}$$ and normalized to unity at $\xi = 0$.
Thus, the cross section for the solar proton burning is given by $$\begin{aligned}
\label{label3.6}
&&\sigma_{\rm pp}(T_{\rm pp}) = \frac{e^{\displaystyle
- \pi/r_C\sqrt{M_{\rm N}T_{\rm pp}}}}{v^2}\,
\alpha\,\frac{9g^2_{\rm A} G^2_{\rm V} Q_{\rm D}
M^3_{\rm N}}{320\,\pi^4}\,G^2_{\rm \pi NN}\,(W + T_{\rm
pp})^5\,f\Bigg(\frac{m_{\rm e}}{W + T_{\rm pp}}\Bigg)\nonumber\\
&&\times\,\frac{\displaystyle F^2_{\rm D}(M_{\rm N}T_{\rm pp})}{
\displaystyle \Big[1 - \frac{1}{2}\,a^{\rm e}_{\rm pp}
r^{\rm e}_{\rm pp}M_{\rm N}T_{\rm pp} + \frac{a^{\rm e}_{\rm
pp}}{r_C}\,h(2 r_C\sqrt{M_{\rm N}T_{\rm pp}})\Big]^2 + (a^{\rm e}_{\rm
pp})^2M_{\rm N}T_{\rm pp} C^4_0(\sqrt{M_{\rm N}T_{\rm pp}})}
=\nonumber\\ &&\hspace{1.2in} = \frac{S_{\rm pp}(T_{\rm pp})}{T_{\rm
pp}}\,e^{\displaystyle - \pi/r_C\sqrt{M_{\rm N}T_{\rm pp}}}.\end{aligned}$$ The astrophysical factor $S_{\rm pp}(T_{\rm pp})$ reads $$\begin{aligned}
\label{label3.7}
\hspace{-0.5in}&&S_{\rm pp}(T_{\rm pp}) =
\alpha\,\frac{9g^2_{\rm A}G^2_{\rm V}Q_{\rm
D}M^4_{\rm N}} {1280\pi^4}\,G^2_{\rm \pi NN}\,(W + T_{\rm
pp})^5\,f\Bigg(\frac{m_{\rm e}}{W + T_{\rm pp}}\Bigg)\nonumber\\
\hspace{-0.5in}&&\times\, \frac{\displaystyle F^2_{\rm D}(M_{\rm
N}T_{\rm pp})} {\displaystyle
\Big[1 - \frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}M_{\rm
N}T_{\rm pp} + \frac{a^{\rm e}_{\rm pp}}{r_C}\,h(2 r_C\sqrt{M_{\rm
N}T_{\rm pp}})\Big]^2 + (a^{\rm e}_{\rm pp})^2M_{\rm N}T_{\rm pp}
C^4_0(\sqrt{M_{\rm N}T_{\rm pp}})}.\end{aligned}$$ At zero kinetic energy of the relative movement of the protons $T_{\rm
pp} = 0$ the astrophysical factor $S_{\rm pp}(0)$ is given by $$\begin{aligned}
\label{label3.8}
\hspace{-0.5in}S_{\rm pp}(0) =\alpha\,\frac{9g^2_{\rm A}G^2_{\rm V}Q_{\rm
D}M^4_{\rm N}}{1280\pi^4}\,G^2_{\rm \pi NN}\,W^5\,f\Bigg(\frac{m_{\rm
e}}{W}\Bigg) = 4.08\,\times 10^{-25}\,{\rm MeV\,\rm b}.\end{aligned}$$ The value $S_{\rm pp}(0) = 4.08 \times 10^{-25}\,{\rm MeV\,\rm b}$ agrees good with the recommended value $S_{\rm pp}(0) = 4.00 \times
10^{-25}\,{\rm MeV\,\rm b}$ \[4\]. Insignificant disagreement with the result obtained in Ref.\[11\] where we have found $S_{\rm pp}(0) = 4.02
\times 10^{-25}\,{\rm MeV\,\rm b}$ is due to the new value of the constant $g_{\rm A} = 1.260 \to 1.267$ \[18\] (see Ref.\[13\]).
Unlike the astrophysical factor obtained by Kamionkowski and Bahcall \[5\] the astrophysical factor given by Eq.(\[label3.8\]) does not depend explicitly on the S–wave scattering wave of pp scattering. This is due to the normalization of the wave function of the relative movement of two protons. After the summation of an infinite series and by using the relation Eq.(\[label2.19\]) we obtain the wave function of two protons in the form $$\begin{aligned}
\label{label3.9}
\psi_{\rm pp}(k)= e^{\textstyle i\,\delta^{\,\rm e}_{\rm
pp}(k)}\,\frac{\sin\,\delta^{\,\rm e}_{\rm pp}(k)}{-a^{\rm e}_{\rm pp}kC_0(k)},\end{aligned}$$ that corresponds the normalization of the wave function of the relative movement of two protons used by Schiavilla [*et al.*]{} \[6\]. For the more detailed discussion of this problem we relegate readers to the paper by Schiavilla [*et al.*]{} \[6\][^5].
Unfortunately, the value of the astrophysical factor $S_{\rm pp}(0) =
4.08 \times 10^{-25}\,{\rm MeV\,\rm b}$ does not confirm the enhancement by a factor of 1.4 obtained in the modified version of the RFMD in Ref.\[14\].
Neutrino disintegration of the deuteron induced by charged weak current
=======================================================================
The evaluation of the amplitude of the process $\nu_{\rm e}$ + D $\to$ ${\rm e}^-$ + p + p has been given in details in Ref.\[10\]. The result can be written in the following form $$\begin{aligned}
\label{label4.1}
\hspace{-0.3in}&& i{\cal M}({\rm p} + {\rm p} \to {\rm D} + {\rm e}^+
+ \nu_{e}) = g_{\rm A} M_{\rm N} \frac{G_{\rm
V}}{\sqrt{2}}\,\frac{3g_{\rm V}}{2\pi^2}\, G_{\rm \pi NN}
\,e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\rm e^-})\gamma^{\mu}(1-\gamma^5)
u(k_{\nu_{\rm e}})]\,{\cal F}^{\rm e}_{\rm
ppe^-}\nonumber\\\hspace{-0.3in} &&\times
\,\frac{C_0(k)}{\displaystyle1 - \frac{1}{2}\,a^{\rm e}_{\rm pp}
r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm e}_{\rm pp}}{r_C}\,h(2 k r_C) +
i\,a^{\rm e}_{\rm pp}\,k\,C^2_0(k) }\,[\bar{u^c}(p_2) \gamma^5
u(p_1)]\,F_{\rm D}(k^2),\end{aligned}$$ where ${\cal F}^{\rm e}_{\rm ppe^-}$ is the overlap factor which we evaluate below, and $F_{\rm D}(k^2)$ is the universal form factor Eq.(\[label2.17\]) describing a spatial smearing of the deuteron \[14\].
The amplitude Eq.(\[label4.1\]) squared, averaged over polarizations of the deuteron and summed over polarizations of the final particles reads $$\begin{aligned}
\label{label4.2}
&&\overline{|{\cal M}(\nu_{\rm e} + {\rm D} \to {\rm e}^- + {\rm p} +
{\rm p})|^2} = g^2_{\rm A}M^6_{\rm N}\frac{144 G^2_{\rm V}Q_{\rm
D}}{\pi^2}\,G^2_{\rm \pi NN}\,|{\cal F}^{\rm e}_{\rm
ppe^-}|^2\,\,F^2_{\rm D}(k^2)\,F(Z, E_{\rm e^-})\nonumber\\ &&\times
{\displaystyle \frac{\displaystyle C^2_0(k)}{\displaystyle \Big[1 -
\frac{1}{2}a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp} k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_{\rm C}}\,h(2kr_{\rm C})\Big]^2 + (a^{\rm e}_{\rm
pp})^2k^2 C^4_0(k)}} \Bigg( E_{{\rm e^-}}E_{{\nu}_{\rm e}} -
\frac{1}{3}\vec{k}_{{\rm e^-}}\cdot \vec{k}_{{\nu}_{\rm e}}\Bigg),\end{aligned}$$ where $F(Z,E_{\rm e^-}$ is the Fermi function \[21\] describing the Coulomb interaction of the electron with the nuclear system having a charge $Z$. In the case of the reaction $\nu_{\rm e}$ + D $\to$ e$^+$ + p + p we have $Z = 2$. At $\alpha^2 Z^2 \ll 1$ the Fermi function $F(Z,E_{\rm e^-})$ reads \[21\] $$\begin{aligned}
\label{label4.3}
F(Z,E_{\rm e^-}) = \frac{2\pi \eta_{\rm e^-}}{\displaystyle 1 -
e^{\textstyle -2\pi \eta_{\rm e^-}}},\end{aligned}$$ where $\eta_{\rm e^-} = Z \alpha/v_{\rm e^-} = Z \alpha E_{\rm
e^-}/\sqrt{E^2_{\rm e^-} -m^2_{\rm e^-} }$ and $v_{\rm e^-}$ is a velocity of the electron.
The r.h.s. of Eq.(\[label4.2\]) can be expressed in terms of the astrophysical factor $S_{\rm pp}(0)$ for the solar proton burning brought up to the form $$\begin{aligned}
\label{label4.4}
\hspace{-0.5in}&&\overline{|{\cal M}(\nu_{\rm e} + {\rm D} \to {\rm
e}^- + {\rm p} + {\rm p})|^2} = S_{\rm
pp}(0)\,\frac{2^{12}5\pi^2}{\Omega_{\rm D e^+ \nu_{\rm
e}}}\,\frac{r_{\rm C}M^3_{\rm N}}{m^5_{\rm e}}\,\frac{|{\cal F}^{\rm e}_{\rm
ppe^-}|^2}{|{\cal F}^{\rm e}_{\rm
pp}|^2}\,F^2_{\rm
D}(k^2)\,F(Z, E_{\rm e^-})\nonumber\\
\hspace{-0.5in}&&\times \frac{\displaystyle C^2_0(k)}{\displaystyle
\Big[1 - \frac{1}{2}a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp} k^2 +
\frac{a^{\rm e}_{\rm pp}}{r_{\rm C}}\,h(2kr_{\rm C})\Big]^2 + (a^{\rm
e}_{\rm pp})^2k^2 C^4_0(k)}\, \Bigg( E_{{\rm e^-}}E_{{\nu}_{\rm e}} -
\frac{1}{3}\vec{k}_{{\rm e^-}} \cdot \vec{k}_{{\nu}_{\rm e}}\Bigg).\end{aligned}$$ We have used here the expression for the astrophysical factor $$\begin{aligned}
\label{label4.5}
S_{\rm pp}(0) = \frac{9g^2_{\rm A}G^2_{\rm V}Q_{\rm D}M^3_{\rm
N}}{1280\pi^4r_{\rm C}}\,G^2_{\rm \pi NN}\,|{\cal F}^{\rm e}_{\rm
pp}|^2\,m^5_{\rm e}\,\Omega_{\rm D e^+ \nu_{\rm e}},\end{aligned}$$ where $m_{\rm e} = 0.511\,{\rm MeV}$ is the electron mass, and $\Omega_{\rm D e^+ \nu_{\rm e}} = (W/m_{\rm e})^5 f(m_{\rm e}/W) =
4.481$ at $W = 0.932\,{\rm MeV}$. The function $f(m_{\rm e}/W)$ is defined by Eq.(\[label3.5\]).
In the rest frame of the deuteron the cross section for the process $\nu_{\rm e}$ + D $\to$ ${\rm e}^-$ + p + p is defined as $$\begin{aligned}
\label{label4.6}
&&\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}) = \frac{1}{4M_{\rm
D}E_{\nu_{\rm e}}}\int\,\overline{|{\cal M}(\nu_{\rm e} + {\rm D} \to
{\rm e}^- + {\rm p} + {\rm p})|^2}\nonumber\\
&&\frac{1}{2}\,(2\pi)^4\,\delta^{(4)}(k_{\rm D} + k_{\nu_{\rm e}} -
p_1 - p_2 - k_{\rm e^-})\, \frac{d^3p_1}{(2\pi)^3 2E_1}\frac{d^3
p_2}{(2\pi)^3 2E_2}\frac{d^3k_{{\rm e^-}}}{(2\pi)^3 2E_{{\rm e^-}}},\end{aligned}$$ where $E_{\nu_{\rm e}}$, $E_1$, $E_2$ and $E_{{\rm e^-}}$ are the energies of the neutrino, the protons and the electron. The abbreviation (cc) means the charged current. The integration over the phase volume of the (${\rm p p e^-}$)–state we perform in the non–relativistic limit and in the rest frame of the deuteron, $$\begin{aligned}
\label{label4.7}
&&\frac{1}{2}\,\int\frac{d^3p_1}{(2\pi)^3 2E_1}\frac{d^3p_2}{(2\pi)^3
2E_2} \frac{d^3k_{\rm e}}{(2\pi)^3 2E_{\rm
e^-}}(2\pi)^4\,\delta^{(4)}(k_{\rm D} + k_{\nu_{\rm e}} - p_1 - p_2 -
k_{\rm e^-})\,\nonumber\\ &&{\displaystyle \frac{\displaystyle
C^2_0(\sqrt{M_{\rm N}T_{\rm pp}})\,F^2_{\rm D}(M_{\rm N}T_{\rm
pp})\,F(Z, E_{\rm e^-})}{\displaystyle \Big[1 - \frac{1}{2}a^{\rm
e}_{\rm pp} r^{\rm e}_{\rm pp}M_{\rm N}T_{\rm pp} + \frac{a^{\rm
e}_{\rm pp}}{r_{\rm C}}\, h(2 r_{\rm C}\sqrt{M_{\rm N}T_{\rm
pp}})\Big]^2 + (a^{\rm e}_{\rm pp})^2 M_{\rm N}T_{\rm pp}
C^4_0(\sqrt{M_{\rm N}T_{\rm pp}})}}\nonumber\\ &&\Bigg( E_{\rm e^-}
E_{\nu_{\rm e}} - \frac{1}{3} \vec{k}_{\rm e^-} \cdot
\vec{k}_{\nu_{\rm e}}\Bigg)\, =\frac{E_{\bar{\nu}_{\rm e}}M^3_{\rm
N}}{128\pi^3}\,\Bigg(\frac{E_{\rm th}}{M_{\rm N}}
\Bigg)^{\!\!7/2}\Bigg(\frac{2 m_{\rm e}}{E_{\rm
th}}\Bigg)^{\!\!3/2}\frac{1}{E^2_{\rm th}}\nonumber\\ &&\int\!\!\!\int
dT_{\rm e^-} dT_{\rm pp}\delta(E_{\nu_{\rm e}}- E_{\rm th} - T_{\rm
e^-} - T_{\rm pp}) \sqrt{T_{\rm e^-}T_{\rm pp}}\Bigg(1 + \frac{T_{\rm
e^-}}{m_{\rm e}}\Bigg)\,{\displaystyle \sqrt{1 + \frac{T_{\rm e^-}}{2
m_{\rm e}}}}\nonumber\\ &&{\displaystyle \frac{\displaystyle
C^2_0(\sqrt{M_{\rm N}T_{\rm pp}}) \,F^2_{\rm D}(M_{\rm N}T_{\rm
pp})\,F(Z, E_{\rm e^-})}{\displaystyle \Big[1 - \frac{1}{2}a^{\rm
e}_{\rm pp} r^{\rm e}_{\rm pp}M_{\rm N}T_{\rm pp} + \frac{a^{\rm
e}_{\rm pp}}{r_{\rm C}}\,h(2 r_{\rm C}\sqrt{M_{\rm N}T_{\rm
pp}})\Big]^2 + (a^{\rm e}_{\rm pp})^2 M_{\rm N}T_{\rm pp}
C^4_0(\sqrt{M_{\rm N}T_{\rm pp}})}} \nonumber\\ &&= \frac{E_{\nu_{\rm
e}}M^3_{\rm N}}{128\pi^3} \,\Bigg(\frac{E_{\rm th}}{M_{\rm N}}
\Bigg)^{\!\!7/2}\Bigg(\frac{2 m_{\rm e}}{E_{\rm
th}}\Bigg)^{\!\!3/2}\,(y-1)^2\,\Omega_{\rm p p e^-}(y),\end{aligned}$$ where $T_{\rm e^-}$ is the kinetic energy of the electron, $E_{\rm th}$ is the neutrino energy threshold of the reaction $\nu_{\rm e}$ + D $\to$ ${\rm
e}^-$ + p + p, and is given by $E_{\rm th}= \varepsilon_{\rm D} + m_{\rm e}
- (M_{\rm n} - M_{\rm p}) = (2.225 + 0.511 - 1.293) \, {\rm MeV} =
1.443\,{\rm MeV}$. The function $\Omega_{\rm p p e^-}(y)$, where $y=E_{\nu_{\rm
e}}/E_{\rm th}$, is defined as $$\begin{aligned}
\label{label4.8}
\hspace{-0.5in}&&\Omega_{\rm p p e^-}(y) = \int\limits^{1}_{0} dx
\sqrt{x (1 - x)} \Bigg(1 + \frac{E_{\rm th}}{m_{\rm
e}}(y-1)(1-x)\Bigg) \sqrt{1 + \frac{E_{\rm th}}{2 m_{\rm
e}}(y-1)(1-x)}\nonumber\\
\hspace{-0.5in}&&C^2_0(\sqrt{M_{\rm N}E_{\rm th}\,(y - 1)\,x})\,F^2_{\rm
D}(M_{\rm N}E_{\rm th}\,(y - 1)\,x)\,F(Z,m_{\rm e} + E_{\rm th}(y -
1)\,(1-x))\nonumber\\
\hspace{-0.5in}&&\Bigg\{\Bigg[1 - \frac{1}{2}a^{\rm e}_{\rm pp} r^{\rm
e}_{\rm pp}M_{\rm N}E_{\rm th}\,(y - 1)\,x + \frac{a^{\rm e}_{\rm
pp}}{r_{\rm C}}\,h(2 r_{\rm C}\sqrt{M_{\rm N}E_{\rm th}\,(y -
1)\,x})\Bigg]^2 \nonumber\\
\hspace{-0.5in}&&\hspace{0.2in} + (a^{\rm e}_{\rm pp})^2\,M_{\rm N}E_{\rm
th}\,(y - 1)\,x C^4_0(\sqrt{M_{\rm N}E_{\rm th}\,(y - 1)\,x})
\Bigg\}^{-1}\!\!\!\!\!,\end{aligned}$$ where we have changed the variable $T_{\rm pp} = (E_{\nu_{\rm e}} -
E_{\rm th})\,x$.
The cross section for $\nu_{\rm e}$ + D $\to$ ${\rm e}^-$ + p + p is defined $$\begin{aligned}
\label{label4.9}
\hspace{-0.5in}\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}) &=&
S_{\rm pp}(0)\, \frac{640 r_{\rm C}}{\pi \Omega_{\rm D e^+\nu_{\rm
e}}}\Bigg(\frac{M_{\rm N}}{E_{\rm th}}\Bigg)^{3/2}\Bigg(\frac{E_{\rm
th}}{2m_{\rm e}}\Bigg)^{7/2}\frac{|{\cal F}^{\rm e}_{\rm
ppe^-}|^2}{|{\cal F}^{\rm e}_{\rm
pp}|^2}\,(y-1)^2\,\Omega_{\rm p p
e^-}(y)=\nonumber\\
&=&3.69\times 10^5\,S_{\rm
pp}(0)\,\frac{|{\cal F}^{\rm e}_{\rm
ppe^-}|^2}{|{\cal F}^{\rm e}_{\rm
pp}|^2}\,(y-1)^2\,\Omega_{\rm p p e^-}(y),\end{aligned}$$ where $S_{\rm pp}(0)$ is measured in ${\rm MeV}\,{\rm cm}^2$. For $S_{\rm pp}(0) = 4.08\times 10^{-49}\,{\rm MeV}\,{\rm cm}^2$ Eq.(\[label3.8\]) the cross section $\sigma^{\nu_{\rm e} D}_{\rm
cc}(E_{\nu_{\rm e}})$ reads $$\begin{aligned}
\label{label4.10}
\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}) =
1.50\,\frac{|{\cal F}^{\rm e}_{\rm
ppe^-}|^2}{|{\cal F}^{\rm e}_{\rm
pp}|^2}\,(y-1)^2\,\Omega_{\rm p p e^-}(y)\,10^{-43}\,{\rm cm}^2.\end{aligned}$$ In order to make numerical predictions for the cross section Eq.(\[label4.10\]) we should evaluate the overlap factor ${\cal
F}^{\rm e}_{\rm ppe^-}$. This evaluation can be carried out in analogy with the evaluation of ${\cal F}^{\rm e}_{\rm pp}$. By using the results obtained in Ref.\[10\] we get $$\begin{aligned}
\label{label4.11}
\hspace{-0.2in}{\cal F}^{\rm e}_{\rm ppe^-} = 1 +
\frac{32}{9}\int\limits^{\infty}_0 dv\,v^2\,\Bigg[e^{\textstyle -
\alpha\pi/4v}\,\Gamma\Bigg(1 - \frac{i\alpha}{2v}\Bigg)-
1\Bigg]\Bigg[\frac{1}{(1 + v^2)^{5/2}} - \frac{1}{16}\,\frac{1}{(1 +
v^2)^{3/2}}\Bigg].\end{aligned}$$ The perturbative evaluation of the integral gives $$\begin{aligned}
\label{label4.12}
{\cal F}^{\rm e}_{\rm ppe^-} = 1 -
\alpha\,\Bigg(\frac{13\pi}{54} - i\,\frac{13\gamma}{27}\Bigg) + O(\alpha^2).\end{aligned}$$ Thus, the overlap factor ${\cal F}^{\rm e}_{\rm ppe^-}$ differs slightly from unity as well as the overlap factor ${\cal F}^{\rm
e}_{\rm pp}$ of the solar proton burning. The ratio of the overlap factors is equal to $$\begin{aligned}
\label{label4.13}
\frac{|{\cal F}^{\rm e}_{\rm ppe^-}|^2}{|{\cal F}^{\rm e}_{\rm pp}|^2}
= 1 - \alpha\,\frac{2\pi}{3} + O(\alpha^2) = 1 +
(-1.53 \times 10^{-2}) \simeq 1.\end{aligned}$$ Setting $|{\cal F}^{\rm e}_{\rm ppe^-}|^2/|{\cal F}^{\rm e}_{\rm
pp}|^2 = 1$ we can make numerical predictions for the cross section Eq.(\[label4.10\]) and compare them with the PMA ones.
The most recent PMA calculations the cross section for the reaction $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p have been obtained in Refs.\[22,23\] and tabulated for the neutrino energies ranging over the region from threshold up to 160$\,{\rm MeV}$. Since our result is restricted by the neutrino energies from threshold up to 10$\,{\rm
MeV}$, we compute the cross section only for this energy region $$\begin{aligned}
\label{label4.14}
\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}= 4\,{\rm MeV}) &=&
2.46\,(1.86/1.54)\times 10^{-43}\,{\rm cm}^2,\nonumber\\
\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}= 6\,{\rm MeV}) &=&
9.60\,(5.89/6.13)\times 10^{-43}\,{\rm cm}^2,\nonumber\\
\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}= 8\,{\rm MeV}) &=&
2.38\,(1.38/1.44)\times 10^{-42}\,{\rm cm}^2,\nonumber\\
\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}}= 10\,{\rm MeV}) &=&
4.07\,(2.55/2.66)\times 10^{-43}\,{\rm cm}^2,\end{aligned}$$ where the data in parentheses are taken from Refs.\[22\] and \[23\], respectively. Thus, on the average our numerical values for the cross section $\sigma^{\nu_{\rm e} D}_{\rm cc}(E_{\nu_{\rm e}})$ by a factor of 1.5 are larger compared with the PMA ones.
Our predictions for the cross section Eq.(\[label4.14\]) differ from the predictions of Ref.\[14\]. This is related to (i) the value of the astrophysical factor which is by a factor 1.4 larger in Ref.\[14\] and (ii) the form factor describing a spatial smearing of the deuteron which is $F^2_{\rm D}(k^2)$ is this paper (see Ref. \[13\]) and $F_{\rm
D}(k^2)$ in Ref.\[14\].
Astrophysical factor for pep process
====================================
In the RFMD the amplitude of the reaction p + e$^-$ + p $\to$ D + $\nu_{\rm e}$ or the pep–process is related to the effective Lagrangian Eq.(\[label2.3\]) and reads $$\begin{aligned}
\label{label5.1}
&& i{\cal M}({\rm p} + {\rm e}^- + {\rm p} \to {\rm D} + \nu_{e}) =
G_{\rm V}\,g_{\rm A} M_{\rm N}\,G_{\rm \pi NN}\,\frac{3g_{\rm
V}}{4\pi^2}\,e^*_{\mu}(k_{\rm D})\,[\bar{u}(k_{\nu_{\rm
e}})\gamma^{\mu} (1-\gamma^5) u(k_{\rm e^-})]\,{\cal F}^{\rm e}_{\rm
pp}\nonumber\\ &&\times \,\frac{\displaystyle C_0(k)}{\displaystyle1 -
\frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_C}\,h(2 k r_C) + i\,a^{\rm e}_{\rm pp}\,k\,C^2_0(k)
}\,[\bar{u^c}(p_2) \gamma^5 u(p_1)]\,F_{\rm D}(k^2),\end{aligned}$$ where we have described low–energy elastic pp scattering in analogy with the solar proton burning and the neutrino disintegration of the deuteron.
The amplitude Eq.(\[label5.1\]) squared, averaged and summed over polarizations of the interacting particles is defined $$\begin{aligned}
\label{label5.2}
&&\overline{|{\cal M}({\rm p} + {\rm e}^- + {\rm p} \to {\rm D} +
\nu_{e})|^2} = G^2_{\rm V}\, g^2_{\rm A} M^6_{\rm N}\,G^2_{\rm \pi
NN}\,\frac{27 Q_{\rm D}}{\pi^2}\,|{\cal F}^{\rm e}_{\rm
pp}|^2\,\,F^2_{\rm D}(k^2)\,F(Z, E_{\rm e^-})\nonumber\\
&&\times\,\frac{\displaystyle C^2_0(k)} {\displaystyle \Big[1 -
\frac{1}{2}\,a^{\rm e}_{\rm pp} r^{\rm e}_{\rm pp}k^2 + \frac{a^{\rm
e}_{\rm pp}}{r_C}\,h(2 k r_C)\Big]^2 + (a^{\rm e}_{\rm pp})^2 k^2
C^4_0(k)}\,\Bigg( E_{\rm e^+} E_{\nu_{\rm e}} -
\frac{1}{3}\vec{k}_{\rm e^+}\cdot \vec{k}_{\nu_{\rm e}}\Bigg),\end{aligned}$$ where $F(Z, E_{\rm e^-})$ is the Fermi function given by Eq.(\[label4.3\]).
At low energies the cross section $\sigma_{\rm pep}(T_{\rm pp})$ for the pep–process can be determined as follows \[24\] $$\begin{aligned}
\label{label5.3}
&&\sigma_{\rm pep}(T_{\rm pp}) = \frac{1}{v}\frac{1}{4M^2_{\rm N}}\int
\frac{d^3k_{\rm e^-}}{(2\pi)^3 2 E_{\rm e^-}}\,g\, n(\vec{k}_{\rm
e^-})\int \overline{|{\cal M}({\rm p} + {\rm e}^- + {\rm p} \to {\rm
D} + \nu_{\rm e})|^2}\nonumber\\ &&(2\pi)^4 \delta^{(4)}(k_{\rm D} +
k_{\nu_{\rm e}} - p_1 - p_2 - k_{\rm e^-}) \frac{d^3k_{\rm
D}}{(2\pi)^3 2M_{\rm D}}\frac{d^3k_{\nu_{\rm e}}}{(2\pi)^3
2E_{\nu_{\rm e}}},\end{aligned}$$ where $g = 2$ is the number of the electron spin states and $v$ is a relative velocity of the protons. The electron distribution function $n(\vec{k}_{\rm e^-})$ can be taken in the form \[21\] $$\begin{aligned}
\label{label5.4}
n(\vec{k}_{\rm e^-}) = e^{\displaystyle \bar{\nu} - T_{\rm e^-}/kT_c},\end{aligned}$$ where $k = 8.617\times 10^{-11}\,{\rm MeV\,K^{-1}}$, $T_c$ is a temperature of the core of the Sun. The distribution function $n(\vec{k}_{\rm e^-})$ is normalized by the condition $$\begin{aligned}
\label{label5.5}
g\int \frac{d^3k_{\rm e^-}}{(2\pi)^3}\,n(\vec{k}_{\rm e^-}) = n_{\rm e^-},\end{aligned}$$ where $n_{\rm e^-}$ is the electron number density. From the normalization condition Eq.(\[label5.5\]) we derive $$\begin{aligned}
\label{label5.6}
e^{\displaystyle \bar{\nu}} = \frac{\displaystyle 4\,\pi^3\, n_{\rm
e^-}}{\displaystyle (2\pi\,m_{\rm e}\,kT_c)^{3/2}}.\end{aligned}$$ The astrophysical factor $S_{\rm pep}(0)$ is then defined by $$\begin{aligned}
\label{label5.7}
S_{\rm pep}(0) = S_{\rm pp}(0)\,\frac{15}{2\pi}\,\frac{1}{\Omega_{\rm D
e^+ \nu_{\rm e}}}\,\frac{1}{m^3_{\rm e}}\,\Bigg(\frac{E_{\rm
th}}{m_{\rm e}}\Bigg)^2\,e^{\displaystyle \bar{\nu}}\,\int d^3k_{\rm
e^-} \,e^{\displaystyle - T_{\rm e^-}/kT_c}\,F(Z, E_{\rm e^-}).\end{aligned}$$ For the ratio $S_{\rm pep}(0)/S_{\rm pp}(0)$ we obtain $$\begin{aligned}
\label{label5.8}
\frac{S_{\rm pep}(0)}{S_{\rm pp}(0)} = \frac{2^{3/2}\pi^{5/2}}{f_{\rm
pp}(0)}\,\Bigg(\frac{\alpha Z n_{\rm e^-}}{m^3_{\rm
e}}\Bigg)\,\Bigg(\frac{E_{\rm th}}{m_{\rm
e}}\Bigg)^2\,\sqrt{\frac{m_{\rm e}}{k T_c}}\,I\Bigg(Z\sqrt{\frac{2
m_{\rm e}}{k T_c}}\Bigg).\end{aligned}$$ We have set $f_{\rm pp}(0) = \Omega_{\rm D e^+ \nu_{\rm e}}/30 =
0.149$ \[21\] and the function $I(x)$ having been introduced by Bahcall and May \[21\] reads $$\begin{aligned}
\label{label5.9}
I(x) = \int\limits^{\infty}_0 {\displaystyle \frac{\displaystyle
du\,e^{\displaystyle -u}}{\displaystyle 1 - e^{\displaystyle
-\pi\alpha\,x/\sqrt{u}}}}.\end{aligned}$$ The relation between the astrophysical factors $S_{\rm pep}(0)$ and $S_{\rm pp}(0)$ given by Eq.(\[label5.8\]) is in complete agreement with that obtained by Bahcall and May \[21\]. The ratio Eq.(\[label5.8\]) does not depend on whether the astrophysical factor $S_{\rm pp}(0)$ is enhanced with respect to the recommended value or not.
Conclusion
==========
We have shown that the contributions of low–energy elastic pp scattering in the ${^1}{\rm S}_0$–state with the Coulomb repulsion to the amplitudes of the reactions p + p $\to$ D + e$^+$ + $\nu_{\rm e}$, $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p and p + e$^-$ + p $\to$ D + $\nu_{\rm e}$ can be described in the RFMD in full agreement with low–energy nuclear phenomenology in terms of the S–wave scattering length and the effective range. The amplitude of low–energy elastic pp scattering has been obtained by summing up an infinite series of one–proton loop diagrams and the evaluation of the result of the summation in leading order in the large $N_C$ expansion. This takes away fully the problem pointed out by Bahcall and Kamionkowski \[17\] that in the RFMD with the effective local four–nucleon interaction Eq.(\[label2.1\]) one cannot describe low–energy elastic pp scattering in the ${^1}{\rm S}_0$–state with the Coulomb repulsion in agreement with low–energy nuclear phenomenology.
The obtained numerical value of the astrophysical factor $S_{\rm
pp}(0) = 4.08\times 10^{-25}\,{\rm MeV\, b}$ agrees with the recommended value $S_{\rm pp}(0) = 4.00\times 10^{-25}\,{\rm MeV\, b}$ and recent estimate $S_{\rm pp}(0) = 4.20\times 10^{-25}\,{\rm MeV\,
b}$ \[9\] obtained from the helioseismic data.
Unfortunately, the value of the astrophysical factor $S_{\rm pp}(0) =
4.08 \times 10^{-25}\,{\rm MeV\,\rm b}$ does not confirm the enhancement by a factor of 1.4 obtained in the modified version of the RFMD in Ref.\[14\] which is not well defined due to a violation of Lorentz invariance of the effective four–nucleon interaction describing N + N $\to$ N + N transitions. This violation has turned out to be incompatible with a dominance of one–nucleon loop anomalies which are Lorentz covariant.
The cross section for the neutrino disintegration of the deuteron has been evaluated with respect to $S_{\rm pp}(0)$. We have obtained an enhancement of the cross section by a factor of order 1.5 on the average for neutrino energies $E_{\nu_{\rm e}}$ varying from threshold to $E_{\nu_{\rm e}} \le 10\,{\rm MeV}$. It would be important to verify our results for the reaction $\nu_{\rm e}$ + D $\to$ e$^-$ + p + p in solar neutrino experiments planned by SNO. In fact, first, this should provide an experimental study of $S_{\rm pp}(0)$ and, second, the cross sections for the anti–neutrino disintegration of the deuteron caused by charged $\bar{\nu}_{\rm e}$ + D $\to$ e$^+$ + n + n and neutral $\bar{\nu}_{\rm e}$ + D $\to$ $\bar{\nu}_{\rm e}$ + n + p weak currents have been found in good agreement with recent experimental data obtained by the Reines’s experimental group \[26\].
The evaluation of the astrophysical factor $S_{\rm pep}(0)$ for the reaction p + e$^-$ + p $\to$ D + $\nu_{\rm e}$ or pep–process in the RFMD has shown that the ratio $S_{\rm pep}(0)/S_{\rm pp}(0)$, first, agrees fully with the result obtained by Bahcall and May \[21\] and, second, does not depend on whether $S_{\rm pp}(0)$ is enhanced with respect to the recommended value or not.
Concluding the paper we would like to emphasize that our model, the RFMD, conveys the idea of a dominant role of one–fermion loop (one–nucleon loop) anomalies from elementary particle physics to the nuclear one. This is a new approach to the description of low–energy nuclear forces in physics of finite nuclei. In spite of almost 30 year’s history after the discovery of one–fermion loop anomalies and application of these anomalies to the evaluation of effective Lagrangians of low–energy interactions of hadrons, in nuclear physics fermion–loop anomalies have not been applied to the analysis of low–energy nuclear interactions and properties of nuclei. However, an important role of $N\bar{N}$ fluctuations for the correct description of low–energy properties of finite nuclei has been understood in Ref.\[16\]. Moreover, $N\bar{N}$ fluctuations have been described in terms of one–nucleon loop diagrams within quantum field theoretic approaches, but the contributions of one–nucleon loop anomalies have not been considered in the papers of Ref.\[16\].
The RFMD strives to fill this blank. Within the framework of the RFMD we aim to understand, in principle, the possibility of the description of strong low-energy nuclear forces in terms of one–nucleon loop anomalies. Of course, our results should be quantitatively compared with the experimental data and other theoretical approaches. Nevertheless, at the present level of the development of our model one cannot demand at once to describe, for example, the astrophysical factor $S_{\rm pp}(0)$ with accuracy better than it has been carried out by Schiavilla [*et al.*]{} \[6\], where only corrections not greater than 1$\%$ are allowed. It is not important for our approach at present. What is much more important is in the possibility to describe without free parameters in quantitative agreement with both the experimental data and other theoretical approaches all multitude of low–energy nuclear reactions of the deuteron coupled to nucleons and other particles. In Ref.\[13\] we have outlined the procedure of the evaluation of chiral meson–loop corrections in the RFMD. The absence of free parameters in the RFMD gives the possibility to value not only the role of these corrections but also the corrections of other kind mentioned recently by Vogel and Beacom \[25\].
The justification of the RFMD within QCD and large $N_C$ expansion \[12\] implies that one–nucleon loop anomalies might be natural objects for the understanding of low-energy nuclear forces. The real accuracy of the approach should be found out for the process of the development.
Acknowledgement
===============
We would like to thank Prof. Kamionkowski and Prof. Beacom for reading manuscript and useful comments.
[9]{} J. N. Bahcall, in [*NEUTRINO ASTROPHYSICS*]{}, Cambridge University Press, Cambridge, 1989. C. E. Rolfs and W. S. Rodney, in [*CAULDRONS IN THE COSMOS*]{}, the University of Chicago Press, Chicago and London, 1988. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 67 (1995) 781; Bahcall et al., Phys. Rev. C54 (1996) 411. E. G. Adelberger [*et al.*]{}, Rev. Mod. Phys. 70 (1998) 1265. M. Kamionkowski and J. N. Bahcall, ApJ. 420 (1994) 884. R. Schiavilla [*et al.*]{}, Phys. Rev. C58 (1998) 1263. T.–S. Park, K. Kubodera, D.–P. Min and M. Rho, ApJ. 507 (1998) 443. X. Kong and F. Ravndal, [*Proton–Proton Fusion in Leading order of Effective Field Theory*]{}, nucl–th/9902064, March 1999; [*Effective–Range Corrections to the Proton–Proton Fusion Rate*]{}, nucl–th/9904066, April 1999. H. Schlattl, A. Bonanno and L. Paterno, [*Signature of the efficiency of solar nuclear reactions in the neutrino experiments*]{}, astro–ph/9902354, July 1999. A. N. Ivanov, N. I. Troitskaya, M. Faber and H. Oberhummer, Phys. Lett. B361 (1995) 74; Nucl. Phys. A617 (1997) 414, [*ibid.*]{} A625 (1997) 896 (Erratum). A. N. Ivanov, H. Oberhummer, N. I. Troitskaya and M. Faber, [*Solar proton burning, photon and anti–neutrino disintegration of the deuteron in the relativistic field theory model of the deuteron*]{}, nucl–th/9810065, October 1998. A. N. Ivanov, H. Oberhummer, N. I. Troitskaya and M. Faber, [*The relativistic field theory model of the deuteron from low–energy QCD*]{}, nucl–th/9908029, August 1999. A. N. Ivanov, H. Oberhummer, N.I. Troitskaya $\&$ M. Faber, [*Neutron–proton radiative capture, photo–magnetic and anti–neutrino disintegration of the deuteron in the relativistic field theory model of the deuteron*]{}, nucl–th/9908080, August 1999. A. N. Ivanov, H. Oberhummer, N. I. Troitskaya and M. Faber, [*Solar neutrino processes in the relativistic field theory model of the deuteron*]{}, nucl–th/9811012, November 1998. R. Jackiw, [*Field theoretic investigations in current algebra*]{}, [*Topological investigations of quantized gauge theories*]{}, in [*CURRENT ALGEBRA AND ANOMALIES*]{}, S. B. Treiman, R. Jackiw, B. Zumino and E. Witten (eds), World Scientific, Singapore, p.81 and p.211; N. S. Manton, Ann. of Phys. (NY) 159 (1985) 220; N. Ogawa, Progr. Theor. Phys. 90 (1993) 717; R. A. Bertlemann, in [*ANOMALIES IN QUANTUM FIELD THEORY*]{}, Oxford Science Publications, Clarendon Press–Oxford, 1996, pp.227–233 and references therein. J. D. Walecka, Ann. Phys. (NY) 83 (1974) 121; C. J. Horowitz and B. D. Scrot, Nucl. Phys. A368 (1981) 503; Phys. Lett. B140 (1984) 181; R. J. Perry, Phys. Lett. B182 (1986) 269. J. N. Bahcall and M. Kamionkowski, Nucl. Phys. A625 (1997) 893. C. Caso [*et al.*]{}, Eur. Phys. J. C3 (1998) 1. K. B. Mather and P. Swan, in [*NUCLEAR SCATTERING*]{}, Cambridge University Press 1958, pp.212–235. J. R. Bergervoet, P. C. van Campen, W. A. van der Sande and J. J. de Swart, Phys. Rev. C38 (1988) 15. J. N. Bahcall, ApJ. 139 (1964) 318; J. N. Bahcall and R. M. May, ApJ. 155 (1969) 501. S. Ying, W. C. Haxton and E. M. Henley, Phys. Rev. C45 (1992) 1982. M. Doi and K. Kubodera, Phys. Rev. C45 (1992) 1988. M. L. Goldberger and K. M. Watson, in [*COLLISION THEORY*]{}, John Wiley $\&$ Sons, Inc., New York–London–Sydney, 1964. P. Vogel and J. F. Beacom, Phys. Rev. D60 (1999) 053003. S. P. Riley, Z. D. Greenwood, W. R. Kroop, L. R. Price, F. Reines, H. W. Sobel, Y. Declais, A. Etenko and M. Skorokhvatov, Phys. Rev. C59 (1999) 1780.
[^1]: E–mail: ivanov@kph.tuwien.ac.at, Tel.: +43–1–58801–14261, Fax: +43–1–58801–14299
[^2]: E–mail: ohu@kph.tuwien.ac.at, Tel.: +43–1–58801–14251, Fax: +43–1–58801–14299
[^3]: Permanent Address: State Technical University, Department of Nuclear Physics, 195251 St. Petersburg, Russian Federation
[^4]: E–mail: faber@kph.tuwien.ac.at, Tel.: +43–1–58801–14261, Fax: +43–1–58801–14299
[^5]: See the last paragraph of Sect.3 and the first paragraph of Sect.5 of Ref.\[6\].
|
---
abstract: 'In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high dimensional eigenvalue problems arising from quantum many-body problems. Under the proposed framework, we establish the convergence theorems for several recently proposed randomized algorithms, including the full configuration interaction quantum Monte Carlo (FCIQMC) and the fast randomized iteration (FRI). The analysis is consistent with numerical experiments for physical systems such as Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.'
address:
- 'Department of Mathematics, Department of Physics and Department of Chemistry, Duke University'
- 'Department of Mathematics, Duke University'
author:
- Jianfeng Lu
- Zhe Wang
bibliography:
- 'qmc.bib'
title: The full configuration interaction quantum Monte Carlo method in the lens of inexact power iteration
---
[^1]
Introduction
============
In recent years, following the work of full configuration interaction quantum Monte Carlo (FCIQMC) [@BoothThomAlavi:09; @ClelandBoothAlavi:10], the idea of using randomized or truncated power method to solve the full configuration interaction (FCI) eigenvalue problem has become quite popular in quantum chemistry literature. From a mathematical point of view, the FCI calculation essentially asks for the smallest eigenvalue of a real symmetric matrix (for ground state calculation) or a few low-lying eigenvalues (for low-lying excited state calculation). The computational challenge lies in the fact that the size of the matrix grows exponentially fast with respect to the number of orbitals / electrons in the chemical system, and thus a brute-force numerical diagonalization method (such as power method or Lanczos method) does not work except for very small molecules.
The goal of this work is two folds: On the one hand, we want to establish a general framework to understand these recently proposed randomized algorithms. As we shall see, from the angle of numerical linear algebra, these recent methods can be understood as generalizations of conventional power method when inexact matrix-vector product is used. As a result, the convergence of these methods can be dealt with by a simple extension of the usual proof of convergence of power method. A natural consequence of this understanding is that, to compare the various approaches, the crucial part is to understand the error caused by different strategies of inexact matrix-vector multiplication. Using this insights, we will compare a few of the recently proposed randomized or truncated FCI methods analytically and also numerically using Hubbard model and some small chemical molecules as toy examples.
While the motivation of the study is from FCI calculation in quantum chemistry, these methods can be understood on the general setting of numerical linear algebra, and hence except in the numerical section, we will not restrict ourselves to the FCI Hamiltonian. For a given real symmetric positive definite matrix $A\in \mathbb{R}^{N\times N}$, we are interested in numerically obtaining the largest eigenvalue and corresponding eigenvector. It is possible to extend the method to leading $k$ eigenvalues where $k$ is on the order of $1$ based on the subspace iteration method, generalization of the power method. In the sequel, we denote the eigenvalues of $A$ as $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_N \geq 0$, and corresponding orthonormal eigenvectors are $u_1, u_2, \cdots, u_N$ (viewed as column vectors).
To obtain the largest eigenvalue and the corresponding eigenvector, one of the simplest algorithm is standard power iteration, given by $$\begin{aligned}
& y_{t+1} = Ax_{t}; \\
& x_{t+1} = y_{t+1}/ {\lVerty_{t+1}\rVert}_2 \end{aligned}$$ with some initial guess $x_0$ and iterate till convergence. The algorithm is simple to understand: The matrix multiplication $A x_t$ amplifies $x_t$ in the leading eigenspace. The convergence of the algorithm is also well known: As long as the initial vector satisfies that $u_1^{\top} x_0$ is nonzero and there exists an eigengap ($\lambda_1 > \lambda_2$), the subspace $\operatorname{span}{x_t}$ converges to the eigenspace $\operatorname{span}{u_1}$ linearly as $t\to\infty$ with rate proportional to $\lambda_2 / \lambda_1$.
Since only the convergence of subspace is of interest, the norm of the vector $x_t$ plays no role. Hence the normalization step of power iteration may be omitted $$v_{t+1} = A v_t.$$ This is equivalent with the original power method. Of course, in practical computations, the normalization is important to avoid issues like arithmetic overflow.
Motivated by the recent proposed algorithms in quantum chemistry literature, in this work, we take the view point that we cannot afford (or choose not to perform) the matrix-vector multiplication $y_{t+1}=Ax_t$ exactly. Among other applications, such a scenario naturally arises when the dimension of the matrix $A$ is extremely large, so that even storage of the vector $y_{t+1}$ (even in sparse format) is too expensive. For example, this is a common situation for FCI calculations in quantum chemistry since the dimension of the matrix $A$ grows exponentially with respect to the number of electrons in chemical system.
Thus, in power iteration, we would replace the matrix multiplication step by a map $$\label{eq:inexact_iteration}
y_{t+1} = F_m(A, x_t).$$ Here, given the matrix $A$ and the current iterate $x_t$, the map $F_m$, either deterministic or stochastic, outputs an approximation of the product $y_{t+1} \approx A x_t$. Different choices of $F_m$ corresponds to various recently proposed algorithms, as will be discussed below. We have used the index $m$ to indicate the “complexity” (computational cost) of $F_m$, the specific meaning depends on the choice of the family of maps. Replacing the matrix-vector multiplication by , we get Inexact Power Iteration Algorithm \[alg:inexact\_power\] and its unnormalized version \[alg:inexact\_power\_no\].
Initialization: Choose a normalized vector $x_0\in \mathbb{R}^N$, ${\lVertx_0\rVert}_2 = 1$, $u_1^\top x_0 \neq 0$.\
Initialization: Choose a vector $v_0\in \mathbb{R}^{N}$, $u_1^\top v_0 \neq 0$.\
Notice that the two versions of inexact power iteration Algorithm \[alg:inexact\_power\] and \[alg:inexact\_power\_no\] are equivalent if the function $F_m(A, \cdot)$ is homogeneous; we will make this as a standing assumption in our analysis.
Various inexact matrix-vector multiplication has been proposed in the literature for configuration interaction calculations, either deterministic or stochastic, see e.g., earlier attempts in [@Huron1973; @Buenker1974; @Harrison1991; @Illas1991; @Daudey1993; @Greer1995; @Greer1998; @Ivanic2003; @Abrams2005], the FCIQMC approach [@BoothThomAlavi:09; @BoothGruneisKresseAlavi:13; @ClelandBoothAlavi:10; @Booth2011; @Schwarz2017], the semi-stochastic approach [@PetruzieloHolmesChanglaniNightingaleUmrigar:12; @Blunt2015; @HolmesChanglaniUmrigar:16; @Sharma2017], other stochastic approaches [@Ten-no2013; @Giner2013; @LimWeare:17], and various deterministic strategies for compressed or truncated representation of the wave functions [@Rolik2008; @Roth2009; @Ma2011a; @Evangelista2014; @Knowles2015; @Zhang2016; @TubmanLeeTakeshitaHeadGordonWhaley:16; @Liu2016; @Schriber2016; @Zimmerman2017; @Zimmerman2017a]. In this work, we will focus on two of such strategies: the full configuration-interaction quantum Monte Carlo (FCIQMC) [@BoothThomAlavi:09; @ClelandBoothAlavi:10] and the fast randomized iteration (FRI) [@LimWeare:17]. In some sense, these methods represent two ends of the spectrum of the possibilities; so that the analysis of those can be easily extended to other methodologies. The FCIQMC uses interacting particles to represent the vector $v_t$ and stochastic evolution of particles to represent the action of the matrix $A$ on the vector $v_t$. The FRI on the other hand is based on exact matrix-vector multiplication and stochastic schemes to compress the resulting vectors into sparse ones with given number of nonzero entries. These algorithms will be discussed and analyzed in Section \[sec:algorithms\], following the general analysis framework we establish in Section \[sec:conv\].
The rest of the paper is organized as the following. In Section \[sec:conv\], we provide a convergence analysis for a generic class of inexact power iteration. In Section \[sec:algorithms\], we give more details of FCIQMC and FRI and analyze them following the convergence analysis established in Section \[sec:conv\]. In Section \[sec:numerics\], we perform numerical tests on 2D Hubbard model and some chemical molecules to compare the various algorithms and to verify the analysis results.
General convergence analysis of inexact power iteration {#sec:conv}
=======================================================
As an advantage of taking a unified framework of various algorithms, the convergence of those can be understood in a fairly generic way, which also facilitates comparison of different proposed strategies. In this section, we establish a general convergence theorem of the inexact power iteration.
The convergence of the iteration to the desired eigenvector will be measured in the angle of the vectors. Recall that the angle between two vectors $v$ and $w$ is given by $$\theta(v,w) = \arccos
\biggl( \frac{{\lvert\langle v, w \rangle\rvert}}{{\lVertv\rVert}_2{\lVertw\rVert}_2} \biggr).$$ From the definition, it is obvious that $\theta(v,w) = \theta(a v, b w)$ for any vectors $v,w$ and real numbers $a, b$. In view of this insensitivity of the constant multiple of vectors in the error measure, if the inexact matrix-vector multiplication $F_m(A,v_t)$ satisfies the homogeneity assumption below, the two versions of inexact power iterations with or without normalization Algorithm \[alg:inexact\_power\] and \[alg:inexact\_power\_no\] are equivalent.
\[asmp:F\_homo\] $$F_m(A,cv) = c F_m(A,v),$$ for all vectors $v\in \mathbb{R}^N$ and real number $c\in \mathbb{R}$.
More precisely, if the initial vectors of the two algorithms are the same up to a number $x_0 = c_0 v_0$, then there exist numbers $c_t$ such that $v_t = c_tx_t$ for all $t$. Therefore, $\theta(u_1,v_t) = \theta(u_1,x_t)$. In the following, when we analyze the algorithm, we will always use $v_t$ for the unnormalized iterate and $x_t$ for the normalized version $x_t = v_t / {\lVertv_t\rVert}_2$.
To analyze the effect of the inexact matrix-vector multiplication, we write $F_m(A, v_{t-1})$ as a sum of the exact matrix-vector product with an error term $$\label{eq:inexactproduct}
v_{t} = F_m(A,v_{t-1}) = Av_{t-1} + \xi_{t},$$ where $\xi_t$ is the error of the inexact multiplication at step $t$, and the dependence on $m$ is suppressed to keep the notation simple. Note that $\xi_t$ can be either deterministic or stochastic depending on the choice of $F_m$. For example, $\xi_t$ is deterministic for the hard thresholding compression and stochastic for both FCIQMC and FRI methods. While we will proceed viewing $v_t$ as a stochastic process, the results apply to the deterministic case as well.
Denote $\mathcal{F}_t = \sigma(v_1, v_2, \cdots, v_t)$ the $\sigma$-algebra generated by $v_1, v_2, \cdots, v_t$. We assume that error $\xi_t$ satisfies the following properties. Note that this assumption holds for both FCIQMC and FRI algorithms, as we will prove in Section \[sec:algorithms\].
\[asmp:error\] The error $\xi_t$ in the inexact matrix-vector product satisfies
a) Martingale difference sequence condition $$\label{eq:martingale_difference}
{\mathbb{E}\,}(\xi_t \mid \mathcal{F}_{t-1}) = 0.$$
b) Expectation $2$-norm bound $$\label{eq:2_norm_bound}
{\mathbb{E}\,}({\lVert\xi_t\rVert}_2^2 \mid \mathcal{F}_{t-1}) \leq C_e\frac{{\lVertA\rVert}_1^2 {\lVertv_{t-1}\rVert}_1^2}{m},$$ where $C_e$ is a constant that is scale invariant of $A$ (*i.e.*, it does not depend on the norm of $A$).
c) Growth of expectation $1$-norm bound $$\label{eq:1_norm_control}
{\mathbb{E}\,}({\lVertv_t\rVert}_1 \mid \mathcal{F}_{t-1}) \leq {\lVertA\rVert}_1 {\lVertv_{t-1}\rVert}_1.$$
A few remarks are in order to help appreciate the Assumption \[asmp:error\]. The martingale difference sequence property is just assumed here for convenience, in fact the convergence result extends to the biased case as we will see in Corollary \[coro:biased\]. The other two assumptions are more essential. Assumption \[asmp:error\]b indicates that the error of the inexact matrix-vector product $F_m(A, v_{t-1})$ can be controlled by the sparsity of the $v_{t-1}$, as the $1$-norm of $v_{t-1}$ is a sparsity measure. This is a natural assumption considering that the compression of a vector would be easier if the vector is more sparse. The bound depends proportional to inverse of $m$, so that one could control the error of the inexact matrix-vector multiplication at the price of increasing the complexity. Note that $1/m$ dependence can be understood as a standard Monte Carlo error scaling. More detailed discussions can be found in Section \[sec:algorithms\] when specific algorithms are analyzed. Assumption \[asmp:error\]c then assumes that the sparsity is not destroyed by the error in the iteration; since otherwise we will lose control of the accuracy of the inexact matrix-vector multiplication.
We now state the convergence theorem for the inexact power iteration Algorithms \[alg:inexact\_power\] and \[alg:inexact\_power\_no\]. The theorem provides a convergence guarantee with high probability given that the complexity parameter is sufficiently large with the number of iteration steps $T$ chosen properly. Note that the logarithmic dependence of $T$ on the spectral gap $\lambda_1 / \lambda_2$ and the error criteria $\delta$ and $\varepsilon$ is expected from the standard power method. The dependence of $m_0$, the complexity parameter, on the ratio of the $1$-norm and $2$-norm of $A$ is due to the competition between the $1$- and $2$-norm growth of the iterate, where the $1$-norm matters for the control of the error of the inexact matrix-vector product.
\[thm:main\_conv\] For the inexact power iteration Algorithm \[alg:inexact\_power\_no\], under Assumption \[asmp:error\], for any precision $\varepsilon > 0$ and small probability $\delta \in (0,1)$, there exist time $$\label{eq:choiceT}
T = \log (\lambda_1 / \lambda_2)^{-1} \log \left( \frac{2\sqrt{2}}{\sqrt{\delta}\varepsilon\cos\theta(u_1,v_0)} \right)$$ and measure of complexity $$\label{eq:choicem0}
m_0 = \frac{4C_e}{\delta \varepsilon^2 \bigl(\cos\theta(u_1,v_0)\bigr)^2} \frac{{\lVertv_0\rVert}_1^2}{{\lVertv_0\rVert}_2^2} \, T \biggl( \frac{{\lVertA\rVert}_1}{{\lVertA\rVert}_2}\biggr)^{2T},$$ such that with probability at least $1 - 2\delta$, for any $m \geq m_0$, it holds $$\tan\theta(u_1,v_T) \leq \varepsilon$$ Moreover, if Assumption \[asmp:F\_homo\] is satisfied, the same result holds for Algorithm \[alg:inexact\_power\].
Before proving the theorem, let us collect a few immediate consequences of Assumption \[asmp:error\]. The proof is obvious and will be omitted.
\[lem:main\_conv\] If the error $\xi_t$ satisfies Assumption \[asmp:error\], we have
a) The error is unbiased $${\mathbb{E}\,}\xi_t = 0.$$
b) The error at different step is uncorrelated, in particular $${\mathbb{E}\,}\xi_t^{\top} A^r \xi_s = 0$$ for any $t \neq s$ and for all non-negative integer $r$.
c) The expected $2$-norm of the error can be controlled as $$\label{eq:2_norm_control}
{\mathbb{E}\,}{\lVert\xi_t\rVert}_2^2 \leq C_e {\lVertA\rVert}_1^{2t} \frac{{\lVertv_0\rVert}_1^2}{m}.$$
From the iteration of Algorithm \[alg:inexact\_power\_no\], we obtain $$\begin{aligned}
v_t &= Av_{t-1} + \xi_t \\
&= A^t v_0 + A^{t-1} \xi_1 + \cdots + A\xi_{t-1} + \xi_t.
\end{aligned}$$ Since the error $\xi_t$ is unbiased, we have $${\mathbb{E}\,}v_t = A^t v_0.$$ We now control the variance of $v_t$. Since $\xi_t$ is uncorrelated, we have $${\mathbb{E}\,}v_t^{\top}v_t = v_0^{\top}A^{2t}v_0 + {\mathbb{E}\,}\xi_1^{\top} A^{2t-2} \xi_1 + \cdots {\mathbb{E}\,}\xi_{t-1}^{\top} A^2 \xi_{t-1} + {\mathbb{E}\,}\xi_t^{\top}\xi_t.$$ Thus, $${\mathbb{E}\,}v_t^{\top}v_t - {\mathbb{E}\,}v_t^{\top}{\mathbb{E}\,}v_t = {\mathbb{E}\,}\xi_1^{\top} A^{2t-2} \xi_1 + \cdots {\mathbb{E}\,}\xi_{t-1}^{\top} A^2 \xi_{t-1} + {\mathbb{E}\,}\xi_t^{\top}\xi_t.$$ Using Lemma \[lem:main\_conv\], we estimate $$\begin{aligned}
{\lvert{\mathbb{E}\,}v_t^{\top}v_t - {\mathbb{E}\,}v_t^{\top}{\mathbb{E}\,}v_t\rvert} &= {\lvert{\mathbb{E}\,}\xi_1^{\top} A^{2t-2} \xi_1 + \cdots + {\mathbb{E}\,}\xi_{t-1}^{\top} A^2 \xi_{t-1} + {\mathbb{E}\,}\xi_t^{\top}\xi_t\rvert} \cr
&\leq {\mathbb{E}\,}{\lvert\xi_1^{\top} A^{2t-2} \xi_1\rvert} + \cdots + {\mathbb{E}\,}{\lvert \xi_{t-1}^{\top} A^2 \xi_{t-1}\rvert} + {\mathbb{E}\,}{\lvert\xi_t^{\top}\xi_t\rvert} \cr
&\leq \lambda_1^{2t-2} {\mathbb{E}\,}{\lvert\xi_1^{\top}\xi_1\rvert} + \cdots + \lambda_1^2 {\mathbb{E}\,}{\lvert \xi_{t-1}^{\top}\xi_{t-1}\rvert} + {\mathbb{E}\,}{\lvert\xi_t^{\top}\xi_t\rvert} \cr
&\leq C_e \frac{{\lVertA\rVert}_1^2 {\lVertv_0\rVert}_1^2}{m} \Bigl(\lambda_1^{2t-2} + \cdots + \lambda_1^2 {\lVertA\rVert}_1^{2t-4} + {\lVertA\rVert}_1^{2t-2}\Bigr) \cr
& = C_e \frac{{\lVertA\rVert}_1^2 {\lVertv_0\rVert}_1^2}{m}
\frac{{\lVertA\rVert}_1^{2t} - {\lVertA\rVert}_2^{2t}}{{\lVertA\rVert}_1^{2} - {\lVertA\rVert}_2^{2}},
\end{aligned}$$ where in the last step, we used the fact that $\lambda_1 = {\lVertA\rVert}_2$ since $\lambda_1$ is the largest eigenvalue. Recall that for symmetric matrix, ${\lVertA\rVert}_1 \geq {\lVertA\rVert}_2$, hence we have $$\label{eq:varest1}
{\lvert{\mathbb{E}\,}v_t^{\top}v_t - {\mathbb{E}\,}v_t^{\top}{\mathbb{E}\,}v_t\rvert}
\leq C_e \frac{{\lVertv_0\rVert}_1^2}{m} \; t {\lVertA\rVert}_1^{2t}.$$ By analogous arguments for $${\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t{\mathbb{E}\,}v_t^{\top} = {\mathbb{E}\,}A^{t-1} \xi_1 \xi_1^{\top} A^{t-1} + \cdots + {\mathbb{E}\,}A \xi_{t-1} \xi_{t-1}^{\top} A + {\mathbb{E}\,}\xi_t\xi_t^{\top},$$ we get $$\label{eq:varest2}
\bigl\lVert {\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t{\mathbb{E}\,}v_t^{\top} \bigr\rVert_2 \leq C_e \frac{ {\lVertv_0\rVert}_1^2}{m} \; t {\lVertA\rVert}_1^{2t}.$$
Let us now estimate the angle between $v_t$ and $u_1$ – the eigenvector associated with the largest eigenvalue. By definition, $$\label{eq:tan}
\bigl( \tan\theta(u_1,v_t) \bigr)^2 = \frac{{\lVertv_t\rVert}_2^2 - (u_1^{\top} v_t)^2 }{(u_1^{\top} v_t)^2}.$$ For the denominator, we know the expectation $${\mathbb{E}\,}u_1^{\top} v_t = u_1^{\top} A^t v_0
= \lambda_1^t (u_1^{\top} v_0 ),$$ and the variance $$\begin{aligned}
\operatorname{Var}(u_1^{\top} v_t) &= u_1^{\top} ({\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t{\mathbb{E}\,}v_t^{\top}) u_1 \cr
&\leq {\lVert{\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t{\mathbb{E}\,}v_t^{\top}\rVert}_2
\stackrel{\eqref{eq:varest2}}{\leq} C_e \frac{{\lVertv_0\rVert}_1^2}{m}\; t {\lVertA\rVert}_1^{2t}.
\end{aligned}$$ The Chebyshev inequality implies that $$\mathbb{P}\biggl({\left\lvertu_1^{\top}v_t - \lambda_1^t u_1^{\top}v_0\right\rvert} \geq \sqrt{\frac{C_e t}{m\delta}} {\lVertv_0\rVert}_1 {\lVertA\rVert}_1^t\biggr) \leq \delta,$$ and hence, as ${\lvert\lambda_1^t u_1^{\top}v_0\rvert} - {\lvertu_1^{\top}v_t\rvert} \leq
{\lvertu_1^{\top}v_t - \lambda_1^t u_1^{\top}v_0\rvert}$, $$\mathbb{P} \biggl( {\left\lvertu_1^{\top}v_t\right\rvert} \leq {\left\lvert\lambda_1^t u_1^{\top}v_0\right\rvert} - \sqrt{\frac{C_e t}{m\delta}} {\lVertv_0\rVert}_1 {\lVertA\rVert}_1^{t} \biggr) \leq \delta,$$ or equivalently $$\mathbb{P} \biggl( {\left\lvertu_1^{\top}v_t\right\rvert}^2 \leq \biggl({\left\lvert\lambda_1^t u_1^{\top}v_0\right\rvert} - \sqrt{\frac{C_e t}{m\delta}} {\lVertv_0\rVert}_1 {\lVertA\rVert}_1^{t}\biggr)^2 \biggr) \leq \delta,$$ For the numerator of , the expectation is $$\begin{aligned}
{\mathbb{E}\,}\bigl( {\lVertv_t\rVert}_2^2 - (u_1^{\top} v_t)^2\bigr) &= \sum_{i=2}^N u_i^{\top} {\mathbb{E}\,}v_tv_t^{\top} u_i \\
&= \sum_{i=2}^N u_i^{\top} ({\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t {\mathbb{E}\,}v_t^{\top}) u_i + \sum_{i=2}^N (u_i^{\top} {\mathbb{E}\,}v_t)^2 \\
&\leq \operatorname{tr}({\mathbb{E}\,}v_tv_t^{\top} - {\mathbb{E}\,}v_t {\mathbb{E}\,}v_t^{\top}) + \lambda_2^{2t} {\lVertv_0\rVert}_2^2 \\
&= ({\mathbb{E}\,}v_t^{\top}v_t - {\mathbb{E}\,}v_t^{\top}{\mathbb{E}\,}v_t) + \lambda_2^{2t} {\lVertv_0\rVert}_2^2 \\
&\stackrel{\eqref{eq:varest1}}{\leq} C_e \frac{ {\lVertv_0\rVert}_1^2}{m} \; t {\lVertA\rVert}_1^{2t} + \lambda_2^{2t} {\lVertv_0\rVert}_2^2.
\end{aligned}$$ By Markov inequality, for any $\delta \in (0,1)$ $$\mathbb{P} \left( {\lVertv_t\rVert}_2^2 - (u_1^{\top} v_t)^2 \geq \frac{1}{\delta} \Bigl( C_e \frac{ {\lVertv_0\rVert}_1^2}{m} \; t {\lVertA\rVert}_1^{2t} + \lambda_2^{2t} {\lVertv_0\rVert}_2^2 \Bigr)\right) \leq \delta.$$ Therefore, $$\mathbb{P} \left(
\bigl( \tan \theta(u_1, v_t) \bigr)^2
\leq \frac{1}{\delta} \frac{ C_e \frac{ {\lVertv_0\rVert}_1^2}{m} \; t {\lVertA\rVert}_1^{2t} + \lambda_2^{2t} {\lVertv_0\rVert}_2^2 }{ \Bigl(
{\left\lvert\lambda_1^t u_1^{\top}v_0\right\rvert} - \sqrt{\frac{C_e t}{m\delta}} {\lVertv_0\rVert}_1 {\lVertA\rVert}_1^{t} \Bigr)^2}
\right) \geq 1-2\delta.$$ We can explicitly check then with the choices and , for $m \geq m_0$, we have $$\mathbb{P} \bigl(\tan\theta(u_1, v_T) \leq \varepsilon\bigr) \geq 1-2\delta,$$ thus the claim of the theorem.
As mentioned above, it is possible to drop the martingale difference sequence condition in Assumption \[asmp:error\] and get a similar result. The reason is that the second moment bound can be used to control the bias of $\xi_t$. We state this as the following theorem.
\[coro:biased\] For the inexact power iteration Algorithm \[alg:inexact\_power\_no\], under the Assumptions \[asmp:error\](b) and \[asmp:error\](c), for any precision $\varepsilon>0$ and small probability $\delta \in (0,1)$, there exist time $$T = \log (\lambda_1 / \lambda_2)^{-1} \log \left( \frac{4}{\sqrt{\delta}\varepsilon\cos\theta(u_1,v_0)} \right)$$ and measure of complexity $$m_0 = \frac{8C_e}{\delta \varepsilon^2 \bigl(\cos\theta(u_1,v_0)\bigr)^2} \frac{{\lVertv_0\rVert}_1^2}{{\lVertv_0\rVert}_2^2} \, T^2 \biggl( \frac{{\lVertA\rVert}_1}{{\lVertA\rVert}_2}\biggr)^{2T},$$ such that with probability at least $1 - 2\delta$, for any $m \geq m_0$, it holds $$\tan\theta(u_1,v_T) \leq \varepsilon$$ Moreover, if Assumption \[asmp:F\_homo\] is satisfied, the same result holds for Algorithm \[alg:inexact\_power\].
Note that $$\begin{aligned}
u_1^\top v_t &= u_1^\top A^t v_0 + u_1^\top A^{t-1} \xi_1 + \cdots + u_1^\top A \xi_{t-1} + u_1^\top \xi_t \\
&= \lambda_1^t u_1^\top v_0 + \lambda_1^{t-1} u_1^\top \xi_1 + \cdots + \lambda_1 u_1^\top \xi_{t-1} + u_1^\top \xi_t,
\end{aligned}$$ so we get $$\begin{aligned}
{\mathbb{E}\,}(u_1^\top v_t - \lambda_1^t u_1^\top v_0)^2 &= {\mathbb{E}\,}(\lambda_1^{t-1} u_1^\top \xi_1 + \cdots + \lambda_1 u_1^\top \xi_{t-1} + u_1^\top \xi_t)^2 \cr
&= \sum_{i,j=1}^t \lambda_1^{2t-i-j} {\mathbb{E}\,}u_1^\top \xi_i \xi_j^\top u_1 \cr
&\leq \sum_{i,j=1}^t \lambda_1^{2t-i-j} \bigl({\mathbb{E}\,}{\lVert\xi_i\rVert}_2^2 {\mathbb{E}\,}{\lVert\xi_j\rVert}_2^2\bigr)^{1/2} \cr
&\leq C_e t^2 {\lVertA\rVert}_1^{2t} \frac{{\lVertv_0\rVert}_1^2}{m}.
\end{aligned}$$ Moreover, $$\begin{aligned}
{\mathbb{E}\,}\bigl( {\lVertv_t\rVert}_2^2 - (u_1^{\top} v_t)^2\bigr)^2 &= \sum_{i=2}^N (u_i^\top A^t v_0)^2 + 2\sum_{i=2}^N\sum_{b=1}^t {\mathbb{E}\,}u_i^\top A^t v_0 \xi_b^\top A^{t-b} u_i + \sum_{i=2}^N \sum_{a=1}^t\sum_{b=1}^t {\mathbb{E}\,}u_i^\top A^{t-a}\xi_a\xi_b^\top A^{t-b} u_i \\
&\leq \lambda_2^{2t} {\lVertv_0\rVert}_2^2 + 2t \sqrt{C_e} \lambda_2^t {\lVertv_0\rVert}_2 {\lVertA\rVert}_1^t \frac{{\lVertv_0\rVert}_1}{\sqrt{m}} + t^2 C_e {\lVertA\rVert}_1^{2t} \frac{{\lVertv_0\rVert}_1^2}{m} \\
& \leq 2 \lambda_2^{2t} {\lVertv_0\rVert}_2^2 + 2 t^2 C_e {\lVertA\rVert}_1^{2t} \frac{{\lVertv_0\rVert}_1^2}{m},
\end{aligned}$$ where the Cauchy-Schwarz inequality is used in the last line. Thus we can again use the Markov inequality to bound both numerator and denominator on the right hand side of to obtain the claimed result.
Algorithms {#sec:algorithms}
==========
In this section, we will review two stochastic power iteration methods recently proposed in the literature: full configuration-interaction quantum Monte Carlo (FCIQMC) [@BoothThomAlavi:09] and fast randomized iteration (FRI) [@LimWeare:17]. They can be analyzed in the same framework we established in the previous section. In particular, we prove the convergence of the two algorithms using Theorem \[thm:main\_conv\]. We focus on these two methods since in some sense they represent two opposite ends of strategies inexact matrix-vector multiplications. It is possible to combine the ideas and get a zoo of different approaches, which possibly yield better results; and our analysis can be extended to these as well. We will also comment on two variants: *i*FCIQMC and hard thresholding (HT), closely related to the FCIQMC and FRI approaches.
Without loss of generality, we will assume the matrix $A$ is close to the identity matrix and thus the eigenvalues $\lambda_i$ are close to $1$ (we can always scale and center the original matrix so that this is true).
Full configuration-interaction quantum Monte Carlo
--------------------------------------------------
### Algorithm Description
FCIQMC is an algorithm originated in quantum chemistry literature to calculate the ground energy of a many-body electron system by a Monte Carlo algorithm for the full configuration-interaction of the many-body Hamiltonian [@BoothThomAlavi:09].
Let the Hamiltonian be a real symmetric matrix $H \in \mathbb{R}^{N\times N}$ under the Slater determinant basis. To find the ground state (the smallest eigenvector) of $H$, we write $A = I - \delta H$ for $\delta > 0$ sufficiently small and hence focus on the largest eigenvalue of $A$; this can be viewed as a first order truncation of the Taylor series of $e^{-\delta H}$. It is also possible to construct other variants of $A$ from $H$, which we will not go here.
The FCIQMC can be viewed as a stochastic inexact power iteration for finding the largest eigenvector of $A$, which corresponds more naturally to the unnormalized version of the inexact power iteration (Algorithm \[alg:inexact\_power\_no\]). In the algorithm, the vector $v_t$ is not stored as a vector, but represented as a collection of “signed particles” $\{\alpha_t^{(i)}\}_{i=1}^{M_t}$, where $M_t$ is the number of signed particles at iteration step $t$. Each signed particle $\alpha$ has two attributes: location $l_\alpha \in \{1,2,\cdots,N\}$ and sign $s_\alpha \in \{1,-1\}$. Denote $e_l \in \mathbb{R}^{N}$ the standard basis vector with value $1$ at its $l$-th component and $0$ at every other component. Then each signed particle $\alpha$ represents a signed unit vector $\alpha = s_\alpha e_{l_\alpha}$. The vector $v_t \in \mathbb{Z}^N$ is given by the sum of all signed particles at time $t$: $$\label{eq:vectorsum}
v_t = \sum_{i=1}^{M_t} \alpha_t^{(i)}.$$ With some ambiguity of notation, we refer to both the set of particles and the corresponding vector as $v_t$, connected by . As we always assume that the particles with opposite signs on the same location are annihilated (see the annihilation step in the algorithm description below), the vector $v_t$ uniquely determines the set of particles.
In FCIQMC, the inexact matrix-vector multiplication $F_m(A,v_t)$ consists of three steps of particle evolution: spawning, diagonal death/cloning and annihilation. Write $A = A_d + A_o$ with $A_d$ the diagonal part and $A_o$ the off-diagonal part. The spawning step approximates $A_o v_t$; the diagonal death / cloning step approximates $A_d v_t$, and the annihilation step sums up the results from the previous two steps and approximates the summation $A v_t = A_o v_t + A_d v_t$. The three steps will be described in more details below.
**Spawning.** Each signed particle $\alpha$ (we suppress the index of $\alpha_t^{(i)}$ to simplify notation) is allowed to spawn a child particle to another location, corresponding to a nonzero component of $A_o\alpha = s_{\alpha} A_o(:,l_\alpha)$.[^2] The location of spawning is chosen at random, with probability $p_{\text{loc}}(l \mid l_{\alpha})$, which is chosen in the original FCIQMC algorithm to be uniformly random over all nonzero components of $A_o\alpha$ for some simple Hamiltonian $H$. In general, $p_{\text{loc}}(\cdot \mid l_{\alpha})$ can be more complicated; we refer readers to [@BoothThomAlavi:09] for more details. In the following of the paper, $p_{\text{loc}}(\cdot \mid l_{\alpha})$ is assumed to be uniform distribution over all nonzero components of $A_o\alpha$, while our analysis can be extended to other choice of $p_{\text{loc}}(\cdot \mid l_{\alpha})$.
Once the location $l$ is chosen, $n$ (possibly $0$) children particles are spawned with the same sign $s = \operatorname{sgn}(A_o(l, l_\alpha) s_\alpha)$ determined by the sign of vector entry $(A_o\alpha)(l)$ and the particle $\alpha$. The location $l$ and number $n$ are stochastically chosen such that the overall step gives an unbiased estimate of $A_o\alpha$: $$\label{eq:fciqmc_spawn_unbiased}
{\mathbb{E}\,}(n s e_l \mid \alpha) = A_o\alpha.$$ Please refer Algorithm \[alg:fciqmc\_spawn\] for details.
$M^{\text{sp}} = 0$;
**Diagonal cloning / death.** This step represents $A_d v_t$ as a collection of particles in an analogous way to the spawning step. For every signed particle $\alpha$, we would consider children particles on the location $l_{\alpha}$ (i.e., the location of the new particles is chosen to be $l_{\alpha}$) and obtain an unbiased representation $$\label{eq:diagonal_unbiased}
{\mathbb{E}\,}(n s e_{l_\alpha} \mid \alpha) = A_d\alpha.$$ The details can be found in Algorithm \[alg:fciqmc\_diagonal\_death\], the key steps are similar to Algorithm \[alg:fciqmc\_spawn\].
$M^{\text{diag}} = 0$;
**Annihilation.** The annihilation step merges the children particles from the previous two steps and remove all pairs of particles with the same location and opposite signs. If we denote $v^{\text{sp}}$ and $v^{\text{diag}}$ the corresponding vector representation of the particles are the spawning and diagonal cloning / death steps, the annihilation steps create a collection of particles representing the new vector $v = v^{\text{sp}} + v^{\text{diag}}$. Applying the three steps above to the particles representing $v_t$, we obtain the new set of particles $v_{t+1}$ at time $t+1$. Since by construction $${\mathbb{E}\,}( v^{\text{sp}} \mid v_t) = A_o v_t, \quad \text{and} \quad
{\mathbb{E}\,}( v^{\text{diag}} \mid v_t) = A_d v_t,$$ we have on expectation $${\mathbb{E}\,}( v_{t+1} \mid v_t) = A v_t.$$ In terms of the notations used in the framework of inexact power iteration, $v_{t+1}$ represented using particles can be viewed as the approximate matrix-vector product $F_m(A, v_t)$: $$\label{eq:fciqmc_inexact_power_iteration}
F_m(A, v_t) := v_{t+1} = \sum_{i=1}^{M_{t+1}} \alpha_{t+1}^{(i)} = Av_t + \xi_{t+1},$$ where $\xi_{t+1}$ is introduced in the last equality to denote the error from the approximate matrix-vector multiplication through the stochastic particle representation. As we will show in the analysis below, the accuracy of FCIQMC iteration is controlled by the number of particles $M_t$; and thus it plays the role of the complexity parameter $m$ in our general framework. We would drop the subscript $m$ for $F_m$ in the sequel for FCIQMC, as the complexity parameter is implicit.
Now that we have defined the inexact matrix-vector multiplication $F(A,v_t)$ in FCIQMC, we may apply this in the inexact power iteration as Algorithm \[alg:inexact\_power\_no\]. However, this can be problematic in practice. Recall that $A = I - \delta H$ is assumed to be a perturbation of identity so its eigenvalue is around $1$. If the largest eigenvalue of $A$ is strictly larger than $1$, when the signed particles become a good approximation to the leading eigenvector, the number of particles $M_t$ will grow exponentially with rate $\lambda_1$, which quickly increases the computational cost and memory requirement. It is also possible (while the probability is tiny) that the number of particles may decrease to $0$ due to the randomness.
In practice, it is desirable to have controls on the number of particles to make the algorithm more stable. One such strategy is to introduce a shift $s_t \in \mathbb{R}$ and use matrix $$\label{eq:dynshift}
\widetilde{A} = A + \delta s_t I = I - \delta ( H - s_t I)$$ instead of $A$ at the $t$-th step. Notice that $s_t$ only shifts the eigenvalues while not changing the eigenspace. The shift $s_t$ is adjusted dynamically to control the number of particles. With such shifts, the full FCIQMC algorithm is presented in Algorithm \[alg:fciqmc\].
Initialization: $t=0$ and set initial particles $v_0$.\
Set $s_t = s_0$;
The Algorithm \[alg:fciqmc\] contains two phases for different strategies of choosing the shifts and thus controlling the particle population. In Phase 1, the shift is fixed to be $s_0$, which is chosen such that ${\lvertA(i,i) - s_0\rvert} \geq 1$ for all $i$ so that the particle number is most likely to grow exponentially till the target population $M^{\text{target}}$. In the second phase, the shift is dynamically adjusted, so to control the growth of the population by a negative feedback loop. The target number of population $M^{\text{target}}$ is chosen to be sufficiently large that the variance is small enough to ensure convergence. It plays the role as the ‘complexity’ $m$ in Theorem \[thm:main\_conv\]. $\eta$ and $q$ are two parameters to control the fluctuation of number of particles. For the details of the parameter choices, we refer the readers to the original paper on FCIQMC [@BoothThomAlavi:09] for details.
**Energy Estimator**. Several estimators can be used to estimate the smallest eigenvalue of $H$ based on the FCIQMC Algorithm \[alg:fciqmc\], which is just a linear transformation of the largest eigenvalue $\lambda_1$ of $A$. One estimator is simply the shift $s_t$. When the algorithm converges, $v_t$ is approximately proportional to the eigenvector $u_1$. Since $s_t$ is adjusted to control the number of particles steady, the largest eigenvalue of $A + \delta s_t I$ is approximately $1$, hence connecting $s_t$ with the desired eigenvalue estimate, cf. . The other estimator we will consider is the projected energy estimator $$E_t = \frac{v_*^{\top} Hv_t}{v_*^{\top} v_t}.$$ Here $v_*$ is some fixed vector, for example the Hartree-Fock state of the system. It is clear that when $v_t$ becomes a good approximation of the eigenvector $u_1$, $E_t$ gives a good estimate of the leading eigenvalue. In the numerical examples, we will focus on the projected energy estimator, since it can be applied to all algorithms we consider in this work (while shift estimator is unique for FCIQMC, in practice, it gives similar results compared to the projected energy estimator).
### Convergence Analysis
Since FCIQMC can be viewed as an inexact power iteration as in , we apply Theorem \[thm:main\_conv\] to analyze the convergence of FCIQMC. For simplicity, we will focus on the case that the shift is constantly $0$, $s_t = 0$, since the shift does not affect the eigenvector which is the main focus of Theorem \[thm:main\_conv\]. The probability distribution in the spawning step $p_{\textrm{loc}}(\cdot\mid l_\alpha)$ is assumed to be uniform distribution over all the nonzero entries of $A_o(:,l_\alpha)$. To avoid some degenerate case, we will assume that each diagonal entry of $A$ is non-zero and each column of $A$ has more than $2$ nonzero entries (so there is at least one possible location for children particles in the spawning step).
We now check the three conditions in Assumption \[asmp:error\]. The unbiasedness is guaranteed by construction as discussed above for the FCIQMC algorithm, we have $${\mathbb{E}\,}(v_{t+1} \mid \mathcal{F}_t) = A v_t,$$ or equivalently, the error $\xi_t$ is a martingale difference sequence: $${\mathbb{E}\,}(\xi_{t+1} \mid \mathcal{F}_t) = 0.$$ The expectation $2$-norm bound is established in the following proposition.
For the inexact matrix-vector multiplication in FCIQMC Algorithm \[alg:fciqmc\], the error $\xi_t$ satisfies $${\mathbb{E}\,}\bigl( {\left\lVert\xi_{t+1}\right\rVert}_2^2 \mid \mathcal{F}_t\bigr) \leq \left(\max_{1\leq k \leq n}(\|a_k\|_0 - 2)\|a_{o, k}\|_2^2 + \frac{1}{2} \right) \frac{{\left\lVertv_t\right\rVert}_1^2}{M_t},$$ where $a_k = A(:,k)$ is the $k$-th column vector of $A$, and $a_{o, k}$ is the $k$-th column vector of $A_o$, thus $a_{o, k}$ equals $a_k$ except for the $k$-th entry $a_{o, k}(k) = 0$.
Since each particle evolves independently, $$F(A,v_t) = F\Bigl(A,\sum_{i=1}^{M_t} \alpha_t^{(i)}\Bigr) = \sum_{i=1}^{M_t} F(A,\alpha_t^{(i)}).$$ Moreover $F(A,\alpha_t^{(i)})$ and $F(A,\alpha_t^{(j)})$ are independent for $i\neq j$ conditioned on $\mathcal{F}_t$.
By construction, $F(A,\alpha_t^{(i)})$ is unbiased, *i.e.*, $${\mathbb{E}\,}\bigl(F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)} \mid \mathcal{F}_t\bigr) = 0.$$ Therefore, $$\begin{aligned}
{\mathbb{E}\,}( {\left\lVert\xi_{t+1}\right\rVert}_2^2 \mid\mathcal{F}_t) &= {\mathbb{E}\,}\biggl( \biggl\lVert \sum_{i=1}^{M_t} (F(A, \alpha_t^{(i)}) - A\alpha_t^{(i)}) \biggr\rVert_2^2 \mid\mathcal{F}_t \biggr) \cr
&= {\mathbb{E}\,}\biggl( \Bigl(\sum_{i=1}^{M_t} F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)}\Bigr)^{\top} \Bigl(\sum_{j=1}^{M_t} F(A,\alpha_t^{(j)}) - A\alpha_t^{(j)}\Bigr) \mid\mathcal{F}_t\biggr) \cr
&= \sum_{i=1}^{M_t} {\mathbb{E}\,}\bigl( (F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)})^{\top} (F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)}) \mid\mathcal{F}_t\bigr) \cr
&\qquad + 2 \sum_{1\leq i < j \leq M_t} {\mathbb{E}\,}\bigl( (F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)})^{\top} \mid\mathcal{F}_t\bigr)\; {\mathbb{E}\,}\bigl( (F(A,\alpha_t^{(j)}) - A\alpha_t^{(j)}) \mid\mathcal{F}_t\bigr) \cr
&= \sum_{i=1}^{M_t} {\mathbb{E}\,}\Bigl(\bigl\lVert F(A,\alpha_t^{(i)}) - A\alpha_t^{(i)} \bigr\rVert_2^2 \mid\mathcal{F}_t\Bigr).
\end{aligned}$$ Hence, it suffices to consider each particle individually. To simplify the notation, without loss of generality, let us consider a particle with $\alpha_t^{(i)} = e_k$ for some $k$. Since the spawning and diagonal cloning/death steps are independent and unbiased, we have the decomposition $${\mathbb{E}\,}( {\left\lVert F(A,e_k) - A e_k\right\rVert}_2^2 \mid\mathcal{F}_t) = {\mathbb{E}\,}( {\left\lVert F(A_o,e_k) - A_o e_k\right\rVert}_2^2 \mid\mathcal{F}_t) + {\mathbb{E}\,}( {\left\lVert F(A_d,e_k) - A_d e_k\right\rVert}_2^2 \mid \mathcal{F}_t).$$
For the spawning step, since $A_o e_k = a_{o, k}$, there are ${\lVerta_{o, k}\rVert}_0$ locations to spawn. Remind that $p_{\textrm{loc}}(\cdot\mid k)$ is assumed to be uniform distribution, so each location is chosen with probability $\frac{1}{{\lVerta_{o,k}\rVert}_0}$. Following the Algorithm \[alg:fciqmc\_spawn\], we calculate that $$F(A_o, e_k) =
\begin{cases}
\lfloor{\left\lVerta_{o,k}\right\rVert}_0 |a_k(j)|\rfloor \operatorname{sgn}(a_k(j)) e_j, &
\text{w.p.} \quad \bigl(1 - \bigl( {\left\lVerta_{o,k}\right\rVert}_0
|a_k(j)| - \lfloor {\left\lVerta_{o,k}\right\rVert}_0 |a_k(j)|\rfloor \bigr) \bigr) / {\left\lVerta_{o,k}\right\rVert}_0, \\
\bigl(\lfloor{\left\lVerta_{o,k}\right\rVert}_0 |a_k(j)|\rfloor + 1\bigr)
\operatorname{sgn}(a_k(j)) e_j, & \text{w.p.} \quad \bigl( {\left\lVerta_{o,k}\right\rVert}_0
|a_k(j)| - \lfloor {\left\lVerta_{o,k}\right\rVert}_0 |a_k(j)|\rfloor \bigr) /
{\left\lVerta_{o,k}\right\rVert}_0,
\end{cases}$$ for each $j$ such that $a_{o,k}(j) \neq 0$. Straightforward calculation yields $$\begin{aligned}
{\mathbb{E}\,}{\left\lVertF(A_o, e_k) - A_o e_k\right\rVert}_2^2
& = ({\left\lVerta_{o,k}\right\rVert}_0 - 1) {\left\lVerta_{o,k}\right\rVert}_2^2 \cr
& \qquad + \frac{1}{{\left\lVerta_{o,k}\right\rVert}_0} \sum_{j,\, a_{o,k}(j) \neq 0} \bigl({\left\lVerta_{o,k}\right\rVert}_0 a_k(j) - \lfloor
{\left\lVerta_{o,k}\right\rVert}_0 a_k(j) \rfloor\bigr) \times \cr
& \hspace{10em} \times \Bigl(1 - \bigl({\left\lVerta_{o,k}\right\rVert}_0 a_k(j) - \lfloor
{\left\lVerta_{o,k}\right\rVert}_0 a_k(j) \rfloor\bigr) \Bigr) \cr
&\leq ({\left\lVerta_{o,k}\right\rVert}_0 - 1) {\left\lVerta_{o,k}\right\rVert}_2^2 + \frac{1}{4}.
\end{aligned}$$ For the diagonal cloning/death step, we have $$F(A_d, e_k) =
\begin{cases}
\lfloor |a_k(k)| \rfloor \operatorname{sgn}(a_k(k)) e_k, & \text{w.p.}
\quad 1 - \bigl( {\lverta_k(i)\rvert} - \lfloor {\lverta_k(i)\rvert} \rfloor \bigr); \\
(\lfloor|a_k(k)| \rfloor + 1) \operatorname{sgn}(a_k(k)) e_k, & \text{w.p.}
\quad {\lverta_k(i)\rvert} - \lfloor {\lverta_k(i)\rvert} \rfloor.
\end{cases}$$ Therefore $${\mathbb{E}\,}( {\left\lVertF(A_d,e_k) - A_d e_k \right\rVert}_2^2 \mid\mathcal{F}_t) =
\bigl( {\lverta_k(i)\rvert} - \lfloor {\lverta_k(i)\rvert} \rfloor \bigr) \Bigl(1 -
\bigl( {\lverta_k(i)\rvert} - \lfloor {\lverta_k(i)\rvert} \rfloor \bigr) \Bigr) \leq \frac{1}{4}.$$ Summing up the contribution from the two steps, we arrive at $${\mathbb{E}\,}( {\lVertF(A, e_k) - A e_k\rVert}_2^2 \mid\mathcal{F}_t) \leq ({\lVerta_k\rVert}_0 - 2){\left\lVerta_{o,k}\right\rVert}_2^2 + \frac{1}{2},$$ where we used ${\lVerta_{o, k}\rVert}_0 = {\lVerta_k\rVert}_0 - 1$. Thus $${\mathbb{E}\,}( {\lVertF(A, v_t) - A v_t\rVert}_2^2 \mid\mathcal{F}_t) \leq M_t \Bigl(\max_{1\leq k\leq n} ({\lVerta_k\rVert}_0 - 2){\left\lVerta_{o,k}\right\rVert}_2^2 + \frac{1}{2}\Bigr).$$ Since $M_t = {\lVertv_t\rVert}_1$, we can rewrite the above estimate as $${\mathbb{E}\,}({\lVertF(A, v_t) - A v_t\rVert}_2^2 \mid\mathcal{F}_t) \leq \frac{{\lVertv_t\rVert}_1^2}{M_t} \Bigl(\max_{1\leq k\leq n} (\|a_k\|_0 - 2)\|a_{o,k}\|_2^2 + \frac{1}{2}\Bigr).$$
Here we emphasize the important role of the annihilation step in FCIQMC reflected in the error analysis above. Only with the annihilation step is $M_t = {\lVertv_t\rVert}_1$ true, so that the growth of error is controlled as in the last step of the proof. In general, without annihilation, the error will be exponentially larger as $\frac{M_t}{{\lVertv_t\rVert}_1}$ grows exponentially even when $v_t$ is close to the eigenvector $u_1$. Suppose $v_t$ is approximately $u_1$. Then $v_{t+1} \approx \lambda_1 v_t$. Therefore, ${\lVertv_{t+1}\rVert}_1 \approx {\lVertA\rVert}_2 {\lVertv_t\rVert}_1$. However for the number of particles $M_t$ without annihilation, $M_{t+1} \approx {\lVert{\lvertA\rvert}\rVert}_2 M_t$, where ${\lvertA\rvert}$ is the entry-wise absolute value of $A$. To see this, let us denote $v_t^+$ the vector represented by all the particles with positive sign and $-v_t^-$ the vector represented by all the particles with negative sign. Then $v_t = v_t^+ - v_t^-$. Denote $\tilde{v}_t = v_t^+ + v_t^-$. Then $M_t = {\lVert\tilde{v}_t\rVert}_1$ without annihilation. We can easily check that $\tilde{v}_t$ evolves according to $\tilde{v}_{t+1} = {\lvertA\rvert} \tilde{v}_t$. So finally, $\tilde{v}_t$ will converge to the eigenvector of ${\lvertA\rvert}$, and $M_{t+1} \approx {\lVert{\lvertA\rvert}\rVert}_2 M_t$. Noticing that ${\lVertA\rVert}_2 \leq {\lVert{\lvertA\rvert}\rVert}_2 \leq {\lVertA\rVert}_1$, we know $\frac{M_t}{{\lVertv_t\rVert}_1}$ grows exponentially at rate $\frac{{\lVert{\lvertA\rvert}\rVert}_2}{{\lVertA\rVert}_2}$ after convergence. Therefore if the number of particles $M_t$ has an upper bound, which is always true in practice due to computational resource constraint, ${\lVertv_t\rVert}_1$ will decay to zero exponentially, which means the algorithm will not converge to the correct eigenvector. Also comment that if the spawning distribution $p_{\text{loc}}(\cdot \mid l_{\alpha})$ is not exactly uniform distribution, then ${\mathbb{E}\,}( {\left\lVert F(A_o,e_k) - A_o e_k\right\rVert}_2^2 \mid\mathcal{F}_t)$ will be bound by another constant depending on $A_o$. Therefore the bound of ${\mathbb{E}\,}\bigl( {\left\lVert\xi_{t+1}\right\rVert}_2^2 \mid \mathcal{F}_t\bigr)$ in the Proposition will only differ by a constant multiplier.
Compared with Assumption \[asmp:error\], we observe that the particle number $M_t$ plays the role of the “complexity” parameter. The more particles we have, the smaller the error is. We have the following corollary assuming the particle number is bounded from below by $m$
If the particle number satisfies $M_t \geq m$, $${\mathbb{E}\,}({\lVertF(A, v_t) - A v_t\rVert}_2^2 \mid\mathcal{F}_t, M_t \geq m) \leq C_e \frac{{\lVertA\rVert}_1^2 {\lVertv_t\rVert}_1^2}{m},$$ where $C_e = \dfrac{\max_{k} (\|a_k\|_0 - 2)\|a_{o,k}\|_2^2 +
\frac{1}{2}}{{\lVertA\rVert}_1^2}$ is a parameter scale-invariant of $A$.
In summary, FCIQMC satisfies Assumption \[asmp:error\]b, as long as the particle number is not too small. Note that in practice the particle number can be controlled by the dynamic shift $s_t$ to ensure that it does not drop below the required lower bound.
The Assumption \[asmp:error\]c, the growth of expectation $1$-norm bound, can also be checked easily, since we have $$\begin{aligned}
{\mathbb{E}\,}({\lVertv_{t+1}\rVert}_1 \mid \mathcal{F}_t) &= {\mathbb{E}\,}\Bigl(\biggl\lVert F(A, \sum_{i=1}^{M_t} \alpha_t^{(i)}) \biggr\rVert_1 \mid \mathcal{F}_t\Bigr)
= {\mathbb{E}\,}\Bigl( \biggl\lVert\sum_{i=1}^{M_t} F(A,\alpha_t^{(i)}) \biggr\rVert_1 \mid \mathcal{F}_t\Bigr) \cr
&\leq \sum_{i=1}^{M_t} {\mathbb{E}\,}({\lVertF(A,\alpha_t^{(i)})\rVert}_1 \mid \mathcal{F}_t)
= \sum_{i=1}^{M_t} {\lVertA\alpha_t^{(i)}\rVert}_1
\leq \sum_{i=1}^{M_t} {\lVertA\rVert}_1 {\lVert\alpha_t^{(i)}\rVert}_1
= {\lVertA\rVert}_1 {\lVertv_t\rVert}_1.\end{aligned}$$
In conclusion, we have verified the assumptions of Theorem \[thm:main\_conv\], and thus it can be applied for the convergence and error analysis of FCIQMC.
### Remarks on *i*FCIQMC
*i*FCIQMC (initiator FCIQMC) [@ClelandBoothAlavi:10] is a modified version of FCIQMC. It can be viewed as a bias-variance tradeoff strategy to reduce the computational cost and error of the FCIQMC approach, by restricting the spawning step.
The $n$ locations are divided into two sets: the initiators $L_i$ and non-initiators $L_n$ with $L_i \cap L_n = \emptyset$, $L_i \cup L_n = \{1,2,\cdots, N\}$. The rule of *i*FCIQMC is that for any particle $\alpha$ at a non-initiator location $l_\alpha \in L_n$, it is only allowed to spawn children particles at locations already occupied by some other particles. If $\alpha$ spawns particles to a location unoccupied, then the children particles are discarded. An exception rule is that if at least two particles at non-initiator locations spawn children particles with the same sign at one unoccupied location, then the children particles are kept. There are no restrictions for spawning steps for particles in initiators. In the case that all the locations are initiators $L_n = \emptyset$, *i*FCIQMC reduces to FCIQMC.
The initiators $L_i$ are chosen at the beginning according to some prior knowledge. The initiators are then updated at each step of iteration. Suppose $n_{i,thre}\in\mathbb{N}$ is a fixed threshold. As soon as the number of particles at a non-initiator location is greater than the threshold $n_{i,thre}$, then the location becomes an initiator. Intuitively, initiators are more important locations for the eigenvector since they are occupied by many particles. The restrictions on the spawning ability of non-initiators reduce the computational cost and the variance of the inexact matrix-vector product while only introducing small bias since there are few particles on non-initiators. Therefore, *i*FCIQMC can be viewed as a variance control technique for FCIQMC.
Fast Randomized Iteration
-------------------------
In this section, we provide a numerical analysis based on our general framework for the convergence of the fast randomized iteration (FRI), recently proposed in the applied mathematics literature [@LimWeare:17], inspired by FCIQMC type algorithms. The basic idea of the FRI method is to first apply the matrix $A$ on the vector of current iterate, and then employ a stochastic compression algorithm to reduce the resulting vector to a sparse representation. The original convergence analysis [@LimWeare:17] uses a norm motivated by viewing the vectors as random measures. In comparison, as we have seen in the proof of Theorem \[thm:main\_conv\], our viewpoint and analysis is closer in spirit to numerical linear algebra, in particular the standard convergence analysis of power method.
### Algorithm Description
The fast randomized iteration (FRI) algorithm is based on a choice of random compression function $\Phi_m : \mathbb{R}^N \to \mathbb{R}^N$, which maps a full vector $v$ to a sparse vector $\Phi_m(v)$ with approximately only $m$ nonzero components. The sparsity of $\Phi_m(v)$ reduces the storage cost of the vector and associated computational cost. To combine FRI with the inexact power iteration, define $$\label{eq:fri_inexact}
F_m(A, v_t) = \Phi_m(A v_t)$$ in Algorithm \[alg:inexact\_power\] and \[alg:inexact\_power\_no\]. The error is $\xi_{t+1} = \Phi_m(A v_t) - A v_t$.
Thus the FRI algorithm is completely characterized by the choice of compression function $\Phi_m$, about which we assume the following properties. These are adaptations of the Assumptions \[asmp:F\_homo\] and \[asmp:error\] in the context of a compression function.
\[asmp:fri\_error\] For any vector $v \in \mathbb{R}^N$, the compression function $\Phi_m$ satisfies:
a) Homogeneity: For all $c \in \mathbb{R}$, $$\Phi_m(cv) = c\Phi_m(v);$$
b) Unbiasedness $${\mathbb{E}\,}(\Phi_m(v) \mid v) = v;$$
c) Variance bound. For some constant $C_{\Phi}$ independent of $m$ and $v$, $${\mathbb{E}\,}({\lVert\Phi_m(v) - v\rVert}_2^2 \mid v) \leq C_{\Phi} \frac{{\lVertv\rVert}_1^2}{m};$$
d) Expectation $1$-norm bound $${\mathbb{E}\,}({\lVert\Phi_m(v)\rVert}_1 \mid v) = {\lVertv\rVert}_1.$$
The compression function $\Phi_m$ introduced in [@LimWeare:17] is as follows. For a given vector $v\in \mathbb{R}^N$, first we sort the entries as ${\lvertv(q_1)\rvert} \geq {\lvertv(q_2)\rvert} \geq \cdots \geq {\lvertv(q_N)\rvert}$, where $q: [N] \to [N]$ is a permutation. The compression function consists of two parts. In the first part, large components of the vector are preserved exactly. Define $$\tau = \max_{ 1\leq i\leq N} \left\{ i : |v(q_i)| \geq \frac {\sum_{j=i}^N |v(q_j)|}{m+1-i} \right\},$$ with the convention $\max \{\emptyset\} = 0$, so $0\leq \tau \leq m$. The compression function keeps the entries $v(q_i)$ for any $1 \leq i \leq \tau$, $$\bigl(\Phi_m(v)\bigr)(q_i) = v(q_i), \qquad \forall i \leq \tau.$$ Note that if ${\lVertv\rVert}_0 \leq m$, all components are ‘large’ and $\Phi_m(v) = v$, the input vector is kept without compression. The remaining $n - \tau$ components are considered ‘small’. Under the compression we only keep a few entries so the resulting vector $\Phi_m(v)$ has about $m$ nonzero entries, as in Algorithm \[alg:fri\_compression\]; the details are further discussed below.
$B = \{1,2,\cdots,N\}$;
$s = {\lVertv\rVert}_1$;
$\displaystyle i' = \arg\max_{i\in B} {\lvertv(i)\rvert}$;
$\tau = 0$;
In the second part of Algorithm \[alg:fri\_compression\], the set $B = \{q_{\tau+1}, q_{\tau+2}, \cdots, q_N\}$ consists of the indices of all ‘small’ components to be compressed. Note that for the integer random variable $N_i$, $i \in B$, only its expectation ${\mathbb{E}\,}N_i \in (0,1)$ is specified, so there is still freedom to choose the probability distribution of $\{N_i\}_{i\in B}$. Here we only discuss independent Bernoulli (which is easy to understand) and systematic sampling (which we use in the numerical examples) approaches, while other choices are possible. Let us focus on the entries in $B$ and define $v' \in \mathbb{R}^n$ such that $v'(i) = v(i) {\boldsymbol{1}}_{\{ i\in B \}}$. It follows that ${\lVertv'\rVert}_1 = {\lVertv\rVert}_1 - \sum_{i=1}^{\tau } {\lvertv(q_i)\rvert}$.
For the independent Bernoulli, $N_i$ is independent for each $i \in B$ and follows the Bernoulli distribution as $$N_i =
\begin{cases}
0, & \text{w.p.} \quad 1 - \frac{|v(i)|}{\|v'\|_1 /(m - \tau)}\,; \\
1, & \text{w.p.} \quad \frac{|v(i)|}{\|v'\|_1 /(m - \tau)}\,.
\end{cases}$$ Note that the probability is well defined due to the choice of $\tau$. The number of nonzero components of the compressed vector is ${\lVert\Phi_m(v)\rVert}_0 = \tau + \sum_{i\in B} N_i$. From the choice of $N_i$, ${\mathbb{E}\,}({\lVert\Phi_m(v)\rVert}_0 \mid v) = m$; so $m$ is the expected sparsity of $\Phi_m(v)$.
Another choice is the systematic sampling [@LimWeare:17]: Take a random variable $U$ uniformly distributed in $(0,1)$. Then for $k=1,2,\cdots,m - \tau$, define $$U_k = \frac{U+k-1}{m - \tau}.$$ Given $\{q'_1, q'_2, \cdots, q'_{N-\tau}\}$ any permutation of indices in $B$, define $$I_k = \max_{1\leq i\leq N-\tau} \biggl\{i: \sum_{j=1}^{i-1} |v(q'_i)| \leq U_k \|v'\|_1 < \sum_{j=1}^i |v(q'_i)| \biggr\},$$ then $N_i$ is given by $$N_i =
\begin{cases}
1, & \text{if}\quad i = q'_{I_k} \ \text{for some}\ k, \\
0, & \text{otherwise}.
\end{cases}$$ Notice that by construction, the number of nonzero $N_i$s is exactly $m - \tau$, therefore ${\lVert\Phi_m(v)\rVert}_0 = m$. The $N_i$s generated by systematic sampling is obviously correlated as only one random number $U$ drives the generation. The two approaches will be analyzed in the next section in the framework of inexact power iteration.
### Convergence Analysis
We now apply Theorem \[thm:main\_conv\] to analyze the convergence of the FRI algorithm with either independent Bernoulli or systematic sampling. Notice that we have the immediate result
Assumption \[asmp:fri\_error\] implies Assumptions \[asmp:F\_homo\] and \[asmp:error\].
Therefore it suffices to check Assumption \[asmp:fri\_error\] for the compression function $\Phi_m$. Homogeneity is obvious. From the construction of $\Phi_m$, the unbiasedness is guaranteed by the expectation of $N_i$s, no matter which particular distribution is used for $N_i$. $${\mathbb{E}\,}(\Phi_m(v) \mid v) = v.$$ The variance bounds are proved in the following lemmas.
For FRI compression with either independent Bernoulli or systematic sampling, $${\mathbb{E}\,}({\lVert\Phi_m(v) - v\rVert}_2^2 \mid v) \leq \frac{{\lVertv'\rVert}_1^2}{m - \tau} \leq \frac{{\lVertv\rVert}_1^2}{m}.$$ Moreover, we have the almost sure bound for systematic sampling, $${\lVert\Phi_m(v) - v\rVert}_2^2 \leq \frac{2 {\lVertv'\rVert}_1^2}{m - \tau} \leq \frac{2 {\lVertv\rVert}_1^2}{m}, \quad \text{a.s.}$$
It is not possible to obtain an almost sure bound as above for independent Bernoulli, since for example it is possible that all the Bernoulli variables are $1$, which gives large error ${\lVert\Phi_m(v)-v\rVert}_2^2 \geq
\frac{(N-2m+\tau){\lVertv'\rVert}_1^2}{(m-\tau)^2}$. This Lemma thus implies the advantage of using the systematic sampling strategy, which in practice gives smaller variance in general. We will only show numerical results using the systematic sampling strategy in the numerical examples later.
Since large components of $v$ are kept exactly by $\Phi_m(\cdot)$, we have $${\lVert\Phi_m(v) - v\rVert}_2^2 = \sum_{i=\tau+1}^N (\Phi_m(v)(q_i) - v(q_i))^2
= \sum_{i=\tau+1}^N \bigl( (\Phi_m(v)(q_i))^2 + v(q_i)^2 - 2\Phi_m(v)(q_i)v (q_i) \bigr).$$ Take the expectation $${\mathbb{E}\,}\bigl({\lVert\Phi_m(v)-v\rVert}_2^2 \mid v\bigr) = {\mathbb{E}\,}\biggl( \sum_{i=\tau+1}^N (\Phi_m(v)(q_i))^2 \mid v \biggr) + \sum_{i=\tau+1}^N v(q_i)^2 - 2\sum_{i=\tau+1}^N v(q_i) {\mathbb{E}\,}(\Phi_m(v)(q_i) \mid v).$$ Since both independent Bernoulli and systematic sampling are unbiased, $${\mathbb{E}\,}\Phi_m(v)(q_i) = v(q_i).$$ Moreover, because there are $\sum_{i=\tau+1}^N N_i$ number of $\frac{{\lVertv'\rVert}_1}{m - \tau}$ and $n-\tau-\sum_{i=\tau+1}^N N_i$ number of $0$ in $\{{\lvert\Phi_m(v)(q_i)\rvert}\}_{i\in B}$, we have $${\mathbb{E}\,}\biggl( \sum_{i=\tau+1}^N (\Phi_m(v)(q_i))^2 \mid v \biggr) = {\mathbb{E}\,}\biggl({\mathbb{E}\,}\biggl( \sum_{i=\tau+1}^N (\Phi_m(v)(q_i))^2 \mid \sum_{i=\tau+1}^N N_i \biggr) \mid v \biggr)
= \frac{{\lVertv'\rVert}_1^2}{(m - \tau)^2} {\mathbb{E}\,}\biggl( \sum_{i=\tau+1}^N N_i \mid v\biggr).$$ For independent Bernoulli, ${\mathbb{E}\,}\bigl( \sum_{i=\tau+1}^N N_i \mid v \bigr) = m-\tau$ and for systematic sampling, $\sum_{i=\tau+1}^N N_i = m-\tau$, so $${\mathbb{E}\,}\biggl( \sum_{i=\tau+1}^N (\Phi_m(v)(q_i))^2 \mid v\biggr) = \frac{{\lVertv'\rVert}_1^2}{m - \tau}.$$ Finally, $${\mathbb{E}\,}\bigl({\lVert\Phi_m(v)-v\rVert}_2^2 \mid v\bigr) = \frac{{\lVertv'\rVert}_1^2}{m - \tau} - {\lVertv'\rVert}_2^2 \leq \frac{{\lVertv'\rVert}_1^2}{m - \tau}.$$
We now show that $\frac{{\lVertv'\rVert}_1}{m - \tau} \leq \frac{{\lVertv\rVert}_1}{m}$, which follows from the fact that $\frac {\sum_{j=i}^N |v(q_j)|}{m+1-i}$ is nonincreasing in $i$ for $i \leq \tau$. Indeed, recall from the choice of $\tau$ that for $i \leq \tau$, $|v(q_i)| \geq \frac{ \sum_{j=i}^N |v(q_j)|}{m+1-i}$, which is equivalent to $$\frac{\sum_{j=i}^N |v(q_j)|}{m+1-i} \geq \frac{\sum_{j=i+1}^N |v(q_j)|}{m-i}.$$ Thus, combined with ${\lVertv'\rVert}_1 \leq {\lVertv\rVert}_1$, we arrive at $${\mathbb{E}\,}({\lVert\Phi_m(v) - v\rVert}_2^2 \mid v) \leq \frac{{\lVertv'\rVert}_1^2}{m - \tau}
\leq \frac{{\lVertv\rVert}_1^2}{m}.$$
Next we give the almost sure bound for systematic sampling. Note that if $N_i \neq 0$ for $i \in B$, since $\bigl(\Phi_m(v)\bigr)(q_i)$ and $v(q_i)$ have the same sign, we have $$(\Phi_m(v)(q_i) - v(q_i))^2 \leq \Phi_m(q_i)^2 = \frac{{\lVertv'\rVert}_1^2}{(m - \tau)^2}.$$ Since there are exactly $m-\tau$ nonzero $N_i$s, we can estimate $$\begin{aligned}
{\lVert\Phi_m(v) - v\rVert}_2^2 &= \sum_{i=\tau+1}^N (\bigl(\Phi_m(v)\bigr)(q_i) - v(q_i))^2 \cr
&\leq \sum_{i = \tau + 1}^N v(q_i)^2 {\boldsymbol{1}}_{N_i = 0} + \bigl(\Phi_m(v)\bigr)(q_i)^2 {\boldsymbol{1}}_{N_i \neq 0}\cr
&\leq {\lVertv'\rVert}_2^2 + (m - \tau)\frac{{\lVertv'\rVert}_1^2}{(m - \tau)^2} \cr
&\leq \frac{{\lVertv'\rVert}_1^2}{m - \tau} + \frac{{\lVertv'\rVert}_1^2}{m - \tau} =
\frac{2{\lVertv'\rVert}_1^2}{m - \tau} \leq \frac{2{\lVertv\rVert}_1^2}{m}.
\end{aligned}$$
The expectation $1$-norm bound can be easily checked from the definition.
For FRI with independent Bernoulli compression, $${\mathbb{E}\,}({\lVert\Phi_m(v)\rVert}_1 \mid v) = {\lVertv\rVert}_1.$$ For FRI with systematic sampling compression, $${\lVert\Phi_m(v)\rVert}_1 = {\lVertv\rVert}_1, \quad \text{a.s.}$$
Therefore, the compression function $\Phi_m$ satisfies Assumption \[alg:fri\_compression\], and thus the convergence follows Theorem \[thm:main\_conv\].
### Deterministic compression by hard thresholding
Another way to choose the compression function $\Phi_m$ is by simple hard thresholding, which means $\Phi_m = \Phi_m^{\text{HT}}$ keeps the $m$ largest entries (in absolute value) and drops the remaining ones. Compared to the previously discussed approaches of compression, the hard thresholding obviously has smaller variance since it is deterministic, as a price to pay, it introduces bias to the inexact matrix-vector multiplication. The bias-variance tradeoff between hard thresholding and FRI type algorithm is similar to that between *i*FCIQMC and FCIQMC.
Numerical Results {#sec:numerics}
=================
In this section, we give some numerical tests of the FCIQMC and FRI algorithms, and their variance *i*FCIQMC and Hard Thresholding to compare the performance. The numerical problem is to compute the ground energy of a Hamiltonian $H$ for a quantum system. As discussed before, we define $A = I - \delta H$ for $\delta$ small so the problem is equivalent to find the largest eigenvalue of $A$. We will test these methods with two types of model systems: the 2D fermionic Hubbard model and small chemical molecules under the full CI discretization. The Hamiltonians for these have the same structure. Each electron lives in a finite dimensional one-particle Hilbert space. The vectors in the basis set of the one-particle Hilbert space are called orbitals. The number of orbitals $N^{\text{orb}}$ is the dimension of the one-particle space. We denote $N^{\text{elec}}$ the total number of electrons in the system in total. Due to the Pauli exclusion principle, there are at most two electrons with opposite spins in one orbital. In our test examples, we choose the total spin $S^{\text{tot}} = 0$. Therefore the dimension of the space is $\binom{N^{\text{orb}}}{N^{\text{elec}}/2}^2$, neglecting other constraints like symmetry. The dimension grows exponentially as $N^{\text{orb}}$ and $N^{\text{elec}}$ grows. Here we summarize the system in our numerical tests in the Table \[tb:system\]:
System $N^{\text{orb}}$ $N^{\text{elec}}$ dimension HF energy Ground energy
--------------------- ------------------ ------------------- ------------------ ----------- ---------------
$4\times 4$ Hubbard 16 10 $1.2\times 10^6$ -17.7500 -19.5809
, aug-cc-pVDZ 23 10 $1.4\times 10^8$ -128.4963 -128.7114
, cc-pVDZ 24 10 $4.5\times 10^8$ -76.0240 -76.2419
: Test Systems[]{data-label="tb:system"}
The exact ground energy of the Hubbard model and are computed using exact power iteration, and the ground energy of is from the paper [@Olsen:98]. We use [@Spencer2015] (<http://www.hande.org.uk/>), an open source stochastic quantum chemistry program written in Fortran, for FCIQMC and *i*FCIQMC calculation. FRI and HT subroutines are implemented in Fortran based on . The Hamiltonian of the Hubbard model is included in the package and the Hamiltonians of and are calculated using RHF (restricted Hartree Fock) by (<http://www.psicode.org/>), an open source ab initio electronic structure package. The code to generate the entries of Hamiltonian $H$ is the same for all four algorithms, so the comparison among algorithms is fair in terms of computational time. The four algorithms are tested on a computer with 6 Core Xeon CPU at 3.5GHz and 64GB RAM.
Note that our comparison is mostly for illustrative purpose and should not be taken as benchmark tests for the various algorithms especially for large scale calculations, which would depend on parallel implementation, hardware infrastructure, etc. On the other hand, even for small problems, the numerical results still offer some suggestions on further development of inexact power iteration based solvers for many-body quantum systems.
Hubbard Model
-------------
The Hubbard model is a standard model used in condensed matter physics, which describes interacting particles on a lattice. In real space, the Hubbard Hamiltonian is $$\label{eq:hubbard_real}
H = -\sum_{\langle r,r'\rangle,\sigma} \hat{c}_{r,\sigma}^\dagger \hat{c}_{r',\sigma} + U\sum_r \hat{n}_{r\uparrow}\hat{n}_{r\downarrow},$$ where we have scale the hopping parameter to be $1$ and so the on-site repulsion parameter $U$ gives the ratio of interaction strength relative to the kinetic energy. We choose an intermediate interaction strength $U=4$ in our test.
In the $d$ dimensional Hubbard Hamiltonian , $r$ is a $d$-dimensional vector representing a site in the lattice, $\langle r,r'\rangle$ means $r$ and $r'$ are the nearest neighbor, and $\sigma$ takes values of $\uparrow$ and $\downarrow$, which is the spin of the electron. $\hat{c}_{r,\sigma}$ and $\hat{c}_{r,\sigma}^\dagger$ are the annihilation and creation operator of electrons at site $r$ with spin $\sigma$. They satisfy the commutation relations $$\{\hat{c}_{r,\sigma}, \hat{c}_{r',\sigma'}^\dagger \} = \delta_{r,r'}\delta_{\sigma, \sigma'}, \qquad
\{\hat{c}_{r,\sigma}, \hat{c}_{r',\sigma'} \} = 0, \quad \text{and} \quad
\{\hat{c}_{r,\sigma}^\dagger, \hat{c}_{r',\sigma'}^\dagger \} = 0,$$ where $\{A, B\} = AB + BA$ is the anti-commutator. $\hat{n}_{r,\sigma}$ is the number operator and defined as $\hat{n}_{r,\sigma} = \hat{c}_{r,\sigma}^\dagger \hat{c}_{r,\sigma}$. We will consider Hubbard model on a finite 2D lattice with periodic boundary condition.
When the interaction strength $U$ is small, it is better to work in the momentum space instead of the real space, since the planewaves are the eigenfunctions of the kinetic part of the Hamiltonian. The annihilation operator in momentum space is $\hat{c}_{k,\sigma} = \frac{1}{\sqrt{N^{\text{orb}}}} \sum_r
e^{ik\cdot r} \hat{c}_{r,\sigma}$, where $k=(k_1,k_2)$ is the wave number and $N^{\text{orb}}$ is the total number of orbitals or sites. The Hubbard Hamiltonian in momentum space is then $$\label{eq:hubbard_k}
H = \sum_{k,\sigma} \varepsilon(k)n_{k,\sigma} + \frac{U}{N^{\text{orb}}} \sum_{k,p,q}c_{p-q,\uparrow}^\dagger c_{k+q,\downarrow}^\dagger c_{k,\downarrow}c_{p,\uparrow},$$ where $\varepsilon(k) = -2\sum_{i=1}^2 \cos(k_i)$.
Written as a matrix, the Hubbard Hamiltonian in the momentum space is just a real symmetric matrix with diagonal entries $\varepsilon(k)$ and off-diagonal either $0$ or $\pm\frac{U}{N^{\text{orb}}}$. For inexact power iteration, we take $A = I - \delta H$ with $\delta = 0.01$. In our numerical test, we will use the projected energy estimator for the smallest eigenvalue of $H$; the projected vector $v_*$ is chosen to be the Hartree-Fock state. The initial iteration of all methods is also chosen as the Hartree-Fock state (a vector whose only nonzero entry is at the Slater determinant corresponding to the Hartree-Fock ground state of the system).
Figure \[fig:convergence\] plots the error of projected energy of each iteration versus wall-clock time (first $1500$ seconds) for a typical realization. The error is defined as the difference between the projected energy estimate and the exact ground energy. The complexity parameters of the algorithms are shown in Table \[tb:4410hubbard\], which are chosen such that FRI and FCIQMC use about the same amount of memory (e.g., the particle number in FCIQMC is roughly equal to the non-zero entries of the matrix-vector product in FRI or HT before compression), and also chosen so large that all the algorithms converge. The time per iteration listed in Table \[tb:4410hubbard\] is averaged over several realizations and is used in the Figure \[fig:convergence\].
![Convergence of the projected energy with respect to time for System 1, a $4\times 4$ Hubbard model with $10$ electrons, $5$ spin up and $5$ spin down, and interaction strength $U=4$.[]{data-label="fig:convergence"}](figure/hubbard.eps){width="90.00000%"}
$m$ ${\lVertAv_t\rVert}_0$ avg. error std. MSE $\tau_{\text{auto}}$ compr. error time/iter.(s)
----------- ------------------ ------------------------ --------------------- --------------------- --------------------- ---------------------- --------------------- --------------- --
FCIQMC $1.7\times 10^6$ - $4.4\times 10^{-4}$ $3.0\times 10^{-4}$ $2.6\times 10^{-7}$ $14.1$ $4.6\times 10^{-2}$ $1.1$
*i*FCIQMC $1.7\times 10^6$ - $3.2\times 10^{-4}$ $2.2\times 10^{-4}$ $1.8\times 10^{-7}$ $13.8$ $2.6\times 10^{-2}$ $0.91$
FRI $3.0\times 10^4$ $9.4\times 10^5$ $1.2\times 10^{-4}$ $6.1\times 10^{-5}$ $2.8\times 10^{-8}$ $13.7$ $1.6\times 10^{-1}$ $3.6$
HT $3.0\times 10^4$ $7.2\times 10^5$ $1.6\times 10^{-2}$ - $2.5\times 10^{-4}$ - $4.5\times 10^{-3}$ $3.5$
: Parameters and numerical results for $4\times 4$ Hubbard Model with $10$ electrons, $5$ spin up and $5$ spin down, and interaction strength $U=4$.[]{data-label="tb:4410hubbard"}
As shown in Figure \[fig:convergence\], all four algorithms converge to result close to the exact eigenvalue and the estimated value from each iteration stays around the eigenvalue for a long time. FCIQMC and *i*FCIQMC take much less time to converge, thanks to their much lower-cost inexact matrix-vector multiplication compared to FRI and HT, but the variance is also larger. In terms of iteration number, the convergence of the four algorithms is similar, which can be understood from our analysis since it is the same eigenvalue gap of the Hamiltonian that drives the convergence. As we mentioned already, per iteration, the FCIQMC and *i*FCIQMC is much cheaper in comparison. The reason is that FRI and HT need to access all nonzero elements of $A$ for each column associated with a non-zero entry in the current iterate (for multiplying $A$ with the sparse vector), while FCIQMC and *i*FCIQMC just need to randomly pick some, without accessing the others. The number of non-zero entries per row is large and accessing elements of $A$ is quite expensive for FCI type problems. More quantitatively, we see in Table \[tb:4410hubbard\] that for a sparse vector of $3\times 10^4$ non-zero entries in FRI, after multiplication by $A$ before compression, the number of non-zero entries increases to roughly $10^6$. Thus for this problem, on average, each column has about $40$ nonzero entries that FRI needs to access, while FCIQMC algorithm only needs access of few entries after the random choice.
After convergence, the projected energy of FCIQMC and *i*FCIQMC fluctuate around the exact ground state energy. Although *i*FCIQMC is biased, the bias is not large for the current problem, while the variance is smaller than FCIQMC. So *i*FCIQMC is an effective strategy for bias-variance trade-off. The projected energy of FRI also varies around the true energy, and the variance is much smaller than FCIQMC or *i*FCIQMC. HT is deterministic and the projected energy shows no variance. However the bias is also quite visible.
We can average the projected energy over the path to get a better estimate. The variance of the estimator will decay to zero as we include longer time period in the average. Thus, due to unbiasedness, the error of FCIQMC and FRI can be made smaller if we run for long enough. In Table \[tb:4410hubbard\], we give more quantitative comparison of the results of the algorithms. The quantities in the table are defined as below $$\begin{aligned}
& \text{avg. error} && \frac{1}{w} \sum_{i=i_0}^{i_0+w-1} {\lvertE_i - E^{\text{true}}\rvert} \\
& \text{std.} && \sqrt{\frac{1}{w-1} \sum_{i=i_0}^{i_0+w-1} \Bigl(E_i - \frac{1}{w} \sum_{j=i_0}^{i_0+w-1} E_i\Bigr)^2} \sqrt{\frac{1+2\tau_{\text{auto}}}{W}} \\
& \text{MSE} && \text{avg. error}^2 + \text{std.}^2 \\
& \tau_{\text{auto}} && \sum_{t=1}^{w-1} \frac{\frac{1}{w-1} \sum_{i=i_0}^{i_0+w-t-1} \Bigl(E_i - \frac{1}{w} \sum_{j=i_0}^{i_0+w-1} E_i\Bigr)\Bigl(E_{i+t} - \frac{1}{w} \sum_{j=i_0}^{i_0+w-1} E_i\Bigr)}{\frac{1}{w-1} \sum_{i=i_0}^{i_0+w-1} \Bigl(E_i - \frac{1}{w} \sum_{j=i_0}^{i_0+w-1} E_i\Bigr)^2}\\
& \text{compr. error} && \frac{1}{w} \sum_{i=i_0}^{i_0+w-1} \frac{{\lVert\xi_{t+1}\rVert}_2}{{\lVertAv_t\rVert}_2} \\\end{aligned}$$ Here $E^{\text{true}}$ is the true ground energy obtained by exact power iteration, $i_0$ is a burn-in parameter and $w$ is the window size of the average. For FCIQMC and *i*FCIQMC, $w = 1600$ and $i_0 = 2400$. For FRI and HT, $w = 400$ and $i_0 = 600$. The numerical tests show that the quantities above are insensitive to the choice of $w$ and $i_0$, as long as the algorithms indeed converge after $i_0$ steps and the window size $w$ is not too small. $\tau_{\text{auto}}$ is the integrated autocorrelation time and $W$ is the number of iterations averaged. The std. is short for the standard deviation of the sample mean $\bar{E}^{(W)}$ defined as $\bar{E}^{(W)} = \frac{1}{W} \sum_{i = i_0}^{i_0+W-1} E_i$. Since the time cost per iteration of different algorithms is quite different, to make a fair comparison, we take $W = \frac{10000}{\text{time per iter.}}$ for each algorithm. It gives the standard error of the sample mean if we run each algorithm for $10000$ seconds after convergence. The mean square error (MSE) is simply defined to incorporate the variance and bias together.
[0.45]{} ![$4\times 4$ Hubbard model. (left) Relative compression error ${\lVert\xi_{t+1}\rVert}_2 / {\lVertAv_t\rVert}_2$ as a function of iteration steps; (right) Angle between the iterate and the exact ground state $\tan\theta(v_t,
u_1)$ \[fig:tot2\]](figure/err.eps "fig:"){width="\textwidth"}
[0.45]{} ![$4\times 4$ Hubbard model. (left) Relative compression error ${\lVert\xi_{t+1}\rVert}_2 / {\lVertAv_t\rVert}_2$ as a function of iteration steps; (right) Angle between the iterate and the exact ground state $\tan\theta(v_t,
u_1)$ \[fig:tot2\]](figure/tan.eps "fig:"){width="\textwidth"}
To further obtain insights of the interplay between the error per step of inexact power iteration and the convergence, we plot in Figure \[fig:tot2\] the relative compression error and the tangent of the angle between $v_t$ and the exact eigenvector $u_1$. We observe that that FRI and HT reach convergence after about $100$ steps and FCIQMC and *i*FCIQMC converge after about $350$ steps; the more steps of FCIQMC and *i*FCIQMC are related with the first phase of the algorithm where the particle number is exponentially growing. This can be seen from Figure \[fig:tot2\](left) as the huge error growth of the initial stage of the iterations. Only when the particle number reaches a certain level, the compression error becomes small and the power iteration convergence kicks in.
After convergence, FRI has the largest compression error and HT has the smallest. The compression error of *i*FCIQMC is also smaller than the one of FCIQMC. It is reasonable since HT and *i*FCIQMC reduce variance and thus compression error compared with the fully stochastic FRI and FCIQMC. As shown in Figure \[fig:tot2\], in this example with the parameter choice, FCIQMC has smaller compression error than FRI; and the larger the compression error is, the further $v_t$ is away from the true eigenvector $u_1$. This agrees with the theoretical results we obtain in Theorem \[thm:main\_conv\], because $\tan\theta(v_t,u_1)$ is controlled by the error $\xi_t$ at each step.
We remark that the $\tan\theta(v_t, u_1)$ error measure does not directly translate to the error of the projected energy estimator using say the Hartree-Fock state. In fact, we observe in Figure \[fig:convergence\] and Table \[tb:4410hubbard\] that per iteration, the projected energy estimated by FRI is smaller than FCIQMC and *i*FCIQMC. As an explanation, in our parameter regime, the exact ground state has a large overlap with the Hartree-Fock state, so in FRI, that component is kept unchanged in the compression, while for FCIQMC and *i*FCIQMC, the stochastic error is more uniformly distributed over all the entries. This behavior seems more problem dependent though, as we will see in the chemical molecular examples that the MSE of FRI become comparable with FCIQMC.
Molecules
---------
We also tested the four algorithms for some molecule examples. The FCI Hamiltonian is obtained by a Hartree-Fock calculations in a chosen chemical basis (for single-particle Hilbert space), such as cc-PVDZ. We choose and at equilibrium geometry as examples, which is described in Table \[tb:system\]. The time step is taken as $\delta = 0.01$.
The convergence of projected energy error versus wall-clock time is shown in Figure \[fig:ne\] and Figure \[fig:h2o\] respectively. The parameter choice of the algorithms and more quantitative comparison are shown in Table \[tb:ne\] and Table \[tb:h2o\]. The four algorithms also work well for molecule systems. The convergence behavior is similar to the Hubbard case.
The complexity parameter $m$ needed to achieve convergence depends on the system. The ratio $m/N$ of is smaller than . The time cost of FRI and HT is much larger than FCIQMC and *i*FCIQMC, because they require the exact matrix-vector multiplication $Av_t$, which is still expensive although $v_t$ is sparse. Unlike the Hubbard case where FRI gives much smaller error, the MSE of FRI is similar to FCIQMC and *i*FCIQMC in these cases.
![Convergence of the projected energy with respect to time for in aug-cc-pVDZ basis[]{data-label="fig:ne"}](figure/ne.eps){width="95.00000%"}
$m$ ${\lVertAv_t\rVert}_0$ avg. error std. MSE $\tau_{\text{auto}}$ time/iter.(s)
----------- ------------------ ------------------------ --------------------- --------------------- --------------------- ---------------------- ---------------
FCIQMC $1.8\times 10^6$ - $1.1\times 10^{-4}$ $3.8\times 10^{-5}$ $1.8\times 10^{-8}$ $12.4$ $1.5$
*i*FCIQMC $1.8\times 10^6$ - $7.7\times 10^{-5}$ $2.4\times 10^{-5}$ $9.6\times 10^{-9}$ $15.3$ $1.1$
FRI $1.0\times 10^4$ $7.9\times 10^6$ $8.0\times 10^{-5}$ $4.8\times 10^{-5}$ $1.1\times 10^{-8}$ $12.3$ $11.6$
HT $1.0\times 10^4$ $3.3\times 10^6$ $1.7\times 10^{-3}$ - $2.8\times 10^{-6}$ - $7.9$
: Comparison of algorithms for in aug-cc-pVDZ basis[]{data-label="tb:ne"}
![Convergence of the projected energy with respect to time for in cc-pVDZ basis[]{data-label="fig:h2o"}](figure/h2o.eps){width="95.00000%"}
$m$ ${\lVertAv_t\rVert}_0$ avg. error std. MSE $\tau_{\text{auto}}$ time/iter.(s)
----------- ------------------ ------------------------ --------------------- --------------------- ---------------------- ---------------------- ---------------
FCIQMC $6.0\times 10^7$ - $4.1\times 10^{-5}$ $1.7\times 10^{-4}$ $2.1\times 10^{-9}$ $25.6$ $54.1$
*i*FCIQMC $6.0\times 10^7$ - $1.4\times 10^{-5}$ $5.3\times 10^{-5}$ $2.7\times 10^{-10}$ $119$ $36.6$
FRI $1.2\times 10^5$ $1.6\times 10^8$ $2.2\times 10^{-5}$ $1.2\times 10^{-4}$ $8.9\times 10^{-10}$ $12.8$ $379.3$
HT $1.2\times 10^5$ $3.4\times 10^7$ $1.1\times 10^{-3}$ - $1.2\times 10^{-6}$ - $227.0$
: Comparison of algorithms for in cc-pVDZ basis[]{data-label="tb:h2o"}
In summary, the numerical examples show that the FCIQMC, FRI and their variants can achieve convergence using much less memory and computational time compared to the standard power iteration. The stochastic algorithms FCIQMC, iFCIQMC and FRI give better estimates than the deterministic method HT in general. The numerical test also points out directions to further improve these inexact power iterations, including variance and memory cost reduction of the inexact matrix-vector multiplication and efficient parallel implementation to overcome the memory bottleneck. These will be leaved for future works.
[^1]: This research is supported in part by National Science Foundation under award DMS-1454939. We thank useful discussions with George Booth, Yingzhou Li, Jonathan Weare, Stephen Wright, and Lexing Ying during various stages of the work.
[^2]: MATLAB notation $A(:, l)$ is used to denote the $l$-th column of $A$.
|
---
abstract: 'How to model a pair of sentences is a critical issue in many NLP tasks such as answer selection (AS), paraphrase identification (PI) and textual entailment (TE). Most prior work (i) deals with one individual task by fine-tuning a specific system; (ii) models each sentence’s representation separately, rarely considering the impact of the other sentence; or (iii) relies fully on manually designed, task-specific linguistic features. This work presents a general $\textbf{A}$ttention $\textbf{B}$ased $\textbf{C}$onvolutional $\textbf{N}$eural $\textbf{N}$etwork (ABCNN) for modeling a pair of sentences. We make three contributions. (i) ABCNN can be applied to a wide variety of tasks that require modeling of sentence pairs. (ii) We propose three attention schemes that integrate mutual influence between sentences into CNN; thus, the representation of each sentence takes into consideration its counterpart. These interdependent sentence pair representations are more powerful than isolated sentence representations. (iii) ABCNN achieves state-of-the-art performance on AS, PI and TE tasks.'
author:
- |
Wenpeng Yin, Hinrich Schütze\
Center for Information and Language Processing\
LMU Munich, Germany\
[wenpeng@cis.lmu.de]{}\
Bing Xiang, Bowen Zhou\
IBM Watson\
Yorktown Heights, NY, USA\
[@us.ibm.com]{}\
bibliography:
- 'acl2012.bib'
title: |
ABCNN: Attention-Based Convolutional Neural Network\
for Modeling Sentence Pairs
---
Introduction
============
How to model a pair of sentences is a critical issue in many NLP tasks such as answer selection (AS) [@yu2014deep; @feng2015applying], paraphrase identification (PI) [@madnani2012re; @yinnaacl], textual entailment (TE) [@marelli2014semeval; @bowman2015large] etc.
Most prior work derives each sentence’s representation separately, rarely considering the impact of the other sentence. This neglects the mutual influence of the two sentences in the context of the task. It also contradicts what humans do when comparing two sentences. We usually focus on key parts of one sentence by extracting parts from the other sentence that are related by identity, synonymy, antonymy and other relations. Thus, human beings model the two sentences together, using the content of one sentence to guide the representation of the other.
demonstrates that each sentence of a pair partially determines which parts of the other sentence we should focus on. For AS, correctly answering $s_0$ requires putting attention on “gross”: $s_1^+$ contains a corresponding unit (“earned”) while $s_1^-$ does not. For PI, focus should be removed from “today” to correctly recognize ($<\!s_0,s_1^+\!>$) as paraphrases and ($<\!s_0,s_1^-\!>$) as non-paraphrases. For TE, we need to focus on “full of people” (to recognize TE for $<\!s_0,s_1^+\!>$) and on “outdoors” / “indoors” (to recognize non-TE for $<\!s_0,s_1^-\!>$). These examples show the need for an architecture that computes different representations of $s_i$ for different $s_{1-i}$’s .
Convolutional Neural Network (CNN, [@lecun1998gradient]) is widely used to model sentences [@kalchbrenner2014convolutional; @kim2014convolutional] and sentence pairs [@yu2014deep; @socher2011dynamic; @yinnaacl], especially in classification tasks. CNN is supposed to be good at extracting robust and abstract features of input. This work presents ABCNN, an attention-based convolutional neural network, that has a powerful mechanism for modeling a sentence pair by taking into account the interdependence between the two sentences. ABCNN is a general architecture that can handle a wide variety of sentence pair modeling tasks.
Some prior work proposes simple mechanisms that can be interpreted as controlling varying attention; e.g., employ word alignment to match related parts of the two sentences. In contrast, our attention scheme based on CNN is able to model relatedness between two parts fully automatically. Moreover, attention at multiple levels of granularity, not only at the word level, is achieved as we stack multiple convolution layers that increase abstraction.
Prior work on attention in deep learning mostly addresses LSTMs (long short-term memory, ). LSTM achieves attention usually in word-to-word scheme, and the word representations mostly encode the *whole context* within the sentence [@bahdanau2015neural; @entail2016]. But it is not clear whether this is the best strategy; e.g., in the AS example in , it is possible to determine that “how much” in $s_0$ matches “\$161.5 million” in $s_1$ without taking the entire remaining sentence contexts into account. This observation was also investigated by where an information retrieval system retrieves sentences with tokens labeled as DATE by named entity recognition or as CD by part-of-speech tagging if there is a “when” question. However, labels or POS tags require extra tools. CNNs benefit from incorporating attention into representations of *local phrases* detected by filters; in contrast, LSTMs encode the *whole context* to form attention-based word representations – a strategy that is more complex than the CNN strategy and (as our experiments suggest) performs less well for some tasks.
Apart from these differences, it is clear that attention has as much potential for CNNs as it does for LSTMs. As far as we know, this is the first NLP paper that incorporates attention into CNNs. Our ABCNN gets state-of-the-art in AS and TE tasks, and competitive performance in PI, then obtains further improvements over all three tasks when linguistic features are used.
Section \[sec:relatedwork\] discusses related work. Section \[sec:bcnn\] introduces BCNN, a network that models two sentences in parallel with shared weights, but without attention. presents three different attention mechanisms and their realization in ABCNN, an architecture that is based on BCNN. Section \[sec:experiments\] evaluates the models on AS, PI and TE tasks and conducts visual analysis for our attention mechanism. Section \[sec:sum\] summarizes the contributions of this work.
Related Work {#sec:relatedwork}
============
Non-NN Work on Sentence Pair Modeling
-------------------------------------
Sentence pair modeling has attracted lots of attention in the past decades. Most tasks can be reduced to a semantic text matching problem. Due to the variety of word choices and inherent ambiguities in natural languages, bag-of-word approaches with simple surface-form word matching tend to produce brittle results with poor prediction accuracy [@bilotti2007structured]. As a result, researchers put more emphasis on exploiting syntactic and semantic structure. Representative examples include methods based on deeper semantic analysis [@shen2007using; @moldovan2007cogex], tree edit-distance [@punyakanok2004mapping; @heilman2010tree] and quasi-synchronous grammars [@wang2007jeopardy] that match the dependency parse trees of the two sentences. Instead of focusing on the high-level semantic representation, turn their attention to improving the shallow semantic component, lexical semantics, by performing semantic matching based on a latent word-alignment structure (cf. ). explore more fine-grained word overlap and alignment between two sentences using negation, hypernym/hyponym, synonym and antonym relations. extend word-to-word alignment to phrase-to-phrase alignment by a semi-Markov CRF. However, such approaches often require more computational resources. In addition, employing syntactic or semantic parsers – which produce errors on many sentences – to find the best match between the structured representation of two sentences is not trivial.
NN Work on Sentence Pair Modeling
---------------------------------
To address some of the challenges of non-NN work, much recent work uses neural networks to model sentence pairs for AS, PI and TE.
For AS, present a bigram CNN to model question and answer candidates. extend this method and get state-of-the-art performance on the WikiQA dataset (). test various setups of a bi-CNN architecture on an insurance domain QA dataset. explore bidirectional LSTM on the same dataset. Our approach is different because we do not model the sentences by two independent neural networks in parallel, but instead as an interdependent sentence pair, using attention.
For PI, form sentence representations by summing up word embeddings. use recursive autoencoder (RAE) to model representations of local phrases in sentences, then pool similarity values of phrases from the two sentences as features for binary classification. present a similar model in which RAE is replaced by CNN. In all three papers, the representation of one sentence is not influenced by the other – in contrast to our attention-based model.
For TE, use recursive neural networks to encode entailment on SICK [@marelli2014sick]. present an attention-based LSTM for the Stanford natural language inference corpus [@bowman2015large]. Our system is the first CNN-based work on TE.
Some prior work aims to solve a general sentence matching problem. present two CNN architectures, ARC-I and ARC-II, for sentence matching. ARC-I focuses on sentence representation learning while ARC-II focuses on matching features on phrase level. Both systems were tested on PI, sentence completion (SC) and tweet-response matching. propose the MultiGranCNN architecture to model general sentence matching based on phrase matching on multiple levels of granularity and get promising results for PI and SC. try to match two sentences in AS and SC by multiple sentence representations, each coming from the local representations of two LSTMs. Our work is the first one to investigate attention for the general sentence matching task.
Attention-Based NN in Non-NLP Domains
-------------------------------------
Even though there is little if any work on attention mechanisms in CNNs for NLP, attention-based CNNs have been used in computer vision for visual question answering [@chen2015abc], image classification [@xiao2015application], caption generation [@xu2015show], image segmentation [@hong2015learning] and object localization [@cao2015look].
apply attention in recurrent neural network (RNN) to extract information from an image or video by adaptively selecting a sequence of regions or locations and only processing the selected regions at high resolution. combine a spatial attention mechanism with RNN for image generation. investigate attention-based RNN for recognizing multiple objects in images. and use attention in RNN for speech recognition.
Attention-Based NN in NLP
-------------------------
Attention-based deep learning systems are studied in NLP domain after its success in computer vision and speech recognition, and mainly rely on recurrent neural network for end-to-end encoder-decoder system for tasks such as machine translation [@bahdanau2015neural; @luong2015effective] and text reconstruction [@li2015hierarchical; @rush2015neural]. Our work takes the lead in exploring attention mechanism in CNN for NLP tasks.
BCNN: Basic Bi-CNN {#sec:bcnn}
==================
We now introduce our basic (non-attention) CNN that is based on Siamese architecture [@bromley1993signature], i.e., it consists of two weight-sharing CNNs, each processing one of the two sentences, and a final layer that solves the sentence pair task. See . We refer to this architecture as *BCNN*. The next section will then introduce ABCNN, an attention architecture that extends BCNN. gives our notational conventions.
In our implementation and also in the mathematical formalization of the model given below, we pad the two sentences to have the same length $s=\max(s_0,s_1)$. However, in the figures we show different lengths because this gives a better intuition of how the model works.
symbol description
------------------- ----------------------------------------
$s$, $s_0$, $s_1$ sentence or sentence length
$v$ word
$w$ filter width
$d_i$ dimensionality of input to layer $i+1$
$\mathbf{W}$ weight matrix
: Notation[]{data-label="tab:notation"}
![BCNN: ABCNN without Attention[]{data-label="fig:arc-0"}](ABCNN-ARC0){width="\colfigfactor\columnwidth"}
BCNN has four types of layers: input layer, convolution layer, average pooling layer and output layer. We now describe each in turn.
**Input layer.** In the example in the figure, the two input sentences have 5 and 7 words, respectively. Each word is represented as a $d_0$-dimensional precomputed word2vec [@mikolov2013distributed] embedding,[^1] $d_0=300$. As a result, each sentence is represented as a feature map of dimension $d_0 \times s$.
**Convolution layer.** Let $v_1,v_2,\ldots,v_s$ be the words of a sentence and $\mathbf{c}_i\in\mathbb{R}^{w\cdot d_0}$, $0< i <s+w$, the concatenated embeddings of $v_{i-w+1},\ldots,v_{i}$ where embeddings for $v_i$, $i<1$ and $i>s$, are set to zero. We then generate the representation $\mathbf{p}_i\in\mathbb{R}^{d_1}$ for the *phrase* $v_{i-w+1},\ldots,v_{i}$ using the convolution weights $\mathbf{W}\in\mathbb{R}^{d_1\times
wd_0}$ as follows: $$\mathbf{p}_i=\mathrm{tanh}(\mathbf{W}\cdot\mathbf{c}_i+\mathbf{b})$$ where $\mathbf{b}\in\mathbb{R}^{d_1}$ is the bias. We use *wide convolution*; i.e., we apply the convolution weights $\mathbf{W}$ to words $v_i$, $i<1$ and $i> s$, because this makes sure that each word $v_i$, $1 \leq i \leq s$, can be detected by all weights in $\mathbf{W}$ – as opposed to only the rightmost (resp. leftmost) weights for initial (resp. final) words in narrow convolution.
**Average pooling layer.** Pooling, including min pooling, max pooling and average pooling, is commonly used to extract robust features from convolution. In this paper, we introduce attention weighting as an alternative, but use average pooling as a baseline as follows.
For the output feature map of the last convolution layer, we do column-wise averaging over *all columns*, denoted as *all-ap*. This will generate a representation vector for each of the two sentences, shown as the top “Average pooling (*all-ap*)” layer below “Logistic regression” in Figure \[fig:arc-0\]. These two representations are then the basis for the sentence pair decision.
For the output feature map of non-final convolution layers, we do column-wise averaging over *windows of $w$ consecutive columns*, denoted as $w$-*ap*; shown as the lower “Average pooling ($w$-*ap*)” layer in Figure \[fig:arc-0\]. For filter width $w$, a convolution layer transforms an input feature map of $s$ columns into a new feature map of $s+w-1$ columns; average pooling transforms this back to $s$ columns. This architecture supports stacking an arbitrary number of convolution-pooling blocks to extract increasingly abstract features. Input features to the bottom layer are words, input features to the next layer are short phrases and so on. Each level generates more abstract features of higher granularity.
**Output layer.** The last layer is an output layer, chosen according to the task; e.g., for binary classification tasks, this layer is logistic regression (see Figure \[fig:arc-0\]). Other types of output layers are introduced below.
We found that in most cases, performance is boosted if we provide the output of *all pooling layers* as input to the output layer. For each non-final average pooling layer, we perform $w$-*ap* (pooling over windows of $w$ columns) as described above, but we also perform *all-ap* (pooling over all columns) and forward the result to the output layer. This improves performance because representations from different layers cover the properties of the sentences at different levels of abstraction and all of these levels can be important for a particular sentence pair.
ABCNN: Attention-Based BCNN
===========================
We now describe three architectures based on BCNN, ABCNN-1, ABCNN-2 and ABCNN-3, that each introduce an attention mechanism for modeling sentence pairs; see .
ABCNN-1 {#sec:abcnn-1}
-------
ABCNN-1 () employs an attention feature matrix $\mathbf{A}$ to influence convolution. Attention features are intended to weight those units of $s_i$ more highly in convolution that are relevant to a unit of $s_{1-i}$ ; we use the term “unit” here to refer to words on the lowest level and to phrases on higher levels of the network. shows two *unit representation feature maps* in red: this part of ABCNN-1 is the same as in BCNN (see ). Each column is the representation of a unit, a word on the lowest level and a phrase on higher levels. We first describe the attention feature matrix $\mathbf{A}$ informally (layer “Conv input”, middle column, in ). $\mathbf{A}$ is generated by matching units of the left representation feature map with units of the right representation feature map such that the attention values of row $i$ in $\mathbf{A}$ denote the attention distribution of the $i$-th unit of $s_0$ with respect to $s_1$, and the attention values of column $j$ in $\mathbf{A}$ denote the attention distribution of the $j$-th unit of $s_1$ with respect to $s_0$. $\mathbf{A}$ can be viewed as a new feature map of $s_0$ (resp. $s_1$) in row (resp. column) direction because each row (resp. column) is a new feature vector of a unit in $s_0$ (resp. $s_1$). Thus, it makes sense to combine this new feature map with the representation feature maps and use both as input to the convolution operation. We achieve this by transforming $\mathbf{A}$ into the two blue matrices in Figure \[fig:a\] that have the same format as the representation feature maps. As a result, the new input of convolution has two feature maps for each sentence (shown in red and blue). Our motivation is that the attention feature map will guide the convolution to learn “counterpart-biased” sentence representations.
More formally, let $\mathbf{F}_{i,r}\in\mathbf{R}^{d\times
s}$ be the *representation feature map* of sentence $i$ . Then we define the attention matrix $\mathbf{A}\in\mathbf{R}^{s\times s}$ as follows: $$\label{equ:att}
\mathbf{A}_{i,j} = \mbox{match-score}(\mathbf{F}_{0,r}[:,i], \mathbf{F}_{1,r}[:,j])$$ The function match-score can be defined in a variety of ways. We found that $1/(1+|x-y|)$ works well where $| \cdot |$ is Euclidean distance.
Given attention matrix $\mathbf{A}$, we generate the *attention feature map* $\mathbf{F}_{i,a}$ for $s_i$ as follows: $$\begin{aligned}
\mathbf{F}_{0,a} &=& \mathbf{W}_0 \cdot \mathbf{A}^\top \label{equ:l}\\
\mathbf{F}_{1,a} &=& \mathbf{W}_1 \cdot \mathbf{A} \label{equ:r}\end{aligned}$$ The weight matrices $\mathbf{W}_0\in\mathbf{R}^{d\times s}$, $\mathbf{W}_1\in\mathbf{R}^{d\times s}$ are parameters of the model to be learned in training.[^2]
We stack the representation feature map $\mathbf{F}_{i,r}$ and the attention feature map $\mathbf{F}_{i,a}$ as an order 3 tensor and feed it into convolution to generate a higher-level representation feature map for $s_i$ . In , $s_0$ has has 5 units, $s_1$ has 7. The output of convolution (shown in the top layer, filter width $w=3$) is a higher-level representation feature map with 7 columns for $s_0$ and 9 columns for $s_1$.
ABCNN-2 {#sec:abcnn-2}
-------
ABCNN-1 computes attention weights *directly on the input representation* with the aim of *improving the features computed by convolution*. ABCNN-2 () instead computes attention weights *on the output of convolution* with the aim of *reweighting this convolution output*. In the example shown in , the feature maps output by convolution for $s_0$ and $s_1$ (layer marked “Convolution” in ) have 7 and 9 columns, respectively; each column is the representation of a unit. The attention matrix $\mathbf{A}$ compares all units in $s_0$ with all units of $s_1$. We sum all attention values for a unit to derive a single attention weight for that unit. This corresponds to summing all values in a row of $\mathbf{A}$ for $s_0$ (“col-wise sum”, resulting in the column vector of size 7 shown) and summing all values in a column for $s_1$ (“row-wise sum”, resulting in the row vector of size 9 shown).
More formally, let $\mathbf{A}\in\mathbf{R}^{s\times s}$ be the attention matrix, $a_{0,j}=\sum\mathbf{A}[j,:]$ the attention weight of unit $j$ in $s_0$, $a_{1,j}=\sum\mathbf{A}[:,j]$ the attention weight of unit $j$ in $s_1$ and $\mathbf{F}^c_{i,r}\in\mathbf{R}^{d\times(s_i+w-1)}$ the output of convolution for $s_i$. Then the $j$-th column of the new feature map $\mathbf{F}^p_{i,r}$ generated by $w$-*ap* is derived by: $$\begin{aligned}
\mathbf{F}^p_{i,r}[:,j] = \sum_{k=j:j+w}a_{i,k}\cdot
\mathbf{F}^c_{i,r}[:,k], \enspace\enspace j=1 \ldots s_i\end{aligned}$$ Note that $\mathbf{F}^p_{i,r}\in\mathbf{R}^{d\times s_i}$, i.e., ABCNN-2 pooling generates an output feature map of the same size as the input feature map of convolution. This allows us to stack multiple convolution-pooling blocks to extract features of increasing abstraction.
There are three main differences between ABCNN-1 and ABCNN-2. (i) Attention in ABCNN-1 impacts *convolution indirectly* while attention in ABCNN-2 influences *pooling* through *direct* attention weighting. (ii) ABCNN-1 requires the two matrices $\mathbf{W}_i$ to convert the attention matrix into attention feature maps; and the input to convolution has two times as many feature maps. Thus, ABCNN-1 has more parameters than ABCNN-2 and is more vulnerable to overfitting. (iii) As pooling is performed after convolution, pooling handles larger-granularity units than convolution; e.g., if the input to convolution has word level granularity, then the input to pooling has phrase level granularity, the phrase size being equal to filter size $w$. Thus, ABCNN-1 and ABCNN-2 implement attention mechanisms for linguistic units of different granularity. The complementarity of ABCNN-1 and ABCNN-2 motivates us to propose ABCNN-3, a third architecture that combines elements of the two.
ABCNN-3
-------
ABCNN-3 combines ABCNN-1 and ABCNN-2 by stacking them. See Figure \[fig:c\]. ABCNN-3 combines the strengths of ABCNN-1 and ABCNN-2 by allowing the attention mechanism to operate (i) both on the convolution and on the pooling parts of a convolution-pooling block and (ii) both on the input granularity and on the more abstract output granularity.
Experiments {#sec:experiments}
===========
We test the proposed architectures on three tasks: answer selection (AS), paraphrase identification (PI) and textual entailment (TE).
Common Training Setup
---------------------
For all tasks, words are initialized by 300-dimensional word2vec embeddings and not changed during training. A single randomly initialized embedding[^3] is created for all unknown words by uniform sampling from \[-.01,.01\]. We employ Adagrad [@duchi2011adaptive] and $L_2$ regularization.
### Network configuration
Each network in the experiments below consists of (i) an initialization block $b_1$ that initializes words by word2vec embeddings, (ii) a stack of $k-1$ convolution-pooling blocks $b_2,\ldots, b_k$, computing increasingly abstract features, and (iii) one final *LR layer* (logistic regression layer) as shown in .
The input to the LR layer consists of $kn$ features – each block provides $n$ similarity scores, e.g., $n$ cosine similarity scores. shows the two sentence vectors output by the final block $b_k$ of the stack (“sentence representation 0”, “sentence representation 1”); this is the basis of the last $n$ similarity scores. As we explained in the final paragraph of , we perform *all-ap* pooling for *all blocks*, not just for $b_k$. Thus we get one sentence representation each for $s_0$ and $s_1$ for each block $b_1, \ldots, b_k$. We compute $n$ similarity scores for each block (based on the block’s two sentence representations). Thus, we compute a total of $kn$ similarity scores and these scores are input to the LR layer.
Depending on the task, we use different methods for computing the similarity score: see below.
### Layerwise training
In our training regime, we first train a network consisting of just one convolution-pooling block $b_2$. We then create a new network by adding a block $b_3$, initialize its $b_2$ block with the previously learned weights for $b_2$ and train $b_3$ keeping the previously learned weights for $b_2$ fixed. We repeat this procedure until all $k-1$ convolution-pooling blocks are trained. We found that this training regime gives us good performance and shortens training times considerably. Since similarity scores of lower blocks are kept unchanged once they have been learned, this also has the nice effect that “simple” similarity scores (those based on surface features) are learned first and subsequent training phases can focus on complementary scores derived from more complex abstract features.
### Classifier
We found that performance increases if we do not use the output of the LR layer as the final decision, but instead train linear SVM or logistic regression with default parameters[^4] directly on the input to the LR layer (i.e., on the $kn$ similarity scores that are generated by the $k$-block stack after network training is completed). Direct training of SVMs/LR seems to get closer to the global optimum than gradient descent training of CNNs.
shows the values of the hyperparameters. Hyperparameters were tuned on dev.
### Shared Baselines
We use addition and LSTM as two *shared baselines* for all three tasks, i.e., for AS, PI and TE. We now describe these two shared baselines.
\(i) **Addition**. We sum up word embeddings element-wise to form each sentence representation, then concatenate the two sentence representation vectors as classifier input. (ii) **A-LSTM**. Before this work, most attention mechanisms in NLP domain are implemented in recurrent neural networks for text generation tasks such as machine translation (e.g., , ). present an attention-LSTM for natural language inference task. Since this model is the pioneering attention based RNN system for sentence pair classification problem, we consider it as a baseline system (“A-LSTM”) for all our three tasks. A-LSTM has the same configuration as our ABCNN systems in terms of word initialization (300-dimensional word2vec embeddings) and the dimensionality of all hidden layers (50).
Answer Selection
----------------
We use WikiQA,[^5] an open domain question-answer dataset. We use the subtask that assumes that there is at least one correct answer for a question. The corresponding dataset consists of 20,360 question-candidate pairs in train, 1,130 pairs in dev and 2,352 pairs in test where we adopt the standard setup of only considering questions that have correct answers for evaluation. Following , we truncate answers to 40 tokens.
The task is to rank the candidate answers based on their relatedness to the question. Evaluation measures are mean average precision (MAP) and mean reciprocal rank (MRR).
### Task-Specific Setup
We use cosine similarity as the similarity score for this task. In addition, we use sentence lengths, *WordCnt* (count of the number of non-stopwords in the question that also occur in the answer) and *WgtWordCnt* (reweight the counts by the IDF values of the question words). Thus, the final input to the LR layer has size $k+4$: one cosine for each of the $k$ blocks and the four additional features.
We compare with eleven **baselines**. The first seven are considered by : (i) WordCnt; (ii) WgtWordCnt; (iii) LCLR [@yih2013question] makes use of rich lexical semantic features, including word/lemma matching, WordNet [@miller1995wordnet] and distributional models; (iv) PV: Paragraph Vector [@le2014distributed]; (v) CNN: bigram convolutional neural network [@yu2014deep]; (vi) PV-Cnt: combine PV with (i) and (ii); (vii) CNN-Cnt: combine CNN with (i) and (ii). Apart from the baselines considered by , we compare with two Addition baselines and two LSTM baselines. Addition and A-LSTM are the baselines described in . We also combine both with the four extra features; this gives us two additional baselines that we refer to as Addition(+) and A-LSTM(+).
method MAP MRR
-- ------------- ---------------- ----------------
WordCnt 0.4891 0.4924
WgtWordCnt 0.5099 0.5132
LCLR 0.5993 0.6086
PV 0.5110 0.5160
CNN 0.6190 0.6281
PV-Cnt 0.5976 0.6058
CNN-Cnt
Addition 0.5021 0.5069
Addition(+) 0.5888 0.5929
A-LSTM 0.5347 0.5483
A-LSTM(+) 0.6381 0.6537
one-conv 0.6629 0.6813
two-conv 0.6593 0.6738
one-conv 0.6810$^*$ 0.6979$^*$
two-conv 0.6855$^*$ 0.7023$^*$
one-conv 0.6885$^*$ 0.7054$^*$
two-conv 0.6879$^*$ 0.7068$^*$
one-conv 0.6914$^*$ **0.7127**$^*$
two-conv **0.6921**$^*$ 0.7108$^*$
: Results on WikiQA. Best result per column is bold. Significant improvements over state-of-the-art baselines (underlined) are marked with $*$ ($t$-test, p $<$ .05).
### Results
shows performance of the baselines, of BCNN and of the three ABCNN architectures. For CNNs, we test one (one-conv) and two (two-conv) convolution-pooling blocks.
The non-attention network BCNN already performs better than the baselines. If we add attention mechanisms, then the performance further improves by several points. Comparing ABCNN-2 with ABCNN-1, we find ABCNN-2 is slightly better even though ABCNN-2 is the simpler architecture. If we combine ABCNN-1 and ABCNN-2 to form ABCNN-3, we get further improvement.[^6]
This can be explained by ABCNN-3’s ability to take attention of more fine-grained granularity into consideration in each convolution-pooling block while ABCNN-1 and ABCNN-2 consider attention only at convolution input or only at pooling input, respectively. We also find that stacking two convolution-pooling blocks does not bring consistent improvement and therefore do not test deeper architectures.
Paraphrase Identification
-------------------------
We use the Microsoft Research Paraphrase (MSRP) corpus [@dolan2004unsupervised]. The training set contains 2753 true / 1323 false and the test set 1147 true / 578 false paraphrase pairs. We randomly select 400 pairs from train and use them as dev set; but we still report results for training on the entire training set. For each triple (label, $s_0$, $s_1$) in the training set, we also add (label, $s_1$, $s_0$) to the training set to make best use of the training data. Systems are evaluated by accuracy and $F_1$.
### Task-Specific Setup
In this task, we add the 15 MT features from [@madnani2012re] and the lengths of the two sentences. In addition, we compute ROUGE-1, ROUGE-2 and ROUGE-SU4 [@lin2004rouge], which are scores measuring the match between the two sentences on (i) unigrams, (ii) bigrams and (iii) unigrams and skip-bigrams (maximum skip distance of four), respectively. In this task, we found transforming Euclidean distance into similarity score by $1/(1+|x-y|)$ performs better than cosine similarity. Additionally, we use dynamic pooling [@yinnaacl] of the attention matrix $\mathbf{A}$ in Equation \[equ:att\] and forward pooled values of all blocks to the classifier. This gives us better performance than only forwarding sentence-level matching features.
We compare our system with a number of alternative approaches, both with representative neural network (NN) approaches and non-NN approaches: (i) A-LSTM; (ii) A-LSTM(+): A-LSTM plus handcrafted features; (iii) RAE [@socher2011dynamic], recursive autoencoder; (iv) Bi-CNN-MI [@yinnaacl], a bi-CNN architecture; and (v) MPSSM-CNN [@he2015multi], the state-of-the-art NN system for PI. We consider the following four non-NN systems: (vi) Addition (see ); (vii) Addition(+): Addition plus handcrafted features; (viii) MT [@madnani2012re], a system that combines machine translation metrics;[^7] (ix) MF-TF-KLD [@ji2013discriminative], the state-of-the-art non-NN system.
acc $F_1$
-- ----------------- ---------- ----------
majority voting 66.5 79.9
RAE 76.8 83.6
Bi-CNN-MI 78.4 84.6
MPSSM-CNN
MT 76.8 83.8
MF-TF-KLD 84.6
Addition 70.8 80.9
Addition (+) 77.3 84.1
A-LSTM 69.5 80.1
A-LSTM (+) 77.1 84.0
one-conv 78.1 84.1
two-conv 78.3 84.3
one-conv 78.5 84.5
two-conv 78.5 84.6
one-conv 78.6 84.7
two-conv 78.8 84.7
one-conv 78.8 **84.8**
two-conv **78.9** **84.8**
: Results for PI on MSRP[]{data-label="tab:msrp"}
### Results
shows that BCNN is slightly worse than the state-of-the-art whereas ABCNN-1 roughly matches it. ABCNN-2 is slightly above the state-of-the-art. ABCNN-3 outperforms the state-of-the-art in accuracy and $F_1$.[^8] Two convolution layers only bring small improvements over one.
Textual Entailment
------------------
SemEval 2014 Task 1 [@marelli2014semeval] evaluates system predictions of textual entailment (TE) relations on sentence pairs from the SICK dataset [@marelli2014sick]. The three classes are entailment, contradiction and neutral. The sizes of SICK train, dev and test sets are 4439, 495 and 4906 pairs, respectively. We call this dataset ORIG.
We also create NONOVER, a copy of ORIG in which *the words that occur in both sentences have been removed*. A sentence in NONOVER is denoted by the special token $<$empty$>$ if all words are removed. shows three pairs from ORIG and their transformation in NONOVER. We observe that focusing on the non-overlapping parts provides clearer hints for TE than ORIG. In this task, we run two copies of each network, one for ORIG, one for NONOVER; these two networks have a single common LR layer.
Following , we train our final system (after fixing of hyperparameters) on train and dev (4,934 pairs). Our evaluation measure is accuracy.
### Task-Specific Setup
We found that for this task forwarding two similarity scores from each block (instead of just one) is helpful. We use cosine similarity and Euclidean distance. As we did for paraphrase identification, we add the 15 MT features for each sentence pair for this task as well; our motivation is that entailed sentences resemble paraphrases more than contradictory sentences do.
We use the following linguistic features.
**Negation**. Negation obviously is an important feature for detecting contradiction. Feature <span style="font-variant:small-caps;">neg</span> is set to $1$ if either sentence contains “no”, “not”, “nobody”, “isn’t” and to $0$ otherwise.
**Nyms**. Following , we use WordNet to detect synonyms, hypernyms and antonyms in the pairs. But we do this on NONOVER (not on ORIG) to focus on what is critical for TE. Specifically, feature <span style="font-variant:small-caps;">syn</span> is the number of word pairs in $s_0$ and $s_1$ that are synonyms. <span style="font-variant:small-caps;">hyp0</span> (resp. <span style="font-variant:small-caps;">hyp1</span>) is the number of words in $s_0$ (resp. $s_1$) that have a hypernym in $s_1$ (resp. $s_0$). In addition, we collect all *potential antonym pairs* (PAP) in NONOVER. We identify the matched chunks that occur in *contradictory* and *neutral*, but not in *entailed* pairs. We exclude synonyms and hypernyms and apply a frequency filter of $n = 2$. In contrast to [@lai2014illinois], we constrain the PAP pairs to cosine similarity above 0.4 in word2vec embedding space as this discards many noise pairs. Feature <span style="font-variant:small-caps;">ant</span> is the number of matched PAP antonyms in a sentence pair.
**Length.** As before we use sentence length, both ORIG – <span style="font-variant:small-caps;">len0o</span> and <span style="font-variant:small-caps;">len1o</span> – and NONOVER lengths: <span style="font-variant:small-caps;">len0n</span> and <span style="font-variant:small-caps;">len1n</span>.
On the whole, we have 24 extra features: 15 MT metrics, <span style="font-variant:small-caps;">neg</span>, <span style="font-variant:small-caps;">syn</span>, <span style="font-variant:small-caps;">hyp0</span>, <span style="font-variant:small-caps;">hyp1</span>, <span style="font-variant:small-caps;">ant</span>, <span style="font-variant:small-caps;">len0o</span>, <span style="font-variant:small-caps;">len1o</span>, <span style="font-variant:small-caps;">len0n</span> and <span style="font-variant:small-caps;">len1n</span>.
Apart from the Addition and LSTM baselines, we further compare with the top-3 systems in SemEval and TrRNTN [@bowman2015recursive], a recursive neural network developed for this SICK task.
### Results
shows that our CNNs outperform A-LSTM (with or without linguistic features added) as well as the top three systems of SemEval. Comparing ABCNN with BCNN, attention mechanism consistently improves performance. ABCNN-1 roughly has comparable performance as ABCNN-2 while ABCNN-3 has bigger improvement: a boost of 1.6 points compared to the previous state of the art.[^9]
[ll|l]{} & acc\
&[@jimenez2014unal] & 83.1\
& [@zhao2014ecnu] & 83.6\
&[@lai2014illinois] &\
TrRNTN & [@bowman2015recursive]& 76.9\
&no features& 73.1\
& plus features & 79.4\
&no features& 78.0\
& plus features & 81.7\
&one-conv& 84.8\
& two-conv & 85.0\
&one-conv& 85.6\
& two-conv & 85.8\
&one-conv& 85.7\
& two-conv & 85.8\
&one-conv& 86.0$^*$\
& two-conv & **86.2**$^*$
Visual Analysis
---------------
In , we visualize the attention matrices for one TE sentence pair in ABCNN-2 for blocks $b_1$ (unigrams), $b_2$ (first convolutional layer) and $b_3$ (second convolutional layer). Darker shades of blue indicate stronger attention values.
In Figure \[fig:uni\], each word corresponds to exactly one row or column. We can see that words in $s_i$ with semantic equivalents in $s_{1-i}$ get high attention while words without semantic equivalents get low attention, e.g., “walking” and “murals” in $s_0$ and “front” and “colorful” in $s_1$. This behavior seems reasonable for the unigram level.
Rows/columns of the attention matrix in Figure \[fig:bi\] correspond to phrases of length three since filter width $w=3$. High attention values generally correlate with close semantic correspondence: the phrase “people are” in $s_0$ matches “several people are” in $s_1$; both “are walking outside” and “walking outside the” in $s_0$ match “are in front” in $s_1$; “the building that” in $s_0$ matches “a colorful building” in $s_1$. More interestingly, looking at the bottom right corner, both “on it” and “it” in $s_0$ match “building” in $s_1$; this indicates that ABCNN is able to detect some coreference across sentences. “building” in $s_1$ has two places in which higher attentions appear, one is with “it” in $s_0$, the other is with “the building that” in $s_0$. This may indicate that ABCNN recognizes that “building” in $s_1$ and “the building that” / “it” in $s_0$ refer to the same object. Hence, coreference resolution across sentences as well as within a sentence both are detected. For the attention vectors on the left and the top, we can see that attention has focused on the key parts: “people are walking outside the building that” in $s_0$, “several people are in” and “of a colorful building” in $s_1$.
Rows/columns of the attention matrix in Figure \[fig:tri\] (second layer of convolution) correspond to phrases of length 5 since filter width $w=3$ in both convolution layers ($5=1+2*(3-1)$). We use “$\ldots$” to denote words in the middle if a phrase like “several...front” has more than two words. We can see that attention distribution in the matrix has focused on some local regions. As granularity of phrases is larger, it makes sense that the attention values are smoother. But we still can find some interesting clues: at the two ends of the main diagonal, higher attentions hint that the first part of $s_0$ matches well with the first part of $s_1$; “several murals on it” in $s_0$ matches well with “of a colorful building” in $s_1$, which satisfies the intuition that these two phrases are crucial for making a decision on TE in this case. This again shows the potential strength of our system in figuring out which parts of the two sentences refer to the same object. In addition, in the central part of the matrix, we can see that the long phrase “people are walking outside the building” in $s_0$ matches well with the long phrase “are in front of a colorful building” in $s_1$.
Summary {#sec:sum}
=======
In this work, we presented three mechanisms to integrate attention into convolutional neural networks for general sentence pair modeling tasks.
Our experimental results on AS, PI and TE show that attention-based CNNs perform better than CNNs without attention mechanisms. ABCNN-2 generally outperforms ABCNN-1 and ABCNN-3 surpasses both.
In all tasks, we did not find any big improvement of two layers of convolution over one layer. This is probably due to the limited size of training data. We expect that, as larger training sets become available, deep ABCNNs will show even better performance.
In addition, linguistic features contribute in all three tasks: improvements by 0.0321 (MAP) and 0.0338 (MRR) for AS, improvements by 3.8 (acc) and 2.1 ($F_1$) for PI and an improvement by 1.6 (acc) for TE. But our ABCNN can still reach or surpass state-of-the-art even without those features in AS and TE tasks. This shows that ABCNN is generally a strong NN system.
As we discussed in , attention-based LSTMs have been especially successful in tasks that have a strong generation component like machine translation and summarization. CNNs have not been used for this type of task. This is an interesting area of future work for attention-based CNN systems.
[^1]: <https://code.google.com/p/word2vec/>
[^2]: The weights of the two matrices are shared in our implementation to reduce the number of parameters of the model.
[^3]: This worked better than discarding unknown words.
[^4]: <http://scikit-learn.org/stable/> for both.
[^5]: <http://aka.ms/WikiQA> [@yang2015wikiqa]
[^6]: If we limit the input to LR layer to the $k$ similarity scores in ABCNN-3 (two-conv), results are .660 (MAP) / .677 (MRR).
[^7]: For better comparability of approaches in our experiments, we use a simple SVM classifier, which performs slightly worse than ’s more complex meta-classifier.
[^8]: The improvement of .3 in accuracy and .1 in $F_1$ over the state-of-the-art is not significant. If we run ABCNN-3 (two-conv) without the 15+3 “linguistic” features (i.e., MT and ROUGE), performance is 75.1/82.7.
[^9]: If we run ABCNN-3 (two-conv) without the 24 linguistic features, the performance is 84.6.
|
---
abstract: 'Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behaviour of the algorithm is investigated on the uniform electron gas and the water dimer at CCSD, CCSDT and CCSDTQ levels. We also describe two improvements to the original sampling algorithm, *full non-composite* and *multi-spawn* sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.'
author:
- 'J. S. Spencer'
- 'V. A. Neufeld'
- 'W. A. Vigor'
- 'R. S. T. Franklin'
- 'A. J. W. Thom'
title: Large Scale Parallelization in Stochastic Coupled Cluster
---
Introduction {#sec:intro}
============
Coupled cluster (CC) methods[@BartlettMusial_07RMP] are of crucial importance in electronic structure and have been used to explore a variety of systems, including atoms and molecular systems[@BartlettMusial_07RMP; @KowalskiPiecuch_00JCP; @HardingStanton_08JCP; @HattigTew_12CR; @Karton_16CMS], the uniform electron gas[@Freeman1977; @Bishop1978; @Bishop1982; @Shepherd2013; @Roggero2013; @Spencer2016; @McClain2016a; @Shepherd2016a; @Neufeld2017] and solids/other periodic systems[@Hirata2001a; @Hirata2004; @Manby2006; @Nolan2009; @Gruneis2011; @Booth2013; @Gruneis2015; @Liao2016; @Schwerdtfeger2016; @McClain2017; @Gruber2018]. CCSD(T)[@Raghavachari1989], where single and double excitations are included in the wavefunction ansatz and supplemented with the pertubative treatment of triple excitations, is commonly regarded as the “gold standard” of quantum chemistry and can frequently achieve[@Lee1995] chemical accuracy of 1 kcal/mol.
Despite these successes, coupled cluster is not without its drawbacks. Coupled cluster is systematically improvable, at least in principle, by increasing the excitation level included in the CC wavefunction ansatz. Doing so makes the conventional CC equations vastly more complicated and hence computational demanding. As a result, treating higher truncation levels is possibly only in specialist codes[@MRCC]. Conventional implementations of coupled cluster also rely heavily upon dense linear algebra, which does not scale well with increasing numbers of processors on parallel or heterogeneous computer architectures, though recent work in linear algebra and tensor libraries are making impressive progress[@Agullo2009; @Solomonik2014].
One avenue for improving the computational efficiency of coupled cluster is to exploit the nearsighted nature of electron correlation and use local approximations[@FlockeBartlett_04JCP; @Ziokowski2010; @RiplingerNeese_13JCP; @RiplingerNeese_13JCP2]. Another approach, of increasing use in quantum chemistry and the broader electronic structure community, is to use stochastic methods; these have proven to provide low-scaling algorithms for electronic structure methods and typically exhibit excellent scaling with increasing processor count[@Foulkes2001; @Thom2007; @Willow2012; @Willow2013; @Baer2013]. Local and stochastic methods may also be easily combined via a localisation transformation of the mean-field single-particle orbitals.[^1].
The full configuration interaction quantum Monte Carlo (FCIQMC) method[@BoothAlavi_09JCP; @ClelandAlavi_12JCTC] has been a major development in quantum chemistry. By sampling the action of the Hamiltonian, FCIQMC has been able to calculate exact properties for quantum systems inaccessible to conventional diagonalisation techniques[@BoothAlavi_09JCP; @ShepherdAlavi_12PRB; @Overy2014; @Blunt2015; @Blunt2017]. The computational advantage of FCIQMC is largely through a representation of the FCI wavefunction which is significantly more compact than the full wavefunction, though still scaling factorially with the size of the Hilbert space sampled.
One of us (AJWT) subsequently used a similar approach to formulate a Monte Carlo approach to coupled cluster theory (CCMC)[@Thom_10PRL], inheriting the benefits of more compact storage, and now scaling with the polynomial size of the truncated CC space. The initiator approximation can substantially improve the stochastic sampling of the wavefunction in both FCIQMC[@ClelandAlavi_12JCTC] and CCMC[@Spencer2016], though the latter requires careful extrapolation. The stochastic sampling of the coupled cluster wavefunction can be further improved by sampling only linked diagrams[@FranklinThom_16JCP] and non-uniform sampling of the coupled cluster expansion [@Scott2017], and improved sampling of the action of the Hamiltonian[@Neufeld2018]. The utility of CCMC has been demonstrated to calculate coupled cluster energies at up to the CCSDTQ56 level for molecular systems[@Thom_10PRL; @Scott2017], for the uniform electron gas[@Spencer2016; @Neufeld2017] and also been used to automatically generate the $P$ subspace in the CC($P;Q$) method[@Deustua2017]. However, due to the non-linearity of the coupled cluster equations, parallelization of the CCMC algorithm is less straightforward than a parallel FCIQMC implementation[@Booth2014].
We present a brief overview of the CCMC algorithm in \[sec:CCMC\] provide context to the problem. In \[sec:CCMC\_MPI\] we show that the CCMC algorithm can be efficiently parallelized by introducing an additional level of Monte Carlo sampling by considering only a subset of terms in the coupled cluster expansion per iteration. The accuracy and performance of this algorithm is investigated using the uniform electron gas and the water dimer. \[sec:sampling\] provides simple improvements to the original CCMC algorithm to improve stability and convergence of the CC wavefunction. We conclude in \[sec:discussion\].
Coupled Cluster Monte Carlo {#sec:CCMC}
===========================
The algorithms used to sample the FCI and coupled cluster wavefunctions have been previously detailed[@BoothAlavi_09JCP; @Spencer2012; @Thom_10PRL; @Spencer2016; @Scott2017] and as such we summarise the key features relevant to this work here.
The coupled cluster wavefunction ansatz can be expressed as $\ket{\Psi} = N e^{\hat{T}} \ket{D_{\ensuremath{\textrm{HF}}}}$, where $\ket{D_{\ensuremath{\textrm{HF}}}}$ is the Hartree–Fock determinant, $N$ controls the (intermediate) normalisation and the cluster operator $\hat{T}$ is $$\hat{T} = \sum_{i,a} t_{i}^{a} \hat{c}_{i}^{a} + \sum_{\substack{i<j\\a<b}} t_{ij}^{ab} \hat{c}_{ij}^{ab} + \cdots,$$ where $\{t_{i\cdots}^{a\cdots}\}$ is the set of amplitudes and $\hat{c}_{i\cdots}^{a\cdots}$ is an *excitor* comprising of a string of creation and annihilation operators. For convenience, we use $\hat{c}_{\bm{i}}$ and $t_{\bm{i}}$, such that $\hat{c}_{\bm{i}}\ket{D_{\ensuremath{\textrm{HF}}}}$ produces $\ket{D_{\bm{i}}}$ (up to a sign, as discussed later) and $t_{\bm{i}}$ is the corresponding amplitude, and rescale the amplitudes with an additional factor, $t_{{\ensuremath{\textrm{HF}}}}$, such that $$\ket{\Psi} = t_{{\ensuremath{\textrm{HF}}}} e^{\hat{T}/t_{{\ensuremath{\textrm{HF}}}}} \ket{D_{\ensuremath{\textrm{HF}}}}.$$ Within this wavefunction ansatz, the coefficient of a given determinant, $\tilde{t}_{\bm{j}}=\braket{D_{\bm{j}}|\Psi}$, contains contributions from all sets of excitors which can be combined to produce that determinant.
As with FCIQMC, CCMC applies an approximate linear propagator, $1-\delta\tau (\hat H-S)$, where $\delta\tau$ is the timestep and $S$ is an adjustable parameter to control proportionality, and which has the same eigenspectrum as $e^{-\delta\tau \hat H}$ for sufficiently small $\delta\tau$[@Spencer2012]. Applying this to $\ket{\Psi}$ and cancelling quadratic and higher-order terms[@Thom_10PRL; @Spencer2016] results in a form reminiscent of the propagation equation for FCIQMC[@BoothAlavi_09JCP], $$t_{\bm{i}}(\tau+\delta\tau)=t_{\bm{i}}(\tau) - \delta\tau \sum_{\bm{j}} (H_{\bm{ij}} - S\delta_{\bm{ij}}) \tilde{t}_{\bm{j}}(\tau).
\label{eqn:CCMC}$$ The similarity-transformed Hamiltonian can also be used in the projection the coupled cluster wavefunction[@FranklinThom_16JCP] and the approaches presented here equally apply to that formulation. The amplitudes are stochastically sampled by representing them using either particles with integer[@BoothAlavi_09JCP; @Thom_10PRL] or real (as opposed to integer) weights[@Petruzielo2012; @Overy2014], which has been shown to reduce stochastic error within FCIQMC and readily applies to CCMC.
In FCIQMC, the particles on each occupied determinant are explicitly evolved; in CCMC this would require one to first evaluate all possible $\tilde{t}_{\bm{j}}$, which is computationally painful. Instead, we exploit the fact that Monte Carlo is a powerful tool for sampling high dimensional spaces and, in addition to stochastically sampling the action of the Hamiltonian, also sample the wavefunction ansatz. The algorithm used to sample the cluster expansion has been shown to have a significant impact on computational and statistical efficiency[@Scott2017]; here we consider only the simplest approach. The cluster size, $s=[0, l+2]$, is selected according to an exponential distribution, i.e. ${\ensuremath{p_{\textrm{size}}}}(s) = 2^{-(s+1)}$, where $l$ is the highest order term in the cluster expansion[^2]. The cluster is then generated by selecting $s$ excitors from the current distribution, each with probability $|t_{\bm{i}}| / ({\ensuremath{N_{\textrm{total}}}}-t_{{\ensuremath{\textrm{HF}}}})$, where ${\ensuremath{N_{\textrm{total}}}}$ is the total current population. A cluster containing the same excitor more than once is discarded. Alternative approaches for sampling the cluster expansion are discussed in \[sec:sampling\].
The dynamics for evolving the particles on a cluster are essentially identical to those in FCIQMC[@BoothAlavi_09JCP]. A simulation starts with a number of particles of unit weight on the Hartree–Fock determinant. The particles are then evolved by sampling the action of the Hamiltonian on each particle, allowing new particles to be created (‘spawned’), and the particle to die (due to the sign of the Hamiltonian operator). At the end of each iteration particles on the same excitor with opposite signs are removed (‘annihilated’) from the simulation, which aids the sign problem[@Spencer2012] and is a statistically exact process. Note that for clusters of size 2 and higher, the death step amounts to creating a particle of opposite sign on the corresponding excitor. Events which create particles on excitors which are not within the desired truncation level of the cluster operator are simply discarded in our current CCMC implementation.
Anti-commutation relationships in strings of creation and annihilation operators must be handled with care. A given excitor is required to be unique and hence an arbitrary excitor $\{\hat{c}_{ij\cdots l}^{ab\cdots e}\}$ must satisfy $i<j<\cdots<l$ and $a<b<\cdots<e$. Defining $\hat{c}_i^{\vphantom{}}$ ($\hat{c}_i^\dagger$) to annihilate (create) an electron in the $i$-th spin-orbital, an excitor and a determinant can be expressed as $$\begin{gathered}
\hat{c}_{ij\cdots l}^{ab\cdots e} = \hat{c}_a^\dagger \hat{c}_b^\dagger \cdots \hat{c}_e^\dagger \hat{c}_l^{\vphantom{}} \cdots \hat{c}_j^{\vphantom{}} \hat{c}_i^{\vphantom{}} \\
\ket{D_{\bm{i}}} = \ket{i_1 i_2 i_3 \cdots i_N} = \hat{c}_{i_1}^\dagger \hat{c}_{i_2}^\dagger \hat{c}_{i_3}^\dagger \cdots \hat{c}_{i_N}^\dagger \ket{0},\end{gathered}$$ where $\ket{0}$ is the vacuum state and $i_1<i_2<\cdots<i_N$. Therefore, when collapsing a cluster, $\hat{c}_{\bm{i}}\hat{c}_{\bm{j}}\ldots\hat{c}_{\bm{k}}$, to a single excitor, $\hat{c}_{\bm{l}}$, a negative sign must be included as required by anticommutativity in order for the operators in the cluster to match the order in the single excitor. Similarly when an excitor is applied to the Hartree–Fock determinant, the resultant set of creation operators must be permuted in order to achieve the required ordering: $$s_D(\bm{i}) = \braket{D_{\bm{i}} | \hat{c}_{\bm{i}} | D_{\ensuremath{\textrm{HF}}}}.$$ The sign from collapsing a cluster is conveniently absorbed into the amplitude of the cluster and the sign from converting to/from a determinant in the spawning step, such that the sign of the spawned particle is determined by $-\operatorname{sgn}\left(H_{\bm{i}\bm{j}}s_D(\bm{i})s_D(\bm{j})\right)$.
The energy shift, $S$, is not known *a priori*. In keeping with other QMC methods[@Umrigar1993; @BoothAlavi_09JCP], $S$ is updated to keep the population stable. In a simulation $S$ is initially held constant (typically at the Hartree–Fock energy) to allow the population to grow and is only adjusted once the population has reached a desired value. It is important to take the non-linear wavefunction ansatz into account during the constant-shift phase in order to ensure correct normalisation[@FranklinThom_16JCP].
The energy at a given time can, as with FCIQMC, be evaluated with a projected estimator. Again, it is simpler to sample the the wavefunction using the same set of clusters, $\{\hat{c}_{\bm{j}}\}$, chosen above: $$E_{\textrm{proj.}} = t_{{\ensuremath{\textrm{HF}}}}^{-1} \sum_{\{\hat{c}_{\bm{j}}\}} \braket{D_{\ensuremath{\textrm{HF}}}|\hat{H}|D_{\bm{j}}} \frac{t_{\bm{j}} s_D(\bm{j})}{{\ensuremath{p_{\textrm{cluster}}}}(\bm{j})}.$$
Computational Methods {#computational-methods .unnumbered}
=====================
All CCMC calculations are performed using a development version of HANDE[@HANDEpaper; @Spencer2018c]. Most one- and two-body molecular integrals were obtained from restricted Hartree–Fock calculations performed in Psi4[@Psi4], except for the study of three water molecules at large distances from each other where the integrals were obtained with PySCF [@Sun2018] and localised with a Boys method[@Foster1960] by PySCF. Floating-point weights were used to improve stochastic efficiency. Input files and raw data are available under a Creative Commons license at <https://doi.org/10.17863/CAM.30359>. Estimates of the stochastic error in CCMC simulations were obtained via a reblocking analysis[@Flyvbjerg1989]. QMC energies were verified to be unaffected by a population control bias by comparison to those obtained using a reweighting analysis[@Umrigar1993; @Vigor2015].
All data was analysed using numpy[@Oliphant2015], pandas[@Mckinney2010] and pyblock[@pyblock] and plots produced using matplotlib[@Hunter2007] and seaborn[@seaborn].
Parallelisation {#sec:CCMC_MPI}
===============
Using distributed computer architectures is advantageous both in terms of reducing the runtime of a calculation and in being able to treat larger systems due to the corresponding increase in available memory. The memory usage on a given processor of a QMC calculation in Slater determinant space is proportional to the number of states stored on that processor whilst the computational workload is a function of both the number of states and the total population on the processor[@Booth2014]. Ideally both would be evenly balanced across all processors whilst the annihilation step requires all particles on the same determinant to be placed on the same processor. The size of the Hilbert space precludes a lookup table and simply dividing the Hilbert space into chunks and assigning chunk(s) to a processor yields poor load balancing as the distribution of ‘important’ states tends to be highly irregular in many chemical systems. Booth *et al.*[@Booth2014] proposed a deterministic mapping of a determinant to a processor, $p(\ket{D})$, in a time- and space-efficient manner: $$p(\ket{D}) = \operatorname{hash}(\ket{D}) \bmod {\ensuremath{N_{\mathrm{p}}}}\label{eq:fciqmc_dist}$$ where ${\ensuremath{N_{\mathrm{p}}}}$ is the number of processors and $\operatorname{hash}$ is a function which maps an arbitrary amount of data (here a representation of a determinant) to an integer over a fixed range. Crucially a good hash function returns different values for similar inputs and hence determinants which are close in excitation space are mapped to different processors.[^3]
CCMC introduces the additional complication that the cluster expansion must be sampled. One option, which we exploit, is to use a shared-memory paradigm (implemented in HANDE using OpenMP), where the cluster selection and evolution are distributed over threads. Distributing the set of states over multiple nodes[^4], as done in FCIQMC, is not helpful as either each spawning event would involve communication between nodes in order to randomly generate clusters. Instead, we again exploit Monte Carlo sampling: a node only samples the subset of clusters that can be formed from the excitors residing on that node. Crucially, the subset of clusters changes such that all clusters have an equal chance of being selected within a few timesteps.
Concretely, \[eq:fciqmc\_dist\] is extended to periodically change the processor of a given excitor: $$\begin{gathered}
o(\ket{D},i_\tau) = (\operatorname{hash}(\ket{D}) + i_\tau) \gg {\ensuremath{{\nu_{\textrm{move}}}}}\\
p(\ket{D},i_\tau) = \operatorname{hash}\left(\ket{D} \oplus o(\ket{D},i_\tau)\right) \bmod {\ensuremath{N_{\mathrm{p}}}}\label{eq:ccmc_dist}\end{gathered}$$ where $i_\tau$ is the iteration index at time $\tau$ (i.e. $\tau/\delta\tau$), $2^{{\ensuremath{{\nu_{\textrm{move}}}}}}$ is a constant termed the ‘move frequency’ and is discussed below, $a \gg b$ represents the right-shift bit operation on $a$, where the bits in $a$ are moved to the right and the $b$ least significant bits are removed, and $\oplus$ is the exclusive or bit operation. The offset function, $o(\ket{D}, i_\tau)$, discards the lower ${\ensuremath{{\nu_{\textrm{move}}}}}$ bits and hence changes value every $2^{{\ensuremath{{\nu_{\textrm{move}}}}}}$ iterations, where the iteration at which it first changes is determined by the value of $\operatorname{hash}(\ket{D})$. Hence the processor index of an excitor can change every $2^{{\ensuremath{{\nu_{\textrm{move}}}}}}$ iterations. Given a good hash function, $p$ returns a even distribution of values in the range $[0,{\ensuremath{N_{\mathrm{p}}}})$ and all clusters can still be sampled over a number of iterations. The probability of selecting a cluster of size $s$ is scaled by ${\ensuremath{N_{\mathrm{p}}}}^{-s}$ to account for the probability of each excitor in the cluster being on the same processor. Excitors are efficiently redistributed at the same time as newly created particles are communicated to the appropriate node.
Parallelization Scaling
-----------------------
The scaling of the parallelization algorithm is demonstrated on the water dimer in \[fig:scaling\]. All calculations were run with a different number of MPI processes divided into 12 OpenMP threads giving a total number of cores used. They were all restarted from a calculation that was run on 384 cores (32 MPI processes $\times$ 12 OpenMP threads) and used the same parameters. The speed-up was then evaluated as the ratio of time taken per iteration when using 384 cores over the current number of cores.
![Scaling of hybrid MPI+OpenMP CCSDT calculations on the water dimer using a jun-cc-pVDZ basis set performed using even selection[@Scott2017]. 12 OpenMP threads were used per MPI process. Timings were taken from an equilibrated calculation on 384 cores restarted on different numbers of cores. Error bars are only visible for 1152 cores. []{data-label="fig:scaling"}](H2O_dim/scaling/plot){width="1.0\linewidth"}
Up to about 500 cores, the ‘strong scaling’ is approximately ideal. After 1000 cores, over 90% of ideal scaling is still achieved. The calculations used about 1.5$\times 10^7$ excips and had about 8$\times 10^6$ occupied states/excitors. The scaling depends upon the effect of load-balancing, and the ratio of calculation to communication time, both of which reduce efficiency as the number of occupied excitors per core decreases. For calculations with over $10^4$ excitors per core we find no loss of computational efficiency upon parallelization. As system size increases, the number of excitors grows polymonially, so in this ‘weak scaling’ regime the algorithm displays perfect parallelization, over 1000 cores can be employed for sufficient calculation size.
Parallelization Bias
--------------------
{width="1.0\linewidth"}
{width="1.0\linewidth"}
The parallel CCMC algorithm can produce a biased estimate of the energy as not all excitors can form clusters with *all* available exitors in the spawn step and so a subspace is sampled each iteration. If the number of MPI processes is 1, then the complete CC space is sampled each iteration and there is no bias. Conversely if more than one MPI process is employed and ${\ensuremath{{\nu_{\textrm{move}}}}}= \infty$, then \[eq:ccmc\_dist\] reduces to \[eq:fciqmc\_dist\] and clearly the CC wavefunction cannot be sampled as excitors are fixed on specfic MPI processes and hence clusters involving excitors on different processes can never be sampled. Whilst the dominant factor controlling the accessible subspace of clusters per iteration is the number of MPI processes, the timestep, $\delta\tau$, and (log of the) move frequency, ${\ensuremath{{\nu_{\textrm{move}}}}}$, are also important, as decreasing either amounts to increasing the available subspace per unit of imaginary time.
To demonstrate these effects, we have evaluated the correlation between the locations of excitors on a trial system of 3813 determinants. The probability that a single excitor is on a specific processor may be regarded as a random event with probability $\frac1{\ensuremath{N_{\mathrm{p}}}}$, and the long-time distribution of such events is expected therefore to follow a binomial distribution with this probability, giving an unbiased mean of $\frac1{\ensuremath{N_{\mathrm{p}}}}$ and a variance which therefore scales with $\frac{{\ensuremath{N_{\mathrm{p}}}}-1}{{\ensuremath{N_{\mathrm{p}}}}^2}$. Similarly, a move frequency of ${\ensuremath{{\nu_{\textrm{move}}}}}$ moves an excitor’s processor every $2^{{\ensuremath{{\nu_{\textrm{move}}}}}}$ iterations, so decreases the variance by a factor of $2^{{\ensuremath{{\nu_{\textrm{move}}}}}}$. Both of these scaling effects translate directly to the correlated probability of two excitor locations. Figure \[fig:excitdistrib\] shows no notable bias in the time-averaged mean probability of two excitors coinciding, and a standard deviation following the above scaling relationships. Any bias present due to the instanteous probability distribution of excitors is can therefore be reduced by decreasing ${\ensuremath{N_{\mathrm{p}}}}$ and ${\ensuremath{{\nu_{\textrm{move}}}}}$.
As a test system to see a bias, we consider the three-dimensional uniform electron gas (3D UEG)[@MartinUEGChapter; @Giuliani2005; @Loos2016a], which conveniently allows for an easily-adjustable Hilbert space and degree of correlation. Specifically, we calculate the CCSDT energy of the 14-electron 3D UEG with 66 plane-wave spin-orbitals at ${\ensuremath{r_{\textrm{s}}}}=0.5{\ensuremath{\mathrm{a_0}}}$ and ${\ensuremath{r_{\textrm{s}}}}=5{\ensuremath{\mathrm{a_0}}}$, for which parallelization unbiased results using solely OpenMP parallelization are available[@Neufeld2017]. The full non-composite cluster selection algorithm (\[sec:fnc\]) was used to aid convergence. A discrepancy with magnitude of $0.01$eV/electron is similar in magnitude to chemical accuracy[@Foulkes2001; @Wagner2016; @Neufeld2017] and represents an upper bound on any bias, which would preferably be negligible. Note that this is far from a production-level calculation: the CISDT Hilbert space for this system contains only 22969 determinants.
\[fig:MPIbias\] shows the dependence of the bias in the CCSDT projected energy as a function of the number of MPI processes for the UEG. The bias increases with the number of MPI processes and is larger for ${\ensuremath{r_{\textrm{s}}}}=5{\ensuremath{\mathrm{a_0}}}$ than ${\ensuremath{r_{\textrm{s}}}}=0.5{\ensuremath{\mathrm{a_0}}}$. At 240 MPI processes, each MPI process has fewer than 100 excitors (assuming perfect load balancing) and so the subspace spanned by each MPI process at any given timestep is very small. The degree of correlation is also important: 17% of excips are on the reference for ${\ensuremath{r_{\textrm{s}}}}=0.5{\ensuremath{\mathrm{a_0}}}$ compared to just 2% for ${\ensuremath{r_{\textrm{s}}}}=5{\ensuremath{\mathrm{a_0}}}$. As such the relative importance of products of clusters increases with correlation. We have also performed a hybrid MPI-OpenMP calculation for ${\ensuremath{r_{\textrm{s}}}}=5{\ensuremath{\mathrm{a_0}}}$ using 20 MPI processes with 12 OpenMP threads per process for ${\ensuremath{r_{\textrm{s}}}}=5{\ensuremath{\mathrm{a_0}}}$ which agrees within error bars to the expected result and to the corresponding 20 MPI process calculation.
The bias can be reduced by decreasing the imaginary time a given subspace is sampled. \[fig:timebias,fig:mfbias\] show reducing the timestep and move frequency respectively reduces the bias in the CCSDT energy using 240 MPI processes for the 3D 14-electron UEG system. The bias remains smaller for the smaller values of ${\ensuremath{r_{\textrm{s}}}}$ value with otherwise identical parameters.
We wish to emphasise this is a contrived setup to show that it is possible to obtain biased results in extreme parameter ranges.
Even ignoring computational and parallel efficiency, the small number of excitors per processor results in poor sampling of the cluster expansion and results in a biased sampling of the coupled cluster wavefunction. We typically set the number of MPI processes such that each process contains at least $\mathcal{O}(10^5)$ excitors[^5]. In addition to improved parallel efficiency, hybrid MPI-OpenMP parallelization greatly helps with this issue. As a demonstration of non-trivial calculations, we consider a larger UEG system and the water dimer.
[8.5cm]{} ![[The effects of the number of MPI processes, $\delta\tau$, ${\ensuremath{{\nu_{\textrm{move}}}}}$ on the deviation of the CCSDT projected energy from the unbiased value[@Neufeld2017] for the 14-electron 3D UEG with 66 spin-orbitals. An accuracy of $\pm 0.01$ eV/electron corresponds to 5 m[$\mathrm{E_h}$]{}here as we have 14 electrons. This is outside of the range shown. A horizontal line at zero error is shown to guide the eye.]{}[]{data-label="fig:bias"}](ueg/diff_mpi.pdf "fig:"){width="\linewidth"}
[8.5cm]{} ![[The effects of the number of MPI processes, $\delta\tau$, ${\ensuremath{{\nu_{\textrm{move}}}}}$ on the deviation of the CCSDT projected energy from the unbiased value[@Neufeld2017] for the 14-electron 3D UEG with 66 spin-orbitals. An accuracy of $\pm 0.01$ eV/electron corresponds to 5 m[$\mathrm{E_h}$]{}here as we have 14 electrons. This is outside of the range shown. A horizontal line at zero error is shown to guide the eye.]{}[]{data-label="fig:bias"}](ueg/diff_time.pdf "fig:"){width="\linewidth"}
[8.5cm]{} ![[The effects of the number of MPI processes, $\delta\tau$, ${\ensuremath{{\nu_{\textrm{move}}}}}$ on the deviation of the CCSDT projected energy from the unbiased value[@Neufeld2017] for the 14-electron 3D UEG with 66 spin-orbitals. An accuracy of $\pm 0.01$ eV/electron corresponds to 5 m[$\mathrm{E_h}$]{}here as we have 14 electrons. This is outside of the range shown. A horizontal line at zero error is shown to guide the eye.]{}[]{data-label="fig:bias"}](ueg/diff_mf.pdf "fig:"){width="\linewidth"}
The CISDTQ Hilbert space size of the 14-electron UEG in a basis of 358 spin-orbitals is 2.6(1)$\times10^8$. Using 96 MPI processes (pure MPI parallelisation) gave an estimate of the CCSDTQ correlation energy at $r_s=1\,{\ensuremath{\mathrm{a_0}}}$ to be -0.51875(7)[$\mathrm{E_h}$]{}; using 8 MPI processes each with 12 OpenMP threads (i.e. the same total resources) gave an estimate of -0.51866(7)[$\mathrm{E_h}$]{}. A previous study exploiting only OpenMP parallelization found the correlation energy to be -0.51856(7)[$\mathrm{E_h}$]{}[@Neufeld2017]. The hybrid calculation agrees with the previous result within 2 standard errors (individual standard errors added in quadrature) whereas the pure MPI calculation is close but does not agree within 2 standard errors. However, while the Hilbert space is of the order of $3\times10^8$, the number of occupied excitors relevant to the calculation is only about $7\times10^6$–$8\times10^6$. This means that in the pure MPI case, significantly less than $10^5$ excitors are on the same MPI process. We have used even selection [@Scott2017] for this CCSDTQ calculation.
The CCSDT energy of the dimer at its CCSDTQ optimized geometry obtained by Lane[@Lane_13JCTC] in the jun-cc-pVDZ basis set[@PapajakTruhlar_11JCTC] is compared to deterministic results calculated using MRCC[@MRCC; @Rolik2013] in \[tab:prodbias\]. The CCMC calculation employed the heat bath excitation generator[@Holmes2016a] with slight modifications[@Neufeld2018] and was run on 32 MPI processes each using 12 OpenMP threads. The water dimer in the jun-cc-pVDZ basis has a Hilbert space of $1.16\times10^7$ at the CCSDT level. The stochastic wavefunction contained $\approx 7.9\times10^6$ excitors, resuling in $\approx 5.3\times10^4$ excitors per MPI process, assuming perfect load balancing. Both CCSDT results agree well with each other. The CCMC CCSDT result is resolvably different from the CCSD and CCSDTQ energy within error bars, and no bias is visible.
Method $E_\mathrm{tot.}$/[$\mathrm{E_h}$]{} $E_\mathrm{corr.}$/[$\mathrm{E_h}$]{}
------------------------------ -------------------------------------- ---------------------------------------
Hartree–Fock -152.0804195 0
CCSD deterministic -152.5158272 -0.435407682
CCSDT deterministic -152.5241192 -0.443699666
CCSDT QMC $12\times32$ cores -0.44369(7)
CCSDTQ deterministic -152.5251164 -0.444696884
: Total energy $E_\mathrm{tot.}$ and the correlation energy $E_\mathrm{corr.}$ of the dimer in the jun-cc-pVDZ basis set using Hartree–Fock, deterministic coupled cluster from CCSD to CCSDTQ and CCSDT with CCMC using 32 MPI processes threaded into 12 OpenMP threads each. Even selection[@Scott2017] has been used. The units are hartrees.
\[tab:prodbias\]
To demonstrate the capabilities and indicate the future possibilities of parallelized CCMC, we have also studied a system of three water molecules separated at a large distance at the CCSDTQ level in a cc-pVDZ basis[@Dunning1989]. Using three molecules that are — for all practical purposes — infinitely separated has the advantage that the total energy can be calculated by other means, as three times the energy of a single water molecule at CCSDTQ. This is necessary as we have the deterministic calculation has proven too computationally expensive to be performed in a reasonable time.[^6]
The calculation was restarted from a CCSDT QMC calculation and run with 400 MPI processes using 12 OpenMP threads each for part of the calculation that we are analysing. Even selection[@Scott2017] and the heat bath uniform singles[@Holmes2016a] excitation generator [^7] have been used. In the CCMC calculation, the projected energy oscillates in the range -0.6507 to -0.6524 [$\mathrm{E_h}$]{}. The true correlation energy, as found by using MRCC[@MRCC] scaling up from one molecule, is -0.6515 [$\mathrm{E_h}$]{}which is included at around the middle of our range. The CCSDT correlation energy, found by the same method, is -0.6501 [$\mathrm{E_h}$]{}, which is outside of the quoted range for CCSDTQ with CCMC. To give more than merely a range or to give a smaller, more certain, range, the calculation would have to be run for longer. However, the intent of this study is not to find a known CCSDTQ value but to act as a demonstration that (parallelized) CCMC can give coupled cluster energies that are not feasible with deterministic coupled cluster codes. Current developments of CCMC will enable more precise large calculations in the future.
Improved stochastic sampling {#sec:sampling}
============================
The efficiency of a stochastic coupled cluster calculation is highly dependent upon the algorithm used to sample the various steps within the algorithm. The original implementation used a simple and easy-to-implement algorithm for selecting clusters from which to spawn. This has proved to be increasingly inefficient as system size increases. Scott and Thom have shown that an ‘even-selection’ algorithm which selects clusters with probabilities more closely corresponding to their amplitude dramatically increases the stability of calculations[@Scott2017]. In this section we describe some alternative approaches we have explored to improve the stochastic sampling, [^8] and first consider what metric we may use to measure this.
![Shoulder plots for frozen-core CCSD on benzene in a 6-31G basis using the original (dashed) and full non-composite (solid) algorithm. The initial population can be read from the intercept with the abscissa. ([*top*]{}) The arrows indicate the best estimate of the shoulder, showing that a full non-composite calculation has lower shoulder ($5.6\times10^5$) than the original algorithm ($1.6\times10^6$). ([*bottom*]{}) Multispawn, using $A_{\mathrm{thresh}}=1$, lowering the shoulder to $1.5\times10^5$, showing stability for many fewer initial particles.\
Calculations used a timestep of $5\times10^{-4}$ and the renormalized excitation generators. $r_{\mathrm{CH}}=1.084$Å and $r_{\mathrm{CC}}=1.397$Å exploiting $D_{2h}$ symmetry. []{data-label="fig:fnc"}](C6H6/plot "fig:"){width="1.0\linewidth"}\
![Shoulder plots for frozen-core CCSD on benzene in a 6-31G basis using the original (dashed) and full non-composite (solid) algorithm. The initial population can be read from the intercept with the abscissa. ([*top*]{}) The arrows indicate the best estimate of the shoulder, showing that a full non-composite calculation has lower shoulder ($5.6\times10^5$) than the original algorithm ($1.6\times10^6$). ([*bottom*]{}) Multispawn, using $A_{\mathrm{thresh}}=1$, lowering the shoulder to $1.5\times10^5$, showing stability for many fewer initial particles.\
Calculations used a timestep of $5\times10^{-4}$ and the renormalized excitation generators. $r_{\mathrm{CH}}=1.084$Å and $r_{\mathrm{CC}}=1.397$Å exploiting $D_{2h}$ symmetry. []{data-label="fig:fnc"}](C6H6/ms/plot "fig:"){width="1.0\linewidth"}
Shoulder heights
----------------
The stability of a stochastic coupled cluster calculation can be determined by whether its population has overcome a plateau or shoulder in its dynamics[@Spencer2012], where the rate of total particle growth (with imaginary time) slows or stops for some time while the correct wavefunction is evolved. A plateau is commonly visible in FCIQMC calculations whereas CCMC calculations typically only contain a shoulder in the total population growth. After this point, a calculation emerges with a stable growth rate of both the total and reference particle populations. These factors are conveniently described on a shoulder plot[@Spencer2016], which show a maximum in the ratio of total and reference populations at the shoulder. The population at the shoulder is indicative of the relative difficulty of a calculation, and is affected by the parameters used to run it, and a best estimate is given with a low initial reference population and small timestep; using larger timesteps causes the shoulder population to increase. As the population on the reference also dictates the normalization of the calculation, too low an initial population leads to unstable calculations which do not experience a shoulder and merely ‘blow up’. Examples of shoulder plots are given in \[fig:fnc\].
In a stochastic coupled cluster calculation, the number of particles spawned at any step is directly proportional to the timestep, and after a certain threshold, larger timesteps will give a higher shoulder population. To therefore determine the effects of any algorithmic changes we have used the parameters which give the lowest estimate of the shoulder, namely a very small timestep, and reducing the initial population, following a study in Ref. which compared selection algorithms with appropriate timesteps and initial populations.
Non-composite clusters {#sec:fnc}
----------------------
We first look at reducing the noise due to stochastic sampling of the cluster expansion. The total number of cluster selections to be made has previously been chosen to equal the total amplitude of excips, in analogy with FCIQMC, where each discrete psip individually undergoes spawning and death events. In stochastic coupled cluster, clusters built up from single excips as well as multiple excips (known as composite clusters) are sampled, so the single-excip (non-composite) clusters have fewer samples taken than the total number of excips. The sampling of (rather than explicit iteration through) the non-composite clusters proves to be an additional source of stochastic noise. We therefore introduce a modification to the algorithm, [*full non-composite*]{} sampling, which explicitly iterates through the list of excips, performing spawning and death events on these individually. It is still necessary to sample the composite clusters, and these are sampled with the same number of samples as total excips. The effect of this sampling change is to reduce the number of particles at the shoulder, as shown in figure \[fig:fnc\], and so calculations require fewer excips to be stable. Even though the computational effort increases since we are doing twice as many cluster selections, the number of minimum excips required is reduced by more than a factor of two when using the full non-composite cluster algorithm as shown by the shoulder positions in the *top* part of figure \[fig:fnc\]. Figure 2 in Ref. shows that at higher timesteps, the memory cost as a function of number of attempts per unit imaginary time is lower when using the full non-composite algorithm compared to the original algorithm.
Multiple spawning events {#sec:multispawn}
------------------------
The small timestep regime reduces the plateau height because at sufficiently small timestep all spawning attempts produce no more than one particle — there are no ‘blooms’, in which a single (rare) spawning event creates a large number of new particles on the same excitor. Such rare events are undesirable both because of the inefficient exploration of the space (especially if the spawned particles do not have the same sign as the ground-state wavefunction) and the impact on population control. Unfortunately the computational efficiency of using such small timesteps is low, as the majority of spawning attempts produce no particles at all. Once a cluster has been selected and collapsed to produce determinant $\bm{j}$, the number of spawned particles depends on the amplitude of this cluster from which they are spawned. As this is selected stochastically, the amplitude is unbiased by dividing by the probability of cluster selection to give an effective amplitude, $A(\bm{j})$. The probability of spawning a particle from this cluster to determinant $\bm{i}$ is then given by $$p_{\mathrm{spawn}}(\bm{i}\leftarrow\bm{j})=\delta\tau\frac{|A(\bm{j})H_{\bm{ij}}|}{p_{\mathrm{gen}}(\bm{i}|\bm{j})}$$ It is therefore possible to use the amplitude $A(\bm{j})$ to decide on the number of spawning attempts from that cluster. A larger number of attempts to spawn will proportionately increase $p_{\mathrm{gen}}(\bm{i}|\bm{j})$, the probability that spawning onto $\bm{i}$ is attmepted from the excips on $\bm{j}$, and consequently decrease the spawning probability, reducing any blooms. The choice of when to change the number of spawning attempts should depend on $A(\bm{j})$, as merely keeping it at a constant, say $n$, would have a similar effect as using a timestep $\delta\tau/n$ and not account for the impact of rare events from clusters with large amplitudes. We have chosen to introduce a threshold $A_{\mathrm{thresh}}$, such that the number of attempts is given by $$n_{\mathrm{attempts}}=\max\left(1,\left\lfloor\frac{A(\bm{j})}{A_{\mathrm{thresh}}}\right\rfloor\right)$$ The effect of these multiple spawning changes is shown in the lower panel of figure \[fig:fnc\], and in combination with the full non-composite sampling, shows a significant reduction in shoulder heights and a corresponding decrease in memory requirements for calculations. We finish with a note of caution when such sampling is used in combination with large-scale parallelization. If the cluster sampling is such that there are occasional amplitudes significantly larger in magnitude than $A_{\mathrm{thresh}}$, a correspondingly large number of spawning attempts are made. In some calculations $n_{\mathrm{attempts}}$ can be occasionally of the order of $10^6$. Should such events be unevenly distributed over processors, it can lead to a significant load-imbalance and reduction of parallel efficiency. In practice, we have only seen this effect when the number of excitors per process is under $10^5$. In general such effects are averted by the use of even selection[@Scott2017] which ensures $|A(\bm{j})|$ takes an approximately constant value for all clusters. Even selection was used for the scaling plot in \[fig:scaling\].
Discussion {#sec:discussion}
==========
Overall we have shown that, despite the non-linearity of the coupled cluster ansatz, by introducing a stochastic algorithm, it is possible to perform massively parallel calculations at arbitrary orders of coupled cluster theory with great parallel efficiency, and approximately ideal strong scaling up to 500 cores. Our parallelization scheme exploits the stochastic nature of the algorithm to sample combinations of excitors averaged over multiple cycles, and we have shown that the bias introduced can be made minimal (provided the number of MPI processes and other parameters are chosen sensibly), well below the intrinsic accuracy of the calculations themselves. Furthermore, the bias can be systematically reduced and so confidence can be had in its magnitude and as systems studied become larger, the parallelization bias becomes smaller.
We contrast our parallelization scheme to that of the Cyclops tensor framework[@Solomonik2014], which requires explicit knowledge of the sparsity within the tensors of excitor amplitudes, and performs deterministic CCSD and CCSDT calculations. While such a scheme produces numerically precise results, it cannot easily take advantage of the natural sparsity of amplitudes within excitation space, so requires significantly more storage for amplitudes. While the polynomial scaling of the number of amplitudes with system size allows a good weak scaling behaviour, the requirement to communicate all of these results in relatively poor strong scaling behaviour. Efforts are underway[@JagodeDongarra2017_IJHPCA17] to redesign deterministic algorithms using a dataflow paradigm, however these require significant manual reorganisations of the code which appears infeasible for higher levels of coupled cluster theory.
We have also shown that the stochastic sampling can be improved using the *full non-composite* or the *multi-spawn* additions. The later *even selection* sampling[@Scott2017] was inspired by and was built on top of the *full non-composite* algorithm and when using multiple MPI processes is more efficient than *multi-spawn* sampling which was a first step on the way to improving shoulder heights.
We close by noting that the approach described here is not restricted to just coupled cluster; rather the idea of sampling both the action of the Hamiltonian and the wavefunction ansatz is applicable to many other methods in quantum chemistry.
J.S.S. acknowledges the research environment provided by the Thomas Young Centre under Grant No. TYC-101. V.A.N. acknowledges the EPSRC Centre for Doctoral Training in Computational Methods for Materials Science for funding under grant number EP/L015552/1 and the Cambridge Philosophical Society for a studentship. W.A.V. is grateful to EPSRC for a studentship. R.S.T.F. acknowledges CHESS for a studentship and A.J.W.T. the Royal Society for a University Research Fellowship under grants UF110161 and UF160398. This work used the ARCHER UK National Supercomputing Service (<http://www.archer.ac.uk>) under grant e507 and the UK Research Data Facility (<http://www.archer.ac.uk/documentation/rdf-guide>) under grant e507. This work was also performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (<http://www.csd3.cam.ac.uk/>), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/P020259/1), and DiRAC funding from the Science and Technology Facilities Council ([www.dirac.ac.uk](www.dirac.ac.uk)).
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[^1]: Our implementation does not require canonical orbitals, so we have the full freedom to choose any transformation of the single particle orbitals, for example the study in Ref.
[^2]: We set ${\ensuremath{p_{\textrm{size}}}}(s) = 1 - \sum_{i=0}^{s-1} {\ensuremath{p_{\textrm{size}}}}(i)$ in order to ensure a normalised probability.
[^3]: Booth *et al.* used a custom hash function based upon the list of occupied orbitals. We find hashing the bit string representation of the determinant simpler and computationally more efficient, whilst giving at least as good distribution over processors when a hash function of sufficient quality is used, and therefore use the MurmurHash2 function.
[^4]: A node may consist of a single processor or multiple processors. Within the MPI paradigm, we distribute over MPI ranks, where each rank contains one or more threads.
[^5]: We note that available computational resources rarely grow polynomially with system size!
[^6]: Using MRCC[@MRCC], a single iteration was not completed within a week on a 32-core node on the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (http://www.csd3.cam.ac.uk/). This provides a lower estimate of 2 node-months, or 46000 core hours.
[^7]: Terminology from Ref. has been used. Single excitations were sampled uniformly, double excitations with the heat bath excitation generator. Idea by and mentioned in Holmes et al. [@Holmes2016a].
[^8]: These approaches were conceived and implemented some time before the work in Ref. , and briefly referred to in the same, and are reported in fuller detail now for completeness.
|
---
abstract: 'We present a novel method for the evaluation of shot noise in quasi one-dimensional field-effect transistors, such as those based on carbon nanotubes and silicon nanowires. The method is derived by using a statistical approach within the second quantization formalism and allows to include both the effects of Pauli exclusion and Coulomb repulsion among charge carriers. In this way it extends Landauer-Büttiker approach by explicitly including the effect of Coulomb repulsion on noise. We implement the method through the self-consistent solution of the 3D Poisson and transport equations within the non-equilibrium Green’s function framework and a Monte Carlo procedure for populating injected electron states. We show that the combined effect of Pauli and Coulomb interactions reduces shot noise in strong inversion down to 23% of the full shot noise for a gate overdrive of 0.4 V, and that neglecting the effect of Coulomb repulsion would lead to an overestimation of noise up to 180%.'
author:
- |
Alessandro Betti, Gianluca Fiori and Giuseppe Iannaccone\
Dipartimento di Ingegneria dell’Informazione: Elettronica, Informatica, Telecomunicazioni,\
via Caruso 16, 56100 Pisa, Italy\
email: [{alessandro.betti, g.fiori, g.iannaccone}@iet.unipi.it]{}, Tel. +39 050 2217639
title: 'Shot noise suppression in quasi-one dimensional Field Effect Transistors'
---
[**[Keywords]{}**]{} - Shot noise, FETs, nanowire transistors, carbon nanotube transistors.
Introduction
============
In the last few years, a huge collective effort has been directed to assess potential performance of quasi-1D Field Effect Transistors (FETs) based on Carbon Nanotubes [@Martel; @JGuo; @GKlimeck] (CNTs), Silicon NanoWires [@YCui] (SNWs), Graphene nanoribbons (GNRs) versus the International Technology Roadmap for Semiconductors [@ITRS] (ITRS) requirements, both from an experimental and a theoretical point of view. However, attention has been focused on electrical quantities like $I_{\rm on}/I_{\rm off}$, subthreshold slope, mobility, transconductance [@Yuzvinsky; @VPerebeinos; @JKnoch], while an accurate investigation of electrical noise has been often neglected. Although the 1/f noise represents the major noise source affecting CNT-FETs performance [@YMLin; @JAppenzeller], the intrinsic shot noise is not only critical from an analog and digital design point of view, but can also provide relevant information regarding interactions among carriers [@ik1prb] [@ik1] [@BlanterBut], electron energy distribution [@Gramespacher] [@Bul2] and electron kinetics [@Land].
Due to the limited device dimensions, even in strong inversion only few electrons take part to transport, so that drain current fluctuations can heavily affect device electrical behavior. Pauli and Coulomb interactions play an important role in noise analysis, through fluctuations of the occupation number of injected states and fluctuations of the potential barrier along the channel.
From a numerical point of view, a self-consistent solution of the electrostatics and transport equations is mandatory in order to properly consider such effects. An analysis of this kind has been performed for example in double gate MOSFETs [@YNaveh] and in nanoscale ballistic MOSFETs [@ik1_2], where a strong shot noise suppression, mostly due to Pauli exclusion principle, has been observed.
A different approach, based on quantum trajectories within the De Broglie-Bohm framework has been instead presented in [@XOriols1], where resonant tunneling diodes have been studied and heavy approximations have been adopted in order to easily consider electron-electron correlation in the many-body problem.
Actually, a complete understanding of the mechanism of suppression of shot noise in CNT and SNW-FETs is still a debated issue. Indeed, the significant suppression of current fluctuations by more than a factor 100 obtained at low temperature for suspended ropes 0.4 $\mu$m long of single wall carbon nanotubes [@PRoche] has not been supported by a comprehensive theoretical analysis. Recent experiments of shot noise in CNT-based Fabry Perot interferometers [@Hermann] show that by only including Pauli exclusion one is able to explain most of the dependence of shot noise on the backgate bias, but in some bias conditions additional mechanisms of electron-electron interaction might be needed to explain the observed noise suppression. Theoretical efforts have been mainly addressed to model the electrical noise in SNW-FETs, where, within a scattering approach with the limitation of excluding space-charge effects on electron transmission, Pauli exclusion reduces electrical noise in strong inversion down to 0.6% of the full value for a gate overdrive of 0.3 V [@Park1], whereas an interesting increase of noise is observed by including electron-phonon scattering processes [@Park2].
Here, we present a new method to compute the shot noise power spectral density in ballistic CNT and SNW-FETs based on Monte Carlo (MC) simulations of randomly injected electrons from the reservoirs. In order to consider correlations between fermions, an analytical formula for the noise power spectral density has been computed by means of a statistical approach within the second quantization formalism. The derived formula has then been implemented in the self-consistent solution of the 3D Poisson and Schrödinger equations, within the NEGF formalism.
Theory
======
The average current in a mesoscopic conductor can be expressed by means of Landauer’s formula: $$\label{eqn:Landauer}
\langle I \rangle= \frac{e}{\pi\hbar} \int d\!E \left\{ \mathrm{Tr}\left[\mathbf{t^\dagger t}(E)\right]\left[f_S(E)-f_D(E)\right] \right\}$$ where $\mathbf{t}$ is the transmission amplitude matrix for states emitted from the source (S) and collected at the drain (D) and $f_S$ and $f_D$ are the Fermi-Dirac statistics of the S and D, respectively.
The zero-frequency noise power spectral density for a two-terminal conductor - the so-called Landauer-Büttiker noise formula - reads [@MBut2] [@TMartin]: $$\begin{aligned}
\label{eqn:noise}
S(0) \!\! \! \! \! \! &=& \! \!\!\! \! \!\frac{2\,e^2}{\pi\hbar}\!\! \int\! \!d\!E \left\{ \left[f_S (1\!-\!f_S)+f_D (1\!-\!f_D)\right]\mathrm{Tr}\!\left[\mathbf{t^\dagger t t^\dagger t}\right] \right. \nonumber \\
&+& \!\!\!\!\!\!\left. \left[f_S (1\!-\!f_D)+f_D (1\!-\!f_S )\right] \!\left( \mathrm{Tr}\!\left[\mathbf{t^\dagger t} \right]\! -\!\mathrm{Tr}\!\left[\mathbf{t^\dagger t t^\dagger t} \right]\right)\! \right\} \, , \nonumber \\\end{aligned}$$ where $\mathbf{t^\dagger}$ is the conjugate transpose of the matrix $\mathbf{t}$. However, eq. (\[eqn:noise\]) holds only if one assumes that fluctuations of the potential profile do not occur, i.e. that Coulomb interaction between carriers is completely neglected. Actually, the potential barrier along the channel fluctuates in time, since randomly injected electrons modify the height of the barrier through long-range Coulomb interaction, which in turn affects carriers transmission and eventually leads to the suppression of the drain current fluctuations.
In order to compute the expression of the power spectral density in the general case, we take advantage of the second quantization formalism. In particular, at zero magnetic field, the time-dependent current operator at the source can be expressed as the difference between the occupation numbers of carriers moving inward and outward the source contact in each quantum channel [@MBut2] ($n_{Sm}^+$ and $n_{Sm}^-$, respectively): $$\begin{aligned}
\label{eqn:current}
I(t) \!\! \! \! &=& \! \!\! \!\frac{e}{2 \pi\hbar} \sum_{m \in S} \int d\!E \left[n_{Sm}^+(E,t) - n_{Sm}^-(E,t)\right] \, ,\end{aligned}$$ where $$\begin{aligned}
\label{eqn:n+}
n_{Sm}^+(E,t) \!\! \! \! &=& \! \!\! \!\int d(\hbar \omega) a_{Sm}^+(E)\,a_{Sm}(E+\hbar \omega)\, e^{-i\omega t} \, , \nonumber \\
n_{Sm}^-(E,t) \!\! \! \! &=& \! \!\! \!\int d(\hbar \omega) b_{Sm}^+(E)\,b_{Sm}(E+\hbar \omega)\, e^{-i\omega t} \, .\end{aligned}$$ The operators $a_{Sm}^{\dagger}(E)$ and $a_{Sm}(E)$ create and annihilate, respectively, incident electrons in the source lead with total energy $E$ in the transverse channel $m$. In the same way, the creation $b_{Sm}^{\dagger}(E)$ and annihilation $b_{Sm}(E)$ operators refer to electrons in the source lead for outgoing states. For the CNT case, the channel index $m$ runs over all the transverse modes and different spin, whereas for SNW, it also runs along the six minima of the conduction band in the $\mathbf{k}$ space. In addition, the operators $a_S$ and $b_S$ are related through the unitary transformation $$\begin{aligned}
\label{eqn:relationab}
b_{Sm}(E)=\sum_{\alpha = S,D}
\sum_{n \in \alpha }^{N_{\alpha}}
\mathbf{s}_{S\alpha;mn}(E)a_{\alpha n}(E) \, ,\end{aligned}$$ where the scattering matrix $\mathbf{s}$ has dimensions $(N_S + N_D)\times (N_S + N_D)$ and $N_S$ and $N_D$ are the number of quantum channels in the source and drain contacts, respectively. In the following, time dependence will be neglected, since we are interested to the zero frequency case.
If $\mid \! \sigma \rangle$ is a many-particle (antisymmetrical) state, the occupation number $\sigma_{\alpha m}(E) $ in the reservoir $\alpha$ ($\alpha= S,D $) in the channel $m$ can be either 0 or 1, and can be expressed as $\sigma_{\alpha m}(E)= \langle a_{\alpha m}^{\dagger}(E)a_{\alpha m}(E)\rangle_{\sigma}$. Since we are interested to current fluctuations, we need to consider an ensemble of many electrons states $\{ \mid \sigma_1 \rangle,\mid \sigma_2 \rangle,\mid \sigma_3 \rangle,\cdots,\mid \sigma_N \rangle \}$ and to compute statistical averages $\langle \, \rangle_s$. By assuming no correlations between states at different energy or injected from different reservoirs, the statistical average of $\sigma_{\alpha m}(E)$ reads $$\begin{aligned}
\label{eqn:sigma}
\langle \sigma_{\alpha m}(E) \rangle_s= \,\langle \langle a_{\alpha m}^{\dagger}(E) a_{\alpha m}(E)\rangle_{\sigma}\rangle_s = \, f_{\alpha}(E) \end{aligned}$$ In the following, we identify $\langle \langle \,\,\, \rangle_{\sigma}\rangle_{s}$ with $\langle \,\,\, \rangle$. By means of (\[eqn:relationab\]), we obtain the mean current: $$\begin{aligned}
\label{eqn:meancurrent}
\langle I \rangle \!\!\!\!&=&\!\!\!\! \frac{e}{2 \pi \hbar} \int d\!E \,\left\{\sum_{n \in S} \langle \left[\mathbf{t^{\dagger} t} (E) \right]_{nn} \sigma_{S n}(E) \rangle_s \right. \nonumber \\
&-&\!\!\! \left. \sum_{k \in D} \langle \left[\mathbf{t'^{\dagger}t'}(E) \right]_{kk} \sigma_{D k} (E) \rangle_s \right\}\end{aligned}$$ where $ \mathbf{t'}$ is the drain-to-source transmission amplitude matrix [@Datta] . Since $ \sigma_{\alpha m}^2= \sigma_{\alpha m} $, $\forall m \in \alpha$ and exploiting the unitarity of the scattering matrix, the mean squared current fluctuation for unit of energy can be expressed as: $$\begin{aligned}
\label{eqn:variance}
\frac{var\left(I \right)}{\Delta E}\!\!\!\!\!\! &=&\!\!\!\!\!\! \left(\frac{e}{h}\right)^2 \!\!\!\int \!\! d\!E \!\!\! \sum_{\alpha = S,D} \sum_{l \in \alpha} \langle \left[\mathbf{\tilde{t}}\right]_{\alpha;ll}\left(1-\left[\mathbf{\tilde{t}}\right]_{\alpha;ll}\right) \! \sigma_{\alpha l}\rangle_s \nonumber \\
&-&\!\!\!\! \left(\frac{e}{h}\right)^2 \! \!\!\int \! d\!E \!\!\! \sum_{\alpha = S,D} \!\!\!\!
\sum_{
\begin{array}{c}
\scriptstyle l,p \in \alpha \\
\scriptstyle l \neq p \\
\end{array}}
\!\!\!\!\langle \left[\mathbf{\tilde{t}}\right]_{\alpha;l p} \left[\mathbf{\tilde{t}}\right]_{\alpha;p l} \sigma_{\alpha l} \sigma_{\alpha p}\rangle_s \nonumber \\
&-& \!\!\!\!2 \left(\frac{e}{h}\right)^2 \!\!\!\!\int \! \!d\!E \! \sum_{k \in D} \sum_{p \in S}\! \langle \left[\mathbf{t'^{\dagger}r}\right]_{kp} \left[\mathbf{r^{\dagger}t'}\right]_{pk} \sigma_{Dk} \sigma_{Sp} \rangle_s \nonumber \\
&+& \!\!\!\!\!\!\frac{1}{\Delta E} var \! \left\{\! \!\frac{e}{h}\! \int \!\!\!d\!E \!\left(\sum_{n \in S} \!\!\left[\mathbf{\tilde{t}} \right]_{S;nn}\! \sigma_{S n} \!
- \!\!\! \sum_{k \in D} \!\!\left[\mathbf{\tilde{t}} \right]_{D;kk}\! \sigma_{D k} \!\! \right)\!\!\right\} \nonumber \\ \end{aligned}$$ where $\left[\mathbf{\tilde{t}}\right]$ is defined as $$\left[\mathbf{\tilde{t}}\right]_{\alpha;lp}=
\left\{
\begin{array}{cc}
\left[\mathbf{t^{\dagger} t}\right]_{lp} \mbox{ \quad if } \alpha=S\\
\left[\mathbf{t'^{\dagger} t'}\right]_{lp} \mbox{ \quad if } \alpha=D \, ,
\end{array}
\right.$$ and $\mathbf{r}$ is the reflection amplitude matrix [@Datta]. $\Delta E$ is our energy step of choice, i.e. the minimum energy separation between injected states.
Eq. (\[eqn:variance\]) is expressed as the sum of four terms: the first, the second and the third terms correspond to the partition noise contribution. In particular, the first term is strictly related on the quantum uncertainty of the transmission process and disappears in the classical limit; the second term is associated to the correlation between transmitted states coming from the same reservoir; the third term to the correlation between transmitted and reflected states in the source lead; the minus sign in the second and third terms is due to exchange pairings, because of the fermionic nature of the electrons. In particular, the second and the third terms provide physical insights on exchange interference effects [@ABetti2]. Finally, the last term represents the injection noise obtained as the variance computed on the ensemble of current samples.
According to the
Milatz Theorem
[@Ziel], the noise power spectral density in the zero frequency limit can be computed as $ S(0)= lim_{f \rightarrow 0}\, S(f) = lim_{\nu \rightarrow 0} \left[2/\nu \cdot var(I)\right] $, where $ \nu $ is the injection rate, which can be expressed as: $$\begin{aligned}
\label{eqn:nu}
\nu = \Delta E/(2 \pi \hbar)
.\end{aligned}$$ Eventually, the power spectral density of shot noise at zero frequency can be expressed as: $$\label{eqn:noisepower}
S(0) = \lim_{\nu \rightarrow 0}\frac{2}{\nu} \, var(I)= \lim_{\Delta E \rightarrow 0} 4 \pi \hbar \frac{var(I)}{\Delta E}$$ It is worth noticing that eqs. (\[eqn:noisepower\]) and (\[eqn:variance\]) are not equivalent to the Landauer-Büttiker’s formula (\[eqn:noise\]), since in eq. (\[eqn:variance\]) the transmission ($\mathbf{t}$, $\mathbf{t'}$) and reflection ($\mathbf{r}$) matrix are expressed as functionals of the statistics of the occupation of injected states from both contacts. In this way we are able to consider the fluctuation in time of the conduction and valence band edge profiles produced by the random injection through long-range Coulomb repulsion, providing a further source of noise suppression not included in eq. (\[eqn:noise\]).
Indeed, from an analitical point of view, eqs. (\[eqn:variance\]) and (\[eqn:noisepower\]) reduce to eq. (\[eqn:noise\]) when transmission and reflection do not depend, through Coulomb interaction, on random occupation numbers of injected states: in that case we can take the terms related to transmission and reflection out of the statistical averages in (\[eqn:variance\]). By means of (\[eqn:sigma\]) and exploiting $ \langle \sigma_{\alpha l}(E) \sigma_{\beta n}(E')\rangle_s = f_{\alpha}(E)f_{\beta}(E')+ \delta(E-E')\delta_{\alpha \beta} \delta_{l n} [f_{\alpha}(E)$ $-f_{\alpha}(E)f_{\beta}(E')]$, the fourth term in (\[eqn:variance\]) becomes: $$\begin{aligned}
\label{eqn:term4}
&&\left(\frac{e}{h}\right)^2\! \int \!d\!E \! \sum_{n \in S} \left[\mathbf{t^{\dagger} t(E)} \right]_{nn}^2 f_{S}(E) \left[1-f_{S}(E)\right] \nonumber \\
&+&\!\!\!\left(\frac{e}{h}\right)^2\! \int \!d\!E \! \sum_{k \in D} \left[\mathbf{t t^{\dagger}(E)} \right]_{kk}^2 f_{D}(E) \left[1-f_{D}(E)\right]\end{aligned}$$ since at zero magnetic field $\mathbf{t'^{\dagger}t'}=\mathbf{t t^{\dagger}}$. The terms $\delta(E-E')$, $\delta_{\alpha \beta}$ and $\delta_{l n}$ are the Kronecker delta. Taking advantage of $\sum_{k \in D} \sum_{p \in S} \left[\mathbf{t'^{\dagger} r}\right]_{kp} \left[\mathbf{r^{\dagger} t'} \right]_{pk}= \mathrm{Tr}\left[\mathbf{t^\dagger t} \right]-\mathrm{Tr}\left[\mathbf{t^\dagger t t^\dagger t} \right] $, $S(0)$ becomes: $$\begin{aligned}
\label{eqn:nonSClimit}
S(0) \!\! \!\!\!\! &=& \!\!\!\! \!\! \lim_{\Delta E \rightarrow 0}
4 \pi \hbar \, \frac{var(I)}{\Delta E} \nonumber \\
&=& \!\! \!\! \!\! \frac{2\,e^2}{\pi\hbar} \!\!\int \!\!d\!E \left\{ \left[f_S (1 \!- \!f_S)+f_D (1 \!- \!f_D)\right]\mathrm{Tr}\left[\mathbf{t^\dagger t t^\dagger t}\right] \right. \nonumber \\
&+& \!\!\!\!\!\!\left. \left[f_S (1 \!\!- \!\!f_D)\!+\!f_D (1 \!\!- \!\!f_S )\right] \!\left( \mathrm{Tr}\left[\mathbf{t^\dagger t} \right]\!-\!\mathrm{Tr}\left[\mathbf{t^\dagger t t^\dagger t} \right]\right) \right\} \, \nonumber \\\end{aligned}$$ which is Landauer-Büttiker’s formula (\[eqn:noise\]).
Let us now point out that eq. (\[eqn:noisepower\]) would also simplify when identical and independent injected modes from the reservoirs are considered. In this case, $\mathbf{t}$, $\mathbf{t'}$ and $\mathbf{r}$ are all diagonal, so that the second term in (\[eqn:variance\]) becomes negligible. By exploiting the reversal time symmetry ($\mathbf{t'}= \, \mathbf{t^t}$, where $\mathbf{t^t}$ is the transpose of $\mathbf{t}$) and the unitarity of the scattering matrix, the power spectral density becomes: $$\begin{aligned}
\label{eqn:noisesimulation}
S(0) \! \!\!\!\! \! &=& \!\!\!\!\!\! \frac{e^2}{\pi\hbar} \!
\left\{\!
\int \!\!d\!E \!\!\!\!\sum_{\alpha = S,D} \sum_{l \in \alpha} \langle \left[\mathbf{\tilde{t}}\right]_{\alpha;ll} \!\left(1\!-\!\left[\mathbf{\tilde{t}}\right]_{\alpha;ll}\right) \sigma_{\alpha l}\rangle_s \right. \nonumber \\
&-&\!\! 2 \int d\!E \sum_{l \in S} \langle \left[\mathbf{\tilde{t}}\right]_{S;ll}
\left(1- \left[\mathbf{\tilde{t}}\right]_{S;ll}\right) \sigma_{Dl} \sigma_{Sl} \rangle_s
\nonumber \\
&+&\!\!\!\!\!\! \frac{1}{\Delta E}\!\! \left. var \! \left[\!\int \!\!\!d\!E \! \left(\sum_{n \in S} \! \left[\mathbf{\tilde{t}}\right]_{S;nn} \sigma_{S n} \! - \!\!\! \sum_{k \in D} \! \left[\mathbf{\tilde{t}}\right]_{D;kk} \sigma_{D k} \right)\!\right]\! \right\} \nonumber \\\end{aligned}$$
Simulation Methodology
======================
In order to properly include the effect of Coulomb interaction, we self-consistently solve the 3D Poisson equation imposing Dirichlet boundary conditions in correspondence of the metal gates, and null Neumann boundary conditions on the ungated surfaces which define the 3D domain. Within a self-consistent scheme, the 3D Poisson equation is coupled with the Schrödinger equation with open boundary conditions, within the Non-Equilibrium Green’s function (NEGF) formalism which has been implemented in our in-house open source simulator [*NanoTCAD ViDES*]{} [@ViDES]. In particular the 3D Poisson equation reads $$\begin{aligned}
\label{eqn:Poisson}
\nabla\left(\epsilon \nabla \phi\left(\vec{r}\right) \right)= -\left(\rho \left(\vec{r}\right)+\rho_{fix}\left(\vec{r}\right)\right) \, ,\end{aligned}$$ where $\phi$ is the electrostatic potential, $\rho_{fix}$ is the fixed charge which accounts ionized impurities in the doped regions, while $\rho$ is the charge density per unit volume $$\begin{aligned}
\label{eqn:density}
\rho \left(\vec{r}\right) \!\!\!&=&\!\!\! - e\!\int_{E_i}^{+\infty}\!\! \!d\!E \!\sum_{\alpha= S,D} \sum_{n \in \alpha} DOS_{\alpha n}\left(\vec{r},E\right) \sigma_{\alpha n}(E) \nonumber \\
&+&\!\!\!e \!\!\int_{-\infty}^{E_i} \!\!\!\!\!d\!E \!\!\!\sum_{\alpha= S,D} \sum_{n \in \alpha}\! DOS_{\alpha n}\left(\vec{r},E\right)\! \left[1\!-\!\sigma_{\alpha n}(E)\right] ,\end{aligned}$$ where $E_i$ is the mid-gap potential, $DOS_{\alpha n}(\vec{r},E)$ is the local density of states associated to channel $n$ injected from contact $\alpha$ and $\vec{r}$ is the 3D spatial coordinate.
From a numerical point of view, in order to model the stochastic injection of electrons from the contacts, a statistical simulation on an ensemble of random configurations of injected electron states from both contacts has been performed. In particular, we have uniformly discretized with step $\Delta$E the whole energy range of integration \[equations (\[eqn:variance\]) and (\[eqn:density\])\]. Each random injection configuration has been obtained by extracting a random number $r$ uniformly distributed between 0 and 1 for each state represented by energy $E$, reservoir $\alpha$, and quantum channel $n$. The state is occupied if $r$ is smaller than the Fermi Dirac factor, i.e. $\sigma_{Sn}(E)$ $[\sigma_{Dn}(E)]$ is 1 for $ r < f_S(E)\, [f_D(E)]$, and 0 otherwise.
Self-consistent simulations for a given actual random statistics in the source and drain contacts have been then performed, and, at convergence, the transmission ($\mathbf{t}$, $\mathbf{t'}$) and reflection ($\mathbf{r}$) matrix have been computed, obtaining an element of the ensemble. In particular, for an actual electron distribution in the contacts, the Schrödinger equation is solved in order to obtain the spatial charge distribution (\[eqn:density\]) along the channel. Then, the latter is included in eq. (\[eqn:Poisson\]) and the electrostatic potential is then computed and, once convergence of the NEGF-Poisson iteration scheme is achieved, the scattering matrix is evaluated, and a new sample to be added to the noise ensemble is obtained. Finally, the power spectral density $S(0)$ can be extracted by means of eqs. (\[eqn:variance\]) and (\[eqn:noisepower\]). From a computational point of view, we have verified that $S(0)$ computed on a record of 500 samples, using the energy step $\Delta E= $ 5 $\times$10$^{-4}$ eV, represents a good tradeoff between computational cost and accuracy of results [@ABetti].
Let us mention the fact that our approach is based on a mean field approximation of the Coulomb interaction, and that therefore the exchange term is not included. In the following, we will refer to self-consistent (SC) simulations when the Poisson-Schrödinger equations are solved considering $f_S$ and $f_D$ in eq. (\[eqn:density\]), while we refer to self-consistent Monte Carlo simulations (SC-MC), when statistical simulations with random occupations $\sigma_{Sn}(E)$ and $\sigma_{Dn}(E)$ inserted in eq. (\[eqn:density\]) are used. SC-MC simulations of randomly injected electrons allow to consider both the effect of Pauli and Coulomb interaction on noise. As a test, we have verified that if we perform MC simulations, keeping the potential profile along the channel fixed and exploiting the one obtained by means of SC simulation, the noise power spectrum computed in this way reduces to the Landauer-Büttiker’s limit (\[eqn:noise\]), as already proved in an analitical way (eq. (\[eqn:nonSClimit\])): we refer to such simulations as non-self consistent Monte Carlo simulations (non SC-MC).
Simulation Results
==================
Considered devices
------------------
The simulated device structures are depicted in Fig. \[fig:struttura\]. We consider a (13,0) CNT embedded in SiO$_{2}$ with oxide thickness equal to 1 nm, an undoped channel of 10 nm and n-doped CNT extensions 10 nm long, with a molar fraction $f=\,5 \times 10^{-3}$. The SNWT has an oxide thickness $t_{ox}$ equal to 1 nm and the channel length $L$ is 10 nm. The channel is undoped and the source and drain extensions (10 nm-long) are doped with $ N_D = \, 10^{20} $ cm$^{-3}$. The device cross section is 4$\times$4 nm$^2$. A p$_z$-orbital tight-binding Hamiltonian has been assumed for CNTs [@GFiori2; @JGuo2], whereas an effective mass approximation has been considered for SNWTs [@GFiori1; @JWang] by means of an adiabatic decoupling in a set of two-dimensional equations in the transversal plane and in a set of one-dimensional equations in the longitudinal direction for each 1D subband.
For both devices, we have developed a quantum fully ballistic transport model with semi-infinite extensions at their ends. A mode space approach has been adopted, since only the lowest subbands take part to transport: we have verified that four modes are enough to compute the mean current both in the ohmic and saturation region. All calculations have been performed at the temperature $T$= 300 K.
\[tbp\]
![3-D structures and transversal cross sections of the simulated CNT (top) and SNW-FETs (bottom).[]{data-label="fig:struttura"}](./fig1.eps){width="8.6cm"}
DC Characteristics
------------------
In Fig. \[fig:transferrandommedio\], the transfer characteristics for different drain-to-source biases $V_{DS}$ computed performing SC and SC-MC simulations are plotted as a function of the gate overdrive $ V_{GS} - V_{th}$ in the logarithmic scale, both for CNT and SNW devices. In particular the threshold voltage $V_{th}$ for the CNT-FET at $V_{DS}=$ 0.05 V and 0.5 V is 0.43 V, whereas we obtain $V_{th}=$ 0.13 V for $V_{DS}=$ 0.05 V and 0.5 V for the SNW-FET. As can be noted, SC and SC-MC simulations give practically the same results for CNT-FET, except in the subthreshold region where an interesting rectifying effect of the statistics emerges in the Monte Carlo simulations for a drain-to-source bias $V_{DS}=$ 0.5 V.
Instead, the rectifying effect is larger for SNW-FET, differences up to 30 % between the drain current $\langle I \rangle $ computed by means of SC-MC and SC simulations can be also observed in the above threshold regime. In particular, for a gate voltage $V_{GS}=$ 0.5 V and a drain-to-source voltage $V_{DS}=$ 0.5 V, the drain current $\langle I \rangle $ holds 2.42 $\times$ 10$^{-5}$ A applying eq. (\[eqn:meancurrent\]), and 1.89 $\times$ 10$^{-5}$ A applying Landauer’s formula (\[eqn:Landauer\]). Current in the CNT-FET transfer characteristics increases for negative gate voltages due to the interband tunneling. Indeed, the larger the negative gate voltage, the higher the number of electrons that tunnel from bound states in the valence band to the drain, leaving positive charge in the channel, which eventually lowers the barrier and increases the off current [@GFiori].
\[tbp\]
![Transfer characteristics computed for $ V_{DS}$= 0.5 V and 0.05 V, obtained by SC-MC and SC simulations, for CNT and SNW-FET. Full dots refer to CNT, empty dots to SNW. Inset: average number of electrons in CNT-FETs and SNW-FETs channel, evaluated for $ V_{DS}$= 0.5 V and 0.05 V.[]{data-label="fig:transferrandommedio"}](./fig2.eps){width="8.6cm"}
In the inset of Fig. \[fig:transferrandommedio\] the average number of electrons inside the channel of a CNT and SNW-FET for two different biases $V_{DS}=$ 0.5 V and 0.05 V is shown. As can be seen, only very few electrons contribute to transport at any give instant, which requires us to attently evaluate the sensitivity of such devices to charge fluctuations: the smaller the drain-to-source voltage, the larger the average number of electrons in the channel, since, for low $V_{DS}$, carriers are injected from both contacts.
Noise
-----
Let us now focus our attention on the Fano factor $F$, defined as the ratio of the computed noise power spectral density $S(0)$ and the full shot noise $2 e \langle I \rangle $, $F=S(0)/(2 e \langle I \rangle)$. In Fig. \[fig:FanoCNTSNWvsI\], the Fano factor for both CNT-FETs and SNW-FETs is shown for $ V_{DS}$= 0.5 V as a function of drain-to-source current $\langle I \rangle$.
Let us discuss separately the effects of Pauli exclusion alone and concurrent Pauli and Coulomb interactions. Triangles in Fig. \[fig:FanoCNTSNWvsI\] refer to $F$ computed by means of non SC-MC simulations on 10$^4$ samples, while diamonds to results obtained by means of Landauer-Büttiker’s formula, applying eq. (\[eqn:noise\]). As expected the two approaches give the same results for both structures. Solid lines refer to $S(0)$ computed by means of eqs. (\[eqn:variance\]) and (\[eqn:noisepower\]) and SC-MC simulations, i.e. Pauli and Coulomb interactions simultaneously taken into account.
In the sub-threshold regime ($\langle I \rangle<$ 10$^{-9}$ A), drain current noise is very close to the full shot noise, since electron-electron correlations are negligible due to the very small amount of mobile charge in the channel.
From the point of view of eq. (\[eqn:noise\]), for energies larger than the top of the barrier, we have $f_D(E)\ll f_S(E)\ll \,1$ and the integrand in (\[eqn:noise\]) reduces to $\mathrm{Tr}\left[\mathbf{t^\dagger t}(E)\right]f_S(E)$. Instead, for energies smaller than the high potential profile along the channel, $\left[\mathbf{t^{\dagger} t}(E)\right]_{n m} \ll 1$ $\forall n,\, m \in S$, so that we can neglect $\mathrm{Tr}\left[\mathbf{t^\dagger t t^\dagger t} \right]$ in (\[eqn:noise\]), with respect to $\mathrm{Tr}\left[\mathbf{t^\dagger t} \right]$. Since $f_D(E) \ll f_S(E)$, the integrand in (\[eqn:noise\]) still reduces to $\mathrm{Tr}\left[\mathbf{t^\dagger t}(E)\right]f_S(E)$. The Fano factor then becomes $$\begin{aligned}
\label{eqn:fullshot}
F = \frac{S(0)}{2 e \langle I \rangle} \approx \frac{ \frac{2\,e^2}{\pi\hbar} \int d\!E \, \mathrm{Tr}\left[\mathbf{t^\dagger t}(E) \right] f_S(E) }
{ 2 e \frac{e}{\pi\hbar} \int d\!E \, \mathrm{Tr}\left[\mathbf{t^\dagger t}(E)\right] f_S(E) }=\, 1\end{aligned}$$
\[tbp\]
![Fano factor as a function of the drain current $\langle I \rangle$ for a) CNT- and b) SNW-FETs for $ V_{DS} = $ 0.5 V. Solid line refers to the Fano factor $F$ obtained by means of SC-MC simulations, dashed line (diamonds) applying eq. (\[eqn:noise\]) and dotted line (triangles) by means of non SC-MC simulations.[]{data-label="fig:FanoCNTSNWvsI"}](./fig3.eps){width="8.6cm"}
In the strong inversion regime instead ($\langle I \rangle>$ 10$^{-6}$ A), the noise is strongly suppressed with respect to the full shot value. In particular for a SNW-FET, at $\langle I \rangle\approx$ 2.4 $\times$ 10$^{-5}$ A ($ V_{GS} - V_{th}\approx$ 0.4 V), combined Pauli and Coulomb interactions suppress shot noise down to 23 % of the full shot noise value, with a significant reduction with respect to the value predicted without including space charge effects as in Ref. [@Park1], while for CNT-FET the Fano factor is equal to 0.27 at $\langle I \rangle\approx$ 1.4 $\times$ 10$^{-5}$ A ($ V_{GS} - V_{th}\approx$ 0.3 V). Indeed, an injected electron tends to increase the potential barrier along the channel, leading to a reduction of the space charge and to a suppression of charge fluctuation. Note that, by only considering Pauli exclusion principle, we would overestimate shot noise by 180 % for SNWT ($\langle I \rangle\approx$ 2.4 $\times$ 10$^{-5}$ A) and by 70 % for CNT-FET ($\langle I \rangle\approx$ 1.4 $\times$ 10$^{-5}$ A).
Shot noise versus thermal channel noise
---------------------------------------
According to the classical approach for the formulation of drain current noise, channel noise is tipically described in terms of a “modified” thermal noise, as $S(0) = \gamma S_T$, where $S_T=4K_BT g_{d0}$ is the thermal noise power spectrum at zero drain-to-source bias $V_{DS}$, $k_B$ is the Boltzmann constant, $\gamma$ is a correction parameter and $g_{d0}= \, \left(\partial \langle I \rangle/\partial V_{DS}\right)_{V_{DS}= \, 0} $ is the source-to-drain conductance at zero $V_{DS}$.
Although the classical formulation accurately predicts drain current noise in long channel MOSFETs, where $\gamma$ is equal to 1 in the ohmic region and 2/3 in saturation [@Ziel], it underestimates noise in short channel devices. In particular, experimental evidences [@Abidi] of an excess noise in short channel MOSFET have been explained in terms of the limited number of scattering events inside the channel which is uneffective in suppressing the non-equilibrium noise component [@RNavid], or in terms of a revised classical formulation by considering short channel effects, such as the carrier heating effect above the lattice temperature [@KHan].
Actually, it can clearly be seen that non equilibrium transport easily provides $\gamma > 1$ and that the cause of $\gamma > 1$ is simply due to the fact that channel noise can be more properly interpreted as shot noise. For example, in the particular case of ballistic transport considered here, we can plot $\gamma$ as $S(0)/ S_T$ as a function of the gate voltage in Fig. \[fig:thermnoise1\]c. As can be seen, values of $\gamma$ larger than 1 can be easily observed in weak and strong inversion. The strange behavior of $\gamma$ as a function of the gate voltage is simply due to the fact that one uses an inadequate model (thermal noise) corrected with the $\gamma$ parameter to describe a qualitatively different type of noise, i.e. shot noise.
\[tbp\]
![(a) noise power spectral density obtained by SC-MC simulations and thermal noise spectral density as functions of the gate voltage for a) CNT-FETs and b) SNW-FETs: the considered drain-to-source biases ($ V_{DS}=$ 0.5 V, 0.05 V) are shown in brackets; c) ratio between the noise power obtained by SC-MC simulations and the thermal noise density as a function of the gate voltage. $ g_{d0} $ is the conductance evaluated for $V_{DS}=$ 0 V: $g_{d0}=\, \left(\frac{\partial \langle I \rangle}{\partial V_{DS}}\right)_{V_{DS}=0} $.[]{data-label="fig:thermnoise1"}](./fig4.eps){width="8.6cm"}
\[tbp\]
![Fano factor as a function of the average number of electrons inside the channel per unit length for three different (13,0) CNT-FETs: (A) $t_{ox}$= 1 nm, $L$= 6 nm, (B) $t_{ox}$= 1 nm, $L$= 10 nm and (C) $t_{ox}$= 2 nm, $L$= 10 nm. In a) only the effect of the Pauli principle is shown \[eq. (\[eqn:noise\])\]; in b) the effect of both Pauli and Coulomb interactions is considered. The drain-to-source bias $V_{DS}$ is 0.5 V.[]{data-label="fig:Fscaling"}](./fig5.eps){width="8.6cm"}
Effect of scaling on noise
--------------------------
Let us now discuss the effect of scaling on noise, focusing our attention on a (13,0) CNT-FET. One would expect that an increase of the oxide thickness would reduce the screening induced by the metallic gate, so that the Coulomb interaction would be expected to produce a larger noise suppression. For example, in the limit of a multimode ballistic conductor without a gate contact, significantly suppression of about two order of magnitude with respect to the full shot value has been shown by Bulashenko et al [@Bul1].
However, Ref. [@Bul1] exploits a semiclassical approach assuming a large number of modes and the conservation of transversal momentum, i.e. the role of the transversal electric field induced by the gate voltage is completely neglected. In our case only four modes contribute to transport, while the top and bottom gates of the simulated devices partially screen the electrostatic repulsion induced by the space charge in the channel on each injected electron, so that a smaller noise suppression than the one achieved in Ref. [@Bul1] can be expected.
The Fano factor as a function of the average number of electrons inside the channel for unit length, computed by means of SC simulation and applying eq. (\[eqn:noise\]), for three CNTs with different oxide thickness $t_{ox}$ and channel length $L$ is shown in Fig. \[fig:Fscaling\]a: it shows results for CNT with $t_{ox}$= 1 nm, $L$= 6 nm (A), CNT with $t_{ox}$= 1 nm, $L$= 10 nm (B), and CNT with $t_{ox}$= 2 nm, $L$= 10 nm (C). Fig. \[fig:Fscaling\]b shows the Fano factor computed by performing SC-MC simulations and applying eqs. (\[eqn:variance\]) and (\[eqn:noisepower\]). As can be seen, if the Fano factor is plotted as a function of the number of electrons per unit length, as in Fig. \[fig:Fscaling\], curves are very close to one another, and effects of scaling are predictable.
Conclusion
==========
We have developed a novel and general approach to study shot noise in nanoscale quasi one-dimensional FETs, such as CNT-FETs and SNW-FETs. Our first important result is the derivation of an analytical formula for the noise power spectral density which exploits a statistical approach and the second quantization formalism. Our formula extends the validity of the Landauer-Buttiker noise formula \[eq. (\[eqn:noise\])\], to include also Coulomb repulsion among electrons. From a quantitative point of view, this is very important, since we show that by only using Landauer-Buttiker noise formula, one can overestimate shot noise by as much as 180%. The second important result is the implementation of the method in a computational code, based on the 3D self-consistent solution of Poisson and Schrödinger equation with the NEGF formalism, and on Monte Carlo simulations over a large ensemble of occurrencies, with random occupation of electronic states incoming from the reservoirs. As a final note, we show that scaling of ballistic onedimensional FETs is expected to weakly affect drain current fluctuations, even in the degenerate injection limit.
Acknowledgment
==============
The work was supported in part by the EC Seventh Framework Program under the Network of Excellence NANOSIL (Contract 216171), and by the European Science Foundation EUROCORES Program Fundamentals of Nanoelectronics, through funds from CNR and the EC Sixth Framework Program, under project DEWINT (ContractERAS-CT-2003-980409).
|
---
abstract: 'Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to study the tropical analogue of the classical linear-fractional programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. We use this idea to design a Newton-like algorithm to solve tropical linear-fractional programming problems, by reduction to a sequence of auxiliary mean payoff game problems.'
address:
- 'INRIA and Centre de Mathématiques Appliquées, École Polytechnique. Postal address: CMAP, École Polytechnique, 91128 Palaiseau Cédex, France'
- 'CONICET. Postal address: Instituto de Matemática “Beppo Levi”, Universidad Nacional de Rosario, Av. Pellegrini 250, 2000 Rosario, Argentina'
- 'INRIA and Centre de Mathématiques Appliquées, École Polytechnique. Postal address: CMAP, École Polytechnique, 91128 Palaiseau Cédex, France'
author:
- Stéphane Gaubert
- 'Ricardo D. Katz'
- Sergeĭ Sergeev
title: 'Tropical linear-fractional programming and parametric mean payoff games'
---
[^1]
Introduction {#s:introduction}
============
Motivation from static analysis
-------------------------------
Tropical algebra is the structure in which the set of real numbers, completed with $-\infty$, is equipped with the “additive” law $\ltr a+b\rtr:=a\vee b=\max(a,b)$ and the “multiplicative” law $\ltr ab\rtr:=a+b$. The max-plus or tropical analogues of convex sets have been studied by a number of authors [@Zim-77; @CG:79; @maxplus97; @LMS-01; @CGQ-04; @DS-04; @BriecHorvath04; @joswig04], under various names (idempotent spaces, semimodules, $\mathbb{B}$-convexity, extremal convexity), with different degrees of generality, and various motivations.
In the recent work [@AGG08], Allamigeon, Gaubert and Goubault have used tropical polyhedra to compute disjunctive invariants in static analysis. A general (affine) tropical polyhedron can be represented as $$\begin{aligned}
\label{e-tpol}
P:=\{ x\in (\R\cup\{-\infty\})^n\mid \big({\mathop{\text{\Large$\vee$}}}_{j\in[n]} (a_{ij}+x_j)\big)\vee c_i\leq \big({\mathop{\text{\Large$\vee$}}}_{j\in[n]} (b_{ij}+x_j)\big)\vee d_i , \forall i\in[m] \} .\end{aligned}$$ Here, we use the notation $[n]:=\{1,\ldots ,n\}$, and the parameters $a_{ij}$, $b_{ij}$, $c_i$ and $d_i$ are given, with values in $\R\cup\{-\infty\}$. The analogy with classical polyhedra becomes clearer with the tropical notation, which allows us to write the constraints as $\ltr Ax+c\leq Bx+d\rtr$, to be compared with classical systems of linear inequalities, $Ax\leq d$ (in the tropical setting, we need to consider affine functions on both sides of the inequality due to the absence of opposite law for addition). The previous representation of $P$ is the analogue of the [*external*]{} representation of polyhedra, as the intersection of half-spaces. As in the classical case, tropical polyhedra have a dual (internal) representation, which involves extreme points and extreme rays. The tropical analogue of Motzkin double description method allows one to pass from one representation to the other [@AGG10].
Disjunctive invariants arise naturally when analyzing sorting algorithms or in the verification of string manipulation programs. The well known [memcpy]{} function of C is discussed in [@AGG08] as a simple illustration: when copying the first [n]{} characters of a string buffer [src]{} to a string buffer [dst]{}, the length [len\_dst]{} of the latter buffer may differ from the length [len\_src]{} of the former, for if [n]{} is smaller than [len\_src]{}, the null terminal character of the buffer [src]{} is not copied. However, the relation $\min(\mathtt{len\_src},\mathtt{n}) = \min(\mathtt{len\_dst},\mathtt{n})$ is valid. This can be expressed geometrically by saying that the vector $(-\mathtt{len\_src},-\mathtt{len\_dst})$ belongs to a tropical polyhedron. Several examples of programs of a disjunctive nature, which are analyzed by means of tropical polyhedra, can be found in [@AGG08; @AllamigeonThesis], in which the tropical analogue of the classical polyhedra-based abstract interpretation method of [@CousotHalbwachs78-POPL] has been developed.
The comparative interest of tropical polyhedra is illustrated in Figure \[f:abstractions\], which gives a simple fragment of code in which the tropical invariant is tighter. Note that there is still an over-approximation in the tropical case, because the transfer function considered here is discontinuous (tropical polyhedra share with classical polyhedra the property of being connected, and therefore cannot represent exactly such discontinuities). Such tropical invariants can be obtained automatically via the methods of [@AGG08; @AllamigeonThesis], which rely on the tropical analogue of the double description algorithm [@AGG10], allowing one to obtain the vertices of a tropical polyhedron from a family of defining inequalities, and vice versa.
As in the case of classical polyhedra, the scalability of the approach is inherently limited by the exponential blow up of the size of representations of polyhedra, since the number of vertices or of defining inequalities can be exponential in the size of the input data [@AGK-09; @AGK-10].
The complexity of earlier polyhedral approaches led Sankaranarayanan, Colon, Sipma and Manna to introduce the method of [*templates*]{} [@Sriram1; @Sriram2]. In a nutshell, a template consists of a finite set $\mathcal{T}=\{{g}_1, \ldots,{g}_m\}$ of linear forms on $\R^n$. The latter define a parametric family of polyhedra $$P_\alpha=\{x\in \R^n\mid {g}_k(x) \leq \alpha_k\enspace ,\enspace k\in [m]\}$$ with precisely $m$ degrees of freedom $\alpha_1,\ldots,\alpha_m\in \R\cup\{+\infty\}$. The classical domains of boxes or the domain of [*zones*]{} (potential constraints) [@PhDMine] are recovered by incorporating in the template the linear forms ${g}(x)=\pm x_i$ or ${g}(x)=x_i-x_j$, respectively. Fixing the template, or changing it dynamically while keeping $m$ bounded, avoids the exponential blow up.
The method of [@Sriram2] relies critically on linear programming, which allows one to evaluate quickly the fixed point functional of abstract interpretation. However, the precision of the invariants remains limited by the linear nature of templates, and it is natural to ask whether the machinery of templates carries over to the non-linear case.
[ccc]{}
if x > y then
z:=x;
else
z:=y+1;
&
&
\
& $\begin{array}{l}
x\leq z\\
y\leq \max(x,z-1)\\
z\leq \max(x,y+1)
\end{array}$ & $\begin{array}{l}
x\leq z\\
y\leq z
\end{array}$
The formalism of templates has been extended to the non-linear case by Adjé, Gaubert and Goubault [@AssaleGG], who considered specially quadratic templates, the linear programming methods of [@Sriram2] being replaced by semidefinite programming, thanks to Shor’s relaxation. More generally, every tractable subclass of optimization problems yields a tractable template.
In order to compute disjunctive invariants, Allamigeon, Gaubert and Goubault suggested to develop a generalization of the template method to the case of tropical polyhedra. As a preliminary step, the relevant results of linear programming must be tropicalized: this is the object of the present paper.
In particular, comparing the expressions of $P$ and $P_\alpha$, we see that the classical linear forms must now be replaced by the differences of tropical affine forms $$\begin{aligned}
\label{theobjective}
{g}(x)=
\big({\mathop{\text{\Large$\vee$}}}_{j\in [n]} (p_j+x_j)\vee r\big)- \big({\mathop{\text{\Large$\vee$}}}_{j\in [n]} (q_j+x_j)
\vee s \big) \end{aligned}$$ where $p_j$, $r$, $q_j$ and $s$ are given parameters with values in $\R\cup\{-\infty\}$.
The problem
-----------
In this paper, we study the following [*tropical linear-fractional programming problem*]{}: $$\begin{aligned}
\label{pblp}
\begin{split}
&\text{minimize }\quad {g}(x) \\
&\text{subject to:}\quad x\in P
\end{split} \end{aligned}$$ where $P$ is given by and ${g}$ is given by . This is the tropical analogue of the classical linear-fractional programming problem $$\begin{split}
&\text{minimize } \quad (px+r)/(qx+s) \\
&\text{subject to:}\quad Ax+c\leq Bx+d \; ,\;
x\geq 0 \; , \; x\in \R^n
\end{split}$$ where $p$, $q$ are nonnegative vectors, $r$, $s$ are nonnegative scalars, and $A$, $B$, $c$, $d$ are matrices and vectors. The constraint $\ltr x\geq 0\rtr$ is implicit in , since any number is “nonnegative” (i.e. $\geq -\infty$) in the tropical world.
Problem includes as special cases $$\begin{aligned}
\label{special1}
{g}(x)={\mathop{\text{\Large$\vee$}}}_{j\in [n]} (p_j+x_j)
\qquad\text{and}\qquad
{g}(x)=- {\mathop{\text{\Large$\vee$}}}_{j\in [n]} (q_j+x_j) \end{aligned}$$ (take $q_j\equiv-\infty$, $r=-\infty$ and $s=0$, or $p_j\equiv-\infty$, $s=-\infty$ and $r=0$). According to the terminology of [@BA-08], the latter may be thought of as the tropical analogues of the linear programming problem. However, optimizing the more general fractional objective function appears to be needed in a number of basic applications. In particular, in static analysis, we need typically to compute the tightest inequality of the form $x_i\leq K+x_j$ satisfied by the elements of $P$. This fits in the general form , but not in the special cases .
Contribution
------------
A basic question in linear programming is to certify the optimality of a given point. This is classically done by exhibiting a feasible solution of the dual problem (i.e. a vector of Lagrange multipliers) with the same value. There is no such a simple result in the tropical setting, because as remarked in [@GK-09], there are (tropically linear) inequalities which can be logically deduced from some finite system of (tropically linear) inequalities but which cannot be obtained by taking (tropical) linear combinations of the inequalities of the system. In other words, the usual statement of Farkas lemma is not valid in the tropical setting. However, recently Allamigeon, Gaubert and Katz [@AGK-10] established a tropical analogue of Farkas lemma, building on [@AGG-10], in which Lagrange multipliers are replaced by strategies of an associated mean payoff game. We use the same idea here, and show in Subsection \[ss:certificates\] (Theorem \[1st-cert\] below) that the optimality of a solution can be (concisely) certified by exhibiting a strategy of a game, having certain combinatorial properties. Similarly, whether the value of the tropical linear-fractional programming problem is unbounded can also be certified in terms of strategies (Theorem \[2nd-cert\]).
The second ingredient is to think of the tropical linear-fractional programming problem as a parametric mean payoff game problem. Then, the tropical linear-fractional programming problem reduces to the computation of the minimal parameter for which the value of the game is nonnegative. As a function of the parameter, this value is piecewise-linear and $1$-Lipschitz, see Subsection \[ss:specf\] for a more detailed description.
The main contribution of this paper is a Newton-like method, where at each iteration we select a strategy playing the role of derivative (whose existence is implied by the fact that the current feasible point is not optimal). This defines a [*one player*]{} parametric game problem, and we show that the smallest value of the parameter making the value zero can be computed in polynomial time for this subgame, by means of shortest path algorithms as described in Subsection \[ss:kleene\]. The master algorithm (Algorithm \[a:pos-newton\]) requires solving at each iteration an auxiliary mean payoff game, see Subsections \[ss:left-optim\] and \[ss:germs\]. Mean payoff games can be solved either by value iteration, which is pseudo-polynomial, or by policy iteration, for which exponential time instances have been recently constructed in [@Friedmann-AnExponentialLowerB], although the algorithm is fast on typical examples. The number of Newton type iterations of the Newton-like algorithm has a trivial exponential bound (the number of strategies), and we show that the algorithm is pseudo-polynomial, see Theorem \[posnewt-comp\]. Although this algorithm seems to behave well on typical examples, see in particular Subsection \[SectionExampleMin\], some further work would be needed to assess its worst case complexity (its behavior is likely to be similar to the one of policy iteration, see Subsection \[ss:NumExp\] for a preliminary account of numerical experiments).
Related work
------------
Butkovič and Aminu [@BA-08] studied the special cases , see also [@But:10 Chapter 10]. At each iteration, they solve a feasibility problem (whether a tropical polyhedron is non-empty), which is equivalent to checking whether a mean payoff game is winning for one of the players. However, their algorithm does not involve a Newton-like iteration, but rather a dichotomy argument. In [@BA-08] the number of calls to a mean payoff oracle, whose implementation relies on the alternating method of [@CGB-03], depends on the size of the integers in the input, whereas the number of calls in the present algorithm can be bounded independently of these, just in terms of strategies.
A different approach to tropical linear programming was developed previously in the works of U. Zimmermann [@Zim:81] and K. Zimmermann [@Zim-05]. This approach, which is based on residuation theory, and works over more general idempotent semirings, identifies important special cases in which the solution can be obtained explicitly (and often, in linear time). However, it cannot be applied to our more general formulation, in which there is little hope to find similar explicit solutions.
Newton methods for finding the least fixed point of nonlinear functions are also closely related to this paper. Such methods were developed in [@Policy1; @ESOP07; @Seidl2] to solve monotone fixed point problems arising in abstract interpretation. Esparza et al. [@esparza:approximative] develop such methods for monotone systems of min-max-polynomial equations. They seek for the least fixed point of a function whose components are max-polynomials or min-polynomials, showing that their Newton methods have linear convergence at least. The class of functions considered there is considerably more general than the tropical linear forms appearing here, but the fixed point problems considered in [@esparza:approximative] appear to be of a different nature. It would be interesting, however, to connect the two approaches.
Further motivation
------------------
Tropical polyhedra have been used in [@katz05] to determine invariants of discrete event systems. Systems of constraints equivalent to the ones defining tropical polyhedra have also appeared in the analysis of delays in digital circuits, and in the study of scheduling problems with both “and” and “or” constraints [@mohring]. Such systems have been studied by Bezem, Nieuwenhuis, and Rodríguez-Carbonell [@bezem2; @bezemjournal], under the name of the “max-atom problem”. The latter is motivated by SAT Modulo theory solving, since conjunctions of max-atoms determine a remarkable fragment of linear arithmetic. Tropical polyhedra also turn out to be interesting mathematical objects in their own right [@DS-04; @joswig04]. A final motivation arises from mean payoff games, the complexity of which is a well known open problem: a series of works show that a number of problems which can be expressed in terms of tropical polyhedra are polynomial time equivalent to mean payoff games problems [@mohring; @DG-06; @AGG-10; @bezemjournal; @AGK-10].
The results of the present paper rely on [@AGG-10; @AGK-10].
Preliminaries
=============
In this section, we recall some definitions and results needed to describe the present tropical linear-fractional programming algorithm.
We start by introducing deterministic mean payoff games played on a finite bipartite digraph, see Subsection \[mpg-start\]. We then summarize some elements of the operator approach to such games, in including a theorem of [@Koh-80], Theorem \[t:inv-half\] below, implying that their dynamic programming operators $f$ have invariant half-lines $(\chi, v)$. These invariant half-lines determine the ultimate growth of the orbits $f^k(x)/k$, known as the cycle-time vector $\chi(f)$ of $f$. They also determine a pair of optimal strategies, and the winning nodes of the players ($\{i\mid \chi_i(f)\geq 0\}$ and $\{i\mid \chi_i(f)<0\}$) in the associated game.
Then, we recall in Subsection \[trop-mpg\] the correspondence between tropical polyhedra and mean payoff games, along the lines of [@AGG-10]: Theorem \[chi-axbx\] shows that $i$ is a winning node $(\chi_i(f)\geq 0)$ if, and only if, the associated tropical two-sided system of inequalities has a solution $x$ whose $i$th coordinate is finite $(x_i\neq -\infty)$. We also recall the max-plus and min-plus representations of min-max functions, together with the combinatorial characterization, in terms of maximal or minimal cycle means, of the cycle time vector for one-player games. This will be used to construct optimality and unboundedness certificates in Subsection \[ss:certificates\].
Mean payoff games and min-max functions {#mpg-start}
---------------------------------------
Consider a two-player deterministic game where the players, called “Max” and “Min”, make alternate moves of a pawn on a weighted bipartite digraph $\Bipdig$. The set of nodes of $\Bipdig$ is the disjoint union of the nodes $[m]$ where Max is active, and the nodes $[n]$ where Min is active. When the pawn is in node $k\in [m]$ of Max, he must choose an arc in $\Bipdig$ connecting node $k$ with some node $l\in [n]$ of Min, and while moving the pawn along this arc, he receives the weight $b_{kl}$ of the selected arc as payment from Min. When the pawn is in node $j\in [n]$ of Min, she must choose an arc in $\Bipdig$ connecting node $j$ with some node $i\in [m]$ of Max, and pays $-a_{ij}$ to Max, where $-a_{ij}$ is the weight of the selected arc. We assume that $b_{kl},a_{ij}\in\R$. Moreover, certain moves may be prohibited, meaning that the corresponding arcs are not present in $\Bipdig$. Then, we set $b_{kl}=-\infty$ and $a_{ij}=-\infty$. Thus, the whole game is equivalently defined by two $m\times n$ matrices $A=(a_{ij})$ and $B=(b_{kl})$ with entries in $\R\cup\{-\infty\}$. We make the following assumptions, which assure that both players have at least one move allowed in each node.
[**Assumption 1.**]{} For all $k\in[m]$ there exists $l\in [n]$ such that $b_{kl}\neq -\infty$.
[**Assumption 2.**]{} For all $j\in[n]$ there exists $i\in[m]$ such that $a_{ij}\neq -\infty$.
A [*general strategy*]{} for a player (Max or Min) is a function that for every finite history of a play ending at some node selects a successor of this node (i.e., a move of the player). A [*positional strategy*]{} for a player is a function that selects a unique successor of every node independently of the history of the play.
A strategy for player Max will be usually denoted by $\sigma$ and a strategy for player Min by $\tau$. Thus, a positional strategy for player Max is a function $\sigma\colon [m] \mapsto [n]$ such that $b_{i\sigma(i)}$ is finite for all $i\in[m]$, and a positional strategy for player Min is a function $\tau\colon [n] \mapsto [m]$ such that $a_{\tau(j)j}$ is finite for all $j\in [n]$.
When player Max reveals his positional strategy $\sigma$, the play proceeds within the digraph $\Bipdig^{\sigma}$ where at each node $i$ of Max all but one arc $(i,\sigma(i))$ are removed. When player Min reveals her positional strategy $\tau$, the play proceeds within the digraph $\Bipdig^{\tau}$ where at each node $j$ of Min all but one arc $(j,\tau(j))$ are removed.
When player Max reveals his positional strategy $\sigma$ and player Min her positional strategy $\tau$, the play proceeds within the digraph $\Bipdig^{\sigma,\tau}$ where each node has a unique outgoing arc $(i,\sigma(i))$ or $(j,\tau(j))$. Thus, $\Bipdig^{\sigma,\tau}$ is a “sunflower” digraph, i.e., such that each node has a unique path to a unique cycle.
The [*infinite horizon version*]{} of this mean payoff game starts at a node $j$ of Min[^2] and proceeds according to the strategies (not necessarily positional) of the players, who are interested in the value of average payment. More precisely, player Min wants to minimize $$\Phi_{A,B}^{\sup}(j,\tau,\sigma):=\limsup_{k\to\infty} \; (\sum_{t=1}^k -a_{i_t j_{t-1}}+b_{i_t j_t} )/k
\enspace, \quad j_0=j \enspace ,$$ while player Max wants to maximize $$\Phi_{A,B}^{\inf}(j,\tau,\sigma):=\liminf_{k\to\infty} \; (\sum_{t=1}^k -a_{i_t j_{t-1}}+b_{i_t j_t} )/k
\enspace, \quad j_0=j \enspace ,$$ where $j_1\in [n]$, $i_1\in [m]$, $j_2\in [n]$, $i_2\in[m]$, $\ldots$ is the infinite sequence of positions of the pawn resulting from the selected strategies $\tau $ and $\sigma $ of the players. The next theorem shows that this game has a value. An analogue of this theorem concerning stochastic games was obtained by [@liggettlippman].
\[value\] For the mean payoff game whose payments are given by the matrices $A,B\in (\R\cup\{-\infty\})^{m\times n}$, where $A,B$ satisfy Assumptions 1 and 2, there exists a vector $\chi\in\R^n$ and a pair of positional strategies $\sigma^*$ and $\tau^*$ such that
(i) $\Phi_{A,B}^{\sup}(j,\tau^*,\sigma)\leq \chi_j$ for all (not necessarily positional) strategies $\sigma$,
(ii) $\Phi_{A,B}^{\inf}(j,\tau,\sigma^*)\geq \chi_j$ for all (not necessarily positional) strategies $\tau$,
for all nodes $j$ of Min.
In other words, player Max has a positional strategy $\sigma^*$ which secures a mean profit of at least $\chi_j$ whatever is the strategy of player Min, and player Min has a positional strategy $\tau^*$ which secures a mean loss of no more than $\chi_j$ whatever player Max does.
A [*finite duration version*]{} of the mean payoff game considered above can be also formulated. Again, it starts at a certain node $j$ of Min and proceeds according to the strategies of the players (not necessarily positional), but stops immediately when a cycle $j_0\in [m] \rightarrow i_1\in [n] \rightarrow j_1\in [m]
\rightarrow \cdots \rightarrow i_k\in [n] \rightarrow j_k=j_0$ is formed. Then, the outcome of the game is the mean weight per turn (so the length of a cycle may be seen as the number of nodes of Max or Min it contains) of that cycle: $$\label{PhiAB}
\Phi_{A,B}(j,\tau,\sigma):=(\sum_{t=1}^k -a_{i_t j_{t-1}}+b_{i_t j_t})/k \; .$$ The ambition of Max is to maximize $\Phi_{A,B}(j,\tau,\sigma)$ while Min is seeking to minimize it.
It was shown in [@EM-79 Theorem 2] that this finite duration version of the game has the same value as the infinite horizon version described above, and that there are positional optimal strategies which secure this value for both versions of the game. It follows by standard arguments that $\chi_j$ is determined uniquely by $$\label{val-exist}
\chi_j=\Phi_{A,B}(j,\tau^*,\sigma^*)=
\min_{\tau}\max_{\sigma} \Phi_{A,B}(j,\tau,\sigma)=
\max_{\sigma}\min_{\tau} \Phi_{A,B}(j,\tau,\sigma) \; ,$$ where $\tau$ and $\sigma$ range over the sets of all strategies (not necessarily positional) for players Min and Max, respectively.
The dynamic programming operator $f: \R^n\mapsto\R^n$ associated with the infinite horizon mean payoff game is defined by: $$\label{minmax-dynamic}
f_j(x)={\mathop{\text{\Large$\wedge$}}}\limits_{k\in[m]}(-a_{kj}+{\mathop{\text{\Large$\vee$}}}\limits_{l\in[n]}(b_{kl}+x_l)) \enspace .$$ This function, combining min-plus and max-plus linearity (see below in Subsection \[trop-mpg\]), is known as a [*min-max function*]{} [@CGG-99]. Min-max functions are isotone ($x\leq y\Rightarrow f(x)\leq f(y)$) and additively homogeneous ($f(\lambda+x)=\lambda+f(x)$). Hence, they are nonexpansive in the sup-norm. Moreover, they are piecewise affine ($\R^n$ can be covered by a finite number of polyhedra on which $f$ is affine). We are interested in the following limit ([*cycle-time vector*]{}): $$\label{chi-limit}
\chi(f)=\lim_{k\to\infty} f^k(x)/k \enspace .$$ The $j$th entry of the vector $\chi(f)$ can be interpreted as the limit of the mean value of the game per turn, as the horizon $k$ tends to infinity, when the starting node is $j$. The existence of $\chi(f)$ follows from a theorem of Kohlberg.
\[t:inv-half\] Let $f:\R^n\mapsto\R^n$ be a nonexpansive and piecewise affine function. Then, there exist $v\in\R^n$ and $\chi\in\R^n$ such that $$f(v+t\chi)=v+(t+1)\chi\enspace,\quad \forall t\geq T \enspace ,$$ where $T$ is a large enough real number.
The function $t\mapsto v+t\chi$ is known as an [*invariant half-line*]{}. Using the nonexpansiveness of $f$, one deduces that the limit exists, is the same for all $x\in\R^n$ and is equal to the growth rate $\chi$ of any invariant half-line.
Given fixed positional strategies $\tau$ and $\sigma$ for players Min and Max, respectively, we can consider the dynamic operators corresponding to the partial digraphs $\Bipdig^{\tau}$ and $\Bipdig^{\sigma}$. These operators are max-only and min-only functions: $$\label{mm-only}
\begin{split}
f_j^{\tau}(x)&=-a_{\tau(j)j}+{\mathop{\text{\Large$\vee$}}}\limits_{l\in [n]} ( b_{\tau(j)l}+x_l ) \enspace ,\\
f_j^{\sigma}(x)&={\mathop{\text{\Large$\wedge$}}}\limits_{i\in [m]}(-a_{ij}+b_{i\sigma(i)}+x_{\sigma(i)})\enspace .
\end{split}$$ They are the main subject of tropical linear algebra, see Subsection \[trop-mpg\], where in particular we recall how their cycle-time vectors can be computed. Theorem \[e:chi-duality-games\] below relates these cycle-time vectors with the cycle time vector of the min-max function .
The following result can be derived as a standard corollary of Kohlberg’s theorem. Indeed, we define a positional strategy $\tau$ of Min and a positional strategy $\sigma$ of Max by the condition that $f(v+t\chi)=f^\sigma(v+t\chi)=f^\tau(v+t\chi)$ for $t$ large enough, where $t\mapsto v+t\chi$ is an invariant half-line. These strategies are easily seen to be optimal for the mean payoff game.
\[value-cycletime\] For $f(x)$ given by , the $j$th coordinate of $\chi(f)$ is the value of the mean payoff game which starts at node $j$ of Min.
In what follows, we shall use the following form of the value existence result , which was proved in [@gg0] as a corollary of the termination of the policy iteration algorithm of [@gg0; @CGG-99], see [@DG-06] for a more recent presentation. Alternatively, it can be quickly derived from [@Koh-80] (the derivation can be found in [@AGG-10 Theorem 2.13]). This result has been known as the “duality theorem” in the discrete event systems literature.
\[chi-duality\] Let $A,B\in (\R\cup\{-\infty\})^{m\times n}$ satisfy Assumptions 1 and 2, and let $S$ and $T$ be the sets of all positional strategies for players Max and Min, respectively. Then, $$\label{e:chi-duality-games}
\max\limits_{\sigma\in S} \chi(f^{\sigma})=\chi(f)=
\min\limits_{\tau\in T} \chi(f^{\tau}) \enspace .$$
This characterization of $\chi(f)$ should be compared with . The latter shows that the infinite horizon version of the game with limsup/liminf payoff has a value, whereas concerns the limit of the value of the finite horizon version. Thus, in loose terms, the “limit” and “value” operations commute.
Tropical linear systems and mean payoff games {#trop-mpg}
---------------------------------------------
Max-only and min-only functions of the form belong to tropical linear algebra. Max-only functions are linear in the [*max-plus semiring*]{} $\Rmax$, which is the set $\R\cup\{-\infty\}$ equipped with the operations of “addition” $\ltr a+b\rtr:=a\vee b$ and “multiplication” $\ltr ab\rtr:=a+b$. For min-only functions, we use the [*min-plus semiring*]{} $\Rmin$, i.e. the set $\R\cup\{+\infty\}$ equipped with the operations of “addition” $\ltr a+b\rtr:=a\wedge b$ and the same “additive” multiplication. The setting in which both structures are considered simultaneously has been called minimax algebra in [@CG:79]. Then, we need to allow the scalars to belong to the enlarged set $\RRbar:=\R\cup\{-\infty\}\cup \{+\infty\}$. Note that in $\RRbar$, $(-\infty)+(+\infty)=-\infty$ if the max-plus convention is considered and $(-\infty)+(+\infty)=+\infty$ if the min-plus convention is considered.
The tropical operations are extended to matrices and vectors in the usual way. In particular, for any matrix $E=(e_{ij})$ and any vector $x$ of compatible dimensions: $$\label{DefMult}
\ltr (Ex)_{i}\rtr={\mathop{\text{\Large$\vee$}}}_j e_{ij}+x_j\quad \text{(max-plus)}\; , \quad
\ltr (Ex)_{i}\rtr={\mathop{\text{\Large$\wedge$}}}_j e_{ij}+x_j\quad \text{(min-plus)}\; .$$ Max-plus and min-plus linear functions are mutually adjoint, or [*residuated*]{}. Recall that for a max-plus linear function from $\RRbar^n$ to $\RRbar^m$, given by $E\in \Rmax^{m\times n}$, the [*residuated operator*]{} $E^{\diez}$ from $\RRbar^m$ to $\RRbar^n$ is defined by $$\label{adiez}
(E^{\diez} y)_j:={\mathop{\text{\Large$\wedge$}}}_{i\in [m]} (-e_{ij}+y_i)\enspace ,$$ with the convention $(-\infty)+(+\infty)=+\infty$. Note that this residuated operator, also known as [*Cuninghame-Green inverse*]{}, is given by the multiplication of $-E^T$ by $y$ with the min-plus operations (here $E^T$ denotes the transposed of $E$), and that it sends $\RR^m$ to $\RR^n$ whenever $E$ does not have columns identically equal to $-\infty$.
In what follows, concatenations such as $Ex$ should be understood as the multiplication of $E$ by $x$ with the max-plus operations, and concatenations such as $E^{\diez}y$ should be understood as the multiplication of $-E^T$ by $y$ with the min-plus operations (and the corresponding conventions for $(-\infty)+(+\infty)$).
The term “residuated” refers to the property $$\label{res-prop}
Ex\leq y\Leftrightarrow
x\leq E^{\diez} y\enspace ,$$ where $\leq$ is the partial order on $\RR^m$ or $\RR^n$, which can be deduced from $$e_{ij} + x_j \leq y_i \; \forall i, j
\Leftrightarrow
x_j \leq -e_{ij} +y_i \; \forall i, j \enspace .$$ As a consequence, the residuated operator is crucial for max-plus two-sided systems of inequalities, because implies: $$\label{ineqs-equiv}
Ax\leq Bx\Leftrightarrow x\leq A^{\sharp}Bx \enspace .$$ Writing the last inequality explicitly, we have $$\label{ineqs-expl}
x_j\leq{\mathop{\text{\Large$\wedge$}}}_{k\in [m]}(-a_{kj}+{\mathop{\text{\Large$\vee$}}}_{l\in [n]}(b_{kl}+x_l))
\enspace , \quad\forall j\in [n]\enspace .$$ Thus, we obtain the same min-max function as in .
Moreover, positional strategies $\sigma\colon [m] \mapsto [n]$ and $\tau\colon [n]\mapsto [m]$ correspond to affine functions $B^{\sigma}$ and $A_{\tau}$ defined by $$\label{ataubsigma}
(A_{\tau})_{ij}=
\begin{cases}
a_{ij} \; &\makebox{ if } i=\tau(j) ,\\
-\infty \; &\makebox { otherwise} ,
\end{cases}\quad
(B^{\sigma})_{ij}=
\begin{cases}
b_{ij}\; &\makebox{ if } j=\sigma(i) ,\\
-\infty \; &\makebox { otherwise} .
\end{cases}$$
Recasting in max(min)-plus algebra, we obtain $$\label{e:chi-duality}
\max\limits_{\sigma\in S}\chi(A^{\sharp}B^{\sigma}) = \chi(A^{\sharp}B) =
\min\limits_{\tau\in T}\chi(A^{\sharp}_{\tau}B) \enspace .$$
The following result, obtained by Akian, Gaubert and Guterman, relates solutions of $Ax\leq Bx$ and nonnegative coordinates of $\chi(A^{\sharp}B)$. These coordinates correspond to [*winning nodes*]{} of the mean payoff game: if the game starts in these nodes, then Max can ensure nonnegative profit with any strategy of Min.
\[chi-axbx\] Let $A,B\in\Rmax^{m\times n}$ satisfy Assumptions 1 and 2. Then, $\chi_i(A^{\sharp}B)\geq 0$ if and only if there exists $x\in\Rmax^n$ such that $Ax\leq Bx$ and $x_i\neq -\infty$.
This is derived in [@AGG-10] from Kohlberg’s theorem. The vector $x$ is constructed by taking an invariant half-line, $t\mapsto v+t\chi$, setting $x_i=v_i+t\chi_i$ for $t$ large enough if $\chi_i\geq 0$, and $x_i=-\infty$ otherwise.
Theorem \[chi-axbx\] shows that to decide whether $Ax\leq Bx$ can be satisfied by a vector $x$ such that $x_i\neq -\infty$, we can exploit a [*mean payoff oracle*]{}, which decides whether $i$ is a winning node of the associated mean payoff game and gives a winning strategy for player Max. This oracle can be implemented either by using the value iteration method, which is pseudo-polynomial [@ZP-96], by the approach of Puri (solving an associated discounted game for a discount factor close enough to $1$ by policy iteration [@puri]), by using the policy iteration algorithm for mean payoff games of [@CGG-99; @gg0; @DG-06], or the one of [@bjorklund].
In tropical linear algebra, there is no obvious subtraction. However, for any $E\in\RRbar^{n\times n}$ we can define the [*Kleene star*]{} $$\label{kls-def}
E^*:=\ltr(I-E)^{-1}\rtr=I\vee E\vee E^2\vee\cdots\quad \text{(max-plus)}\; ,$$ and analogously with $\wedge$ in the min-plus case. In , $I$ is the max-plus identity matrix with $0$ entries on the main diagonal and $-\infty$ off the diagonal, and the powers are understood in the tropical (max-plus) sense. Due to the order completeness of $\RRbar$, the series in is well-defined for all matrices. Note that in $\RRbar^{n\times n}$, $X=E^*$ is a solution of the matrix Bellman equation $X=E X\vee I$. Similarly, $x=E^*h$ is a solution of $x=Ex\vee h$. Indeed, if $z\geq Ez\vee h$, then we also have $$z\geq Ez\vee h \geq E^2 z\vee E h\vee h\geq\cdots
\geq E^{k+1}z\vee E^kh\vee E^{k-1}h\vee\cdots\vee h \enspace ,$$ so that $z\geq E^*h$ for all such $z$. We sum this up in the following standard proposition.
\[bellman\] Let $E\in\RRbar^{n\times n}$ and $h\in\RRbar^n$. Then, $E^*h$ is the least solution of $z\geq Ez\vee h$.
For $E\in\RRbar^{n\times n}$, consider the associated digraph $\digr(E)$, with set of nodes $[n]$ and an arc connecting node $i$ with node $j$ whenever $e_{ij}$ is finite, in which case $e_{ij}$ is the weight of this arc. We shall say that [*node $i$ accesses node $j$*]{} if there exists a path from $i$ to $j$ in $\digr(E)$.
The maximal (minimal) cycle mean is another important object of tropical algebra. For $E\in \Rmax^{n\times n}$ ($E\in \Rmin^{n\times n}$), it is defined as $$\label{mcm-def}
\begin{split}
\mu^{\max}(E)&=\max_{k\in [n]}\; \max_{i_1,\ldots ,i_k} \;
\frac{e_{i_1 i_2} + \cdots + e_{i_k i_1}}{k} \quad\text{(max-plus)}\enspace , \\
\mu^{\min}(E)&=\min_{k\in [n]} \; \min_{i_1,\ldots,i_k} \;
\frac{e_{i_1 i_2} + \cdots + e_{i_k i_1}}{k} \quad\text{(min-plus)}\enspace .
\end{split}$$ Denote by $\mu_i^{\max}(E)$ ($\mu_i^{\min}(E)$) the maximal (minimal) cycle mean of the strongly connected component of $\digr(E)$ to which $i$ belongs. These numbers are given by the same expressions as in , but with $i_1,\ldots ,i_k$ restricted to that strongly connected component. Using $\mu^{\max}_i(E)$ ($\mu^{\min}_i(E)$), we can write explicit expressions for the cycle-time vector of a max-plus (min-plus) linear function $x\mapsto Ex$: $$\label{chi-expl}
\begin{split}
&\chi_i^{\max}(E)=\max\{\mu_j^{\max}(E)\mid \text{$i$ accesses $j$}\}\quad \text{(max-plus)},\\
&\chi_i^{\min}(E)=\min\{\mu_j^{\min}(E)\mid \text{$i$ accesses $j$}\}\quad \text{(min-plus)}.
\end{split}$$ See [@Coc-98] or [@HOW:05] for proofs. Importantly, these cycle-time vectors of max-plus and min-plus linear functions appear in .
Finally, note that and can be deduced from if $\sigma$ or $\tau$ is fixed.
\[r:shortestpaths\] Observe that any entry $(i,j)$ of $E^k$ in max-plus (resp. min-plus) algebra expresses the maximal (resp. minimal) weight of paths with $k$ arcs connecting node $i$ with node $j$ in $\digr(E)$. It follows then from that any entry $(i,j)$ of $E^*$, for $i\neq j$, expresses the maximal (or minimal) weight of paths connecting node $i$ with node $j$ without restrictions on the number of arcs. Further we can add to $\digr(E)$ a new node and, whenever $b_i$ if finite, an arc of weight $b_i$ connecting node $i$ of $\digr(E)$ with this new node. Then, $(E^*b)_i$ provides the maximal (or minimal) weight of paths connecting node $i$ with the new node. Therefore, computing $E^*b$ is equivalent to solving a single destination shortest path problem, which can be done in $O(n^3)$ time (for instance by the Bellman-Ford algorithm).
\[r:chi-karp\] We note that $\chi^{\max}(E)$ and $\chi^{\min}(E)$ can also be computed in $O(n^3)$ time. To do this, decompose first the digraph $\digr(E)$ in strongly connected components, and apply Karp’s algorithm to compute the maximal or minimal cycle mean of each component.
Tropical linear-fractional programming
======================================
This is the main section of the paper. Here we solve the tropical linear-fractional programming problem , i.e. the problem $$\label{mainproblem}
\begin{split}
&\text{minimize } \quad (px\vee r) - (qx\vee s) \\
&\text{subject to:}\quad Ax\vee c\leq Bx\vee d \; ,\;
x\in \Rmax^n
\end{split}$$ where $p,q\in\Rmax^n$, $c,d\in\Rmax^m$, $r,s\in\Rmax$ and $A,B\in\Rmax^{m\times n}$.
In Subsection \[ss:Pformulations\], we apply Theorem \[chi-axbx\] to reduce to the problem of finding the smallest zero of a function giving the value of a parametric game (the [*spectral function*]{}).
In Subsection \[ss:specf\], we show that the spectral function is $1$-Lipschitz and piecewise linear, and that it can be written as a finite supremum or infimum of [*partial*]{} spectral functions, corresponding to one player games. We also prove a number of technical statements about the piecewise-linear structure of the spectral functions, which will be used in the complexity analysis.
In Subsection \[ss:certificates\], we provide certificates of optimality and unboundedness (these certificates are given by strategies for the players). This generalizes the result of [@AGK-10], concerning the tropical analogue of Farkas lemma. We recover as a special case the unboundedness certificates of [@BA-08].
The rest of the section is devoted to finding the least zero of the spectral function. With this aim, we introduce a bisection method, as well as a Newton-type method, in which partial spectral functions play the role of derivatives, see Subsection \[ss:bisnewt\].
Each Newton iteration consists of
(i) Computing a derivative, i.e. choosing a strategy for player Max (or dually Min) which satisfies a local optimality condition;
(ii) Finding the smallest zero of the tangent map, which represents the parametric spectral function of a one-player game in which the strategy for player Max (or dually Min) is already fixed.
The iteration in the space of strategies for player Max has an advantage: the second subproblem can be reduced to a shortest-path problem (Subsection \[ss:kleene\]). The first subproblem is discussed in Subsection \[ss:left-optim\], where the overall worst-case complexity of Newton method is given. Subsection \[ss:germs\], which can be skipped by the reader, gives an alternative approach to the first subproblem in which the computation of (left) optimal strategies is rather algebraic and not relying on the integrality.
The spectral function method {#ss:Pformulations}
----------------------------
In this subsection we recast as a parametric two-sided tropical system and a mean payoff game, introducing the key concept of spectral function. However, before doing this we need to mention special cases in which there exists a feasible $x$ (i.e., satisfying $Ax\vee c\leq Bx\vee d$) such that $px\vee r= -\infty$ or $qx\vee s= -\infty$. For these cases we assume the following rules: $$\label{infty-rules}
\begin{array}{c|c|c}
px\vee r & qx\vee s & (px\vee r) - (qx\vee s) \\
\hline
-\infty & \text{finite} & -\infty \\
\text{finite} & -\infty & +\infty \\
-\infty & -\infty & -\infty
\end{array}$$ which are formally consistent with the rules of $\RRmax$. Then, it is easy to check that $$\label{resid-obs}
(px\vee r) - (qx\vee s) = \min\{\lambda\in \Rmax \mid px\vee r\leq \lambda + (qx\vee s)\} \; .$$
Introducing the notation $$\label{UVnotat}
U=
\begin{pmatrix}
A & c\\
p & r
\end{pmatrix}\; \makebox{ and } \;
V(\lambda )=
\begin{pmatrix}
B & d\\
\lambda +q & \lambda+s
\end{pmatrix},$$ we reformulate the tropical linear-fractional programming problem in terms of a spectral function, which gives the value of a parametric mean payoff game: the payments are given by the matrices $U$ and $V(\lambda)$, and the initial node is $n+1$.
\[pd-specf\] With assumption , the tropical linear-fractional programming problem is equivalent to $$\label{problem-simple}
\min\{\lambda \in\Rmax\mid {\phi}(\lambda)\geq 0\}$$ where the [*spectral function*]{} ${\phi}$ is given by $${\phi}(\lambda):=\chi_{n+1}(U^{\sharp}V(\lambda)) \enspace .$$
We first show that is equivalent to the following problem: $$\label{problem-sets}
\begin{split}
&\text{minimize } \quad \lambda \\
&\text{subject to:}\quad px\vee r\leq \lambda+(qx\vee s)\; ,\;
Ax\vee c\leq Bx\vee d \; , \; x\in\Rmax^n \; ,\; \lambda \in\Rmax
\end{split}$$ Indeed, denoting $P=\left\{x\in \Rmax^n \mid Ax\vee c\leq Bx\vee d\right\}$, we verify that $$\begin{split}
\min\limits_{x\in P} \{ (px\vee r) - (qx\vee s)\}=
&\min\limits_{x\in P}\min\limits_{\lambda}
\{\lambda\mid px\vee r\leq \lambda + (qx\vee s)\}\\
= &\min\limits_{\lambda}\{\exists x\in P\mid px\vee r\leq \lambda +
(qx\vee s)\} \; .
\end{split}$$
Every problem concerning affine polyhedra has an equivalent “homogeneous” version concerning cones, which is obtained by adding to the system of inequalities defining an affine polyhedron a new variable whose coefficients are the free terms of this system. Then, the original polyhedron is recovered by setting this new variable to $0$. The homogeneous equivalent version of reads: $$\label{problem-cones}
\begin{split}
&\text{minimize } \quad \lambda \\
&\text{subject to:}\quad u y\leq \lambda + v y\; ,
\; C y\leq D y\; , \; y_{n+1}\neq -\infty \; , \; y\in\Rmax^{n+1} \; ,
\; \lambda \in\Rmax
\end{split}$$ where we set $u=[p, r]$, $v=[q, s]$, $C=[A,c]$ and $D=[B, d]$.
We can still reformulate in a more compact way: $$\min \{ \lambda \in\Rmax\mid U y\leq V(\lambda )y\; ,
\; y_{n+1}\neq -\infty\; \text{ is solvable}\} \; ,$$ with $U$ and $V(\lambda)$ defined in . Finally, by Theorem \[chi-axbx\], it follows that $Uy\leq V(\lambda )y$ is solvable with finite $y_{n+1}$ if, and only if, $\chi_{n+1}(U^{\sharp}V(\lambda ))\geq 0$.
Butkovič and Aminu [@BA-08] considered the following special cases of : $$\label{problem-straight}
\begin{split}
&\text{minimize } \quad px \quad (\text{resp.\ maximize } \quad qx)\\
&\text{subject to:}\quad Ax\vee c\leq Bx\vee d \; , \; x\in\R^n
\end{split}$$ where $p,q\in\R^n$, $c,d\in\R^m$, $r,s\in\R$ and $A,B\in\R^{m\times n}$ have only finite entries. Clearly, becomes if we set $r=-\infty$, $q\equiv -\infty$ and $s=0$ for minimization, or respectively $s=-\infty$, $p\equiv -\infty$ and $r=0$ for maximization, where the opposite (tropical inverse) of the minimal value of $\lambda$ equals the maximum of $qx$.
In this connection, formulation (or equivalently ) has a good geometric insight, meaning optimization for general tropical half-spaces (or hyperplanes) defined by bivectors $(u,\lambda +v)$, see [@GK-09; @AGK-10] for more background.
\[Example1\] Assume we want to maximize $(1+x_1)\vee (3+x_2)$ over the tropical polyhedron of $\Rmax^2$ defined by the system $Ax\vee c\leq Bx\vee d$, where $$A=
\left(\begin{array}{cc}
-\infty & -1 \\
-2 & -2 \\
-1 & -\infty \\
0 & -\infty
\end{array}\right) \; ,\quad
c=
\left(\begin{array}{c}
-\infty\\
-\infty\\
-\infty\\
-\infty
\end{array}\right) \; ,\quad
B=
\left(\begin{array}{cc}
0 & -\infty \\
-\infty & -\infty \\
-\infty & 0 \\
-\infty & 2
\end{array}\right) \; ,\quad
d=
\left(\begin{array}{c}
0\\
0\\
0\\
0
\end{array}\right) \; .$$ This tropical polyhedron is displayed on the left-hand side of Figure \[fig-max\] below. This maximization problem is equivalent to minimizing $\lambda$ subject to $0\leq \lambda + ((1+x_1)\vee (3+x_2))$, $Ax\vee c\leq Bx\vee d$. Indeed, the value of the latter problem is the opposite (tropical inverse) of the value of the maximization problem. The homogeneous version of this minimization problem reads: $$\label{EqProbMaxExample}
\begin{split}
&\text{minimize } \quad \lambda \\
&\text{subject to:}\quad u y\leq \lambda + v y\; ,
\; C y\leq D y\; , \; y_{3}\neq -\infty \; , \; y\in\Rmax^{3} \; ,
\; \lambda \in\Rmax
\end{split}$$ where $C=[A,c]$, $D=[B, d]$, $u=(-\infty,-\infty,0)$, and $v=(1,3,-\infty)$.
Partial spectral functions and piecewise linearity {#ss:specf}
--------------------------------------------------
We have shown that solving the tropical linear-fractional programming problem is equivalent to finding the least zero of the spectral function ${\phi}(\lambda):=\chi_{n+1}(U^{\sharp}V(\lambda))$. Here we will analyze the graph of this spectral function, after introducing analogues of derivatives, the partial spectral functions.
Given a strategy $\sigma\in S$ for player Max and a strategy $\tau\in T$ for player Min, we respectively define the min-plus linear function $U^{\sharp}V^{\sigma}(\lambda)$ and the max-plus linear function $U_{\tau}^{\sharp}V(\lambda)$, see and . We introduce the [*partial spectral functions*]{} ${\phi}^{\sigma}(\lambda):=\chi_{n+1}(U^{\sharp}V^{\sigma}(\lambda))$ and ${\phi}_{\tau}(\lambda):=\chi_{n+1}(U^{\sharp}_{\tau}V(\lambda))$. With this notation, yields $$\label{sl-duality}
{\phi}(\lambda)=\max\limits_{\sigma\in S} {\phi}^{\sigma}(\lambda)=
\min\limits_{\tau\in T} {\phi}_{\tau}(\lambda) \; .$$ Partial spectral functions can be represented as in , where one of the strategies is fixed, see also : $$\label{phi-Phi}
\phi^{\sigma}(\lambda)=\min_{\tau\in T}\Phi_{U,V(\lambda)}(n+1,\tau,\sigma)\; ,
\quad
\phi_{\tau}(\lambda)=\max_{\sigma\in S}\Phi_{U,V(\lambda)}(n+1,\tau,\sigma)\; .$$
A graphical presentation of and using only ${\phi}_{\tau}$ is given in Figure \[FigureSpecFunction\].
Let $\Bipdig_\lambda $ be the bipartite digraph of the mean payoff game whose payments are given by the matrices $U$ and $V(\lambda )$, see Figures \[BipDigrapfExMin\] and \[FigureBipDigraGen\] below for illustrations. Observe that in this digraph, only the weight of the arcs connecting node $m+1$ of Max with nodes $l\in [n+1]$ of Min depend on $\lambda $. Recall that given a strategy $\sigma$ for player Max (resp. $\tau $ for player Min), $\Bipdig^{\sigma}_\lambda $ (resp. $\Bipdig^{\tau }_\lambda $) denotes the bipartite subdigraph of $\Bipdig_\lambda $ obtained by deleting from $\Bipdig_\lambda $ all the arcs $(i,j)$ such that $i\in[m+1]$ and $j\neq\sigma(i)$ (resp. the arcs $(j,i)$ such that $j\in [n+1]$ and $i\neq\tau (j)$).
We next investigate the properties of spectral functions.
\[t:philambda\] Let $\sigma\colon [m+1]\mapsto [n+1]$ be a positional strategy for player Max and $\tau\colon [n+1]\mapsto [m+1]$ be a positional strategy for player Min. Then,
(i) \[t:philambdaP1\] ${\phi}(\lambda)$, ${\phi}_{\tau}(\lambda)$ and ${\phi}^{\sigma}(\lambda)$ are $1$-Lipschitz nondecreasing piecewise-linear functions, whose linear pieces are of the form $(\alpha+\beta\lambda)/k$, where $k\in [\min(m,n)+1]$ and $\beta\in\{0,1\}$.
(ii) \[t:philambdaP2\] If the absolute values of all the finite coefficients in are bounded by $M$, then $|\alpha/k|\leq 2M$.
(iii) \[t:philambdaP3\] ${\phi}_{\tau}(\lambda)$ is convex and ${\phi}^{\sigma}(\lambda)$ is concave. Both functions consist of no more than $\min(m,n)+2$ linear pieces.
As follows from and , spectral functions are built from the finite number of functions $\lambda \mapsto \Phi_{U,V(\lambda)}(n+1,\sigma,\tau)$, each of which is given by the mean weight per turn of the only elementary cycle in $\Bipdig^{\sigma,\tau}_\lambda$ accessible from node $n+1$ of Min. Recall that $\Bipdig^{\sigma,\tau}_\lambda$ is the subdigraph of $\Bipdig_\lambda$ where all arcs at all nodes except for those chosen by player Max (strategy $\sigma$) and player Min (strategy $\tau$) are removed. As a function of $\lambda$, this mean weight per turn is a line $(\alpha+\beta\lambda)/k$. Here $k\in [\min(m,n)+1]$ since the length (i.e., the number of nodes of Max it contains) of any elementary cycle in the bipartite digraph $\Bipdig_\lambda$ does not exceed both $m+1$ and $n+1$. Also $\beta\in\{0,1\}$, because an elementary cycle can contain node $m+1$ of Max no more than once. If the absolute values of all the payments in the game are bounded by $M$, then $|\alpha/k|\leq 2M$ since the arithmetic mean of payments, counted per turn, does not exceed the greatest sum of two consecutive payments.
Thus, the functions $\lambda \mapsto \Phi_{U,V(\lambda)}(n+1,\sigma,\tau)$ satisfy all the properties of and . Using and we conclude that ${\phi}(\lambda)$, ${\phi}_{\tau}(\lambda)$ and ${\phi}^{\sigma}(\lambda)$ also satisfy these properties.
Convexity (resp. concavity) of ${\phi}_{\tau}$ (resp. ${\phi}^{\sigma}$) follows from . In a convex or concave piecewise-linear function, each slope can appear only once, while the possible slopes are $0$, $1$, $1/2$,$\ldots$, $1/(\min(m,n)+1)$. This shows .
Some useful facts can be deduced further from this description of spectral functions.
\[c:philambda\] Spectral functions satisfy the following properties:
(i) \[c:philambdaP1\] If the absolute values of all coefficients in are either infinite or bounded by $M$, then ${\phi}(\lambda),{\phi}^{\sigma}(\lambda)$ and ${\phi}_{\tau}(\lambda)$ are linear for $\lambda\leq -4M(\min(m,n)+1)^2$ and for $\lambda\geq 4M(\min(m,n)+1)^2$.
(ii) \[c:philambdaP2\] If the absolute values of all coefficients in are either infinite or bounded by $M$, then the solutions to the problems $\min\{\lambda \mid {\phi}(\lambda)\geq 0\}$, $\min\{\lambda \mid {\phi}^{\sigma}(\lambda)\geq 0\}$ and $\min\{\lambda \mid {\phi}_{\tau}(\lambda)\geq 0\}$ lie (if finite) in $[-2M(\min(m,n)+1),2M(\min(m,n)+1)]$. Moreover, if all the finite coefficients are integers, then the solutions to all these problems are integers as well.
(iii) \[c:philambdaP3\] If the finite coefficients in are integers, then the breaking points of ${\phi}(\lambda)$, ${\phi}^{\sigma}(\lambda)$ or ${\phi}_{\tau}(\lambda)$ are rational numbers whose denominators do not exceed $\min(m,n)+1$.
(iv) \[c:philambdaP4\] If the finite coefficients in are integers with absolute values bounded by $M$, then ${\phi}(\lambda)$ consists of no more than $8M(\min(m,n)+1)^4+2$ linear pieces.
Consider the intersection point $\mu$ of one linear piece $(\alpha_1+\beta_1\lambda)/k_1$ with another linear piece $(\alpha_2+\beta_2\lambda)/k_2$. By Theorem \[t:philambda\], $k_1,k_2\leq\min(m,n)+1$ and $| \alpha_1/k_1 | , | \alpha_2/k_2 | \leq 2M$, and we obtain from $$| \beta_1/k_1-\beta_2/k_2 | \geq \frac{1}{(\min(m,n)+1)^2} \; ,
\quad | \alpha_1/k_1-\alpha_2/ k_2| \leq 4M \; ,$$ that $|\mu |\leq 4M(\min(m,n)+1)^2$. This means that ${\phi}(\lambda)$ is linear for $\lambda\geq 4M(\min(m,n)+1)^2$ and $\lambda\leq -4M(\min(m,n)+1)^2$. (Note that this part did not impose the integrality of coefficients.)
Note that due to piecewise-linearity, the solution to each of these problems (if finite) is given by the intersection point of a certain linear piece of the form $(\alpha+\lambda)/k$ with zero. Then, since $|\alpha/k|\leq 2M$ and $k\leq\min(m,n)+1$ by Theorem \[t:philambda\], we conclude that this intersection point $-\alpha$ lies in $[-2M(\min(m,n)+1),2M(\min(m,n)+1)]$. Moreover, if the finite coefficients in are integers, then this solution $-\alpha$ is also integer.
By Theorem \[t:philambda\], spectral functions are piecewise linear and the linear pieces are of the form $(\alpha+\beta\lambda)/k$, where in particular $k\in [\min(m,n)+1]$ and $\beta\in\{0,1\}$. Considering the intersection point $\mu $ of one such piece $(\alpha_1+\beta_1\lambda)/k_1$ with another piece $(\alpha_2+\beta_2\lambda)/k_2$ and assuming the integrity of $\alpha_1,\alpha_2$ we obtain that $\mu =(k_1 \alpha_2-k_2\alpha_1)/(k_2 \beta_1-k_1\beta_2)$ is a rational number with denominator not exceeding $\min(m,n)+1$.
The denominators of breaking points do not exceed $\min(m,n)+1$, and hence the difference between their inverses is not less than $1/(\min(m,n)+1)^2$. This is a lower bound for the difference between two consecutive breaking points. We get the claim applying part .
Note that to determine the slope of $\phi(\lambda)$ at $+\infty$, meaning for $\lambda\geq 4M(\min(m,n)+1)^2$, or at $-\infty$, meaning for $\lambda\leq -4M(\min(m,n)+1)^2$, we can set all the finite coefficients in to $0$. Then, we “play” the mean payoff game at $\lambda=1$ or at $\lambda=-1$, respectively.
Denote by $\MPGinteger(m,n,M)$ the worst-case complexity of an oracle computing the [*value*]{} of mean payoff games with integer payments whose absolute values are bounded by $M$, with $m$ nodes of Max and $n$ nodes of Min. There exist pseudo-polynomial algorithms computing the value of mean payoff games. For instance, in [@ZP-96] the authors describe a value iteration algorithm with $O(mn^4M)$ complexity. Using this we now show that all the linear pieces of a spectral function can be identified in pseudo-polynomial time. Note that we do not require the oracle to compute optimal strategies here.
\[p:reconstr\] Let all the finite coefficients in be integer with absolute values not exceeding $M$. Then, all the linear pieces that constitute the graph of ${\phi}(\lambda)$ can be identified in $$O(M\min(m,n)^4)\times \MPGinteger(m+1,n+1,M(\min(m,n)+1)(1+4(\min(m,n)+1)^2))$$ operations.
By Corollary \[c:philambda\] part , the breaking points of ${\phi}(\lambda)$ are rational numbers whose denominators do not exceed $\min(m,n)+1$. To identify the linear pieces that constitute the graph of the spectral function, we only need to evaluate ${\phi}(\lambda)$ on such rational points in the interval $[-4M(\min(m,n)+1)^2, 4M(\min(m,n)+1)^2 ]$, the number of which does not exceed $O(M\min(m,n)^4)$.
Further, when computing ${\phi}(\lambda)$, the payments in the mean payoff games that the oracle works with are either $a$ or $a+\lambda$, where $a$ is an integer satisfying $|a|\leq M$ and $\lambda$ is a rational number in $[-4M(\min(m,n)+1)^2, 4M(\min(m,n)+1)^2 ]$ whose denominator does not exceed $\min(m,n)+1$. The properties of the game will not change if we multiply all the payments by this denominator, obtaining a new game in which the payments are integers with absolute values bounded by $(\min(m,n)+1)(M+4M(\min(m,n)+1)^2)$. Then, the complexity of the mean payoff oracle will not exceed $\MPGinteger(m+1,n+1,(\min(m,n)+1)(M+4M(\min(m,n)+1)^2))$. Multiplying by $O(M\min(m,n)^4)$ we get the claim.
\[r:pseudopol\] It follows that the tropical linear-fractional programming problem can be solved in pseudo-polynomial time by reconstructing all the linear pieces that constitute the graph of ${\phi}(\lambda)$. However, more efficient methods will be described in Subsection \[ss:bisnewt\].
\[r:parmpg\] A similar spectral function has been introduced in [@GStwosided10] to compute the set of solutions $\lambda$ of the two-sided eigenproblem $Ax=\lambda Bx$. The present approach can be extended to a larger class of parametric games, in which the payments are piecewise affine functions of the parameter $\lambda$, with integer slopes. See [@Ser-lastdep].
Strategies as certificates {#ss:certificates}
--------------------------
In the classical simplex method, the optimality of a feasible solution is certified by the sign of Lagrange multipliers. In the tropical case, following the idea of [@AGK-10], we shall show that the certificate is of a different nature: it is a strategy. We shall also use such strategies to guide the next iteration of Newton method in Subsection \[ss:bisnewt\], when the current feasible solution is not optimal.
\[d:lropt\] A strategy $\sigma-$ for player Max (resp. $\tau-$ for player Min) is [*left optimal*]{} at $\lambda\in\R$, if there exists $\epsilon>0$ such that $${\phi}(\mu )={\phi}^{\sigma-}(\mu )
\qquad (\makebox{resp.\ }{\phi}(\mu ) ={\phi}_{\tau-}(\mu))
\qquad \forall \mu \in [\lambda -\epsilon ,\lambda ] \enspace.$$ Right optimal strategies $\sigma+$ and $\tau+$ are defined in a similar way, replacing $[\lambda -\epsilon ,\lambda ]$ by $[\lambda ,\lambda +\epsilon ]$.
The existence of left and right optimal strategies at each point follows readily from , together with the finiteness of the number of strategies and the piecewise affine character of each function $\phi^\sigma(\lambda)$ and $\phi_\tau(\lambda)$.
\[1st-cert\] The tropical linear-fractional programming problem has the optimal value $\lambda^\ast \in \R$ if, and only if, ${\phi}(\lambda^\ast)\geq 0$ and there exists a strategy $\tau $ for player Min such that the digraph $\Bipdig_{\lambda^\ast }^{\tau}$ satisfies the following conditions:
(i) all cycles accessible from node $n+1$ of Min have nonpositive weight,
(ii) any cycle of zero weight accessible from node $n+1$ of Min passes through node $m+1$ of Max.
Moreover, these conditions are always satisfied when $\tau$ is left optimal at $\lambda^\ast $.
The tropical linear-fractional programming problem has the optimal value $\lambda^\ast $ if, and only if, ${\phi}(\lambda^\ast )=0$ and ${\phi}(\lambda)<0$ for all $\lambda <\lambda^\ast $. If $\tau$ is any left optimal strategy at $\lambda^*$, then the previous conditions are satisfied if, and only if, ${\phi}_{\tau}(\lambda^\ast )=0$ and ${\phi}_{\tau}(\lambda)$ has nonzero left derivative at $\lambda^*$.
By , or and , we know that ${\phi}_{\tau}(\lambda)$ is the maximal cycle mean (per turn) over all cycles in $\Bipdig^{\tau}_\lambda$ accessible from node $n+1$ of Min. It follows that ${\phi}_{\tau}(\lambda^\ast )=0$ if, and only if, all cycles in $\Bipdig^{\tau}_{\lambda^\ast }$ accessible from node $n+1$ of Min have nonpositive weight and at least one of them has zero weight. Moreover, ${\phi}_{\tau}(\lambda )$ has nonzero left derivative at $\lambda^\ast $ if, and only if, any zero-weight cycle in $\Bipdig^{\tau}_{\lambda^\ast }$ accessible from node $n+1$ of Min has arcs with weights depending on $\lambda $, which can only occur if it passes through node $m+1$ of Max. Thus, the conditions of the theorem are necessary and they are satisfied by any left optimal strategy $\tau $ at $\lambda^\ast $.
Assume now that there exists a strategy $\tau$ satisfying the conditions of the theorem. Then, the argument above shows that ${\phi}_\tau(\lambda^\ast)\leq 0$ and ${\phi}_\tau(\lambda)<0$ for all $\lambda < \lambda^\ast$. Since ${\phi}(\lambda^\ast)\geq 0$ and by we have ${\phi}(\lambda)\leq {\phi}_\tau(\lambda)$ for all $\lambda$, it follows that ${\phi}(\lambda^\ast)=0$ and ${\phi}(\lambda)<0$ for all $\lambda < \lambda^\ast$. Therefore, $\lambda^\ast $ is the optimal value of the tropical linear-fractional programming problem .
In the same way, we can certify when the tropical linear-fractional programming problem is unbounded.
\[2nd-cert\] The tropical linear-fractional programming problem is unbounded if, and only if, there exists a strategy $\sigma$ for player Max such that all cycles in the digraph $\Bipdig^{\sigma}_0$ accessible from node $n+1$ of Min do not contain node $m+1$ of Max and have nonnegative weight.
We know that the tropical linear-fractional programming problem is unbounded if, and only if, ${\phi}(\lambda)\geq 0$ for all $\lambda$. By the first equality in , the latter condition is satisfied if, and only if, there exists a strategy $\sigma$ for player Max such that ${\phi}^{\sigma}(\lambda)\geq 0$ for all $\lambda$. Note that the weight of a cycle in $\Bipdig^{\sigma}_\lambda $ that passes through node $m+1$ of Max can be made arbitrarily small by decreasing $\lambda $, because this cycle must contain an arc whose weight depends on $\lambda $. Therefore, using the fact that ${\phi}^{\sigma}(\lambda)$ is the minimal cycle mean (per turn) over all cycles in $\Bipdig^{\sigma}_{\lambda}$ accessible from node $n+1$ of Min (see , or and ), it follows that ${\phi}^{\sigma}(\lambda)\geq 0$ for all $\lambda$ if, and only if, all cycles in $\Bipdig^{\sigma}_0$ accessible from node $n+1$ of Min have nonnegative weight and do not pass through node $m+1$ of Max.
Theorems \[1st-cert\] and \[2nd-cert\] are inspired by Theorem 18 and Corollary 20 of [@AGK-10], in which similar certificates are given for the problem of checking whether an implication of the form $Ax\leq Bx\implies px \leq qx$ holds. The latter can be cast as a special tropical linear-fractional programming problem.
\[Example2\] Consider the tropical linear programming problem given by the minimization of $(2+x_1)\vee (-4+x_2)$ over the tropical polyhedron of $\Rmax^2$ defined by the system of inequalities $Ax\vee c\leq Bx\vee d$, where $$A=
\left(\begin{array}{cc}
-\infty & -\infty \\
-\infty & -\infty \\
-\infty & -\infty \\
-\infty & -3 \\
-\infty & -4 \\
-\infty & -5 \\
-\infty & -6
\end{array}\right) \; , \quad
c=
\left(\begin{array}{c}
0\\
0\\
0\\
0\\
-\infty\\
-\infty\\
-\infty
\end{array}\right) \; ,\quad
B=
\left(\begin{array}{cc}
-2 & 0 \\
0 & -1 \\
1 & -2 \\
2 & -\infty \\
0 & -\infty \\
-2 & -\infty \\
-4 & -\infty
\end{array}\right) \; , \quad
d=
\left(\begin{array}{c}
-\infty\\
-\infty\\
-\infty\\
-\infty\\
0\\
0\\
0
\end{array}\right) \; .$$ This polyhedron is displayed on the left-hand side of Figure \[tropprogs-ex\] below. The direction of minimization of $(2+x_1)\vee (-4+x_2)$ is shown there by a dotted line above the polyhedron, together with the optimal tropical hyperplane $(2+x_1)\vee (-4+x_2)=0$. The bipartite digraph $\Bipdig_\lambda $ corresponding to this problem is depicted in Figure \[BipDigrapfExMin\], where the nodes of Max are represented by squares and the nodes of Min by circles. Note that in this case we have $m=7$ and $n=2$.
The equivalent homogeneous version of this problem (as described in Subsection \[ss:Pformulations\]) is to minimize $\lambda $ subject to $u y\leq \lambda + v y$, $C y \leq D y$, and $y_3\neq -\infty$, where $C=[A,c]$, $D=[B,d]$, $u=(2,-4,-\infty)$ and $v=(-\infty,-\infty,0)$.
Thanks to Theorem \[1st-cert\], it is possible to certify that $\lambda^\ast=0$ is the optimal value of this problem. To show this, consider the strategy $\tau $ for player Min defined by: $\tau (1)=8$, $\tau(2)= 4$ and $\tau(3)=4$, which is represented in bold in Figure \[BipDigrapfExMin\]. Observe that the resulting subdigraph $\Bipdig_{\lambda^\ast }^{\tau}$ contains only one cycle, which is accessible from node $n+1$ of Min (indeed it passes through this node), has zero weight and passes through node $m+1$ of Max. Moreover, by Theorem \[chi-axbx\], we have ${\phi}(\lambda^\ast)\geq 0$ because $y=(-2,2,0)^T$ satisfies $C y \leq D y$ and $u y\leq \lambda^\ast + v y= v y$. Therefore, by Theorem \[1st-cert\], $\lambda^\ast=0$ is the optimal value.
The special cases of the tropical linear-fractional programming problem have been studied in [@BA-08], where necessary and sufficient conditions for these problems to be unbounded were in particular given. We next show that under the assumptions of [@BA-08], which require the entries of all vectors and matrices to be finite, these conditions turn out to be equivalent to the one given in Theorem \[2nd-cert\].
Theorem 3.3 of [@BA-08] shows that, when only finite entries are considered, the minimization problem in is unbounded if, and only if, $c\leq d$. Under the finiteness assumption, this condition is equivalent to the one given in Theorem \[2nd-cert\]. To show this, in the first place observe that in this case the associated digraph $\Bipdig_\lambda $ (see Figure \[FigureBipDigraGen\]) contains arcs connecting any node of Min $[n+1]$ with any the node of Max $[m+1]$, with exception of the arc connecting node $n+1$ with node $m+1$, and arcs connecting any node of Max $[m+1]$ with any node of Min $[n+1]$, with exception of the arcs connecting node $m+1$ with nodes in $[n]$. Thus, if we define the strategy $\sigma $ for player Max by $\sigma (i)=n+1$ for all $i\in [m+1]$, it can be checked that the only cycles in $\Bipdig^{\sigma}_0 $ accessible from node $n+1$ are of the form $n+1\rightarrow i_1 \rightarrow n+1 \rightarrow \cdots \rightarrow n+1 \rightarrow i_k \rightarrow n+1$ for some $i_1,\ldots ,i_k \in [m]$. Since the weight of such a cycle is $d_{i_1}-c_{i_1}+\cdots +d_{i_k}-c_{i_k}$, the strategy $\sigma $ satisfies the conditions in Theorem \[2nd-cert\] if $c\leq d$. Conversely, assume that a strategy $\sigma $ for player Max satisfies the conditions in Theorem \[2nd-cert\]. Then, the only possible value for $\sigma (m+1)$ is $n+1$, and we must also have $\sigma (i)=n+1$ for all $i\in [m]$, because if $\sigma (i)=j\neq n+1$ for some $i\in [m]$, $\Bipdig^{\sigma}_0 $ would contain the cycle $m+1\rightarrow n+1\rightarrow i\rightarrow j\rightarrow m+1$, contradicting the fact that no cycle accessible from node $n+1$ of Min passes through node $m+1$ of Max. Now, since $\Bipdig^{\sigma}_0 $ contains the cycles $n+1\rightarrow i \rightarrow n+1$ for $i\in [m]$, which are accessible from node $n+1$ of Min, the weights of these cycles $d_i-c_i$ must be nonnegative, implying that $c\leq d$.
Regarding the maximization problem in , Theorem 3.4 of [@BA-08] shows that this problem is unbounded if, and only if, the system $A x \leq B x$ has a finite solution. In this case, due to the finiteness assumption, it follows that the associated digraph $\Bipdig_\lambda $ (see Figure \[FigureBipDigraGen\]) contains arcs connecting any node of Max $[m+1]$ with any node of Min $[n+1]$, with exception of the arc connecting node $m+1$ with node $n+1$, and arcs connecting any node of Min $[n+1]$ with any the node of Max $[m+1]$, with exception of the arcs connecting nodes in $[n]$ with node $m+1$. If the system $A x \leq B x$ has a finite solution, from Theorem \[chi-axbx\] and it follows that there exists a strategy $\bar{\sigma}: [m]\mapsto [n]$ such that $\chi(A^{\sharp}B^{\bar{\sigma}}) = \chi(A^{\sharp}B)\geq 0$. By and , this implies that any cycle in $\bar{\Bipdig}^{\bar{\sigma}}$ has nonnegative weight, where $\bar{\Bipdig}$ is the bipartite digraph of the mean payoff game associated with the matrices $A$ and $B$. If we define the strategy $\sigma(i)=\bar{\sigma}(i)$ for all $i\in [m]$ and $\sigma(m+1)=j$ for some $j\in [n]$, then $\sigma $ satisfies the conditions of Theorem \[2nd-cert\] because the cycles accessible from node $n+1$ of Min in $\Bipdig^{\sigma}_0 $ are precisely the cycles in $\bar{\Bipdig}^{\bar{\sigma}}$ and there is no cycle containing node $m+1$ of Max in $\Bipdig^{\sigma}_0 $. Conversely, if a strategy $\sigma $ for player Max satisfies the conditions in Theorem \[2nd-cert\], then necessarily we have $\sigma (i)\in [n]$ for all $i\in [m]$, because if $\sigma (i)=n+1 $ for some $i\in [m]$, $\Bipdig^{\sigma}_0 $ would contain the cycle $n+1\rightarrow m+1 \rightarrow j \rightarrow i \rightarrow n+1$ where $j=\sigma (m+1)\in [n]$, contradicting the fact that there is no cycle in $\Bipdig^{\sigma}_0 $ accessible from node $n+1$ of Min passing through node $m+1$ of Max. Now, if we define $\bar{\sigma}(i)=\sigma(i)$ for all $i\in [m]$, the cycles accessible from node $n+1$ of Min in $\Bipdig^{\sigma}_0 $ are precisely the cycles in $\bar{\Bipdig}^{\bar{\sigma}}$, which therefore have nonnegative weight. Then, by and we have $\chi(A^{\sharp}B^{\bar{\sigma}})\geq 0$, and so from Theorem \[chi-axbx\] and we conclude that the system $A x\leq Bx $ has a finite solution.
If the strategies $\sigma$ or $\tau$ and the scalar $\lambda^*$ are fixed (considered as inputs) the conditions of Theorems \[1st-cert\] and \[2nd-cert\], i.e. the validity of the certificates, can be checked in polynomial time.
To see this, in the first place assume that $\tau$ and $\lambda^*$ are given. Using Karp’s algorithm, compute the maximal cycle mean of each strongly connected component of $\Bipdig_{\lambda^\ast }^{\tau}$ that is accessible from node $n+1$ of Min. The certificate is valid only if these maximal cycle means are nonpositive and one of them is zero. To check the second condition of Theorem \[1st-cert\], delete node $m+1$ of Max (and the arcs adjacent to it) from $\Bipdig_{\lambda^\ast }^{\tau}$ and compute for the resulting digraph (using again Karp’s algorithm) the maximal cycle mean of each strongly connected component accessible from node $n+1$ of Min. To be valid, all these maximal cycle means must be negative. Observe that in Theorem \[1st-cert\] we also assume that $\phi(\lambda^\ast )\geq 0$. By , this can be certified by a strategy $\sigma $ for player Max such that the minimal cycle mean of any strongly connected component of $\Bipdig^{\sigma}_{\lambda^\ast }$ accessible from node $n+1$ of Min is nonnegative, which can be checked by applying Karp’s algorithm to each of these components. By Theorem \[chi-axbx\], another possibility is to exhibit a vector $y$ such that $C y \leq D y$, $u y\leq \lambda^\ast + v y$ and $y_{n+1}\neq -\infty$.
Assume now that $\sigma$ is given. To check the validity of the certificate in Theorem \[2nd-cert\], decompose first $\Bipdig^{\sigma}_0$ in strongly connected components and see whether the component containing node $m+1$ of Max is trivial (i.e. contains just this node) or it is not accessible from node $n+1$ of Min. If this is the case, compute the minimal cycle mean of each strongly connected component of $\Bipdig^{\sigma}_0$ accessible from node $n+1$ of Min by applying Karp’s algorithm. Then, the certificate is valid if each of these minimal cycle means is nonnegative.
Bisection and Newton methods for tropical linear-fractional programming {#ss:bisnewt}
-----------------------------------------------------------------------
In , we need to find the least $\lambda $ such that ${\phi}(\lambda )\geq 0$, where ${\phi}(\lambda )$ is nondecreasing and Lipschitz continuous. Thus, we can consider certain classical methods for finding zeroes of “good enough” functions of one variable. In particular, the bisection method for ${\phi}(\lambda)$ corresponds to the approach of [@BA-08]. More specifically, it can be formulated as follows, when the finite coefficients in are integers.
\[a:bisection\] Bisection method
[**Start.**]{} A point $\overline{\lambda}_0$ such that ${\phi}(\overline{\lambda}_0)\geq 0$ and a point $\underline{\lambda}_0$ such that ${\phi}(\underline{\lambda}_0)< 0$.
[**Iteration $k$.**]{} Let $\lambda =\lceil(\overline{\lambda}_{k-1}+\underline{\lambda}_{k-1})/2\rceil$. If ${\phi}(\lambda)\geq 0$, then set $\overline{\lambda}_{k}=\lambda$ and $\underline{\lambda}_{k}=\underline{\lambda}_{k-1}$. Otherwise, set $\overline{\lambda}_{k}=\overline{\lambda}_{k-1}$ and $\underline{\lambda}_{k}=\lambda$.
[**Stop.**]{} Verify $\overline{\lambda}_{k}-\underline{\lambda}_{k}=1$. If true, return $\overline{\lambda}_{k}$.
For this method, which uses that tropical linear-fractional programming preserves integrity (Corollary \[c:philambda\] part ), it is not important to know the actual value of ${\phi}(\lambda)$, but just whether ${\phi}(\lambda)\geq 0$, i.e. whether $Uy\leq V(\lambda)y$ is solvable with $y_{n+1}\neq -\infty$.
Further, the concept of (left, right) optimal strategy, see Definition \[d:lropt\], yields an analogue of (left, right) derivative, and leads to the following analogue of Newton method, which does not have any integer restriction.
\[a:pos-newton\] Positive Newton method
[**Start.**]{} A point $\lambda_0$ such that ${\phi}(\lambda_0)\geq 0$.
[**Iteration $k$.**]{} Find a left optimal strategy $\sigma $ for player Max at $\lambda_{k-1}$ and compute $\lambda_k=\min\{\lambda \in\Rmax \mid {\phi}^{\sigma}(\lambda)\geq 0\}$.
[**Stop.**]{} Verify $\lambda_k=\lambda_{k-1}$ or $\lambda_k=-\infty$. If true, return $\lambda_{k}$.
It remains to explain how each step of this algorithm can be implemented. We shall see that $\lambda_k$ can be easily computed (reduction to a shortest path problem) and that finding left optimal strategies can be done by existing algorithms for mean payoff games.
For the sake of comparison, we state a dual version of Algorithm \[a:pos-newton\].
\[a:neg-newton\] Negative Newton method
[**Start.**]{} A point $\lambda_0$ such that ${\phi}(\lambda_0)< 0$.
[**Iteration $k$.**]{} Find a (right) optimal strategy $\tau$ for player Min at $\lambda_{k-1}$ and compute $\lambda_k=\min\{\lambda \in\Rmax \mid {\phi}_{\tau}(\lambda)\geq 0\}$.
[**Stop.**]{} Verify ${\phi}(\lambda_k)=0$ or $\lambda_k=+\infty$. If true, return $\lambda_{k}$.
\[r:justopt\] Note that in Algorithm \[a:neg-newton\] we can use optimal strategies instead of right optimal ones, because if $\tau$ is optimal at $\lambda_{k-1}$, we have ${\phi}_{\tau}(\lambda_{k-1})={\phi}(\lambda_{k-1})<0$ and so $\lambda_{k}> \lambda_{k-1}$ by the definition of $\lambda_{k}$ (recall that by Theorem \[t:philambda\] all the spectral functions are nondecreasing and piecewise-linear). This means that all the strategies considered in the iterations of Algorithm \[a:neg-newton\] are different, and as the number of strategies is finite, this algorithm must terminate in a finite number of steps. A similar argument shows that we can also use optimal strategies in Algorithm \[a:pos-newton\] at points $\lambda_{k-1}$ where the spectral function ${\phi}$ is strictly positive, because in that case we have $\lambda_{k} < \lambda_{k-1}$ even if $\sigma $ is just optimal and not left optimal (however, when ${\phi}(\lambda_{k-1})=0$ and $\lambda_{k-1}$ is not optimal, only a left optimal strategy at $\lambda_{k-1}$ guarantees $\lambda_{k} < \lambda_{k-1}$).
\[r:newtstart\] Due to Corollary \[c:philambda\] part , the values $\lambda^+:=2M(\min(m,n)+1)$ and $\lambda^-:=-2M(\min(m,n)+1)$ can be first checked in the case of the positive and negative Newton methods, respectively. We recall that $M$ is a bound on the absolute value of the coefficients in .
If ${\phi}(\lambda^+)<0$ then the problem is infeasible, and if ${\phi}(\lambda^-)>0$ then the problem is unbounded. If ${\phi}(\lambda^+)\geq 0$ and ${\phi}(\lambda^-)<0$, then the problem is both feasible and bounded. The case ${\phi}(\lambda^-)=0$ requires a left optimal strategy for player Max at $\lambda^-$ to decide that either this point is optimal, or the problem is unbounded.
This rule of starting with $\pm 2M(\min(m,n)+1)$, as we shall see, secures pseudo-polynomiality of the instances of the mean payoff games generated by the bisection and Newton methods.
The following logarithmic bound on the complexity of the bisection method is standard and its proof will be omitted.
\[termination-bis\] If the finite coefficients in are integers with absolute values bounded by $M$, then the number of iterations of the bisection method does not exceed $\log(4M(\min(m,n)+1))$ if it is started as in Remark \[r:newtstart\]. Hence, the computational complexity of the bisection method in this case does not exceed $$\log(4M(\min(m,n)+1))\times\MPGinteger(m+1,n+1,M+2M(\min(m,n)+1))\; .$$
\[r:bisbounds\] Butkovič and Aminu [@BA-08] give better initial values $\overline{\lambda}_0$ and $\underline{\lambda}_0$ for the bisection method than $\pm 2M(\min(m,n)+1)$, but only for the special cases , where all the coefficients are assumed to be finite. These initial values depend on the input data and lie in the interval $[-3M,3M]$. As oracle, they exploit the alternating method of [@CGB-03], which requires $O(mn(m+n)M)$ operations, being related to the value iteration of [@ZP-96]. Hence, in this case, the complexity of the bisection method is no more than $O(mn(m+n)M \log M)$. In [@Ser-lastdep], the same kind of initial values were obtained for the general formulation , leading to a similar complexity, but with the same finiteness restriction on the coefficients. The initial values of [@BA-08] and [@Ser-lastdep] will be exploited in the numerical experiments, see Subsection \[ss:NumExp\].
As observed above, in the case of the bisection method the mean payoff oracle is only required to check whether ${\phi}(\lambda)\geq 0$.
In the case when the finite coefficients in are real, the bisection method computes $(\overline{\lambda}_{k-1}+\underline{\lambda}_{k-1})/2$ without rounding and it yields only an approximate solution to the problem. However, Newton methods always converge in a finite number of steps.
\[termination-newt\] Denote by $\stratmax$ and $\stratmin$ the number of available strategies for players Max and Min, respectively. Then,
(i) \[termination-newtP1\] Algorithms \[a:pos-newton\] and \[a:neg-newton\] terminate in a finite number of steps, the number of which does not exceed $\stratmax$ and $\stratmin$, respectively.
(ii) \[termination-newtP2\] If the finite coefficients in are integers with absolute values bounded by $M$, then the values $\lambda_k$ produced by Algorithm \[a:pos-newton\] are also integer, and the number of iterations does not exceed $4M(\min(m,n)+1)$ if it is started as in Remark \[r:newtstart\].
At different iterations of Algorithm \[a:pos-newton\] we have different strategies, because $\lambda_k=\min\{\lambda \in\Rmax \mid {\phi}^{\sigma}(\lambda)\geq 0\}$ are different for all $k$. Similarly for Algorithm \[a:neg-newton\], with $\tau$ instead of $\sigma$. Thus, the number of steps is limited by the number of strategies, which is finite.
The numbers $\lambda_k$ generated by Algorithm \[a:pos-newton\] are integers due to Corollary \[c:philambda\] part . By Remark \[r:newtstart\], we can start the algorithm at $2M(\min(m,n)+1)$, and it will finish before it reaches $-2M(\min(m,n)+1)$.
A worst-case complexity bound, different from the number of strategies, will be given below in Theorem \[posnewt-comp\], and it is worse than that of the bisection method above (see also Remark \[r:bisbounds\]). First, Newton iterations require more sophisticated oracles which compute the value of the game and a left optimal strategy. Second, we have only used the integrality of the method in Proposition \[termination-newt\], so the bound on the number of iterations is rough. However, the positive Newton method is an interesting alternative to the bisection method, since it preserves feasibility. Therefore, it may be more sensitive to the geometry of the feasible set, which is especially convenient if this set has only few generators or its dimension is small. The experiments of Subsection \[ss:NumExp\] indicate that this is indeed the case, and the worst-case complexity bound of Theorem \[posnewt-comp\] (using Proposition \[termination-newt\]) is often too pessimistic. The main reason to give the result of Theorem \[posnewt-comp\] is that it shows the method is pseudo-polynomial.
Not aiming to obtain a better overall worst-case complexity result, in the next subsection we will rather consider the implementation of the positive Newton method, reducing the computation of $\lambda_k$ to a (polynomial-time solvable) shortest path problem. Subsection \[ss:left-optim\] will be devoted to the computation of left optimal strategies in the integer case by means of perturbed mean payoff games. As noticed in Proposition \[termination-newt\], Newton iterations should work also in the case of real coefficients. For this we propose the algebraic approach of Subsection \[ss:germs\], encoding a perturbed game as a game over the semiring of germs.
Newton iterations by means of Kleene star {#ss:kleene}
-----------------------------------------
In this subsection we show that in the case of Algorithm \[a:pos-newton\] the steps of Newton method can be performed by calculating least solutions of inequalities of the form $z \geq E z \vee h$, as in Proposition \[bellman\].
Assume that we are at iteration $k$ of Algorithm \[a:pos-newton\], so that we need to compute $\lambda_k=\min\{\lambda\in\Rmax\mid {\phi}^{\sigma}(\lambda)\geq 0\}$, where $\sigma$ is a left optimal strategy for player Max at $\lambda_{k-1}$. If we set $V^{\sigma}(\lambda)$ instead of $V(\lambda)$ in $Uy\leq
V(\lambda)y$, by Theorem \[chi-axbx\] the minimal zero $\lambda_k$ of ${\phi}^{\sigma}(\lambda)$ is exactly the least value of $\lambda$ for which this system is satisfied by some $y\in \Rmax^{n+1}$ with $y_{n+1}\neq -\infty $, i.e. we have $$\label{DefLambdaK}
\lambda_k=\min\{ \lambda\in\Rmax\mid Uy\leq V^{\sigma}(\lambda)y\;,\; y_{n+1}\neq -\infty \;
\makebox{ is solvable}\} \; .$$ The main idea is to compute this minimal zero by considering the system $Uy\leq V^{\sigma}(\lambda)y$ directly. With this aim, we shall need the following observation.
\[problems-equiv2\] Assume that at iteration $k$ of Algorithm \[a:pos-newton\] we have $l:=\sigma(m+1)\neq n+1$, where $\sigma$ is a left optimal strategy for player Max at $\lambda_{k-1}$. Then, if $$\label{ConditionEq}
\left( Uy\leq V^{\sigma}(\lambda)y\makebox{ and } y_{n+1}\neq -\infty \right)
\implies y_l\neq -\infty \; ,$$ for all $\lambda$, it follows that $$\label{EqDefLambdaK}
\lambda_k=\min\{ \lambda\in\Rmax\mid Uy\leq V^{\sigma}(\lambda)y\;,\; y_{l}\neq -\infty \;
\makebox{ is solvable}\} \; .$$ Otherwise, i.e. if Condition does not hold, $\lambda_k=-\infty$.
Condition implies that the minimum in is less than or equal to that in .
To show the converse, suppose that for some $\lambda\in \R$ there exists $\hat{y}\in \Rmax^{n+1}$ such that $\hat{y}_l\neq -\infty$ and $U\hat{y}\leq V^{\sigma}(\lambda)\hat{y}$. Since ${\phi}^{\sigma}(\lambda_{k-1})={\phi}(\lambda_{k-1})\geq 0$, by Theorem \[chi-axbx\] there exists a solution $\tilde{y}$ of $Uy\leq V^{\sigma}(\lambda_{k-1})y$ (and so, in particular, of the first $m$ inequalities of this system, i.e. $C y\leq D^{\sigma} y$) such that $\tilde{y}_{n+1}\neq -\infty$. Then, for any $\beta \in \R$ the combination $\overline{y}=\hat{y}\vee \Tilde{y}-\beta$ satisfies the first $m$ inequalities in $Uy\leq V^{\sigma} (\lambda)y$ (in other words, we have $C \overline{y}\leq D^{\sigma} \overline{y}$) as a tropical linear combination of solutions of this system of tropically linear inequalities. Moreover, if $\beta$ is sufficiently large, $\overline{y}$ also satisfies the last inequality $uy\leq \lambda+v^{\sigma}y$ of the system $Uy\leq V^{\sigma} (\lambda)y$ because $u\hat{y}\leq \lambda + v^{\sigma} \hat{y}$ and $v^{\sigma} \hat{y}=v_l+\hat{y}_l > -\infty$. But then $\overline{y}$ satisfies $U\overline{y}\leq V^{\sigma}(\lambda)\overline{y}$ and $\overline{y}_{n+1}\geq \tilde{y}_{n+1}-\beta \neq -\infty$. This shows that the minimum in is less than or equal to that in .
Finally, if Condition does not hold, for some $\bar{\lambda}$ there exists a solution $\bar{y}$ of the system $Uy\leq V^{\sigma}(\bar{\lambda})y$ such that $\bar{y}_{n+1}\neq -\infty$ but $\bar{y}_l= -\infty$. Since $u\bar{y}\leq \bar{\lambda }+ v^{\sigma} \bar{y}=\bar{\lambda }+v_l+\bar{y}_l$, this can only happen if $u\bar{y}=-\infty$, which implies that $\bar{y}$ satisfies $U\bar{y}\leq V^{\sigma}(\lambda)\bar{y}$ for any $\lambda \in \R$, and so $\lambda_k=-\infty$.
We next show how to make sure that Condition is satisfied. Note that this condition is not satisfied if, and only if, for some $\lambda $ the system $Uy\leq V^{\sigma}(\lambda)y$ has a solution $\bar{y}$ with $\bar{y}_{n+1}\neq -\infty$ but $\bar{y}_l=-\infty$. The latter implies the existence of a solution $\bar{y}$ of $C y\leq D y$ such that $\bar{y}_i=-\infty$ for all $i\in\supp(u)$, but $\bar{y}_{n+1}\neq -\infty$ (so in particular Condition is satisfied if $n+1\in \supp(u)$). Eliminating from the system $C y\leq D y$ the columns corresponding to the indices in $\supp(u)$, the existence of such a solution is reduced to the solvability of a two-sided homogeneous system with the condition $y_{n+1}\neq -\infty$, which can be decided using a mean payoff game oracle. If this problem has no solution, then Condition is satisfied. Otherwise, the value of the original tropical linear-fractional programming problem is $-\infty$.
As a consequence of the previous discussion, in what follows we assume that it has already been checked that Condition is satisfied, and we explain how to perform Newton iterations in that case.
Suppose that we are at iteration $k$ of Algorithm \[a:pos-newton\], and let $\sigma$ be a left optimal strategy at $\lambda_{k-1}$. Then, if we set $l:=\sigma(m+1)$, by Lemma \[problems-equiv2\] we have $$\begin{aligned}
\lambda_k = \min\{ \lambda \in\Rmax \mid Uy\leq V^{\sigma}(\lambda)y\;,\; y_l\neq -\infty \;
\makebox{ is solvable}\} \; .\end{aligned}$$ Since the system $Uy\leq V^{\sigma}(\lambda)y$ is satisfied by some $\bar{y}$ with $\bar{y}_l\neq -\infty$ if, and only if, it is satisfied by some $\hat{y}$ with $\hat{y}_l=0$ (it is enough to define $\hat{y}_i=\bar{y}_i-\bar{y}_l$ for all $i\in [n+1]$), it follows that $$\begin{aligned}
\lambda_k&=&\min\{ \lambda \in\Rmax \mid Uy\leq V^{\sigma}(\lambda)y\;,\; y_l= 0 \;
\makebox{ is solvable}\} \\
&=& \min \{ \lambda \in\Rmax \mid Cy\leq D^{\sigma}y \; ,\;
uy\leq \lambda + v_l + y_l \; , \; y_l=0 \; \makebox{ is solvable}\}
\; .\end{aligned}$$ Thus, setting $y_l=0$ we are in the situation of problem , because $\lambda_k+v_l$ is given by: $$\label{problem-sigma}
\begin{split}
&\makebox{minimize }\quad p x\vee r \\
&\makebox{subject to:}\quad Ax\vee c \leq B^{\sigma}x\vee d^{\sigma}
\; , \; x\in\Rmax^n
\end{split}$$ where $p\in\Rmax^n$, $r\in\Rmax$, $c,d^{\sigma}\in\Rmax^m$ and $A,B^{\sigma}\in\Rmax^{m\times n}$ are such that $$U=
\begin{pmatrix}
A & c\\
p & r
\end{pmatrix}\quad \makebox{and}\quad
V^{\sigma}(\lambda)=
\begin{pmatrix}
B^{\sigma} & d^{\sigma}\\
-\infty & \lambda+v_l
\end{pmatrix} \enspace .$$ Here, just for the simplicity of the presentation, column $l$ is in the place of column $n+1$ when $l\neq n+1$ (in other words, the columns of $A$ and $B^{\sigma}$ are respectively the columns of $C$ and $D^{\sigma}$ with exception of column $l$, $c$ is column $l$ of $C$, $d^{\sigma}$ is column $l$ of $D^{\sigma}$, $r=u_l$ and $p_i=u_i$ for $i\neq l$).
We claim that the system of constraints (the second line) in has a least solution, which then minimizes $px\vee r$, and we explain how to find it. First note that the constraints in can be written as: $$\label{two-subsystems}
\begin{split}
(Ax)_i\vee c_i&\leq b_{i\sigma(i)}+x_{\sigma(i)}\; ,\quad
\makebox{if} \; \sigma (i) \neq l \; ,\\
(Ax)_i\vee c_i&\leq d_i\enspace , \quad \makebox{if} \; \sigma (i)= l\; .
\end{split}$$
In order to find the least solution of this system, observe that the second subsystem can be dispensed with. Indeed, since ${\phi}^{\sigma}(\lambda_{k-1})={\phi}(\lambda_{k-1})\geq 0$, by Theorem \[chi-axbx\] there exists a solution $\tilde{y}$ of $Uy\leq V^{\sigma}(\lambda_{k-1})y$ such that $\tilde{y}_{n+1}\neq -\infty$, and so this solution also satisfies $\tilde{y}_{l}\neq -\infty$ (recall we assume that Condition holds). Then, if $\tilde{x}\in \Rmax^n$ is the vector defined by $\tilde{x}_i:=\tilde{y}_i-\tilde{y}_l$ for $i\neq l$, it follows that $\tilde{x}$ is a solution of . Hence, if the first subsystem has the least solution $\underline{x}$, we have $\underline{x}\leq \tilde{x}$ and so $\underline{x}$ is also a solution of the second subsystem.
To show that the first subsystem in has a least solution, first note that a system of two inequalities of the form $$\begin{split}
r_{11}+x_1\vee \cdots \vee r_{1n}+x_n&\leq x_1 \\
r_{21}+x_1\vee \cdots \vee r_{2n}+x_n&\leq x_1
\end{split}$$ is equivalent to just one inequality: $$s_1+x_1\vee \cdots \vee s_n+x_n\leq x_1 \; ,$$ where $s_i=r_{1i}\vee r_{2i}$ for $i=1,\ldots, n$. Using this kind of reduction, the first subsystem can be transformed in no more than $m(n+1)$ operations to an equivalent system of the form $$\label{eq-sys}
Ex_I\vee Fx_J\vee h\leq x_I\enspace ,$$ where $x_I$ is the sub-vector whose coordinates appear on the right-hand side of the first subsystem in , and $x_J$ is the sub-vector corresponding to the rest of the coordinates which are present in that system. Since we are interested in the least solution, we can set $x_J\equiv -\infty $, and then the remaining system is just of the form $$\label{kleene-ineq}
E z\vee h\leq z\enspace ,$$ where $z=x_I$. By Proposition \[bellman\], the least solution to this system in $\RRbar^{|I|}$ is given by $$\underline{z}=E^*h=h\vee Eh\vee E^2h\vee E^3h\vee\cdots$$ As $\tilde{x}_I$ satisfies , we have $\underline{z}\leq \tilde{x}_I$ and so $\underline{z}\in\Rmax^{|I|}$.
Thus, we have the following method:
\[a:problem-sigma\] Solving
[**Step 1.**]{} Split the system $Ax\vee c\leq B^{\sigma}x\vee d^{\sigma}$ in two subsystems as in and transform the first subsystem to the form .
[**Step 2.**]{} Compute $\underline{z}=E^*h$. Set $\underline{x}_I=\underline{z}$ and $\underline{x}_J\equiv -\infty$.
[**Step 3.**]{} Return $p\underline{x}\vee r$.
We also conclude the following.
\[prop-newnice2\] The problems $\min\{\lambda \mid {\phi}^{\sigma}(\lambda)\geq 0\}$ can be solved in $O(mn)+O(n^3)$ time.
Note that in general, a system of the form is solvable in $\Rmax^n$ if, and only if, the least solution of the first subsystem belongs to $\Rmax^n$ and satisfies the second subsystem.
\[Example3\] Consider the following tropical linear-fractional programming problem: $$\begin{split}
&\text{minimize } \quad \lambda \\
&\text{subject to:}\quad u y\leq \lambda + v y\; ,
\; C y\leq D y\; , \; y_{4}\neq -\infty \; , \; y\in\Rmax^{4} \; ,
\; \lambda \in\Rmax
\end{split}$$ where $$C=
\left(\begin{array}{cccc}
-3 & -4 & -\infty & -\infty \\
-1 & -\infty & -\infty & 1 \\
-\infty & -\infty & -\infty & 0\\
1 & -\infty & 0 & -\infty
\end{array}\right) \; ,\quad
D=
\left(\begin{array}{cccc}
-\infty & -\infty & -\infty & 0 \\
-\infty & 0 & -\infty & -\infty \\
0 & -\infty & -\infty & -\infty \\
0 & -\infty & -\infty & 3
\end{array}\right) \; ,$$ $u=(-\infty,0,-\infty,-\infty)$ and $v=(3,-\infty,-\infty,-\infty)$, so in this case we have $m=4$ and $n=3$ with the notation of Problem .
Before performing Newton iterations, we need to check whether the system $Cy\leq Dy$ has solutions with $y_2=-\infty$ but $y_4\neq -\infty$ (since $\supp(u)=\{2\}$). By the second inequality of this system, it follows that this is impossible. Hence, Condition is satisfied and so we can use in order to compute $\lambda_k$. Moreover, in this example Newton method requires no more than two iterations, since player Max has only two strategies, which correspond to the two finite entries in the last row of $D$.
Assume that we start with $\lambda_0=0$. Then, an optimal strategy $\sigma$ for player Max at $\lambda_0$ is given by: $\sigma(1)= 4$, $\sigma(2)= 2$, $\sigma(3)= 1$, $\sigma(4)= 4$ and $\sigma(5)= 1$. Since $l=\sigma(m+1)=\sigma(5)=1$, we have $$\lambda_1=\min\{ \lambda \in\Rmax \mid Cy\leq D^{\sigma}y\;,\; y_2=uy\leq \lambda+v_1+y_1\; ,\; y_1= 0 \;
\makebox{ is solvable}\} \; ,$$ and so, setting $y_1=0$, $\lambda_1+v_1=\lambda_1+3$ is given by: $$\label{problem-sigma-Example}
\begin{split}
&\makebox{minimize }\quad y_2 \\
&\makebox{subject to:}\; (y_2-4)\vee (-3) \leq y_4\; , \;
(y_4+1)\vee (-1) \leq y_2 \; ,\;
y_4\leq 0 \; ,\;
y_3 \vee 1 \leq y_4+3 \; .
\end{split}$$ Note that the system of constraints in , obtained by setting $y_1=0$ in $C y\leq D^{\sigma} y$ (i.e., in the first column plays the role of free term), reduces to $$E \left(\begin{array}{cc}
y_2 \\
y_4
\end{array}\right)
\vee F y_3\vee h\leq
\left(\begin{array}{cc}
y_2 \\
y_4
\end{array}\right)
\; ,\;
y_4\leq 0 \; ,$$ where $$E=
\left(\begin{array}{cc}
-\infty & 1 \\
-4 & -\infty
\end{array}\right) \; , \;
F=
\left(\begin{array}{cc}
-\infty \\
-3
\end{array}\right) \; , \;
h=
\left(\begin{array}{cc}
-1 \\
-2
\end{array}\right) \; .$$ Since $$E^* h=
\left(\begin{array}{cc}
\;0 & \;1\; \\
-4 & \;0\;
\end{array}\right)
\left(\begin{array}{cc}
-1 \\
-2
\end{array}\right) =
\left(\begin{array}{cc}
-1 \\
-2
\end{array}\right) \; ,$$ the least solution of the system of constraints in is $(y_2,y_3,y_4)^T=(-1,-\infty,-2)$. Therefore, the value of problem is $-1$, and thus $\lambda_1=-4$. It can be checked that this is the optimal solution of the tropical programming problem (and in particular, that $\sigma$ is still a left optimal strategy for player Max at $\lambda_1=-4$).
Computing left optimal strategies {#ss:left-optim}
---------------------------------
In order to compute left optimal strategies, consider the mean payoff game associated with the tropical linear-fractional programming problem , and let the weights $\lambda+v$ of the arcs connecting node $m+1$ of Max with nodes of Min be replaced by $\lambda-\epsilon+v$, where $\epsilon\in \R$. In this way, we obtain a [*perturbed mean payoff game*]{}. For small enough $\epsilon>0$, the optimal strategies for this game are the left optimal strategies required by the positive Newton iterations. Here, we will require the mean payoff oracle to find optimal strategies, not just the value of the game. The complexity of such oracle will be denoted by $\MPGstrong(m,n,M)$, for mean payoff games with integer payments whose absolute values are bounded by $M$, with $m$ nodes of Max and $n$ nodes of Min. A pseudo-polynomial algorithm for computing optimal strategies of such games is described in [@ZP-96].
\[p:germs3\] If the finite coefficients in are integers with absolute values bounded by $M$, then at each iteration of the positive Newton method, started and finished as in Remark \[r:newtstart\], a left optimal strategy can be found in $$\MPGstrong(m+1,n+1,(\min(m,n)+2)(M+2M(\min(m,n)+1))+1)$$ operations.
By Corollary \[c:philambda\], it follows that each $\lambda_k$ is integer and the breaking points of ${\phi}(\lambda)$ are rational numbers whose denominators do not exceed $\min(m,n)+1$. Therefore, optimal strategies for the game at $\lambda^*=\lambda_k-1/(\min(m,n)+2)$ are left optimal strategies at $\lambda_k$. Then, we only need to apply a mean payoff oracle in order to compute optimal strategies for the perturbed mean payoff game with $\epsilon=1/(\min(m,n)+2)$. Multiplying (in the usual sense) the payments by $\min(m,n)+2$ we obtain a mean payoff game with integer payments and with the same optimal strategies (in this sense, equivalent to the perturbed game). The computation of optimal strategies in the latter game takes no more than $\MPGstrong(m+1,n+1,(\min(m,n)+2)(M+2M(\min(m,n)+1))+1)$ operations, because the payments in the perturbed mean payoff game are either of the form $a$ or $a+\lambda_k -\epsilon $, where $|a|\leq M$ and $|\lambda_k| \leq 2M(\min(m,n)+1)$, and these payments are multiplied by $\min(m,n)+2$.
In order to find a left optimal strategy $\sigma$ for player Max at $\lambda_{k-1}=0$ in Example \[Example3\], we only need to compute an optimal strategy for the associated game at $\lambda^*=\lambda_{k-1}-1/(\min(m,n)+2)=-1/5$. This can be done by solving the game whose payments are given by the matrices $$\left(\begin{array}{cccc}
-15 & -20 & -\infty & -\infty \\
-5 & -\infty & -\infty & 5 \\
-\infty & -\infty & -\infty & 0\\
5 & -\infty & 0 & -\infty \\
-\infty & 0 & -\infty & -\infty
\end{array}\right) \; \makebox { and } \;
\left(\begin{array}{cccc}
-\infty & -\infty & -\infty & 0 \\
-\infty & 0 & -\infty & -\infty \\
0 & -\infty & -\infty & -\infty \\
0 & -\infty & -\infty & 15 \\
14 & -\infty & -\infty & -\infty
\end{array}\right) \; ,$$ which are obtained by multiplying (in the usual sense) the payments for the game at $\lambda^*$ by $5$.
\[r:noleftopt\] Proposition \[p:germs3\] was necessary to establish the pseudo-polynomiality of the positive Newton method, which regularly uses left optimal strategies. However, as observed in Remark \[r:justopt\], the use of left optimal strategies is not necessary when ${\phi}(\lambda_k)>0$. Moreover, when ${\phi}(\lambda_k)=0$ and the coefficients are integers, an alternative to computing a left optimal strategy is to use an optimal strategy, checking whether ${\phi}(\lambda_k-1)<0$ when $\lambda_{k+1}=\lambda_k$. In that case, Corollary \[c:philambda\] part guarantees that $\lambda_k$ is optimal. Otherwise, proceed with $\lambda_{k+1}:=\lambda_k-1$. With this modification, the complexity of the computation of optimal strategies falls to $$\MPGstrong(m+1,n+1,M+2M(\min(m,n)+1)) \; ,$$ instead of the bound of Proposition \[p:germs3\].
We are now ready to sum up the computational complexity of the positive Newton method with left optimal strategies.
\[posnewt-comp\] If the finite coefficients in are integers with absolute values bounded by $M$, then the positive Newton method, started and finished as in Remark \[r:newtstart\], takes no more than $$\begin{split}
O(M &\min(m,n)) \times (O(mn)+O(n^3)+\\
& +\MPGstrong(m+1,n+1,(\min(m,n)+2)(M+2M(\min(m,n)+1))+1))
\end{split}$$ operations. In particular, the positive Newton method is pseudo-polynomial.
As the numbers $\lambda_k$ generated by the positive Newton method are integers and lie within $[-2M(\min(m,n)+1),2M(\min(m,n)+1)]$, the number of iterations does not exceed $4M(\min(m,n)+1)+1$. Combining this with Propositions \[prop-newnice2\] and \[p:germs3\], we get the claim.
Note that Remark \[r:noleftopt\] can be used to reduce the bound of Theorem \[posnewt-comp\], if the left optimal strategies are not used, getting rid of the factor $(\min(m,n)+2)$ in the third argument of $\MPGstrong$.
Perturbed mean payoff games as mean payoff games over germs {#ss:germs}
-----------------------------------------------------------
Next we discuss an alternative to the perturbation technique of the previous subsection: instead of considering the mean payoff game for several values of the perturbation parameter, we may consider a mean payoff game the payments of which belong to a lattice ordered group of [*germs*]{}. In a nutshell, the elements of this group encode infinitesimal perturbations of the payments. This algebraic structure allows one to deal more generally with one-parameter perturbed games (not only the ones arising from tropical linear-fractional programming). A similar structure appeared in [@gg0]. This is somehow analogous to the perturbation methods used to avoid degeneracy in linear programming. We hope to develop this further in a subsequent work. The materials of this subsection are not used in the rest of the paper. However, this alternative can be useful in two respects: 1) to develop the present Newton method in the case of real coefficients, 2) to improve the complexity result of the previous subsection.
Consider a [*mean payoff game over germs*]{}, finite-duration version, where the weights of arcs in $\Bipdig$ are pairs of real numbers $(a,b)$ endowed with lexicographic order: $$\label{lex}
(a^1,b^1)\leqlex (a^2,b^2)\Leftrightarrow
\begin{cases}
a^1<a^2,\ \text{or}\\
a^1=a^2,\ b^1\leq b^2,
\end{cases}$$ and the componentwise addition is used to calculate the weights of paths (or cycles). These games correspond to two-sided tropical linear systems over the semiring of germs $\Germ:=\R^2\cup\{(-\infty,-\infty)\}$, where for $g^1=(a^1,b^1), g^2=(a^2,b^2)\in \Germ$ we define $\ltr g^1+g^2\rtr:=\max(g^1,g^2)$ following and $\ltr g^1g^2\rtr:=(a^1+a^2,b^1+b^2)$.
With a game over germs we associate an [*$\epsilon$-perturbed mean payoff game*]{}, for $\epsilon \geq 0$, in which the weights $(a,b)$ of the arcs are replaced by $a+\epsilon b$. If the payments in a mean payoff game over germs are given by the matrices $A$ and $B$ (with entries in $\Germ$), then the matrices associated with the corresponding $\epsilon$-perturbed mean payoff game will be denoted by $A(\epsilon)$ and $B(\epsilon)$, respectively.
\[p:germs1\] Suppose that the matrices of payments in a mean payoff game over germs (finite-duration version) satisfy Assumptions 1 and 2. Then this game has a value and positional optimal strategies (meaning that holds for mean payoff games over germs). Moreover, if $(\chi_i,\kappa_i)$ is the value of such a game, then there exists $\overline{\epsilon}>0$ such that for any $0<\epsilon<\overline{\epsilon}$ the associated $\epsilon$-perturbed mean payoff game has value $\chi_i+\epsilon\kappa_i$ and these games have common positional optimal strategies.
Note that if the matrices of payments $A$ and $B$ in a mean payoff game over germs satisfy Assumptions 1 and 2, then for any $\epsilon$-perturbed mean payoff game the corresponding matrices $A(\epsilon)$ and $B(\epsilon)$ also satisfy these assumptions.
Let $\delta$ be the minimal absolute value of nonzero differences between cycle means in the mean payoff game with payments given by $A(0)$ and $B(0)$, and let $M$ be the greatest absolute value of the second component of germs. Define $\overline{\epsilon}:=\delta/4M$ and consider any $\epsilon $ such that $0<\epsilon<\overline{\epsilon}$.
By , for the $\epsilon$-perturbed mean payoff game there exist positional strategies $\sigma^*$ and $\tau^*$ such that $$\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma)\leq
\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma^*)\leq
\Phi_{A(\epsilon),B(\epsilon)}(j,\tau,\sigma^*)$$ for all (not necessarily positional) strategies $\sigma$ and $\tau$. Let $\sigma $ be any strategy for player Max and assume that $\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma)=a +\epsilon b$ and $\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma^*)=\chi_j+\epsilon \kappa_j$. If $a=\chi_j$, we have $b\leq \kappa_j$ because $a +\epsilon b \leq \chi_j+\epsilon \kappa_j$. Otherwise (i.e., if $a\neq \chi_j$), since $|b|,|\kappa_j|\leq 2M$, $|a-\chi_j|\geq \delta$, $\epsilon <\overline{\epsilon}=\delta/4M$, and $a +\epsilon b \leq \chi_j+\epsilon \kappa_j$, it follows that $a<\chi_j$. Therefore, we conclude that $\Phi_{A,B}(j,\tau^*,\sigma)=(a,b)\leqlex (\chi_j,\kappa_j)=\Phi_{A,B}(j,\tau^*,\sigma^*)$. The same argument shows that $\Phi_{A,B}(j,\tau^*,\sigma^*)\leqlex \Phi_{A,B}(j,\tau ,\sigma^*)$ for any strategy $\tau$ for player Min. This proves that the finite duration version of the mean payoff game over germs has a value, given by $(\chi_j,\kappa_j)$, and that positional optimal strategies for the $\epsilon$-perturbed mean payoff game (with $\epsilon <\overline{\epsilon}$) are also optimal for the game over germs.
Assume now that $\sigma^*$ and $\tau^*$ are positional optimal strategies for the finite duration version of the mean payoff game over germs, and let $(\chi_j,\kappa_j)$ be its value. Then, $\Phi_{A,B}(j,\tau^*,\sigma)\leqlex (\chi_j,\kappa_j)\leqlex \Phi_{A,B}(j,\tau,\sigma^*)$ for all strategies $\sigma$ and $\tau$ (not necessarily positional).
Let $\sigma $ be any strategy for player Max and assume that $\Phi_{A,B}(j,\tau^*,\sigma)=(a,b)$. If $a=\chi_j$, we conclude $b\leq \kappa_j$ because $\Phi_{A,B}(j,\tau^*,\sigma)\leqlex (\chi_j,\kappa_j)$, and thus $\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma)=a+\epsilon b\leq
\chi_j+\epsilon\kappa_j$ for any $\epsilon \geq 0$. Suppose now that $a<\chi_j$. Since $|b|,|\kappa_j|\leq 2M$ and $\chi_j-a\geq \delta$, it follows that $\Phi_{A(\epsilon),B(\epsilon)}(j,\tau^*,\sigma)=a+\epsilon b\leq
\chi_j+\epsilon\kappa_j$ for any $\epsilon$ such that $0<\epsilon<\overline{\epsilon}$. The same argument shows that $\Phi_{A(\epsilon),B(\epsilon)}(j,\tau,\sigma^*)\geq \chi_j+\epsilon\kappa_j$ for any strategy $\tau$ for player Min and any $\epsilon$ such that $0<\epsilon<\overline{\epsilon}$. Therefore, we conclude that $\sigma^*$ and $\tau^*$ are positional optimal strategies for the $\epsilon$-perturbed mean payoff game and that its value is $\chi_j+\epsilon\kappa_j$. This proves the claim.
\[r:germs-convenience\] Proposition \[p:germs1\] opens the way to using mean payoff games over germs in order to find right or left optimal strategies in the Newton methods. In that case, note that the second component of all finite weights must be set to $0$ except for the arcs connecting node $m+1$ of Max with nodes of Min, where it is set to $1$ (for right optimality) or to $-1$ (for left optimality). This raises the issue of developing a direct combinatorial algorithm to solve mean payoff games over germs. Such an algorithm would avoid the perturbation technique of the previous subsection. This will be discussed elsewhere.
By Proposition \[p:germs1\], in Example \[Example3\] we could find a left optimal strategy $\sigma$ for player Max at $\lambda_{k-1}=0$ computing an optimal strategy for the mean payoff game over germs whose matrices of payments are: $$\left(\begin{array}{cccc}
(-3,0) & (-4,0) & \ZGerm & \ZGerm \\
(-1,0) & \ZGerm & \ZGerm & (1,0) \\
\ZGerm & \ZGerm & \ZGerm & (0,0) \\
(1,0) & \ZGerm & (0,0) & \ZGerm \\
\ZGerm & (0,0) & \ZGerm & \ZGerm
\end{array}\right) \; \makebox { and } \;
\left(\begin{array}{cccc}
\ZGerm & \ZGerm & \ZGerm & (0,0) \\
\ZGerm & (0,0) & \ZGerm & \ZGerm \\
(0,0) & \ZGerm & \ZGerm & \ZGerm \\
(0,0) & \ZGerm & \ZGerm & (3,0) \\
(3,-1) & \ZGerm & \ZGerm & \ZGerm
\end{array}\right) \; ,$$ where $\ZGerm:=(-\infty,-\infty)$.
\[r:algorithms\] Proposition \[p:germs1\] extends to germs. Note that this equation provides a very crude algorithm for computing values and optimal strategies. Further idea is to allow more general algorithms, showing that they can be applied to mean payoff games over germs. This will be investigated elsewhere.
Examples
========
Minimization {#SectionExampleMin}
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![The minimization problem of Example \[Example2\]: the tropical polyhedron and the spectral function.[]{data-label="tropprogs-ex"}](tropprogex){width="0.75\linewidth"} [[![The minimization problem of Example \[Example2\]: the tropical polyhedron and the spectral function.[]{data-label="tropprogs-ex"}](tropprognewtondash "fig:"){width="1\linewidth"}]{}]{}
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We next apply the positive Newton method to the minimization problem of Example \[Example2\]. Recall that the equivalent homogeneous version of this problem is to minimize $\lambda $ subject to $u y\leq \lambda + v y$, $C y \leq D y$, and $y_3\neq -\infty$, with $C=[A,c]$, $D=[B,d]$, $u=(2,-4,-\infty)$ and $v=(-\infty,-\infty,0)$, where the matrices $A,B$ and the vectors $c,d$ are given in Example \[Example2\] (also recall that in this example, $m=7$ and $n=2$). Note that in this case, for any strategy $\sigma $ for player Max, the only possible value for $l:=\sigma(m+1)=\sigma(8)$ is $n+1=3$. Then, at each iteration $k$ of the positive Newton method applied to this problem, in order to compute $\lambda_k+v_l=\lambda_k$ we need to minimize $(2+y_1)\vee (-4+y_2)$ subject to the system obtained by setting $y_3=0$ in $C y \leq D^\sigma y$, as explained in Subsection \[ss:kleene\]. The latter is Problem for this particular case.
We start the positive Newton method with $\lambda_0=15$, where ${\phi}(\lambda_0)=5.5$. The function $\sigma(1)=1$, $\sigma(2)=1$, …, $\sigma(7)=1$ and $\sigma(8)=3$ is an optimal strategy for player Max at $\lambda_0$. To perform the first Newton iteration, we find the minimal solution of the system $$\begin{split}
0\leq x_1-2 \; ,\; 0\leq x_1 \; , \; 0\leq 1+x_1 \; , \; x_2-4\leq x_1 \; , \\
(x_2-3)\vee 0\leq x_1+2
\; ,\; x_2-3\leq x_1 \; ,\; x_2-2\leq x_1 \;,
\end{split}$$ which is $(\underline{x}_1,\underline{x}_2)=(2,-\infty)$. The next value is $\lambda_1=(2+\underline{x}_1)\vee (-4+\underline{x}_2)=4$. Then, ${\phi}(\lambda_1)=1.5$ and $\sigma(1)= 2$, $\sigma(2)= 2$, $\sigma(3)= 1$, $\sigma(4)= 1$, $\sigma(5)= 1$, $\sigma(6)=3$, $\sigma(7)= 3$ and $\sigma(8)=3$ is a new optimal strategy for player Max. For the next Newton iteration, we find the minimal solution of the system $$\begin{split}
0\leq x_2 \; ,\; 0\leq x_2-1 \; ,\; 0\leq 1+x_1 \; ,\;
\; x_2-4\leq x_1 \; , \\
(x_2-3)\vee 0\leq x_1+2
\; ,\; x_2-5\leq 0 \; ,\; x_2-6\leq 0 \; ,
\end{split}$$ which is $(\underline{x}_1,\underline{x}_2)=(-1, 1)$. Then, the next value is $\lambda_2=(2+\underline{x}_1)\vee (-4+\underline{x}_2)=1$. Now ${\phi}(\lambda_2)=0.5$ and $\sigma(1)=2$, $\sigma(2)=2$, $\sigma(3)=2$, $\sigma(4)=1$, $\sigma(5)=3$, $\sigma(6)=3$, $\sigma(7)=3$ and $\sigma(8)=3$ is the optimal strategy for player Max. For the next Newton iteration, we find the minimal solution of the system $$\begin{split}
0\leq x_2 \; ,\; 0\leq x_2-1 \; ,\; 0\leq x_2-2 \; , \;
(x_2-3)\vee 0 \leq x_1+2
\; , \\
x_2-4\leq 0 \; ,\; x_2-5\leq 0 \; ,\; x_2-6\leq 0 \; ,
\end{split}$$ which is $(\underline{x}_1, \underline{x}_2)=(-2, 2)$. This gives $\lambda_3=(2+\underline{x}_1)\vee (-4+\underline{x}_2)=0$, which is the optimal value $\lambda^\ast$. The optimality of $\lambda^\ast=0$ can be certified applying Theorem \[1st-cert\], see Example \[Example2\] above.
The vectors $(2,-\infty)$, $(-1,1)$ and $(-2,2)$ found by the Newton iterations are indicated on the left-hand side of Figure \[tropprogs-ex\] as “1”, “2” and “3”.
The right-hand side of Figure \[tropprogs-ex\] displays the graph of ${\phi}(\lambda)$, together with the Newton iterations. The graphs of partial spectral functions ${\phi}^{\sigma}(\lambda)$ are given by red dashed lines.
Maximization
------------
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![The maximization problem of Example \[Example1\]: the tropical polyhedron and the spectral function.[]{data-label="fig-max"}](ex-max-poly){width="0.75\linewidth"} ![The maximization problem of Example \[Example1\]: the tropical polyhedron and the spectral function.[]{data-label="fig-max"}](ex-max-sf23){width="1\linewidth"}
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Consider the maximization problem of Example \[Example1\]. We next apply the positive Newton method to the homogeneous version of the equivalent minimization problem.
With this aim, firstly observe that in this case $n+1=3\in \supp (u)=\{ 3 \}$, and so Condition is always satisfied. Therefore, as explained in Subsection \[ss:kleene\], at each iteration of the positive Newton method applied to this problem, $\lambda_k+v_l$ can be computed by minimizing $y_3$ subject to the system obtained by setting $y_l=0$ in $C y \leq D^{\sigma} y$, which corresponds to Problem in the case of this example.
Let us take $\lambda_0=3$. Then, we obtain that $\sigma(1)= 3$, $\sigma(2)= 3$, $\sigma(3)= 3$, $\sigma(4)= 3$ and $\sigma(5)= 2$ is an optimal strategy for player Max at $\lambda_0$. To perform the Newton iteration we first notice that $l=\sigma(m+1)=\sigma(5)=2$, which means that $\lambda_1+v_2=\lambda_1+3$ is the minimum of $y_3$ subject to the system obtained by setting $y_2=0$ in $C y \leq D^{\sigma} y$, as explained above. Thus, we have to find the minimal solution of the following system: $$\; x_3\geq -1\; , \; x_3\geq (-2+x_1)\vee (-2)
\; ,\; x_3\geq -1+x_1\;
,\; x_3\geq x_1\; ,$$ which is $(\underline{x}_1,\underline{x}_3)=(-\infty,-1)$. The full vector $y=(-\infty , 0 ,-1)$ is a translate of $y+1=(x_1 , x_2 , 0)$, where $(x_1,x_2)=(-\infty,1)$ is marked as “1” at the left of Figure \[fig-max\]. Meanwhile we obtain $\lambda_1=\underline{x}_3-3=-4$, and $\sigma(1)= 1$, $\sigma(2)= 3$, $\sigma(3)= 2$, $\sigma(4)= 2$ and $\sigma(5)= 2$ is now a new optimal strategy for player Max. Again, here $l=\sigma(m+1)=\sigma(5)=2$ and so the second columns of $C$ and $D^{\sigma}$ are the free terms in . We have to find the minimal solution of the following system: $$\; x_1\geq -1\; ,\; x_3\geq (-2+x_1)\vee (-2)
\; , \; 0\geq -1+x_1\; ,
\; 2\geq x_1\;,$$ which is $(\underline{x}_1,\underline{x}_3)=(-1,-2)$. We obtain $\lambda_2=\underline{x}_3-3=-5$, which is the optimal value, and so the value of the original maximization problem is $5$. The full vector $y=(-1, 0, -2)$ is a translate of $y+2=(x_1, x_2, 0)$, where $(x_1,x_2)=(1,2)$ is marked as “2” at the left of Figure \[fig-max\].
As in the case of Figure \[tropprogs-ex\], the right-hand side of Figure \[fig-max\] displays the graph of ${\phi}(\lambda)$, together with the Newton iterations. The graphs of partial spectral functions ${\phi}^{\sigma}(\lambda)$ are given by red dashed lines.
Numerical experiments {#ss:NumExp}
---------------------
A preliminary implementation of the bisection and Newton methods for tropical linear-fractional programming was developed in MATLAB. We next present some graphs showing how they behave on randomly generated instances of tropical linear-fractional programming problems, in which the entries of matrices and vectors range from $-500$ to $500$. The matrices $A$ and $B$ in and are square, with dimensions ranging from $1$ to $400$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Number of iterations of Newton method (thin blue line) and bisection method (thin red line) in the cases of minimization (left) and maximization (right). Thick blue line: average number of iterations of Newton method for each interval of 20 dimensions. Thick red line: level $\log(2M)\approx 10$.[]{data-label="f:minmax"}](minnewtp0dims1to400mod){width="1.4\linewidth" height="0.887\linewidth"} ![Number of iterations of Newton method (thin blue line) and bisection method (thin red line) in the cases of minimization (left) and maximization (right). Thick blue line: average number of iterations of Newton method for each interval of 20 dimensions. Thick red line: level $\log(2M)\approx 10$.[]{data-label="f:minmax"}](maxnewtp0dims1to400mod){width="\linewidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[f:minmax\] displays the cases of the tropical linear programming , in which all the entries are finite. Here the certificates of unboundedness reduce to the solvability of a two-sided tropical system of inequalities. When a feasible and bounded problem is generated, it is solved by the bisection and Newton methods.
For the bisection method, we use the lower initial values $\underline{\lambda}_0$ of [@BA-08], see also Remark \[r:bisbounds\]. Following [@BA-08], the upper initial values $\overline{\lambda}_0$ for the bisection method come from a solution of $Ax\vee c\leq Bx\vee d$. To find this solution, we use the policy iteration of [@DG-06] instead of the alternating method of [@CGB-03]. Shown by the thin red line (up to $m=n=250$), the bisection method worked similarly in the case of minimization and maximization. In our experiments, the interval between lower and upper initial values never exceeded $2M=1000$, with the number of iterations quickly approaching a constant level of $9$ or $10$ iterations ($\log M\approx 9$ or $\log 2M\approx 10$).
![Number of iterations of Newton method (thin blue line) and bisection method (thin red line) in the case of minimization with $M=500000$. Thick blue line: average number of iterations of Newton method for each interval of 20 dimensions. Thick red line: level $\log(2M)-1\approx 19$.[]{data-label="f:bigentries"}](largecoeffs1){width="0.5\linewidth"}
The thin blue line represents the run of Newton method, and the thick blue line represents their average number calculated for each interval of $20$ dimensions. For the sake of fair comparison with the bisection method, the initial value $\lambda_0$ coincides with the upper initial value $\overline{\lambda}_0$ for the bisection method. This value comes from a solution of $Ax\vee c\leq Bx\vee d$, instead of the theoretical value $2M(n+1)$, which depends on $n$ and may be much greater. In the case of minimization, the average number of Newton iterations slowly grows, being smaller than $10$ before $n\approx 250$, but exceeding $10$ at larger dimensions. Naturally, the number of iterations for the same dimension may be very different, depending on the configuration and complexity of the tropical polytopes (i.e., the solution sets of $Ax\vee c\leq Bx\vee d$). In the case of maximization, the number of iterations is usually below $5$. Note that maximization is resolved immediately if we find the greatest point of the solution set, which suggests that the maximization problem may be simpler. We also remark that there is no correlation between the number of iterations of the bisection and Newton methods. In particular, it is easy to construct instances with large integers in which the number of bisection iterations becomes arbitrarily large, whereas the number of Newton iterations remains bounded. This agrees with Figure \[f:bigentries\], where in comparison to the graph on the left-hand side of Figure \[f:minmax\], $M$ is equal to $500000$ instead of $500$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Number of iterations of Newton method (thin blue line) and bisection method (thin red line) in the cases of linear-fractional programming with finite entries (right) and when the average proportion of $-\infty$ entries is 0.7 (left). Thick blue line: average number of iterations of Newton method for each interval of 20 dimensions. Red line on the left: level $\log(2M)\approx 10$.[]{data-label="f:gtp"}](generalnewtp0){width="\linewidth"} ![Number of iterations of Newton method (thin blue line) and bisection method (thin red line) in the cases of linear-fractional programming with finite entries (right) and when the average proportion of $-\infty$ entries is 0.7 (left). Thick blue line: average number of iterations of Newton method for each interval of 20 dimensions. Red line on the left: level $\log(2M)\approx 10$.[]{data-label="f:gtp"}](generalnewtp0dims1to400mod){width="\linewidth"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure \[f:gtp\] displays the cases of tropical linear-fractional programming with all entries finite (right) and with a $0.7$ frequency of $-\infty$ entries (left). In the case of $-\infty$ entries, as required by Assumptions 1 and 2, we ensure that the set of constraints contains neither $-\infty$ rows on the right-hand side nor $-\infty$ columns on the left-hand side. The case of tropical linear-fractional programming with finite entries shows almost the same picture as in the case of minimization above. The case when $-\infty$ appears with a regular frequency is even more favorable for Newton method, due to the sparsity of $\Bipdig$.
Conclusion
==========
In this paper, we developed an algorithm to solve tropical linear-fractional programming problems. This is motivated by the works [@AGG08] and [@AllamigeonThesis], in which disjunctive invariants of programs are computed by tropical methods: tropical linear-fractional programming problems are needed to tropicalize the method of templates introduced by Sankaranarayanan, Colon, Sipma and Manna [@Sriram1; @Sriram2].
The main technical ingredient, which combines ideas appearing in [@AGG-10; @AGK-10; @GStwosided10] is to introduce a [*parametric*]{} zero-sum two-player game (in which the payments depend on a scalar variable), in such a way that the value of the initial tropical linear-fractional programming problem coincides with the smallest value of the variable for which the game is winning for one of the players. The value of the parametric game, which we call the [*spectral function*]{}, is a piecewise affine function of the variable. Then, the problem is reduced to finding the smallest zero of the spectral function, which we do by a Newton-type algorithm, in which at each iteration, we solve a one-player auxiliary game.
Using this game-theoretic connection, we present concise certificates expressed in terms of the strategies of both players, allowing one to check whether a given feasible solution is optimal, or whether the tropical linear-fractional programming problem is unbounded. This is inspired by [@AGK-10], in which certificates of the same nature were given for the simpler problem of certifying whether a tropical linear inequality is a logical consequence of a finite family of such inequalities.
We also develop a generalization of the bisection method of [@BA-08]. The latter, as well as Newton method, are shown to be pseudo-polynomial. Note that at each iteration, both methods call an oracle solving a mean payoff game problem (for which the existence of a polynomial time algorithm is an open question - only pseudo-polynomial algorithms are known). The pseudo-polynomial bound that we give for Newton method is worse than the one concerning the bisection method, however, for the former we also give a non pseudo-polynomial bound, involving the number of strategies, which is better than the pseudo-polynomial bound if the integers of the instance are very large. This is confirmed by experiments, with a preliminary implementation, which indicates that Newton method scales better as the size of the integers grows. In addition, it has the advantage of maintaining feasibility, and there are significant special instances in which it converges in very few iterations.
The Newton method of this paper appears as a natural product of the game-theoretic connection and the spectral function approach. We further concentrate on the implementation of each Newton step by reduction to optimal path algorithms, and on the proof of pseudo-polynomiality. This method could be also considered in the framework of more abstract Newton methods in a generalized domain, which also means making decent comparison with other Newton schemes, like [@esparza:approximative]. The comparison of Newton and bisection methods, as well as possible alternative approaches, also remain to be further examined.
#### Acknowledgement
The authors thank the anonymous reviewers for numerous important suggestions, which helped us to improve the presentation in this paper. The authors are also grateful to Peter Butkovič for many useful discussions concerning tropical linear programming and tropical linear algebra. The first author thanks Xavier Allamigeon and Éric Goubault for having shared with him their insights on disjunctive invariants and static analysis.
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[^1]: The first author was partially supported by the Arpege programme of the French National Agency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005 and by the Digiteo project DIM08 “PASO” number 3389. The third author was supported by the EPSRC grant RRAH12809 and the RFBR-CNRF grant 11-01-93106. This work was initiated when this author was with the School of Mathematics at the University of Birmingham.
[^2]: The game can start also at a node $i$ of Max, the requirement to be started by Min is for better consistency with min-max functions and two-sided tropical systems, see and .
|
---
abstract: 'We report on the measurement of optical isotope shifts for $^{38,39,42,44,46\text{-}51}$K relative to $^{47}$K from which changes in the nuclear mean square charge radii across the $N=28$ shell closure are deduced. The investigation was carried out by bunched-beam collinear laser spectroscopy at the CERN-ISOLDE radioactive ion-beam facility. Mean square charge radii are now known from $^{37}$K to $^{51}$K, covering all $\nu f_{7/2}$-shell as well as all $\nu p_{3/2}$-shell nuclei. These measurements, in conjunction with those of Ca, Cr, Mn and Fe, provide a first insight into the $Z$ dependence of the evolution of nuclear size above the shell closure at $N=28$.'
address:
- 'Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany'
- 'Instituut voor Kern- en Stralingsfysica, KU Leuven, B-3001 Leuven, Belgium'
- 'Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, B-1050 Brussels, Belgium'
- 'Physics Department, CERN, CH-1211 Geneva 23, Switzerland'
- 'Institut für Kernchemie, Universität Mainz, D-55128 Mainz, Germany'
- 'Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany'
- 'GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany'
- 'Helmholtz-Institut Mainz, 55099 Mainz, Germany'
- 'Department of Physics, New York University, New York, NY 10003, USA'
author:
- 'K. Kreim'
- 'M. L. Bissell'
- 'J. Papuga'
- 'K. Blaum'
- 'M. De Rydt'
- 'R. F. Garcia Ruiz'
- 'S. Goriely'
- 'H. Heylen'
- 'M. Kowalska'
- 'R. Neugart'
- 'G. Neyens'
- 'W. Nörtershäuser'
- 'M. M. Rajabali'
- 'R. Sánchez Alarcón'
- 'H. H. Stroke'
- 'D. T. Yordanov'
bibliography:
- 'K-Paper.bib'
title: Nuclear charge radii of potassium isotopes beyond $N=28$
---
Isotope shift ,Nuclear charge radius ,Potassium ,Collinear laser spectroscopy
Mean square charge radii of nuclei in the calcium region ($Z=20$) have been the subject of extensive investigation, both experimentally [@Touchard1982; @Martensson-Pendrill1990; @Behr1997; @Charlwood10; @Blaum2008; @Avgoulea11] and theoretically [@Zamick1971; @Talmi1984; @Caurier01; @Zamick10; @Blaum2008; @Avgoulea11]. Although existing data cover the full $\nu f_{7/2}$ orbital, very little is known on the nuclei above N=28 with valence neutrons in the $p_{3/2}$ orbital. Furthermore, substantial structural changes are predicted in this region, including the inversion and subsequent re-inversion of the $\pi s_{1/2}$ and $\pi d_{3/2}$ shell-model orbitals [@Touchard1982; @Papuga12] and the development of subshell closures at $N=32$ and $N=34$ [@Honma2005; @Gade2008; @Sorlin08; @Wienholtz2013a; @Steppenbeck13]. Whilst theoretical models of charge radii across the $\nu f_{7/2}$ shell have been successful in describing the trend observed for calcium [@Talmi1984; @Caurier01], little is known about how this observable would be influenced by the anticipated structural evolution in the region beyond $N=28$.
The inversion of the odd-$A$ potassium ground-state configuration from $I=3/2$ ($\pi d_{3/2}$) in $^{39\text{-}45}$K to $I=1/2$ ($\pi s_{1/2}$) for $^{47}$K ($N=28$) was reported by @Touchard1982 in 1982 and has subsequently been reproduced by nuclear shell-model and mean-field calculations [@Gade06; @Otsuka05; @Otsuka06]. The question of how the $\pi d_{3/2}$ orbital evolves whilst filling the $\nu p_{3/2}$ orbital has been addressed in a recent paper by @Papuga12 for the even-$N$ isotopes $^{49}$K and $^{51}$K, having respectively spin $1/2$ and $3/2$. In the present work, the ground-state structure of the odd-$N$ isotopes $^{48}$K and $^{50}$K and a detailed analysis of the spin determination for $^{51}$K are presented. With the spin measurements on $^{48\text{-}51}$K the contradictory assignments found in decay spectroscopy data [@Krolas11; @Broda10; @Baumann98; @Crawford09; @Perrot06] are resolved. These spins in combination with the magnetic moment of $^{48}$K fully define the region of inversion [@Papuga13].
In this letter we report on the measurement of optical hyperfine structure and isotope shifts for $^{38,39,42,44,46\text{-}51}$K relative to $^{47}$K, from which changes in the nuclear mean square charge radii are deduced. The measurements were carried out at the collinear laser spectroscopy setup COLLAPS [@Neugart1981] at ISOLDE-CERN [@Kugler00]. Neutron-rich potassium isotopes were produced by $1.4$-GeV protons impinging onto a thick UC$_{x}$ target. The isotopes were surface ionized, accelerated to $40$keV, mass separated and directed to the ISOLDE cooler-buncher ISCOOL [@Franberg2008]. After ISCOOL the ion bunches were directed to the collinear laser spectroscopy beam line, where they were neutralized in collisions with potassium vapor. The atomic $4s\ ^{2}S_{1/2}\rightarrow 4p\ ^{2}P_{1/2}$ transition ($\lambda=769.9$nm) was excited by light generated from a cw titanium-sapphire ring laser. Frequency scanning was performed by applying an additional acceleration potential to the neutralization cell, thus varying the Doppler-shifted laser frequency in the reference frame of the atoms. A light collection system consisting of four separate photomultiplier tubes (PMTs) with imaging optics placed perpendicular to the beam path was used to detect the resonance fluorescence photons. Counts from the PMTs were only accepted during the time in which the atom bunches passed through the light collection system. The background from scattered photons was thus reduced by a factor of $\sim10^{4}$ [@Nieminen2003; @Flanagan2009].
Optical spectroscopy of K isotopes is hindered by the relatively slow decay rate of the atomic transition ($3.8 \times 10^7$s$^{-1}$) as well as a low (2.5%) quantum efficiency of the PMTs with a high heat-related dark count rate. In order to perform the measurements on a $^{51}$K beam of approximately $4000$ions/s, a new optical detection region was developed. Eight 100-mm diameter aspheric lenses were used to precisely image the fluorescence of the laser-excited K atoms onto four extended-red multialkali PMTs in the arrangement shown in Fig. \[Detect\]. The light-collection efficiency of this system is approximately twice that of the previous standard system described in @Mueller1983, whilst the background from scattered laser light is an order of magnitude lower.
![\[Detect\] (Color online) Cut through the optical detection region (top view). The atom and laser beams enter the detection region through the charge exchange cell (bottom). For details see text.](New-LCR){width="1.0\linewidth"}
The PMTs were maintained at $-40\,^\circ$C using a refrigerant circulator to reduce dark counts and held under vacuum to prevent ice formation. RG715 colored glass filters were placed in front of the PMTs in order to cut the strong visible beam light originating from stable contaminant beams of Ti and Cr excited in collisions in the charge exchange cell.
![(Color online) Optical spectrum of $^{48}$K. Also shown is the fitted hfs (solid line) and the hfs centroid (vertical line). The ($S_{1/2}$) and ($P_{1/2}$) level scheme is shown as an inset. \[48K\]](48K){width="0.98\linewidth"}
Isotope shift measurements were performed relative to $^{47}$K. The spectra were fitted with a hyperfine structure (hfs) using a line shape composed of multiple Voigt profiles [@Klose12; @Bendali1981], which were found to describe most adequately the asymmetric line shape, using $\chi^{2}$ minimization (see Fig. \[48K\] and \[50K\]). For the fit a two-parameter first-order hyperfine splitting [@Kopfermann58] was used. The magnetic hyperfine parameters $A$ and the centroid of each hfs were extracted. A sample spectrum for $^{48}$K is shown in Fig. \[48K\] together with the fitted hfs spectrum. Also shown are the energy levels of the ground state ($S_{1/2}$) and excited state ($P_{1/2}$) with allowed transitions.
In Table \[Results\] we give the ground-state spins deduced from the measured hfs patterns. The spins measured for $^{37\text{-}47}$K, previously reported in [@Touchard1982; @Koepf1969], are confirmed by the present analysis. Our new spins measured for $^{49}$K and $^{51}$K have been published in [@Papuga12]. The procedure to determine the spins of $^{48}$K, $^{50}$K and $^{51}$K will be described in the following.
[ccD[.]{}[.]{}[3.9]{}D[.]{}[.]{}[3.4]{}D[.]{}[.]{}[3.11]{}]{} $A$ & $I^{\pi}$ & & &\
37 & $3/2^{+}$ & & -265.(4) & -0.163(40)\[199\]\
38 & 3$^{+}$ & -985.9(4)\[34\] & & -0.126(3)\[177\]\
& & & -127.0(53) & -0.140(51)\[174\]\
39 & 3/2$^{+}$ & -862.5(9)\[30\] & & -0.037(8)\[153\]\
& & & 0 & -0.082(15)\[151\]\
40 & 4$^{-}$ & & 125.6(3) & -0.066(16)\[129\]\
41 & 3/2$^{+}$ & & 235.3(8) & 0.036(17)\[108\]\
42 & 2$^{-}$ & -506.7(7)\[17\] & & 0.034(6)\[89\]\
& & & 351.7(19) & 0.026(23)\[88\]\
43 & 3/2$^{+}$ & & 459.0(12) & 0.049(19)\[69\]\
44 & 2$^{-}$ & -292.1(5)\[10\] & & 0.036(5)\[51\]\
& & & 564.3(14) & 0.047(20)\[50\]\
45 & 3/2$^{+}$ & & 661.7(16) & 0.072(21)\[33\]\
46 & 2$^{-}$ & -91.6(5)\[3\] & & -0.002(4)\[16\]\
& & & 762.8(15) & 0.026(21)\[16\]\
47 & 1/2$^{+}$ & 0 & 857.5(17) & 0\
48 & 1$^{-}$ & 67.9(4)\[3\] & & 0.186(3)\[16\]\
49 & 1/2$^{+}$ & 135.3(5)\[6\] & & 0.342(4)\[32\]\
50 & 0$^{-}$ & 206.5(9)\[9\] & & 0.434(8)\[47\]\
51 & 3/2$^{+}$ & 273.2(14)\[11\] & & 0.538(13)\[61\]\
For $^{48}$K our spin assignment is based on the different relative intensities of the hfs components for $^{46}$K ($I=2$) and $^{48}$K. In Fig. \[46K-48K\] we show the intensities for the individual hfs components of $^{46}$K and $^{48}$K relative to the $I+1/2 \rightarrow I+1/2$ (see Fig. \[48K\]) component as a function of laser power. Data points are taken at $0.2$mW and $1.2$mW, the relative intensities at $0$mW represent the theoretical Racah intensities [@Magnante1969]. The hfs of both isotopes have been recorded under identical experimental conditions and the only property that can cause a difference in the relative intensities is the nuclear spin. The relative intensities of $^{46}$K and $^{48}$K are significantly different and therefore the spins cannot be the same. From the measured intensity ratios and their extrapolation to zero laser intensity, compared to Racah values, it is clear that the $^{48}$K spin has to be $I=1$. This supports the $1^{-}$ assignment recently proposed by @Krolas11 and excludes the $2^{-}$ adopted value in the 2006 nuclear data sheets [@Burrows06] .
The hfs of $^{50}$K shows only one peak in the spectrum, see plot a) in Fig. \[50K\]. No fluorescence was observed in a broad scan of $1.4$GHz around the single peak. This is only possible for a ground-state spin of $I=0$, thus supporting the $0^{-}$ assignment by @Baumann98 and excluding the $1^-$ configuration proposed by @Crawford09.
![(Color online) Intensities of the hfs components of $^{46}$K and $^{48}$K relative to the $I+1/2 \rightarrow I+1/2$ component. Data points at $0$mW laser power are expected intensities (Racah intensities) for different nuclear spin values. See text for details.\[46K-48K\]](Graphic_1){width="0.88\linewidth"}
![(Color online) Optical spectra of $^{50}$K a) and $^{51}$K b) as in Fig.\[48K\]\[50K\]](50-51K){width="0.98\linewidth"}
For the determination of the ground-state spin of $^{51}$K the situation is more complex than in $^{48}$K. Spin $I=1/2$ can be excluded since four, and not three, peaks are visible in the recorded spectra. Figure \[50K\] b) shows the hyperfine spectrum of $^{51}$K with a fit assuming a spin of $I=3/2$, which yields a good agreement with the data. However, $\chi^2$-analysis alone can not exclude the possibility of spin $5/2$. In our previous work [@Papuga12] spin $3/2$ was assigned mainly on the basis of the magnetic moment. To remove any ambiguity we consider here the relative line intensities, as in the case of $^{48}$K. Due to the exoticity of $^{51}$K, spectra were recorded only at the optimal laser power of $1.2$mW, hence a direct comparison with a particular isotope (e.g. $^{39}$K) for different laser powers is not possible. The solution is to compare the relative intensities of the hyperfine components of $^{51}$K with those of several other isotopes at $P=0$mW (Racah intensities) and $P=1.2$mW.
{width="\linewidth"}
Figure \[fig:51K-spin\] shows these intensities of $^{38,39,46,47,48,51}$K relative to the $I+1/2 \rightarrow I+1/2$ component as a function of laser power. Similar to Fig. \[46K-48K\], the data points are connected to their respective value at $P=0$mW with a solid line. The intensities measured for $^{51}$K are denoted by asterisks. Within the error bars the relative intensities of $^{51}$K agree with those for a ground-state spin of $I=3/2$, corresponding to the spin of $^{39}$K. To define a confidence level for the spin determination the observed relative intensities were plotted against the spin. Figure \[fig:51K-spin-2nd\] shows the intensities of the hyperfine component $I-1/2 \rightarrow I+1/2$ of $^{38,39,46,47,48,51}$K (left plot of Fig. \[fig:51K-spin\]) relative to the $I+1/2 \rightarrow I+1/2$ component. These relative intensities of $^{38,39,46,47,48}$K were fitted with a linear function, represented by the dot-dashed line. The relative intensity determined for $^{51}$K is given as a horizontal line shown with its error band (dashed lines). The intersection of the relative intensity of $^{51}$K with the fitted line projected on the spin axis defines $I=3/2$ as the ground-state spin. The spin $5/2$ (or higher) is more than $2 \sigma$ away, and therefore can be ruled out with a confidence of $95$$\%$. The plots for the two remaining intensity ratios from Fig. \[fig:51K-spin\] confirm this level of confidence. As a result, we determine the ground-state spin of $^{51}$K to be $I=3/2$, which supports the $3/2^+$ assignment made by @Perrot06.
![(Color online) Intensities of the hyperfine component $I-1/2 \rightarrow I+1/2$ of $^{38,39,46,47,48,51}$K relative to the $I+1/2 \rightarrow I+1/2$ component as a function of ground-state spin. For details see text.[]{data-label="fig:51K-spin-2nd"}](51K-spin-2_vs-spin){width="0.98\linewidth"}
With the measured spins for $^{48,50}$K, along with the magnetic moment of $^{48}$K, the discussion on the evolution of the $\pi d_{3/2}$ orbital when filling the $\nu p_{3/2}$ orbital can now be completed. The earlier reported spins and magnetic moments of $^{49}$K and $^{51}$K revealed a re-inversion of the $\pi s_{1/2}$ and $\pi d_{3/2}$ levels back to their normal ordering, as neutrons are added from $N=30$ to $N=32$ [@Papuga12]. A comparison of the measured observables to large-scale shell-model calculations has shown that like in $^{47}$K ($N=28$), the $\pi s_{1/2}^{-1}$ hole configuration is the dominant component in the $^{49}$K ground state wave function, although its magnetic moment suggests that the wave function has a significant admixture of a $\pi d_{3/2}^{-1}$ hole configuration. In $^{51}$K the wave function is dominated by the normal $\pi d_{3/2}^{-1}$ hole and from its magnetic moment a rather pure single particle wave function can be inferred. The wave functions of the odd-odd $^{48,50}$K isotopes have been investigated using the same effective shell-model calculations. A comparison to the experimental data along with a discussion on the occupation of the proton orbitals at $N=29$ and $N=31$ will be presented in a forthcoming paper (@Papuga13).
The measured isotope shifts relative to $^{47}$K and literature values referenced to $^{39}$K [@Touchard1982; @Behr1997] are given in Table \[Results\], where statistical errors are given in round brackets. The systematic errors, arising mainly from the uncertainty of Doppler shifts depending on the beam energy, are given in square brackets.
The isotope shift between isotopes with atomic numbers $A, A'$ is related to the change in the nuclear mean square charge radii through: $$\delta \nu ^{A,A'}=\nu^{A'}-\nu^{A}=K\frac{m_{A'}-m_{A}}{m_{A'}m_{A}}+F \delta \langle r^{2}\rangle^{A,A'}
\label{deltanu}$$ with $\nu^{A}$ and $\nu^{A'}$ representing the transition frequencies with respect to the fine structure levels, $K=K_{\text{NMS}}+K_{\text{SMS}}$ the sum of the normal and the specific mass shift, $m_{A}$ and $m_{A'}$ the atomic masses, $F$ the electronic factor, and $\delta \langle r^{2}\rangle^{A,A'}=\langle r^{2}\rangle^{A'}-\langle r^{2}\rangle^{A}$ the change in the nuclear mean square charge radius. For the extraction of $\delta \langle r^{2}\rangle^{A,A'}$ from the measured isotope shifts, the specific mass shift ($K_{\text{SMS}}=-15.4(3.8)\,$GHzu) and the electronic factor ($F= -110(3)$MHzfm$^{-2}$) calculated in [@Martensson-Pendrill1990] were used. The normal mass shift ($K_{\text{NMS}} =\nu^{47}$m$_{e}$) was calculated to $K_{\text{NMS}} =213.55$GHzu using the D1-frequency of $^{39}$K measured in [@Falke2006]. The masses for $^{37}$K-$^{51}$K were taken from [@AME2012]. The calculated values of $\delta \langle r^{2}\rangle^{47,A}$ are shown in Table \[Results\]. For the isotope shifts from [@Touchard1982; @Behr1997] the same procedure was used to calculate the $\delta \langle r^{2}\rangle^{47,A}$ in order to obtain a consistent set of $\delta \langle r^{2}\rangle^{47,A}$ over the entire potassium chain. The statistical error on $\delta \langle r^{2}\rangle^{47,A}$ (round brackets) originates from the statistical error on the isotope shift whilst the systematic error (square brackets) is dominated by the uncertainty on $K_{\text{SMS}}$ and is correlated for all isotopes.
In Fig. \[radii\] we compare the root mean square (rms) charge radii $\langle r^{2}\rangle^{1/2}$ of argon [@Blaum2008], calcium [@Vermeeren92], scandium [@Avgoulea11], titanium [@Gangrsky04], chromium [@Angeli04], manganese [@Charlwood10], iron [@Benton97], and potassium from this work. The $\langle r^{2}\rangle^{1/2}$ values have been obtained by using the originally published changes in mean square charge radii and absolute reference radii from the compilation of Fricke and Heilig [@Fricke2004]. The trend of the isotopic chains below $N=28$ has been discussed in [@Blaum2008; @Avgoulea11]. In this region the behavior of the rms charge radii displays a surprisingly strong dependence on $Z$ (Fig. \[radii\]) with a general increase as a function of $N$ for Ar developing into a dominantly parabolic behavior for Ca and then progressing towards the anomalous downward sloping trends in Sc and Ti. Such dramatic variations of the radii are not observed in the regions around $N=50,82$ and $126$ [@Angeli09; @Otten89].
![(Color online) Root mean square nuclear charge radii versus neutron number for the isotopes of Ar, K, Ca, Sc, Ti, Cr, Mn and Fe. Data for K results from this work. \[radii\]](radii){width="0.98\linewidth"}
To date, no single theoretical model has fully described the strongly $Z$-dependent behavior of radii across $Z=20$ [@Avgoulea11], although a variety of approaches have shown some success in describing the observed trends for specific isotopic chains.
To discuss the new results for $N>28$ we plot the $\delta \langle r^{2}\rangle$ of potassium given in Table \[Results\] from $N=23$ to $N=32$ together with those of Ar, Mn, Ca, Ti, Cr and Fe referenced to the $N=28$ shell closure, see Fig. \[dradii\]. The correlated systematic error in the potassium data is represented by the gray shaded area surrounding the curve.
Looking at the $\delta \langle r^{2}\rangle$ curves one can see the above discussed $Z$ dependence of the radii in a broad structure below $N=28$. Common to all elements is the strong shell-closure effect at $N=28$. Above $N=28$ the $\delta \langle r^{2}\rangle$ values show a steep increasing slope, which is similar for all the elements and thus nearly independent of the number of protons. Below $N=28$ the $\delta \langle r^{2}\rangle$ values show large differences between the elements illustrating the atypical $Z$ dependence below $N=28$.
What changes at $N=28$? Up to $N=28$ the protons and neutrons in the calcium region ($18\leq Z\leq26$) occupy the same orbitals ($sd$ and $f_{7/2}$) resulting in a complex interplay of proton and neutron configurations. This fact is underlined by the shell-model calculations of @Caurier01, where the characteristic radii trend (see Fig. 2. in [@Caurier01]) in calcium between $N=20$ and $N=28$ is reproduced qualitatively by allowing multi-proton - multi-neutron excitations from the $sd$ to the $fp$ shell. Above $N=28$ the neutrons fill the $p_{3/2}$ orbital. Here the changes in mean square radii show little or no dependence on $Z$, indicating that charge radii are purely driven by a common and collective polarization of the proton distribution by the neutrons in the $p_{3/2}$ orbital with little or no dependence on the specific proton configuration at the Fermi surface. This absence of strong proton configuration dependence is further emphasized by the consistency of the $^{47,49}$K radii, having an inverted ground-state proton configuration, with the radii of the remaining K isotopes as well as all other measured radii in the region. Furthermore the changes in the mean square radii of potassium show no indication of a shell closure at $N=32$, as discussed in [@Honma2005; @Gade2008; @Sorlin08]. Theoretical and experimental evidence of a shell effect at $N=32$ exists for Ca, Ti and Cr from the systematics of the first $2^+$ state energies $E_{x}(2^{+}_{1})$ or two-neutron separation energies $S_{2n}$ with the strongest effect in Ca [@Honma2005; @Sorlin08; @Wienholtz2013a; @Steppenbeck13].
Looking at the branch of $\delta \langle r^{2}\rangle$ of K above $N=28$ one can see that the point for $^{48}$K ($N=29$) deviates from the general odd-even behavior in the K chain as well as the great majority of all nuclei. This general behavior of “normal” odd-even staggering is expressed by a smaller radius of the odd-$N$ compared to the neighboring even-$N$ isotopes. Obviously the value for $^{48}$K is larger, corresponding to an “inverted” odd-even staggering. This feature might be related to the particular nature of the ground state wave function of $^{48}$K. Shell- model calculations [@Papuga13] show that the ground states of $^{47}$K and $^{49}$K are dominated by an $s_{1/2}$ configuration, while for $^{48}$K the dominant part comes from $d_{3/2}$.
The results of the rms charge radii are finally compared to theoretical calculations in the framework of the mean field (MF) model. Both the non-relativistic and relativistic MF approaches are considered because they are known to give rise to different descriptions of the spin-orbit field. The non-relativistic MF traditionally makes use of a phenomenological two-body spin-orbit term in the Skyrme force with a single parameter adjusted to experimental data, such as single-particle level splittings. The form of the spin-orbit term is, however, chosen purely by simplicity. In contrast, in the relativistic MF, the spin-orbit field emerges as a consequence of the underlying Lorentz invariance, and its form does not have to be postulated a priori. This approach, for the first time, led to a proper description of the generally observed charge-radii kinks at magic neutron numbers for the example of Pb [@sharma93]. As shown in Fig. \[radii-theory\], such a kink is also predicted by the relativistic MF calculation, based on the DD-ME2 interaction [@lala05], for the K isotopic chain, though the effect is less pronounced than found in our measurement. In contrast, none of the standard Skyrme forces with the traditional spin-orbit term gives rise to a proper description of the kinks. Such a behavior is illustrated in Fig. \[radii-theory\], where the rms nuclear charge radii calculated with the Skyrme-HFB model [@Goriely2013], HFB-24, are compared with experimental data. The parameters of the Skyrme interaction (BSk24) were in this case determined primarily by fitting measured nuclear masses and the properties of infinite nuclear matter. It has been shown that a generalization of the Skyrme interaction [@rein95; @sharma95] should also be able to map the relativistic spin-orbit field and improve the description of charge radii across the shell closures.
It should be mentioned that neither the relativistic nor the non-relativistic MF models reproduce the parabolic shape between $N=20$ and $28$, nor the odd-even staggering. In particular, both calculations make use of the equal filling approximation and violate the time-reversibility in the treatment of odd nuclei, and consequently are not adequate for a proper description of the odd-even effect.
The hyperfine structure and isotope-shift measurements on $^{38,39,42,44,46\text{-}51}$K performed in the present work offer an insight into the development of the mean square charge radii in potassium beyond the $N=28$ shell closure. The now accessible range of K isotopes with neutron numbers $N=18$-$32$, including the previous data on $^{37\text{-}47}$K [@Touchard1982; @Behr1997], covers the full $f_{7/2}$ and $p_{3/2}$ orbitals. Our measurement contributes substantially to a systematic investigation of mean square nuclear charge radii in the calcium region beyond the $N=28$ shell closure. The most striking effect is the difference in the behavior of the rms charge radii below and above $N=28$. In addition the measured spins of $^{48,50}$K together with the spins reported by @Papuga12 allow now assigning spins and parities of excited levels in these isotopes based on a firm ground and resolve the inconsistency between earlier reported data.
This work was supported by the Max-Planck-Society, BMBF (05 P12 RDCIC), the Belgian IAP-projects P6/23 and P7/10, the FWO-Vlaanderen, the NSF grant PHY-1068217 RC 100629, and the EU Seventh Framework through ENSAR (no. 262010). We thank the ISOLDE technical group for their support and assistance.
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---
abstract: 'We theoretically investigate the superfluid density and Berezinskii-Kosterlitz-Thouless (BKT) transition of a two-dimensional Rashba spin-orbit coupled atomic Fermi gas with both in-plane and out-of-plane Zeeman fields. It was recently predicted that, by tuning the two Zeeman fields, the system may exhibit different exotic Fulde-Ferrell (FF) superfluid phases, including the gapped FF, gapless FF, gapless topological FF and gapped topological FF states. Due to the FF paring, we show that the superfluid density (tensor) of the system becomes anisotropic. When an in-plane Zeeman field is applied along the *x*-direction, the tensor component along the *y*-direction $n_{s,yy}$ is generally larger than $n_{s,xx}$ in most parameter space. At zero temperature, there is always a discontinuity jump in $n_{s,xx}$ as the system evolves from a gapped FF into a gapless FF state. With increasing temperature, such a jump is gradually washed out. The critical BKT temperature has been calculated as functions of the spin-orbit coupling strength, interatomic interaction strength, in-plane and out-of-plane Zeeman fields. We predict that the novel FF superfluid phases have a significant critical BKT temperature, typically at the order of $0.1T_{F}$, where $T_{F}$ is the Fermi degenerate temperature. Therefore, their observation is within the reach of current experimental techniques in cold-atom laboratories.'
author:
- 'Ye Cao$^{1,2}$'
- 'Xia-Ji Liu$^{1}$'
- 'Lianyi He$^{3}$'
- 'Gui-Lu Long$^{2,4,5}$'
- 'Hui Hu$^{1}$'
title: 'Superfluid density and Berezinskii-Kosterlitz-Thouless transition of a spin-orbit coupled Fulde-Ferrell superfluid'
---
Introduction
============
Over the past decade, the technique of manipulating ultracold atomic Fermi gases has been well developed and it offers a physical reality to pursue an exotic pairing mechanism, which is referred to as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [@Fulde1964; @Larkin1964] and has attracted impressive attentions in different physical areas [@Casalbuoni2004; @Uji2006; @Kenzelmann2008; @Liao2010; @Gerber2014]. In spin-imbalanced Fermi gases, the standard Bardeen-Cooper-Schrieffer (BCS) pairing is not favorable against the FFLO pairing with a finite center-of-mass momentum. Although there is no unambiguous experimental conclusion for the FFLO superfluidity, strong evidence has been seen in a Fermi cloud of $^{6}$Li atoms confined in quasi-one-dimensional harmonic traps near a crossover from a Bose-Einstein condensate (BEC) to a BCS superfluid [@Liao2010; @Orso2007; @Hu2007; @Liu2007; @Liu2008; @Guan2007].
The FFLO pairing is also favored by spin-orbit coupling [@YipReview; @Barzykin2002; @Agterberg2007; @Dimitrova2007; @Michaeli2012]. Motivated by the recent experimental realization of a synthetic spin-orbit coupling with equal weight combination of Rashba and Dresselhaus components [@Lin2011; @Wang2012; @Cheuk2012; @Fu2013], FF superfluidity - a specific form of the FFLO superfluidity - has been theoretically investigated in spin-orbit coupled atomic Fermi gases [@reviewSOC; @Dong2013; @Zheng2013; @Wu2013; @Liu2013a; @Dong2013NJP; @Zhou2013; @Iskin2013; @Shenoy2013]. In the case of a Rasbha spin-orbit coupling, topological superfluidity is argued to be achievable [@Zhang2008; @Sato2009; @Sau2010; @Oreg2010; @Zhu2011; @Liu2012a; @Liu2012b; @Wei2012], although the underlying pairing is of *s*-wave character. It turns out that the topological superfluidity and FF superfluidity are compatible. As a result, novel topological FF superfluids have also been proposed [@Chen2013; @Liu2013b; @Qu2013; @Zhang2013; @Cao2014; @Hu2014; @Jiang2014]. In particular, in a recent Letter, some of us have predicted that a *gapless* topological FF superfluid may appear in a two-dimensional (2D) spin-orbit coupled atomic Fermi gas with both in-plane and out-of-plane Zeeman fields [@Cao2014]. The purpose of the present work is to provide more details about such an interesting superfluid phase and to discuss its thermodynamic stability by considering the superfluid density and superfluid transition temperature.
It is well known that at finite temperatures the superfluidity of 2D atomic Fermi gases is characterized by the vortex-antivortex (V-AV) binding. The relevant mechanism is the Berezinskii-Kosterlitz-Thouless (BKT) transition occurring at a characteristic temperature $T_{\mathrm{{BKT}}}$ [@Berezinskii1971; @Kosterlitz1972]. Below the critical BKT temperature, a V-AV binding state has a lower free energy and hence superfluidity emerges. The BKT transition was theoretically investigated long time ago in a 2D fermionic system without spin-orbit coupling [@Randeria1989; @Gusynin1999; @Loktev2001]. Following the recent experimental advances, there have been several theoretical investigations about the superfluid density and critical BKT temperature in 2D spin-orbit coupled Fermi gases with BCS pairing [@He2012; @Gong2012; @He2013]. In the case of a large out-of-plane Zeeman field, the temperature region for experimentally observing topological BCS superfluids and related Majorana fermions has been discussed [@Gong2012; @He2013]. However, the BKT physics of a spin-orbit coupled FF superfluid - which can be either gapped or gapless, topologically trivial or non-trivial - has so far not been addressed.
In this work, we explore this interesting issue and study the superfluid density tensor and BKT transition of a 2D Rasbha spin-orbit coupled Fermi gas in the presence of both in-plane and out-of-plane Zeeman fields. By calculating the superfluid density tensor, we obtain the superfluid phase stiffness as functions of the temperature, spin-orbit coupling strength, binding energy (that characterizes the interatomic interaction strength), in-plane and out-of-plane Zeeman fields. This allows us to determine the critical BKT temperature of the system in four different FF superfluid phases [@Cao2014], with a given set of parameters.
Our main results may be summarized as follows: (i) At zero temperature with an applied in-plane Zeeman field in the *x*-direction, the component $n_{s,xx}$ of the superfluid density tensor always changes discontinuously when the system continuously evolves from a gapped FF into a gapless FF phase. The component $n_{s,yy}$ is larger than $n_{s,xx}$ except for a narrow parameter space where the FF momentum is sufficiently large. The two components of the superfluid density tensor decrease monotonically as the temperature increases. (ii) All the four FF superfluid phases have significant critical BKT temperature, except for the parameter region with very small spin-orbit coupling and/or binding energy, or with very large in-plane and/or out-of-plane Zeeman fields. The critical BKT temperature can be enhanced by increasing the binding energy. But it does not increase monotonically as the spin-orbit coupling strength increases.
The rest of the paper is organized as follows. In the next section, we briefly describe the mean-field theoretical framework, and clarify the BKT physics in two dimensions and the related Kosterlitz-Thouless-Nelson (KT-Nelson) criterion for phase transition. Then, we present the expressions for the superfluid density tensor and superfluid phase stiffness. The critical BKT temperature is determined by applying the KT-Nelson criterion. In Sec. III, we first present the finite-temperature phase diagram of the system and then discuss in detail the results on the superfluid density tensor and critical BKT temperature. Finally, Sec. IV is devoted to the conclusions and outlooks.
Model Hamiltonian and mean-field theory
=======================================
We start by considering a 2D spin-orbit coupled two-component Fermi gas near a broad Feshbach resonance with the Rashba spin-orbit coupling $\lambda\bm{\hat{\sigma}}\cdot\bm{\hat{\mathrm{k}}}$, in-plane ($h_{x}$) and out-of-plane ($h_{z}$) Zeeman fields [@notation]. The system can be well described by the following single-channel Hamiltonian, $$\text{\ensuremath{\mathcal{H}}}=\int d{\bf r}\left[\mathcal{H}_{0}+\mathcal{H}_{int}\right],\label{eq:totHami}$$ where
$${\cal H}_{0}=\psi^{\dagger}(\bm{r})\left(\hat{\xi}_{{\bf k}}+\lambda\bm{\hat{\sigma}}\cdot\bm{\hat{\mathrm{k}}}-h_{z}\hat{\sigma}_{z}-h_{x}\hat{\sigma}_{x}\right)\psi(\bm{r})\label{eq:spHami}$$
is the single-particle Hamiltonian and $$\text{\ensuremath{\mathcal{H}}}_{int}=U_{0}\psi_{\uparrow}^{\dagger}({\bf r})\psi_{\downarrow}^{\dagger}({\bf r})\psi_{\downarrow}({\bf r})\psi_{\uparrow}({\bf r})$$ is the density of interaction Hamiltonian in which the bare interaction strength $U_{0}$ is to be regularized as $$\frac{1}{U_{0}}=-\frac{1}{\mathcal{S}}\sum_{\mathbf{k}}\frac{1}{\hbar^{2}\mathbf{k}^{2}/m+E_{b}},$$ with $\mathcal{S}$ being the area of the system and $E_{b}$ the two-particle binding energy that physically characterizes the interaction strength. In the single-particle Hamiltonian, $\lambda$ is the Rashba spin-orbit coupling strength and we have used the following notations: (1) $\hat{\xi}_{{\bf k}}\equiv-\hbar^{2}\nabla^{2}/(2m)-\mu$ with the atomic mass $m$ and chemical potential $\mu$; (2) $\bm{\mathrm{\hat{k}}}=(\hat{k}_{x},\hat{k}_{y})$, where $\hat{k}_{x}=-i\partial_{x}$ and $\hat{k}_{y}=-i\partial_{y}$ are momentum operators; and (3) $\bm{\hat{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y})$, the Pauli matrices. We have also used $\psi({\bf r})=[\psi_{\uparrow}(\bm{r}),\psi_{\downarrow}(\bm{r})]^{T}$ ($\psi^{\dagger}(\bm{r})=[\psi_{\uparrow}^{\dagger}(\bm{r}),\psi_{\downarrow}^{\dagger}(\bm{r})]$) to collectively denote the fermion field operator for creating (annihilating) an atom at ${\bf r}$ with a specific spin $\sigma=\uparrow,\downarrow$.
Mean-field theory
-----------------
We solve the model Hamiltonian Eq. (\[eq:totHami\]) by using the functional path-integral approach [@reviewSOC; @He2013; @Hu2011; @Jiang2011]. At the inverse finite temperature $\beta=1/(k_{B}T)$, the partition function can be written as,
$$\text{\ensuremath{\mathcal{Z}}}=\int\mathcal{\mathcal{D}}\psi\left(\mathbf{r},\tau\right)\mathcal{D}\bar{\psi}\left(\mathbf{r},\tau\right)\exp\left\{ -\mathcal{A\left[\psi,\bar{\psi}\right]}\right\} ,\label{eq:partition1}$$
where
$$\mathcal{A}\left[\psi,\bar{\psi}\right]=\int_{0}^{\beta}d\tau\int d\bm{r}\bar{\psi}\partial_{\tau}\psi+\int_{0}^{\beta}d\tau\mathcal{H}\left(\psi,\bar{\psi}\right).\label{eq:action}$$
Here, the field operators $\psi$ and $\psi^{\dagger}$ in the model Hamiltonian $\mathcal{H}$ have been replaced with the corresponding Grassmann variables $\psi(\mathbf{r},\tau)$ and $\bar{\psi}(\mathbf{r},\tau)$, respectively. Following the standard procedure [@reviewSOC], the interaction term in the Hamiltonian is decoupled using the Hubbard-Stratonovich transformation. Introducing the auxiliary complex pairing field $\phi(\mathbf{r},\tau)=-U_{0}\psi_{\downarrow}(\mathbf{r},\tau)\psi_{\uparrow}(\mathbf{r},\tau)$, and integrating out the Grassmann fields, the partition function becomes
$$\text{\ensuremath{\mathcal{Z}}}=\int\mathcal{\mathcal{D}\phi}\left(\mathbf{r},\tau\right)\mathcal{D}\bar{\phi}\left(\mathbf{r},\tau\right)\exp\left\{ -\mathcal{A}_{eff}\left[\phi,\bar{\phi}\right]\right\} ,\label{eq:partition2}$$
where in the saddle-point approximation (i.e., mean-field treatment by replacing $\phi(\mathbf{r},\tau)$ with a static pairing field $\Delta(\mathbf{r})$), the effective action $\text{\ensuremath{\mathcal{A}}}_{eff}$ takes the form,
$$\text{\ensuremath{\mathcal{A}}}_{mf}=\beta\sum_{\mathbf{k}}\hat{\xi}_{\mathbf{k}}-\int_{0}^{\beta}d\tau\int d\bm{r}\frac{\left|\Delta\right|{}^{2}}{U_{0}}-\frac{1}{2}\textrm{Tr}\ln\left[-G^{-1}\right].\label{eq:effaction}$$
In the above expression, $G^{-1}\left(\mathbf{r},\tau\right)=-\partial_{\tau}-\mathcal{H}_{BdG}$ is the inverse single-particle Green function in the Nambu-Gorkov representation, with a mean-field Bogoliubov Hamiltonian, $$\mathcal{H}_{BdG}=\left[\begin{array}{cc}
H_{0}(\mathbf{\hat{k}}) & -i\Delta(\mathbf{r})\hat{\sigma}_{y}\\
i\Delta(\mathbf{r})\hat{\sigma}_{y} & -H_{0}^{*}\left(-\mathbf{\hat{k}}\right)
\end{array}\right],$$ where $H_{0}\equiv\hat{\xi}_{{\bf k}}+\lambda\bm{\hat{\sigma}}\cdot\bm{\hat{\mathrm{k}}}-h_{z}\hat{\sigma}_{z}-h_{x}\hat{\sigma}_{x}$. In the presence of the in-plane Zeeman field $h_{x}$, it is known that the pairing field takes the FF form $\Delta(\mathbf{r})=\Delta e^{iQx}$, with a finite center-of-mass momentum of the pairs $\mathbf{Q}=Q\mathbf{e}_{x}$ [@Zheng2013; @Wu2013; @Liu2013a; @Dong2013NJP; @Zhou2013]. This helical phase was earlier studied in the context of noncentrosymmetric superconductors [@Agterberg2007; @Dimitrova2007]. The resulting mean-field thermodynamic potential $\Omega_{mf}=k_{B}T\mathcal{A}_{mf}$ reads,
$$\Omega_{mf}=\sum_{\mathbf{k}}\hat{\xi}_{\mathbf{k}}-\mathcal{S}\frac{\Delta^{2}}{U_{0}}-\frac{k_{B}T}{2}\sum_{\mathbf{k},i\omega_{m}}\ln\det\left[-G^{-1}\left(\mathbf{k},i\omega_{m}\right)\right],\label{eq:thermaldynamic}$$
where $G^{-1}(\mathbf{k},i\omega_{m})$ is the inverse Green function in momentum space and $\omega_{m}=\pi(2m+1)/\beta$ with integer $m$ is the fermionic Matsubara frequency. Making use of the inherent particle-hole symmetry of the BdG Hamiltonian, we find that, $$\det\left[-G^{-1}\left(\mathbf{k},i\omega_{m}\right)\right]=\prod_{\eta=1,2}\left[\left(i\omega_{n}\right){}^{2}-\left(E_{\bm{\mathrm{k}}\eta}^{\nu=+}\right){}^{2}\right],$$ where $E_{\mathrm{\bm{k}\eta}}^{\nu}$ is the quasi-particle energy, obtained by diagonalizing $\mathcal{H}_{BdG}$ with the FF pairing field $\Delta(\mathbf{r})=\Delta e^{iQx}$ [@Liu2013a; @Dong2013NJP]. The superscript $\nu\in(+,-)$ represents the particle ($+$) or hole ($-$) branch and the subscript $\text{\ensuremath{\eta\in}(1,2)}$ denotes the upper ($1$) or lower ($2$) branch split by the spin-orbit coupling [@Liu2013a; @Hu2011; @Jiang2011]. By summing over the Matsubara frequency, the mean-field thermodynamic potential takes the form,
$$\begin{aligned}
\Omega_{mf} & = & \frac{1}{2}\sum_{\mathbf{\bm{k}}}\left(\xi_{\mathbf{\bm{k}}+\mathbf{\bm{Q}}/2}+\xi_{\mathbf{\bm{k}}-\mathbf{\bm{Q}}/2}\right)-\frac{1}{2}\sum_{\mathbf{\bm{k}\eta}}|E_{\bm{\mathrm{\bm{k}}}\eta}^{+}|\nonumber \\
& & -k_{B}T\sum_{\mathbf{k\eta}}\ln\left(1+e^{-|E_{\mathbf{k}\eta}^{+}|/k_{B}T}\right)-\mathcal{S}\frac{\Delta^{2}}{U_{0}}.\end{aligned}$$
Here the term $\sum_{\mathbf{k}}\hat{\xi}_{\mathbf{k}}$ is replaced by $(1/2)\sum_{\mathbf{k}}(\xi_{\mathbf{k}+\mathbf{Q}/2}+\xi_{\mathbf{k}-\mathbf{Q}/2})$, in order to cancel the leading divergence of the term $(1/2)\sum_{\mathbf{\bm{k}\eta}}|E_{\bm{\mathrm{\bm{k}}}\eta}^{+}|$.
For a given set of parameters, for example, the temperature $T$, binding energy $E_{b}$ etc., different superfluid phases can be determined using the self-consistent stationary conditions: $$\begin{aligned}
\frac{\partial\Omega_{mf}}{\partial\Delta} & = & 0,\\
\frac{\partial\Omega_{mf}}{\partial Q} & = & 0,\end{aligned}$$ as well as the conservation of total atom number, $$n=-\frac{1}{\mathcal{S}}\frac{\partial\Omega_{mf}}{\partial\mu},$$ where $n=N/\mathcal{S}$ is the number density. At a given temperature, the ground state has the lowest free energy $F=\Omega_{mf}+\mu N$.
Superfluid density tensor
-------------------------
An important quantity to characterize the anisotropic superfluid properties of a 2D spin-orbit coupled Fermi gas is the superfluid density tensor. In the case of BCS pairing, the superfluid density tensor may be analytically derived within mean-field framework [@He2012; @Gong2012; @Zhou2012], yet the formalism has not been obtained for a FF superfluid. According to the definition of the superfluid density, we calculate it by applying a phase twist to the order parameter, $\Delta_{twist}(\mathbf{v}_{s})=\Delta(\mathbf{r})e^{i\mathbf{q}\cdot\bm{r}}$ , which boosts the system with a uniform superfluid flow at a velocity $\mathbf{v}_{s}=\hbar\mathbf{q}/2m$ [@Zhou2012; @Taylor2006; @Fukushima2007]. Here $\Delta(\mathbf{r})$ is the equilibrium FF order parameter. Physically, only the superfluid component moves under the influence of the superfluid flow. Thus, as the result of this boost, the thermodynamic potential assumes the following form in the limit of small velocity,
$$\Omega\left(\mathbf{v}_{s}\right)\simeq\Omega\left(\mathbf{v}_{s}=0\right)+\frac{1}{2}m\mathcal{S}\sum_{ij}n_{s,ij}v_{si}v_{sj},$$
where $n_{s,ij}$ ($i,j=x,y$) is the superfluid density tensor. Therefore, we immediately obtain [@Zhou2012; @Taylor2006; @Fukushima2007], $$n_{s,ij}=\frac{1}{\mathcal{S}}\frac{4m}{\hbar^{2}}\left[\frac{\partial^{2}\Omega\left(\mathbf{v}_{s}\right)}{\partial q_{i}\partial q_{j}}\right]_{\mathbf{q}=0},\label{eq:ns}$$ where $\Omega\left(\mathbf{v}_{s}\right)$ should be calculated with $\Delta_{twist}(\mathbf{v}_{s})$ in the presence of the phase twist. The above relation for the superfluid density tensor is rigorous. In this work, consistent with the mean-field treatment for thermodynamics, in Eq. (\[eq:ns\]) we shall approximate the thermodynamic potential $\Omega\left(\mathbf{v}_{s}\right)$ by its mean-field value $\Omega_{mf}\left(\mathbf{v}_{s}\right)$.
The KT-Nelson criterion for $T_{BKT}$
-------------------------------------
The BKT transition in 2D is peculiar, associated with the spontaneous vortex formation. A unique feature of such a transition is a universal jump in the superfluid density (tensor), characterized by the KT-Nelson criterion for the critical BKT temperature [@Nelson1977]. It may be explained by using the following simple physical picture for the spontaneous creation of a single vortex at finite temperature $T$.
In the absence of spin-orbit coupling and Zeeman fields, let us consider an *isotropic* Fermi superfluid in a circular disk geometry, with a radius of $R\rightarrow\infty$. The kinetic energy cost for creating a single vortex at the origin $\mathbf{r}=\mathbf{0}$ is simply given by, $$E_{V}\simeq\frac{1}{2}mn_{s}\int_{\xi}^{R}d^{2}\mathbf{r}\left(\frac{\hbar}{2mr}\right)^{2}=\frac{\hbar^{2}\pi}{4m}n_{s}\ln\left(\frac{R}{\xi}\right),$$ where $\xi$ is the size of the vortex core. The associated entropy can be calculated by the number of distinct positions at which the vortex can be placed, $$S_{V}\simeq k_{B}\ln\left(\frac{\pi R^{2}}{\pi\xi^{2}}\right)=2k_{B}\ln\left(\frac{R}{\xi}\right).$$ From these two expressions, we see that the free energy associated with the formation of a single vortex is, $$F_{V}=E_{V}-TS_{V}\simeq2\left(\frac{\pi}{2}\frac{\hbar^{2}}{4m}n_{s}-k_{B}T\right)\ln\left(\frac{R}{\xi}\right).$$ It is clear that the free energy changes its sign at a characteristic temperature $T_{BKT}$ determined by $$k_{B}T_{BKT}=\frac{\pi}{2}\mathcal{J},\label{eq:BKT}$$ where $\mathcal{J}=\hbar^{2}n_{s}/(4m)$ is the superfluid phase stiffness. This is the well-known KT-Nelson criterion [@Nelson1977]. As $\ln(R/\xi)$ diverges in the thermodynamic limit $R\rightarrow\infty$, the temperature $T_{BKT}$ separates two qualitatively different regimes. At $T>T_{BKT}$, the free energy is very large and negative, suggesting the spontaneous creation of a free vortex with either positive or negative circulation. While at $T<T_{BKT}$, vortices with opposite circulation will bind together and generate coherence. The spontaneous creation of free vortex suggests that the loss of the phase coherence of the system occurs suddenly. It leads to a universal jump in the superfluid phase stiffness or superfluid density, as can be seen clearly from the KT-Nelson criterion, Eq. (\[eq:BKT\]).
In the case of an *anisotropic* superfluid, we need to define a superfluid density tensor $$\mathscr{\mathcal{N}}_{s}=\left[\begin{array}{cc}
n_{s,xx} & n_{s,xy}\\
n_{s,yx} & n_{s,yy}
\end{array}\right].$$ The associated superfluid phase stiffness takes the form, $$\mathcal{J}=\frac{\hbar^{2}}{4m}\left(\det\mathscr{\mathcal{N}}_{s}\right)^{1/2}=\frac{\hbar^{2}}{4m}\sqrt{n_{s,xx}n_{s,yy}},$$ where in the last equation, we use the fact that $n_{s,xy}=n_{s,yx}=0$, which holds for the system considered in this work.
It is worth noting that although Eq. (\[eq:BKT\]) is obtained by drawing a simple physical picture, it would be a rigorous criterion for the BKT transition. Indeed, the KT-Nelson criterion was first obtained by using a renormalization group analysis [@Nelson1977]. For a microscopic derivation, we may consider the contribution of the pair fluctuations around the saddle-point solution $\delta\phi\left(\mathbf{q},i\nu_{n}\right)$ to the action $\delta\mathcal{A}$, which, at the *Gaussian* (quadratic) level, is given by [@He2012; @He2013; @Salasnich2013; @Yin2014],
$$\delta\mathcal{A}=\frac{1}{2}\sum_{\mathcal{Q}=\mathbf{q},i\nu_{n}}\left[\delta\phi^{\dagger}\left(\mathcal{Q}\right),\delta\phi\left(-\mathcal{Q}\right)\right]\mathbf{M}\left[\begin{array}{c}
\delta\phi\left(\mathcal{Q}\right)\\
\delta\phi^{\dagger}\left(-\mathcal{Q}\right)
\end{array}\right],$$
where the $2\times2$ matrix $$\mathbf{M}\equiv\left[\begin{array}{cc}
M_{11}\left(\mathcal{Q}\right), & M_{12}\left(\mathcal{Q}\right)\\
M_{21}\left(\mathcal{Q}\right), & M_{22}\left(\mathcal{Q}\right)
\end{array}\right]$$ is the inverse two-particle (pair) propagator and its elements can be evaluated with the mean-field fermionic Green function $G(\mathbf{k},i\omega_{m})$. In the case of BCS pairing without the in-plane Zeeman field, the expression of the inverse pair propagator $\bm{\mathrm{M}}$ can be analytically obtained [@He2013; @Salasnich2013]. In particular, in the limit of long wavelength, the matrix elements of $\bm{\mathrm{M}}$ can be expanded as functions of small $\bm{\mathrm{k}}$ and $\omega$. By separating the phase fluctuation and amplitude (density) fluctuation, the low-energy physics of the system can be found to be governed by the well-known classical spin XY model [@He2013; @Salasnich2013], which is the prototype of the BKT physics. In this way, one microscopically derives the superfluid phase stiffness $\mathcal{J}$ and the KT-Nelson relation. The resulting expression for the superfluid phase stiffness *coincides* with the mean-field phase stiffness obtained, for example, by using the mean-field thermodynamic potential in Eq. (\[eq:ns\]). In our FF case, the expression of the superfluid phase stiffness could be derived in a similar manner. However, in this case, the analytical expression of the inverse pair propagator $\bm{\mathrm{M}}$ is more difficult to obtain, although we can numerically sum over the bosonic Matsubara frequency $i\nu_{n}$. Therefore, to calculate the superfluid phase stiffness, we prefer to directly use Eq. (17) with a mean-field thermodynamic potential.
Pair fluctuations beyond mean-field
-----------------------------------
To close this section, we briefly discuss how to improve the mean-field theory. An immediate idea is to work out the Gaussian correction to the action, $\delta\mathcal{A}$, and then use the improved thermodynamic potential around the saddle point $\Delta(\mathbf{r})=\Delta e^{iQx}$ [@Taylor2006; @Fukushima2007; @Hu2006], $$\Omega_{GPF}=\Omega_{mf}+k_{B}T\sum_{\mathcal{Q}=\mathbf{q},i\nu_{n}}\ln\mathbf{M}\left(\mathcal{Q}\right),$$ to calculate the equation of state through the standard thermodynamic relations and the superfluid density tensor via Eq. (\[eq:ns\]). In this way, the thermodynamics and the superfluid density tensor of the system can be consistently determined at the same level of approximation. Alternatively, we may also consider using $\Omega_{GPF}$ to determine the chemical potential $\mu$ and then calculate the superfluid density tensor using the mean-field expression. However, as the trade-off of this cheap treatment, we may have an inconsistency. The resulting critical BKT temperature could be less reliable. For a detailed discussion, we refer to the recent work by Tempere and Klimin [@Tempere2014].
Results and discussions
=======================
Using the above-mentioned mean-field theoretical framework, we have systematically explored the low-temperature phase diagram and the thermodynamic stability of different exotic Fulde-Ferrell superfluid phases. In our numerical calculations, we take the Fermi wavevector $k_{F}=\sqrt{2\pi n}$ and the Fermi energy $E_{F}=\hbar^{2}k_{F}^{2}/(2m)$ as the units for wavevector and energy, respectively. For a typical set of parameters (i.e., default parameters), we use the interaction parameter $E_{b}=0.2E_{F}$, spin-orbit coupling strength $\lambda=E_{F}/k_{F}$, in-plane Zeeman field $h_{x}=0.4E_{F}$, out-of-plane Zeeman field $h_{z}=0.1E_{F}$ and temperature $T=0.05T_{F}$.
Low-temperature phase diagrams
------------------------------
In the recent Letter [@Cao2014], we have discussed the phase diagram and the appearance of an interesting gapless topological Fulde-Ferrell superfluid at a weak interaction strength parameterized by $E_{b}=0.2E_{F}.$ Experimentally, it is most likely that the measurement will be carried out at a stronger interaction strength, where the superfluid transition temperature is anticipated to be higher. In order to optimize the experimental condition for observing the gapless topological superfluid, here we present a systematic study with varying binding energy, from the weakly interacting BCS side to the strongly interacting BEC-BCS crossover regime.
![(color online) Phase diagrams of a 2D spin-orbit coupled atomic Fermi gas at a broad Feshbach resonance and at a typical low temperature $0.05T_{F}$ with (a) $h_{z}=0.1E_{F}$ or (b) $h_{x}=0.4E_{F}$. The strength of spin-orbit coupling is $\lambda=E_{F}/k_{F}$. There are four superfluid phases: gFF, nFF, tnFF and tgFF (whose phase stiffness $\pi\mathcal{J}/2$ - in units of $E_{F}$ - is illustrated in color), as well as a pseudogap phase (grey area). We treat the system as a normal gas (shown in white) when the pairing gap $\Delta<10^{-3}$. In the gapless topological phase, the notations tnFF$_{1}$, tnFF$_{2}$ and tnFF$_{3}$ distinguish different zero-energy contours in energy spectrum. For details, see the contour plots in Fig. \[fig2\].[]{data-label="fig1"}](fig1_phasediagram){width="48.00000%"}
In Fig. 1, we report two phase diagrams at the typical low temperature $T=0.05T_{F}$ on the plane of $E_{b}$-$h_{x}$ (a) or $E_{b}$-$h_{z}$ (b). The superfluid phase stiffness $\pi\mathcal{J}/2$ in different phases is color illustrated and its detailed behavior will be discussed in the next subsection. The superfluid phases are determined using the KT-Nelson criterion $\pi\mathcal{J}(T=0.05T_{F})/2>k_{B}T=0.05E_{F}$. Obviously, there is a pseudogap regime (shown in grey), in which the pairing order parameter is finite but the superfluid phase stiffness is not large enough to drive the BKT transition. A better understanding of the pseudogap phase requires a careful treatment of strong phase fluctuations. It is out of the scope of the present paper.
### gapless topological transition
It is known from previous studies [@Qu2013; @Zhang2013; @Cao2014] that the combined effect of spin-orbit coupling, in-plane and out-of-plane Zeeman fields may induce several exotic superfluid phases: gapped FF (gFF), gapless FF (nFF), gapless topological FF (tnFF) and gapped topological FF (tgFF), classified by considering whether the system has a bulk-gapped and/or topologically non-trivial energy spectrum. In the literature, the topological superfluidity was firstly studied with an out-of-plane Zeeman field only [@reviewSOC; @Zhang2008]. In that case, topological phase transition can be driven by increasing the out-of-plane Zeeman field $h_{z}$ above a threshold $$h_{z,c}=\sqrt{\Delta^{2}+\mu^{2}},\label{eq:tpt}$$ at which the dispersions of the particle- and hole-branches touch each other at the single point $\bm{\mathrm{\bm{k}}}=0$, meanwhile the bulk excitation gap closes. Afterwards, the topology of the Fermi surface dramatically changes and the excitation gap re-opens [@reviewSOC; @Hasan2010; @Qi2011]. It is straightforward to understand the single-point closure of the excitation gap, since the Fermi surface is always rotationally symmetric. This also implies that the resulting topological superfluid must be gapped in the bulk. However, such a scenario may be greatly altered by the presence of a non-zero in-plane Zeeman field, which favors the FF pairing with a finite center-of-mass momentum and consequently breaks the rotational symmetry of the Fermi surface.
In the case of a small in-plane Zeeman field, the rotational symmetry breaking of the energy spectrum is not significant. Although the system becomes a FF superfluid, its bulk excitation gap still closes at the single point $\mathrm{\bm{\mathrm{k}}=0}$, accompanied by the change of the topology of the Fermi surface. An example is the transition from gFF to tgFF shown in Fig. \[fig1\](b) at large binding energy $E_{b}>0.3E_{F}$, where the in-plane Zeeman field is effectively weak. As a result, the picture of the out-of-plane field induced topological phase transition remains unchanged [@Liu2013b; @Qu2013; @Zhang2013].
When the in-plane Zeeman field keeps increasing over a threshold $h_{x,c1}$, however, the closure of the excitation gap and the change of the topology of the Fermi surface may not occur at the same time. A gapless superfluid phase - referred to as nFF - may emerge in the first place at $\bm{\mathrm{\bm{k}}}\neq0$. The nodal points with $E_{\eta=2}^{\nu}(k_{x},k_{y})=0$ form two disjoint loops (see, for example, the transition from gFF to nFF in Fig. \[fig1\](a)). The topology of the Fermi surface only changes when the in-plane Zeeman field further increases up to another critical value $h_{x,c2}$, at which the two nodal loops connect at $\text{\ensuremath{\bm{\mathrm{k}}}}=0$. We refer to the previous work Ref. [@Cao2014] for a detailed characterization of the gapless topological transition.
### Binding energy dependence of the phase diagram
It can now be understood that both the in-plane and out-of-plane fields can drive the topological phase transition, but the underlying property of the resulting topological phase, in terms of the gapless or gapped bulk spectrum, depends critically on the relative strength of the two fields. The gapless topological FF superfluid (tnFF) intentionally emerges in the parameter regime where $h_{x}$ is larger enough relative to $h_{z}$.
This is particularly clear from Fig. \[fig1\](a), where we have fixed the strength of the out-of-plane Zeeman field to $h_{z}=0.1E_{F}$. The tnFF phase accounts for most of the space for topological phases. It is remarkable that the window of the tnFF superfluid remains very significant when the binding energy increases up to $0.5E_{F}$, suggesting the use of a large interaction strength near Feshbach resonances, for the purpose of having a larger BKT transition temperature to observe the exotic tnFF phase. On the contrary, Fig. \[fig1\](b) - where we have fixed the in-plane Zeeman field to $h_{x}=0.4E_{F}$ - clearly reveals that the gapped topological FF superfluid (tgFF) occupies most of the space for topological phases, when the out-of-plane Zeeman field is larger than the in-plane Zeeman field. In this case, the tnFF phase is restricted to the parameter space with a small out-of-plane Zeeman field and a weak interaction strength, as one may anticipate.
### Different gapless topological superfluid phases
![(color online) Dispersion relation of the lower branch $E_{\eta=2}^{\nu}(k_{x},k_{y}=0)$ (left panel, red curves for particle excitations $\nu=+$ and blue curves for hole excitations $\nu=-$) and the corresponding contour of zero-energy nodes (right panel). (a) and (b) correspond to the red point in Fig. \[fig1\] for the tnFF$_{1}$ phase, (c) and (d) the yellow point for the tnFF$_{2}$ phase and, (c) and (f) the magenta point for the tnFF$_{3}$ phase.[]{data-label="fig2"}](fig2_spectrum){width="48.00000%"}
It is interesting that the gapless topological FF superfluid may be further classified into different categories (tnFF$_{1}$, tnFF$_{2}$ and tnFF$_{3}$), according to the number and position of its disjoint loops of nodal points, as shown in the right panel of Fig. \[fig2\]. The tnFF$_{1}$ superfluid is most common and has two nodal loops, one for the particle branch (red loop) and another for the hole branch (blue loop). The tnFF$_{3}$ superfluid also has two nodal loops. However, the loops for the particle and hole branches exchange their position. It occurs only at large in-plane Zeeman field and binding energy. The tnFF$_{2}$ seems to connect the tnFF$_{1}$ and tnFF$_{3}$ phases. It has four disjoint nodal loops and exists only in a very narrow parameter space (see, for example, Fig. \[fig1\](a)). We note that the two gapless topological phases, tnFF$_{1}$ and tnFF$_{3}$, may also be intervened by a *gapped* topological phase, in which there is no nodal loop at all.
Superfluid density
------------------
Having determined the low-temperature phase diagram, we are in position to understand the superfluid density and the critical BKT temperature of different superfluid phases, which have been only briefly mentioned in our previous Letter [@Cao2014]. In the presence of spin-orbit coupling, it is known that the superfluid density is a tensor [@He2012; @He2013]. We then have to consider both diagonal elements of the superfluid density tensor, $n_{s,xx}$ and $n_{s,yy}$.
![(color online) Diagonal elements of the superfluid density tensor as a function of $h_{x}$ and $h_{z}$ at zero temperature (left panel) and at a finite temperature $T=0.05T_{F}$ (right panel). The superfluid density is measured in units of the total density $n=k_{F}^{2}/(2\pi)$. In (a) and (b), the out-of-plane Zeeman field strength $h_{z}=0.1E_{F}$. In (c) and (d), the in-plane Zeeman field strength $h_{x}=0.4E_{F}$. Other parameters are $E_{b}=0.2E_{F}$ and $\lambda=E_{F}/k_{F}$.[]{data-label="fig3"}](fig3_superfluiddensity_hxhz){width="48.00000%"}
In Fig. \[fig3\], we present the Zeeman field dependence of $n_{s,xx}$ and $n_{s,yy}$ at zero temperature (left panel, a and c) and at a finite temperature $T=0.05T_{F}$ (right panel, b and d). In general, as a consequence of the in-plane Zeeman field applied along the $x$-axis, $n_{s,xx}$ is smaller than $n_{s,yy}$, except at extremely low temperature and sufficiently large Zeeman fields.
At zero temperature, $n_{s,xx}$ initially decreases with increasing Zeeman fields and exhibits a sudden drop when the system evolves from the gFF phase into the nFF phase at the threshold $h_{x,c1}$ (or $h_{z,c1}$). At $h_{x}>h_{x,c1}$ (or $h_{z}>h_{z,c1}$) it then rises up gradually and is always enhanced by the Zeeman field. Apart from the discontinuous jump, similar Zeeman-field dependence of the superfluid density has been reported for a gapped BCS topological superfluid across the topological phase transition [@He2013]. Compared with the non-monotonic field dependence of $n_{s,xx}$, we always find that $n_{s,yy}$ decreases continuously with increasing the Zeeman field. Instead of the sudden drop, a kink is observed at the transition from the gFF phase to the nFF phase.
The behavior of the superfluid density is profoundly affected by a nonzero temperature. Already at $T=0.05T_{F}$, the discontinuous drop in $n_{s,xx}$ is smoothed out, leaving a broad minimum with a width $\Delta h_{x,z}\sim2k_{B}T=0.1E_{F}$. Moreover, at the large Zeeman field $h_{x,z}\sim0.6E_{F}$, $n_{s,xx}$ starts to decrease with increasing the Zeeman field. At even higher temperature (not shown in the figure), the local minimum in $n_{s,xx}$ may disappear.
![(color online) Temperature dependence of the diagonal elements of the superfluid density tensor, at the six points shown in Fig. 1(a): (a) $E_{b}=0.21E_{F}$ and $h_{x}=0.58E_{F}$, the tnFF$_{1}$ phase; (b) $E_{b}=0.21E_{F}$ and $h_{x}=0.71E_{F}$, the tnFF$_{2}$ phase; (c) $E_{b}=0.33E_{F}$ and $h_{x}=0.8E_{F}$, the tnFF$_{3}$ phase; (d) $E_{b}=0.4E_{F}$ and $h_{x}=0.789E_{F}$, the tgFF phase; (e) $E_{b}=0.21E_{F}$ and $h_{x}=0.2E_{F}$, the gFF phase; and (f) $E_{b}=0.1E_{F}$ and $h_{x}=0.33E_{F}$, the nFF phase. The superfluid density is measured in units of the total density $n=k_{F}^{2}/(2\pi)$. Other parameters are $h_{z}=0.1E_{F}$ and $\lambda=E_{F}/k_{F}$.[]{data-label="fig4"}](fig4_superfluiddensity_temperature){width="48.00000%"}
In Fig. \[fig4\], we report the temperature dependence of the superfluid density at six typical sets of parameters, which correspond to different superfluid phases at $T=0.05T_{F}$, as shown in Fig. 1(a). $n_{s,xx}$ and $n_{s,yy}$ decrease as temperature increases, in agreement with the common idea that the superfluid component should be gradually destroyed by thermal excitations. It is remarkable that for the gapless topological tnFF$_{1}$ phase (see Fig. \[fig4\](a)), the superfluid density does not decrease rapidly with increasing temperature, implying a sizable critical BKT transition temperature for its experimental observation, as we shall discuss in greater detail in the next subsection. In contrast, the superfluid density of other two gapless topological phases (tnFF$_{2}$ and tnFF$_{3}$ in Figs. \[fig4\](b) and \[fig4\](c), respectively) is more sensitive to temperature and vanishes at $T\sim0.1T_{F}$, probably due to their large Zeeman fields.
Critical BKT temperature and finite-temperature phase diagrams
--------------------------------------------------------------
![(color online) The critical BKT transition temperature as a function of the binding energy $E_{b}$ at different in-plane Zeeman fields (a) or out-of-plane Zeeman fields (b). Here and in the next two figures, the color green, red, blue and yellow in the curves denote the superfluid phase gFF, nFF, tnFF and tgFF, respectively.[]{data-label="fig5"}](fig6_TcBKT_eb){width="48.00000%"}
We now turn to consider the critical BKT temperature, which is determined by the KT-Nelson criterion, $$k_{B}T_{BKT}=\frac{\pi\hbar^{2}}{8m}\left[n_{s,xx}\left(T_{BKT}\right)n_{s,yy}\left(T_{BKT}\right)\right]^{1/2}.\label{eq:BKT2}$$ In the above equation, we have explicitly written down the temperature dependence of the superfluid density, in order to emphasize the fact that the critical BKT temperature should be solved self-consistently. In Figs. \[fig5\], \[fig6\] and \[fig7\], we show the results as a function of the binding energy, Zeeman fields and spin-orbit coupling strength, respectively. These results should be regarded as finite-temperature phase diagrams, as they show clearly which kind of superfluid phases is preferable when temperature decreases. In the curves, we use different colors to distinguish different *emerging* superfluid phases: green for the gFF phase, red for the nFF phase, blue for the tnFF phase and finally yellow for the tgFF phase. It is clear that all the four FF superfluid phases have significant critical BKT temperature except for the parameter regime with very small binding energy $E_{b}$ and/or spin-orbit coupling strength $\lambda$, or with very large in-plane Zeeman field $h_{x}$ and/or out-of-plane Zeeman field $h_{z}$.
![(color online) The critical BKT transition temperature as a function of the out-of-plane Zeeman field $h_{z}$ (a) or the in-plane Zeeman field $h_{x}$ (b).[]{data-label="fig6"}](fig5_TcBKT_hzhx){width="48.00000%"}
As illustrated in Fig. \[fig5\], the critical BKT temperature $T_{BKT}$ always increases monotonically with increasing the binding energy $E_{b}$, as the pairing and superfluidity are enhanced at strong interatomic interactions. The binding energy is the dominant factor in forming Cooper pairs. With a small binding energy, the system is mainly of fermionic character. On the contrary, with a sufficiently large binding energy, the system tends to act as a gas of bosons. Therefore, with increasing the binding energy up to some points, the system would lose its fermionic character near the BEC-BCS crossover (i.e., $E_{b}\sim0.5E_{F}$) and hence should become topologically trivial. Indeed, at large binding energy we observe that the system always approaches the topologically trivial gFF phase. The topological phase, either gapless (tnFF in blue) or gapped (tgFF in yellow), is favored at small binding energy, where the critical BKT temperature is lower. Nevertheless, we find that by suitably tuning the parameters, it is possible to have a gapless topological tnFF phase with a sizable critical BKT temperature $T_{BKT}\sim0.09T_{F}$ for the binding energy up to $E_{b}\simeq0.4E_{F}$ (see, for example, the dot-dashed line at the bottom of Fig. \[fig5\](a)). This temperature is clearly within the reach in current cold-atom experiments [@Ries2014].
On the other hand, the critical BKT temperature decreases monotonically with increasing the Zeeman field, either in-plane or out-of-plane, as shown in Fig. \[fig6\]. It is readily seen that with decreasing temperature the system would first turn into either the tnFF or tgFF phase at sufficiently large in-plane Zeeman field $h_{x}$ or out-of-plane field $h_{z}$, respectively. While at low Zeeman fields, the topologically trivial gFF phase is preferable. This agrees the observation we made in discussing the low-temperature phase diagrams in Fig. \[fig1\].
![(color online) The critical BKT transition temperature as a function of the spin-orbit coupling strength $\lambda$ at different binding energies (a), in-plane Zeeman fields (b) and out-of-plane Zeeman fields (c). []{data-label="fig7"}](fig7_TcBKT_lambda){width="48.00000%"}
It is worth noting that, one may use the binding energy dependence or the Zeeman field dependence of the critical BKT temperature to identify different emerging superfluid phases. This is particularly clear for the gapless tnFF and nFF phases, as the curvature of the $T_{BKT}$ curve for those phases behaves quite differently. For the tnFF phase, the curve is concave; while for the nFF phase, it is convex. This change in curvature (i.e, from concave to convex) seems to be related to the local minimum in the superfluid density component $n_{s,xx}$ that we have reported earlier in Fig. \[fig3\].
We now discuss the critical BKT temperature as a function of the spin-orbit coupling strength $\lambda$, as shown in Fig. \[fig7\]. Compared with the binding energy dependence and Zeeman field dependence, the dependence of $T_{BKT}$ on the spin-orbit coupling strength is non-monotonic and the emerging superfluid phases can re-appear with increasing the coupling strength. Therefore, the $T_{BKT}$ curve is more subtle to understand. Nevertheless, we may identify that the topologically trivial gFF superfluid phase tends to be favorable at large spin-orbit coupling. This is because the pairing gap is usually enhanced by the spin-orbit coupling, which makes the topological phase transition much more difficult to occur (cf. Eq. (\[eq:tpt\])). At small spin-orbit coupling, on the other hand, the critical BKT temperature may dramatically decrease to zero, particularly at a small binding energy and/or a large Zeeman field. Thus, for the purpose of observing the gapless topological tnFF phase, experimentally it seems better to use an intermediate spin-orbit coupling strength, i.e., $\lambda\sim E_{F}/k_{F}$.
Conclusions
===========
In summary, we have presented a systematic investigation of the Berezinskii-Kosterlitz-Thouless transition in a spin-orbit coupled atomic Fulde-Ferrell superfluid in two dimensions. We have calculated the superfluid density and superfluid transition temperature of various Fulde-Ferrell superfluids. We have paid special attention to an interesting gapless topological Fulde-Ferrell superfluid and have clarified that, by suitably tuning the external parameters - for example, the interatomic interaction strength, in-plane and out-of-plane Zeeman fields, and spin-orbit coupling strength - its observation is within the reach in current cold-atom experimental setups.
Our investigation is based on the mean-field theoretical framework, which is supposed to be applicable to a weakly interacting two-dimensional Fermi gas (i.e., the binding energy $E_{b}\leq0.2E_{F}$). For a more reliable and quantitative description, in future studies it would be useful to take into account the strong phase fluctuations by using many-body *T*-matrix theories [@Hu2006; @Hu2008; @Hu2010].
X.J.L. and H.H. were supported by the Australian Research Council (ARC) (Grants Nos. FT140100003, FT130100815, DP140103231 and DP140100637) and the National Key Basic Research Special Foundation of China (NKBRSFC-China) (Grant No. 2011CB921502). L.H. was supported by US Department of Energy Nuclear Physics Office. G.L.L. was supported by the National Natural Science Foundation of China (NSFC-China) (Grant Nos. 11175094, 91221205) and the NKBRSFC-China (Grant No. 2011CB921602).
*Note added*. Recently, a similar publication by Xu and Zhang became public [@Xu2014]. Results qualitatively agree where applicable.
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|
---
abstract: '[New data on the anomalous magnetic moment of the muon together with the [$b \to X_s \gamma~ $]{}decay rate are considered within the supergravity inspired constrained minimal supersymmetric model. We perform a global statistical $\chi^2$ analysis of these data and show that the allowed region of parameter space is bounded from below by the Higgs limit, which depends on the trilinear coupling and from above by the anomalous magnetic moment $a_\mu$. The newest [$b \to X_s \gamma~ $]{}data deviate 1.7 $\sigma$ from recent SM calculations and prefer a similar parameter region as the 2.6 $\sigma$ deviation from $a_\mu$.]{}'
---
-1cm
IEKP-KA/2001-14\
[hep-ph/0106311]{}
[**A global fit to the anomalous magnetic moment, [$b \to X_s \gamma~ $]{} and Higgs limits in the constrained MSSM**]{}\
[**W. de Boer, M. Huber, C. Sander**]{}\
[*Institut für Experimentelle Kernphysik, University of Karlsruhe\
Postfach 6980, D-76128 Karlsruhe, Germany*]{}\
[**D.I. Kazakov**]{}\
[*Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,\
141 980 Dubna, Moscow Region, Russian Federation*]{}
Introduction
============
Recently a new measurement of the anomalous magnetic moment of the muon became available, which suggests a possible 2.6 standard deviation from the Standard Model (SM) expectation[@BNL]: $\Delta a_\mu=a_\mu^{exp}-a_\mu^{th}=(43\pm 16)\cdot 10^{-10}$. The theoretical prediction depends on the uncertainties in the vacuum polarization and the light-by-light scattering, see e.g. the discussion in [@Y]. However, even with a conservative estimate of the theoretical errors, one has a positive difference $\Delta a_\mu$ of the order of the weak contribution to the anomalous magnetic moment, which opens a window for “new physics”. The most popular explanation is given in the framework of SUSY theories [@CM]-[@baer], since the contribution of superpartners to the anomalous magnetic moment of the muon is of the order of the weak contribution and allows to explain the desired difference $\Delta a_\mu$. It requires the Higgs mixing parameter to be positive[@CN] and the sparticles contributing to the chargino-sneutrino $(\tilde{\chi}^\pm - \tilde{\nu}_\mu)$ and neutralino-smuon $(\tilde{\chi}^0 - \tilde{\mu})$ loop diagrams to be relatively light[@CM].
The positive sign of $\mu_0$ is also preferred by the branching ratio of the b-quark decaying radiatively into an s-quark - [$b \to X_s \gamma~ $]{}- [@we]. Last year the observed value of [$b \to X_s \gamma~ $]{}was close to the SM expectation, so in this case the sparticles contributing to the chargino-squark $(\tilde{\chi}^\pm - \tilde{q})$ and charged Higgs-squark $(H^\pm - \tilde{q})$ loops have to be rather heavy in order [*not*]{} to contribute to [$b \to X_s \gamma~ $]{}.
However, it was recently suggested that in the theoretical calculation one should use the running c-quark mass in the ratio $m_c/m_b$, which reduces this ratio from 0.29 to 0.22 [@misiak]. The SM value for [$b \to X_s \gamma~ $]{}increases from $(3.35\pm 0.30)\times 10^{-4}$ to $(3.73\pm 0.30)\times 10^{-4}$ in this case. This value is 1.7 $\sigma$ above the most recent world average of $(2.96\pm 0.46)\times 10^{-4}$, which is the average from CLEO ($(2.85\pm 0.35_{stat}\pm 0.22_{sys})\times 10^{-4}$) [@CLEO], ALEPH ($(3.11\pm 0.80_{stat}\pm 0.72_{sys})\times 10^{-4}$) [@ALEPH] and BELLE ($(3.36\pm 0.53_{stat}\pm 0.42_{sys}(\pm^{0.50}_{0.54})_{model})\times 10^{-4}$) [@BELLE]. For the error of the world average we added all errors in quadrature.
As will be shown, the small deviations from the SM for both $a_\mu$ and [$b \to X_s \gamma~ $]{}require now very similar mass spectra for the sparticles. In the Constrained Minimal Supersymmetric Model (CMSSM) with supergravity mediated breaking terms all sparticles masses are related by the usually assumed GUT scale boundary conditions of a common mass $m_0$ for the squarks and sleptons and a common mass $m_{1/2}$ for the gauginos. The region of overlap in the GUT scale parameter space, where both $a_\mu$ and [$b \to X_s \gamma~ $]{}are within errors consistent with the data, is most easily determined by a global statistical analysis, in which the GUT scale parameters are constrained to the low energy data by a $\chi^2$ minimization.
In this paper we present such an analysis within the CMSSM assuming common scalar and gaugino masses and radiatively induced electroweak symmetry breaking. We use the full NLO renormalization group equations to calculate the low energy values of the gauge and Yukawa couplings and the one-loop RGE equations for the sparticle masses with decoupling of the contribution to the running of the coupling constants at threshold. For the Higgs potential we use the full 1-loop contribution of all particles and sparticles. For details we refer to previous publications[@ZP; @PL].
In principle, one can also require $b-\tau$ Yukawa coupling unification, which has a solution at low and high values of the ratio of vacuum expectation values of the neutral components of the two Higgs doublets, denoted $\tan\beta=\langle H_2^0\rangle /
\langle H_1^0 \rangle$[@ZP; @PL]. From Fig. \[f1\] one observes that if the third generation Yukawa couplings at the GUT scale are constrained by the low energy top, bottom and tau masses, they become equal for $\mu<0$ at ${\mbox{$ \tan\beta~ $}}\approx 40$, while for $\mu>0$ they never become equal, although the difference between the Yukawa couplings is less than a factor three. Since $\mu>0$ is required by $\Delta a_\mu>0$ (see below), we do not insist on Yukawa coupling unification and consider ${\mbox{$ \tan\beta~ $}}$ to be a free parameter, except for the fact that the present Higgs limit of 113.5 GeV from LEP[@newhiggs] requires ${\mbox{$ \tan\beta~ $}}>4.3$ in the CMSSM[@we].
We found that the allowed area of overlap between [$b \to X_s \gamma~ $]{}and $a_\mu$ can be increased considerably for positive values of the common trilinear coupling $A_0$ at the GUT scale. For $A_0>0$ the present Higgs limit becomes more stringent than for the no-scale models with $A_0=0$, as will be shown.
[$a_\mu$]{} and [$b\to X_s\gamma$]{} in the CMSSM
=================================================
The contribution to the anomalous magnetic moment of the muon from SUSY particles are similar to that of the weak interactions after replacing the vector bosons by charginos and neutralinos. The total contribution to $a_\mu$ can be approximated by [@CM] $$|a_\mu^{SUSY}| \simeq \frac{\alpha(M_Z)}{8\pi \sin^2\theta_W}
\frac{m_\mu^2}{m_{SUSY}^2}\tan\beta\left(1-\frac{4\alpha}{\pi}
\ln\frac{m_{SUSY}}{m_\mu}\right) \simeq 140 \cdot 10^{-11}
\left(\frac{100 \ GeV}{m_{SUSY}}\right)^2 {\mbox{$ \tan\beta~ $}}, \label{1}$$where $m_\mu$ is the muon mass, $m_{SUSY}$ is an average mass of supersymmetric particles in the loop (essentially the chargino mass). In our calculations we use the complete one-loop SUSY contributions from [@CN] with zero phase factors and the additional logarithmic suppression factor as in eq.(\[1\]). The calculated value of $a_\mu$ is shown in Fig. \[f2\] as function of ${\mbox{$ \tan\beta~ $}}$. Clearly, it is approximately proportional to ${\mbox{$ \tan\beta~ $}}$ and its sign depends on the sign of $\mu_0$[^1]. Only positive values of $\mu_0$ are allowed for the positive deviation from the SM and in addition the sparticles have to be rather light. However, light sparticles contribute also substantially to the [$b \to X_s \gamma~ $]{}decay rate. In the past this posed a conflict. However, if one uses in the [$b \to X_s \gamma~ $]{}calculations the running mass for the charm quark, as suggested recently by Gambino and Misiak, the SM prediction is increased by 11%. In this case the newest world average on [$b \to X_s \gamma~ $]{}is $1.7 \sigma$ below the SM, as mentioned in the introduction. Such a deviation is most easily obtained for large and not too heavy sparticles, as shown in Fig. \[f3\]. In the upper part the scale uncertainty of the low energy scale $\mu_b$ is displayed by the width of the theoretical curves, while in the lower part the dependence on the trilinear coupling $A_0$ is shown. The scale $\mu_b$ was varied between 0.5$m_b$ and 2$m_b$. For $\approx 40$ only positive values of the Higgsmixing parameter at the GUT scale $\mu_0$ are allowed in agreement with the preferred sign of $\mu_0$ by the anomalous magnetic moment. For intermediate sparticle masses and $\mu_0>0$ large values of $A_0$ and small values of the low energy scale ($\mu_b\approx 0.5m_b$) bring the calculated values of [$b \to X_s \gamma~ $]{}closest to the data, as can be seen from the left hand side of Fig. \[f3\]. Note that for heavy sparticles (right hand side of Fig. \[f3\]) the effect of the trilinear coupling is small, because the stop mixing is small, if the left and right handed stops are much heavier than the top mass.\
Fig. \[f4\] shows the values of [$b \to X_s \gamma~ $]{}and $a_\mu^{SUSY}$ as function of $m_0$ and $m_{1/2}$ for =35. For [$b \to X_s \gamma~ $]{}the ratio $m_c(\mu)/m_b^{pole}=0.22$ was used, while for the NLO QCD contributions the formulae from Ref. [@DGG] were used. The calculated values have to be compared with the experimental values $BR(b\to X_s\gamma) = (2.96\pm 0.46) \times 10^{-4}$ [@CLEO]-[@BELLE] and $\Delta a_\mu=(43\pm 16)\cdot 10^{-10}$ [@BNL], which shows once more that $b\to X_s\gamma$ and $a_\mu^{SUSY}$ prefer a relatively light supersymmetric spectrum.
To find out the allowed regions in the parameter space of the CMSSM, we fitted both the $b\to X_s\gamma$ and $a_\mu$ data simultaneously. The fit includes the following constraints: i) the unification of the gauge couplings, ii) radiative elctroweak symmetry breaking, iii) the masses of the third generation particles, iv) $b\to X_s\gamma$ and $\Delta a_\mu$, v) experimental limits on the SUSY masses, vi) the lightest superparticle (LSP) has to be neutral to be a viable candidate for dark matter. We do not impose $b-\tau$ unification, since it prefers $\mu_0 <0$, as shown in Fig. \[f1\], while $\Delta a_\mu$ requires $\mu_0 >0$, as shown in Fig. \[f2\]. Yukawa unification for $\mu_0 >0$ can only be obtained by relaxed unification of the gauge couplings and non-universality of the soft terms in the Higgs sector [@R].
The $\chi^2$ contributions of $ b\to X_s\gamma$ and the anomalous magnetic moment $a_\mu$ in the global fit are shown in Fig. \[f5\] for $A_0=0$ and ${\mbox{$ \tan\beta~ $}}=35$. As expected, the $\chi^2$ contribution from $b\to X_s\gamma$ is smallest for heavy sparticles, if [$b \to X_s \gamma~ $]{}is calculated with $m_c/m_b=0.29$, while the minimum $\chi^2$ is obtained for intermediate sparticles, if $m_c/m_b=0.22$ is used. With the newly calculated [$b \to X_s \gamma~ $]{}values, one can see, that [$b \to X_s \gamma~ $]{}and $a_{\mu}$ prefer a similar region of the $m_0,m_{1/2}$ plane. Fig. \[f6\] shows the combined $\chi^2$ contributions from [$b \to X_s \gamma~ $]{}and $a_\mu^{SUSY}$ in the $m_0$, $m_{1/2}$ plane, both in 3D and 2D, for $A_0=0$ (top) and $A_0$ free (bottom). In the latter case the lower $2\sigma$ contour from $b\to X_s\gamma$ moves to the lower left corner, but for the preferred value $A_0\approx 3m_0$, which is the maximum allowed value in the fit in order to avoid negative stop- or Higgs masses and colour breaking minima, the Higgs bound moves up considerably. The total allowed region is similar in both cases, as shown by the light shaded areas in the contour plots. The $2\sigma$ contours from the individual contributions are in good agreement with previous calculations [@FM; @E], but in these paper a simple scan over the parameter space was performed without calculating the combined probability. In addition, $A_0=0$ was assumed.
We repeated the fits for ${\mbox{$ \tan\beta~ $}}=20$ and 50, as shown in Fig. \[f7\]. For smaller values of ${\mbox{$ \tan\beta~ $}}$ the allowed region decreases, since $a_\mu$ becomes too small. At larger ${\mbox{$ \tan\beta~ $}}$ values the region allowed by $a_\mu$ and [$b \to X_s \gamma~ $]{}increases towards heavier sparticles, as expected from Eq. \[1\], but it is cut by the region where the charged stau lepton becomes the Lightest Supersymmetric Particle (LSP), which is assumed to be stable and should be neutral. A charged stable LSP would have been observed by its electromagnetic interactions after being produced in the beginning of the universe. Furthermore, it would not be a candidate for dark matter. The increase of the LSP-excluded area is due to the larger mixing term between the left- and right handed staus at larger ${\mbox{$ \tan\beta~ $}}$.
We conclude that the $a_\mu$ measurement strongly restricts the allowed region of the parameter space in the CMSSM, since it excludes the $\mu_0<$ solution, which was the preferred one from $b-\tau$ Yukawa unification. In addition, it prefers large ${\mbox{$ \tan\beta~ $}}$ with relatively light sparticles, if the present deviation from the SM of $2.6\sigma$ persists.
At large ${\mbox{$ \tan\beta~ $}}$ a global fit including both $b\to X_s\gamma$ and $a_\mu$ as well as the present Higgs limit of 113.5 GeV leaves a quite large region in the CMSSM parameter space. Here we left the trilinear coupling to be a free parameter, which affects both the Higgs limit constraint and the [$b \to X_s \gamma~ $]{}constraint, but in opposite ways, so that the preferred region is similar for the no-scale models with $A_0=0$ and models which leave $A_0$ free.
The 95% lower limit on $m_{1/2}$ is $ 300$ GeV (see Figs. \[f6\]+\[f7\]), which implies that the lightest chargino (neutralino) is above 240(120) GeV. The 95% upper limit on $m_{1/2}$ is determined by the lower limit on $a_\mu^{SUSY}$ and therefor depends on (see Fig. \[f2\]). For =35(50) one finds $m_{1/2}\leq 610(720)$ GeV, which implies that the lightest chargino is below 500(590) GeV and the lightest neutralino is below 260(310) GeV.
Acknowledgements {#acknowledgements .unnumbered}
================
D.K. would like to thank the Heisenberg-Landau Programme, RFBR grant \# 99-02-16650 and DFG grant \# 436/RUS/113/626 for financial support and the Karlsruhe University for hospitality during completion of this work.
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[^1]: Our sign conventions are as in Ref. [@HK].
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---
abstract: 'The 1%-accurate calculations of the van der Waals interaction between an atom and a cavity wall are performed in the separation region from 3nm to 150nm. The cases of metastable He${}^{\ast}$ and Na atoms near the metal, semiconductor or dielectric walls are considered. Different approximations to the description of wall material and atomic dynamic polarizability are carefully compared. The smooth transition to the Casimir-Polder interaction is verified. It is shown that to obtain accurate results for the atom-wall van der Waals interaction at shortest separations with an error less than 1% one should use the complete optical tabulated data for the complex refraction index of the wall material and the accurate dynamic polarizability of an atom. The obtained results may be useful for the theoretical interpretation of recent experiments on quantum reflection and Bose-Einstein condensation of ultracold atoms on or near surfaces of different nature.'
author:
- 'A. O. Caride, G. L. Klimchitskaya,[^1] V. M. Mostepanenko,[^2] and S. I. Zanette'
title: ' Dependences of the van der Waals atom-wall interaction on atomic and material properties '
---
Introduction
============
The van der Waals interaction is the well known example of dispersion forces and there is an extensive literature devoted to this subject (see, e.g., monographs [@1; @2; @3]). These forces are of quantum origin and they become detectable with a decrease of separation distances between atoms, molecules and macroscopic bodies. Further miniaturization, which is the main tendency of microelectronics, brings more and more attention to the investigation of fine properties of the van der Waals interaction.
The van der Waals force between an atom (molecule) and a cavity wall has long been investigated. In Ref. [@4] its interaction potential was found in the form of $V_3(a)=-C_3/a^3$ in nonrelativistic approximation ($a$ is the separation between an atom and a wall). The coefficient $C_3$ was calculated and measured for different atoms and wall materials, both metallic [@5; @6; @7] and dielectric [@8; @9]. The theoretical and experimental results were shown to be in qualitative agreement. More precise measurements were performed in Refs. [@10; @11]. Currently the van der Waals interaction attracts considerable interest in connection with experiments on quantum reflection of ultracold atoms on different surfaces [@12; @13]. With the increase of separation distances up to hundreds nanometers and more to several micrometers, the relativistic and thermal effects become significant changing the dependence of the van der Waals force on separation. At moderate separations up to 1$\mu$m for atoms described by the static atomic polarizability near a wall made of ideal metal at zero temperature the interaction potential was found by Casimir and Polder [@14] in the form $V_4(a)=-C_4/a^4$.
Both the van der Waals and Casimir-Polder interactions are of much importance in connection with the experiments on Bose-Einstein condensation of ultracold atoms confined in a magnetic trap near a surface [@15; @16; @17]. They may influence the stability of a condensate and the effective size of the trap [@17]. Conversely, the Bose-Einstein condensates can be used as sensors of the van der Waals and Casimir-Polder forces. The presence of these forces leads to the shift of the oscillation frequency of the trapped condensate [@18]. Note that in application to ultracold atoms not their temperature but the temperature of the wall is the characteristic parameter of the fluctuating electromagnetic field giving rise to the van der Waals interaction [@18; @19].
It is common knowledge that the precision of frequency shift measurements is very high. Interpretation of these measurements requires accurate theoretical results for the van der Waals and Casimir-Polder interaction beyond the expressions given by the simple asymptotic formulas (in fact coefficients $C_3$ and $C_4$ are not constants but depend on both separation distance and temperature, and there is smooth joining between the formulas at some intermediate separations). In the case of the Casimir-Polder forces such results were obtained in Ref. [@19] for different atoms near a metal wall with account of finite conductivity of a metal, dynamic atomic polarizability and nonzero temperature. In Ref. [@18] the influence of the Casimir-Polder force between Rb atoms and sapphire wall onto the oscillations of a condensate was investigated.
In the present paper we find accurate dependences of the van der Waals atom-wall interaction on the dynamic polarizability of an atom and conductivity properties of wall material. As an example, two different atoms are considered (metastable He${}^{\ast}$ and Na), and metallic (Au), semiconductor (Si) and dielectric (vitreous SiO${}_2$) walls. All calculations are performed within the separation distances $3\,\mbox{nm}\leq a\leq 150\,$nm (for Au at larger separations the accurate theoretical results for the Casimir-Polder interaction were obtained in Ref. [@19]). The theoretical formalism for the exact computation of the van der Waals interaction is given by the Lifshitz formula [@20; @21; @22] adapted for the configuration of an atom near a wall.
At small separations, characteristic for the van der Waals force, it is necessary to use the complete optical tabulated data for the complex index of refraction in order to find the behavior of the dielectric permittivity along the imaginary frequency axis (at separations $a\geq 150\,$nm, as was shown in Ref. [@19], the dielectric function of the free electron plasma model can be used in the case of an Au wall to find the Casimir-Polder interaction). We compare the results obtained by the use of complete data for the dynamic polarizability of an atom and the ones given by the single-oscillator model. This gave the possibility to obtain more accurate results than in Ref. [@23], where the single-oscillator model was used for a hydrogen atom near a silver wall, and also to determine the accuracy of the single-oscillator approximation for the dynamic polarizability in the calculations of the van der Waals interaction. It is shown that to calculate the atom-wall van der Waals interaction with an error less than 1% at a separation of several nanometers both the complete optical tabulated data of the wall material and the accurate atomic dynamic polarizability should be used.
The paper is organized as follows. In Sec. II we briefly present the main formulas and notations for the van der Waals interaction between an atom and a cavity wall. Sec. III contains the accurate theoretical results for van der Waals interaction of He${}^{\ast}$ and Na atoms with an Au wall. In Sec. IV the analogical results are presented for semoconductor (Si) and dielectric (vitreous SiO${}_2$) walls. Sec. V contains our conclusions and discussion.
Lifshitz formula for van der Waals atom-wall interaction
========================================================
The Lifshitz formula for the free energy of atom-wall interaction (wall is at a temperature $T$ at thermal equilibrium) can be presented in the form [@19; @22] $$\begin{aligned}
&&{\cal{F}}(a,T)=-k_BT
\sum\limits_{l=0}^{\infty}
\left(1-\frac{1}{2}\delta_{l0}\right)
\alpha(i\xi_l)
\int_{0}^{\infty}k_{\bot}dk_{\bot}q_le^{-2aq_l}
\label{eq1} \\
&&\phantom{aaaaa}\times
\left\{2r_{\|}(\xi_l,k_{\bot})+
\frac{\xi_l^2}{q_l^2c^2}\left[r_{\bot}(\xi_l,k_{\bot})-
r_{\|}(\xi_l,k_{\bot})\right]\right\},
\nonumber\end{aligned}$$ where $\alpha(\omega)$ is the atomic dynamic polarizability, $k_B$ is the Boltzmann constant, $\xi_l=2\pi k_BTl/\hbar$ are the Matsubara frequencies, $l=0,\,1,\,2,\,\ldots\,$, $\delta_{lk}$ is the Kronecker symbol, and the reflection coefficients for two independent polarizations of the electromagnetic field are $$\begin{aligned}
&&r_{\|}(\xi_l,k_{\bot})=
\frac{\varepsilon_lq_l-k_l}{\varepsilon_lq_l+k_l},
\nonumber \\
&&r_{\bot}(\xi_l,k_{\bot})=\frac{k_l-q_l}{k_l+q_l}.
\label{eq2}\end{aligned}$$ In Eqs. (\[eq1\]) and (\[eq2\]) the notations $$q_l=\sqrt{k_{\bot}^2+\frac{\xi_l^2}{c^2}},\quad
k_l=\sqrt{k_{\bot}^2+\varepsilon_l\frac{\xi_l^2}{c^2}}
\label{eq3}$$ are also introduced, where $\varepsilon_l=\varepsilon(i\xi_l)$ is the dielectric permittivity computed at the imaginary Matsubara frequencies, $k_{\bot}$ is the wave vector in the plane of the wall.
We will apply Eq. (\[eq1\]) in the separation region $3\,\mbox{nm}\leq a\leq 150\,$nm which corresponds to the van der Waals interaction (near the left-hand side of the interval) and transition domain to the Casimir-Polder interaction. In fact, in this region at room temperature $T=300\,$K the temperature effect is negligible. For the sake of convenience in numerical computations we, however, do not make the approximate change of the discrete summation for integration over continuous frequencies and use the original exact Eq. (\[eq1\]).
For further application in computations, we introduce the dimensionless variables $$y=2aq_l,\quad\zeta_l=\frac{2a\xi_l}{c}\equiv\frac{\xi_l}{\omega_c},
\label{eq4}$$ where $\omega_c\equiv\omega_c(a)=c/(2a)$ is the characteristic frequency of the van der Waals interaction.
Separating the zero-frequency term, Eq. (\[eq1\]) can be represented in the form $$\begin{aligned}
&&{\cal{F}}=-\frac{C_3(a,T)}{a^3},
\label{eq5} \\
&&
C_3(a,T)=\frac{k_BT}{8}\left\{
2\alpha(0)\frac{\varepsilon(i0)-1}{\varepsilon(i0)+1}+
\sum\limits_{l=1}^{\infty}
\alpha(i\zeta_l\omega_c)\right.\nonumber \\
&&\phantom{aa}\times\left.
\int_{\zeta_l}^{\infty}dye^{-y}
\left[\vphantom{\sum}
2y^2r_{\|}(\zeta_l,y)+
\zeta_l^2\left[r_{\bot}(\zeta_l,y)-
r_{\|}(\zeta_l,y)\right]\right]
\vphantom{\sum\limits_{l=1}^{\infty}}
\right\}.
\nonumber\end{aligned}$$ Note that for metal $[\varepsilon(i0)-1]/[\varepsilon(i0)+1]=1$, whereas for dielectrics and semiconductors this ratio is equal to $(\varepsilon_0-1)/(\varepsilon_0+1)$, where $\varepsilon_0$ is the static dielectric permittivity.
In terms of the new variables the reflection coefficients (\[eq2\]) are $$\begin{aligned}
&&r_{\|}(\zeta_l,y)=
\frac{\varepsilon_ly-
\sqrt{y^2+\zeta_l^2\left(\varepsilon_l-1\right)}}{\varepsilon_ly+
\sqrt{y^2+\zeta_l^2\left(\varepsilon_l-1\right)}},
\nonumber \\
&&r_{\bot}(\zeta_l,y)=
\frac{\sqrt{y^2+\zeta_l^2\left(\varepsilon_l-1\right)}
-y}{\sqrt{y^2+\zeta_l^2\left(\varepsilon_l-1\right)}+y}.
\label{eq6}\end{aligned}$$
In a nonrelativistic limit Eq. (\[eq5\]) leads to $$C_3(T)=\frac{k_BT}{4}\left[
\alpha(0)\frac{\varepsilon(i0)-1}{\varepsilon(i0)+1}+
2\sum\limits_{l=1}^{\infty}
\alpha(i\xi_l)\frac{\varepsilon(i\xi_l)-1}{\varepsilon(i\xi_l)+1}
\right],
\label{eq7}$$ which gives the usual estimation for the value of the van der Waals constant at the shorter separations. Remind that Eq. (\[eq7\]) practically does not depend on temperature. By using the Abel-Plana formula [@24] it can be approximately represented by $$C_3\approx\frac{\hbar}{4\pi}
\int_{0}^{\infty}
\alpha(i\xi)\frac{\varepsilon(i\xi)-1}{\varepsilon(i\xi)+1}
d\xi.
\label{eq8}$$
In the next two sections Eqs. (\[eq5\])–(\[eq7\]) will be used for accurate calculations of the van der Waals force between different atoms near the surfaces made of metallic, semiconducting and dielectric materials.
van der Waals interaction of H$\mbox{e}{}^{\ast}$ and N$\mbox{a}$ atoms with gold wall
======================================================================================
To calculate the van der Waals free energy of atom-wall interaction one should substitute the values of the dielectric permittivity of the wall material and dynamic polarizability of the atom at imaginary Matsubara frequencies into Eqs. (\[eq5\]) and (\[eq6\]).
We consider the separation distances $a\leq 150\,$nm (at larger separations the analytical representation for [${\cal{F}}$]{} was obtained in Ref. [@19] using the plasma model dielectric function and the single oscillator model for the atomic dynamic polarizability; the agreement up to 1% with the results of numerical computations was achieved). As a lower limit of separations under consideration we fix $a=3\,$nm. At smaller separation distances there are additional physical phenomena, connected with the atomic structure of a wall material, which are not taken into account in Eq. (\[eq5\]) but can influence atom-wall interaction. The most important of them are the repulsive exchange potentials with a range of action up to a few angströms, and the spatially nonlocal interaction due to the surface-plasmon charge fluctuations. The latter contributes essentially at separations of the order of $v_F/\omega_p\sim 1\,$[Å]{}, where $v_F$ is the Fermi velocity and $\omega_p$ is the plasma frequency [@25]. As was proved in Ref. [@25], at much larger separations (in fact, starting from $a\approx 3\,$nm) the usual Lifshitz formula, given by Eqs. (\[eq1\]) and (\[eq5\]) is already applicable.
Within the separation region under consideration the characteristic frequency $\omega_c$ reaches and even exceeds (at the shorter separations) the plasma frequency (for Au we use $\omega_p=1.37\times10^{16}\,$rad/s [@26]). By this reason in our case the plasma or Drude dielectric functions are not good approximations for the dielectric permittivity in all relevant frequency range and one should use the complete tabulated data for the complex index of refraction for Au to calculate the imaginary part of the dielectric permittivity Im$\varepsilon(\omega)$ along the real frequency axis. The dielectric permittivity along the imaginary frequency axis is found by means of the dispersion relation [@27] $$\varepsilon(i\xi)=1+\frac{2}{\pi}
\int_0^{\infty}{\!}d\omega
\frac{\omega\,\mbox{Im}\varepsilon(\omega)}{\omega^2+\xi^2}.
\label{eq9}$$ The available tabulated data for Au extend from 0.125eV to 10000eV ($1\,\mbox{eV}=1.519\times 10^{15}\,$rad/s). At shorter separations, to obtain the values of the van der Waals free energy correct up to four significant figures, one should find the dielectric permittivity at first 1850 Matsubara frequencies. Near the right border of the separation interval ($a=150\,$nm) it would suffice to use only 60–70 first Matsubara frequencies. In fact, to obtain $\varepsilon$ by Eq. (\[eq9\]) with sufficient precision one should extend the available tabulated data for the region $\omega<0.125\,$eV. This is conventially done with the help of the imaginary part of the Drude dielectric function $$\varepsilon(\omega)=1-\frac{\omega_p^2}{\omega(\omega+i\gamma)},
\label{eq10}$$ where $\gamma=0.035\,$eV is the relaxation frequency. It should be reminded also that Eqs. (\[eq1\]), (\[eq2\]), (\[eq10\]) are free from contradiction with the Nernst heat theorem which arise when the Drude dielectric function is substituted into the Lifshitz formula at nonzero temperature in the configuration of two parallel plates made of real metal (see Refs. [@28; @29] for more details).
The computational results for Au are presented in Fig. 1 where $\log_{10}\varepsilon(i\xi)$ is plotted as a function of $\log_{10}\xi$ starting from the first Matsubara frequency (at $T=300\,$K one has $\xi_1\approx2.47\times 10^{14}\,$rad/s and $\log_{10}\xi_1\approx 14.4$).
Other data to be substituted into Eq. (\[eq5\]) are the values of the atomic dynamic polarizability at imaginary Matsubara frequencies. The accurate data (having a relative error of about $10^{-6}$) were taken from Ref. [@30] for the atoms of metastable He${}^{\ast}$ and from Ref. [@31] for Na (see also the graphical representation in Fig. 3 of Ref. [@19]). It is interesting to compare the values of $C_3(a,T)$ obtained by the use of the highly accurate data for the atomic dynamic polarizability and in the framework of the single oscillator model $$\alpha(i\zeta\omega_c)=
\frac{\alpha(0)}{1+\frac{\omega_c^2\zeta^2}{\omega_0^2}},
\label{eq11}$$ where for He${}^{\ast}$ it holds $\alpha(0)=315.63\,$a.u., $\omega_0=1.18\,$eV [@32] and for Na it holds $\alpha(0)=162.68\,$a.u., $\omega_0=1.55\,$eV [@33] (1a.u. of polarizability is equal to $1.48\times 10^{-31}\,\mbox{m}^3$).
The computational results for the van der Waals coefficient $C_3$ in the case of Au wall versus separation are represented in Fig. 2 for metastable He${}^{\ast}$ (a) and Na (b) by solid lines. These lines are obtained by the use of the optical tabulated data for Im$\varepsilon$ and accurate atomic dynamic polarizability. In the same figure the long-dashed lines show the results obtained with the same data for Im$\varepsilon$ but with a single oscillator model (\[eq11\]) for the atomic dynamic polarizability. The short-dashed lines illustrate the dependence of $C_3$ on $a$ in the case of a wall made of ideal metal but with the accurate atomic dynamic polarizability.
As is seen from Fig. 2, the account of the realistic properties of a wall metal is important at all separations under consideration. At the shortest separation $a=3\,$nm the result for an ideal metal differs from the accurate result given by the solid line by about 16% for He${}^{\ast}$ and by 28% for Na. These strong deviations only slightly decrease with the increase of separation.
A few calculated results for the values of $C_3$ are presented in Table I at $T=300\,$K for different separations indicated in the first column. In columns 2 and 3 the values of $C_3$ for He${}^{\ast}$ atom are computed for ideal metal and by the use of the optical tabulated data for Im$\varepsilon$, respectively, and in both cases with an accurate atomic polarizability. In column 4 the optical tabulated data for Im$\varepsilon$ were used in combination with the single oscillator model for the atomic polarizability of He${}^{\ast}$. In column 5 the plasma model dielectric function was used in calculations together with an accurate atomic polarizability of He${}^{\ast}$. In columns 6–9 the calculational results for a Na atom are presented in the same order.
As is seen from Fig. 2 and Table I (columns 3 and 4), the use of the accurate data for the atomic dynamic polarizability (if to compare with the single oscillator model) is of most importance at the shortest separations. Thus, at $a=3\,$nm the relative error of $C_3$ given by the single oscillator model is 4.4% for He${}^{\ast}$ and 2.2% for Na. At $a=15\,$nm the single oscillator model becomes more precise. For He${}^{\ast}$ it leads to only 3.3%, and for Na to 1.6% errors.
It is interesting to compare the calculated results obtained by the use of the complete tabulated data for Im$\varepsilon$ of Au and by the plasma model dielectric function \[Eq. (\[eq10\]) with $\gamma=0$\]. From columns 3 and 5 of Table I for He${}^{\ast}$, and 7 and 9 for Na one can conclude that the error, given by the plasma model, decreases from 6.3% for He${}^{\ast}$ and 10% for Na at $a=3\,$nm to 0.8% for He${}^{\ast}$ and 1% for Na at $a=150\,$nm. This illustrates the smooth joining of our present results for the van der Waals interaction obtained by the use of the optical tabulated data for Au with the analytical results of Ref. [@19] for the Casimir-Polder interaction found by the application of the plasma model.
The nonrelativistic asymptotic values of $C_3$ can be calculated by the immediate use of Eqs. (\[eq7\]) and (\[eq9\]) combined with the optical tabulated data for Im$\varepsilon$ and the accurate atomic polarizability. This leads to the results $C_3\approx 1.61\,$a.u. for He${}^{\ast}$ and $C_3\approx 1.37\,$a.u. for Na in rather good agreement with the data of columns 3 and 7 of Table I computed at the shortest separation $a=3\,$nm. Note, however, that the asymptotic values, achieved at separations $a<3\,$nm, may be already outside of the application region of the used theoretical approach (see discussion in the beginning of this section).
As was shown in Ref. [@19], the account of the atomic dynamic polarizability strongly affects the value of the Casimir-Polder interaction if to compare with the original result [@14] obtained in the static approximation. We emphasize that in the case of the van der Waals interaction the influence of dynamic effects is even greater than in the Casimir-Polder case. Thus, if we restrict ourselves by only static polarizability of He${}^{\ast}$ atom, the values of $C_3$ are found to be 11.6 and 1.64 times greater than those given in column 3 of Table I at separations $a=3\,$nm and $a=150\,$nm, respectively.
van der Waals interaction of H$\mbox{e}{}^{\ast}$ and N$\mbox{a}$ atoms with semiconductor and dielectric walls
===============================================================================================================
In this section we apply the formalism of Sec. II to find the accurate separation dependences of the van der Waals interaction between He${}^{\ast}$ and Na atoms and Si or vitreous SiO${}_2$ wall. The chosen separation interval $3\,\mbox{nm}\leq a\leq 150\,$nm is the same as in Sec. III. In the case of dielectric and semiconductor surfaces there are additional interactions due to the charged dangling bonds at separations 1–1.5nm (see, e.g., Ref. [@34]). This is a further factor restricting the application of the conventional theory of van der Waals forces at very short distances.
The tabulated data for the complex refraction index of Si extend from 0.00496eV to 2000eV [@26]. This permits not to use any extension of data to smaller frequencies when using Eq. (\[eq9\]) in order to find the dielectric permittivity at all contributing imaginary Matsubara frequencies. The computational results for Si are presented in Fig. 3a where $\varepsilon(i\xi)$ is plotted as a function of $\log_{10}\xi$ ($\xi$ is measured in rad/s). The static dielectric permittivity of Si is equal to $\varepsilon_0=11.66$.
Substituting the obtained results for $\varepsilon(i\xi)$ and also the data for the atomic dynamic polarizability of He${}^{\ast}$ and Na (the same as in Sec. III) into Eqs. (\[eq5\]) and (\[eq6\]), one finds the dependences of the van der Waals parameter $C_3$ on separation. The results are shown in Fig. 4a (for He${}^{\ast}$) and Fig. 4b (for Na). The solid lines are obtained by the use of the accurate atomic dynamic polarizabilities, and the long-dashed lines by using the single oscillator model given by Eq. (\[eq11\]). The short-dashed lines are obtained with the accurate dynamic polarizability but on the assumption that the dielectric permittivity does not depend on frequency and is equal to its static value. At the shortest separation $a=3\,$nm the error in $C_3$ due to the use of the static dielectric permittivity is approximately 13% for He${}^{\ast}$ and 24% for Na.
In Table II a few calculated values of $C_3$ at $T=300\,$K are presented at separations listed in column 1. In columns 2 and 3 the values of $C_3$ for He${}^{\ast}$ are computed by the use of a static dielectric permittivity and optical tabulated data for Im$\varepsilon$, respectively, and in both cases with the accurate atomic dynamic polarizability. In column 4 the data for Im$\varepsilon$ were used in combination with the single oscillator model for He${}^{\ast}$ dynamic polarizability. In columns 5–7 the same results for a Na atom are presented.
Table II and Fig. 4 permit to follow the influence of atomic and semiconductor characteristics onto the van der Waals force. Thus, comparing columns 3 and 4 we notice that the use of the single oscillator model leads to 4.4% error at $a=3\,$nm and 3.1% error at $a=15\,$nm for the atom of metastable He${}^{\ast}$. For the atom of Na these errors are 1.8% and 1%, respectively. With the increase of separation distance up to 150nm the errors given by the single oscillator model decrease down to 0.4% for He${}^{\ast}$ and practically to zero for Na. This confirms that at larger separations the single oscillator model is quite sufficient for calculations of the van der Waals interactions with errors below 1%.
Now let us consider the case of a dielectric wall (vitreous SiO${}_2$). The tabulated data for the complex refraction index of SiO${}_2$ extend from 0.0025eV to 2000eV [@26]. This is also quite sufficient to calculate the dielectric permittivity at all contributing Matsubara frequencies by Eq. (\[eq9\]) with no use of any extension of data. The dependence of $\varepsilon(i\xi)$ as a function of $\log_{10}\xi$ for SiO${}_2$ is shown in Fig. 3b. The static dielectric permittivity of SiO${}_2$ is equal to $\varepsilon_0=4.88$.
The obtained results for $\varepsilon(i\xi)$ and the data for the atomic dynamic polarizability of He${}^{\ast}$ and Na are substituted into Eqs. (\[eq5\]) and (\[eq6\]). The resulting dependences of $C_3$ on separation are shown in Fig. 5a (for He${}^{\ast}$) and Fig. 5b (for Na). As in Fig. 4, the solid lines are related to the use of the accurate dynamic polarizabilities, the long-dashed lines to the single oscillator model, and the short-dashed lines to the use of the static dielectric permittivity and an accurate dynamic polarizability.
Table III, containing a few calculated results, is organized in the same way as Table II related to the case of a semiconductor wall. It permits to find errors resulting from the use of the static dielectric permittivity instead of the accurate dependence of $\varepsilon(i\xi)$ on frequency, and a single oscillator model instead of an accurate dynamic polarizability for the atom near the dielectric wall. Thus, at $a=3\,$nm the use of the static dielectric permittivity instead of the optical tabulated data leads to 78% error in the value of the van der Waals coefficient $C_3$ for He${}^{\ast}$ and to 95% error for Na. These errors decrease to 2.1% and 6.9%, respectively, if one uses the dielectric permittivity $\tilde{\varepsilon}\approx 2.13$ corresponding not to the zero frequency but to the frequency region of visible light. With the use of $\tilde{\varepsilon}$ the largest errors in the value of $C_3$ are achieved, however, not at the shortest separation but at the largest separation considered here $a=150\,$nm (15% for He${}^{\ast}$ atom and 12.7% for Na atom). At this separation the use of the static dielectric permittivity $\varepsilon_0$ leads to 56.6% error (for He${}^{\ast}$) and 62% error (for Na).
By the comparison of columns 3 and 4 in Table III we conclude that at a separation $a=3\,$nm the use of the single oscillator model results in 5% error for He${}^{\ast}$ atom and in 3% error for Na atom. At $a=15\,$nm the corresponding errors are 3.6% and 1.2%, respectively. At a separation $a=150\,$nm the errors due to the use of the single oscillator model are 0.6% for He${}^{\ast}$ atom and practically zero for Na atom, i.e., the single oscillator model is sufficient.
Conclusions and discussion
==========================
In the foregoing we have performed accurate calculations of the parameter $C_3$ describing the van der Waals atom-wall interaction for the atoms of metastable He${}^{\ast}$ and Na near metallic, semiconductor and dielectric walls. The separation region from 3nm to 150nm was considered covering the proper nonrelativistic van der Waals interaction and some part of the transition region to the relativistic Casimir-Polder interaction. At $a=150\,$nm the smooth joining of the obtained results with the calculations of Ref. [@19] for the Casimir-Polder case was followed.
It was shown that qualitatively the cases of an atom near metallic, semiconductor and dielectric walls are very similar. The use of approximations of the ideal metal or the static dielectric permittivity leads to the errors in the value of $C_3$ of about (13–28)% at the shortest separation depending on the wall material and the type of atoms. This error slowly decreases with the increase of separation remaining rather large in the case of metallic wall. The more adequate (for metals) plasma model dielectric function results in (6–10)% errors at the shortest separation.
We have compared the results for $C_3$ obtained by the use of the accurate atomic dynamic polarizability with those obtained from the single oscillator model. At the shortest separation the single oscillator model leads to errors of about (1.8–4.4)% in the values of $C_3$. These errors quickly decrease with the increase of separation.
The magnitude of the error, given by one or another approximation used, depends qualitatively on the type of the atom. By way of example, for Na atom the use of a single oscillator model leads to less errors than for He${}^{\ast}$ independently of wall material.
The performed investigation permits to make a conclusion that the accurate calculations of the van der Waals atom-wall interaction at short separations with the error no larger than 1% require the use of both complete optical tabulated data of wall material and accurate dynamic polarizability of an atom. This is distinct from the case of the Casimir-Polder interaction with a metallic wall which can be described with no more than 1% error using the plasma model dielectric function of a wall material and the single oscillator model for the dynamic polarizability of an atom.
The obtained results can be used for theoretical interpretation of the experiments on quantum reflection and Bose-Einstein condensation of ultracold atoms on (near) surfaces of different nature, and also in investigation of other physical, chemical and biological processes, where the precise information on the van der Waals and Casimir-Polder forces is needed.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to J. F. Babb for stimulating discussions and for giving accurate data on the atomic dynamic polarizability of He${}^{\ast}$ and Na. We gratefully acknowledge FAPERJ (Processes E–26/170.132 and 170.409/2004) for financial support. G.L.K. and V.M.M. were partially supported by Finep.
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----- ------ ------- ------- ------- ------- ------- ------- -------
(a) (b) (c) (d) (a) (b) (c) (d)
3 1.85 1.59 1.52 1.49 1.72 1.34 1.31 1.20
5 1.82 1.58 1.51 1.48 1.66 1.32 1.29 1.19
10 1.76 1.54 1.48 1.46 1.55 1.27 1.25 1.16
15 1.71 1.50 1.45 1.43 1.48 1.22 1.20 1.13
20 1.67 1.46 1.42 1.40 1.42 1.17 1.16 1.10
25 1.62 1.43 1.39 1.38 1.36 1.13 1.12 1.07
50 1.45 1.28 1.26 1.25 1.15 0.967 0.965 0.932
75 1.32 1.17 1.16 1.15 0.994 0.844 0.844 0.822
100 1.20 1.08 1.07 1.06 0.871 0.748 0.748 0.734
125 1.11 0.994 0.989 0.985 0.773 0.671 0.671 0.662
150 1.02 0.925 0.922 0.918 0.693 0.608 0.608 0.601
----- ------ ------- ------- ------- ------- ------- ------- -------
: Values of the coefficient $C_3$ of the van der Waals atom-wall interaction at different separations computed for the ideal metal (a) and for real metal (Au) described by the optical tabulated data (b), and the accurate atomic dynamic polarizabilities; in column (c) real metal is described by the optical tabulated data and the dynamic polarizability of an atom is given by the single oscillator model; in column (d) real metal is described by the plasma model and the dynamic polarizability of an atom is accurate.
----- ------- ------- ------- ------- ------- -------
(a) (b) (c) (a) (b) (c)
3 1.54 1.36 1.30 1.41 1.14 1.12
5 1.50 1.35 1.29 1.35 1.13 1.11
10 1.43 1.32 1.26 1.24 1.09 1.07
15 1.38 1.28 1.24 1.17 1.05 1.04
20 1.33 1.25 1.21 1.10 1.01 1.00
25 1.28 1.22 1.18 1.05 0.970 0.965
50 1.11 1.08 1.06 0.856 0.814 0.812
75 0.998 0.965 0.954 0.723 0.698 0.697
100 0.890 0.873 0.866 0.625 0.608 0.608
125 0.809 0.797 0.792 0.549 0.537 0.537
150 0.741 0.732 0.729 0.488 0.480 0.480
----- ------- ------- ------- ------- ------- -------
: Values of the coefficient $C_3$ of the van der Waals atom-wall interaction at different separations computed for the semiconductor (Si) described by the static dielectric permittivity (a) and by the optical tabulated data (b), and the accurate atomic dynamic polarizabilities; in column (c) semiconductor is described by the optical tabulated data and the dynamic polarizability of an atom is given by the single oscillator model.
----- ------- ------- ------- ------- ------- -------
(a) (b) (c) (a) (b) (c)
3 1.20 0.672 0.638 1.10 0.563 0.546
5 1.17 0.666 0.633 1.05 0.553 0.539
10 1.12 0.647 0.620 0.967 0.528 0.519
15 1.07 0.629 0.606 0.906 0.505 0.499
20 1.03 0.611 0.591 0.857 0.484 0.479
25 0.999 0.595 0.577 0.815 0.464 0.461
50 0.862 0.524 0.515 0.659 0.385 0.384
75 0.762 0.470 0.465 0.554 0.329 0.329
100 0.684 0.427 0.424 0.477 0.287 0.287
125 0.620 0.391 0.390 0.418 0.255 0.255
150 0.567 0.362 0.360 0.371 0.229 0.229
----- ------- ------- ------- ------- ------- -------
: Values of the coefficient $C_3$ of the van der Waals atom-wall interaction at different separations computed for the dielectric (vitreous SiO${}_2$) described by the static dielectric permittivity (a) and by the optical tabulated data (b), and the accurate atomic dynamic polarizabilities; in column (c) semiconductor is described by the optical tabulated data and the dynamic polarizability of an atom is given by the single oscillator model.
[^1]: On leave from North-West Technical University, St.Petersburg, Russia.
[^2]: On leave from Noncommercial Partnership “Scientific Instruments”, Moscow, Russia.
|
---
abstract: 'We study the long-term evolution of relativistic jets in collapsars and examine the effects of viewing angle on the subsequent gamma ray bursts. We carry out a series of high-resolution simulations of a jet propagating through a stellar envelope in 2D cylindrical coordinates using the FLASH relativistic hydrodynamics module. For the first time, simulations are carried out using an adaptive mesh that allows for a large dynamic range inside the star while still being efficient enough to follow the evolution of the jet long after it breaks out from the star. Our simulations allow us to single out three phases in the jet evolution: a precursor phase in which relativistic material turbulently shed from the head of the jet first emerges from the star, a shocked jet phase where a fully shocked jet of material is emerging, and an unshocked jet phase where the jet consists of a free-streaming, unshocked core surrounded by a thin boundary layer of shocked jet material. The appearance of these phases will be different to observers at different angles. The precursor has a wide opening angle and would be visible far off axis. The shocked phase has a relatively narrow opening angle that is constant in time. During the unshocked jet phase the opening angle increases logarithmically with time. As a consequence, some observers see prolonged dead times of emission even for constant properties of the jet injected in the stellar core. We also present an analytic model that is able to reproduce the overall properties of the jet and its evolution. We finally discuss the observational implications of our results, emphasizing the possible ways to test progenitor models through the effects of jet propagation in the star. In an appendix, we present 1D and 2D tests of the FLASH relativistic hydrodynamics module.'
author:
- 'Brian J. Morsony, Davide Lazzati and Mitchell C. Begelman'
title: Temporal and Angular Properties of GRB Jets Emerging from Massive Stars
---
Introduction
============
The association of gamma-ray bursts (GRBs) with supernova explosions creates an apparent contradiction. On the one hand, the observed gamma-ray spectra require the emitting material to be outflowing at highly relativistic speed (Lithwick & Sari 2001), limiting its rest mass to a fraction of a solar mass. On the other hand, the jet expands through a dense stellar core, potentially picking up several solar masses in its way. Pioneering work by Woosley and collaborators (MacFadyen & Woosley 1999; MacFadyen et al. 2001) and by Aloy and collaborators (Aloy et al. 2000) showed that a light jet can reach the surface of a star virtually unpolluted if its head travels sub-relativistically inside the star. In this way, the cold and dense stellar material can be pushed aside. Once the stellar surface is reached, the light jet accelerates and reaches the high Lorentz factors required by $\gamma$-ray observations (see also Matzner 2003 for a simplified analytic treatment).
Once this basic question is answered, two main issues remain open. First, what is the jet launching mechanism at the base of the star and what are the properties of the injected outflow? Second, how does the propagation of the jet influence its properties in the radiative phase, which takes place when the jet has traveled far from its birth site? The first question has been addressed with MHD numerical simulations (Proga et al. 2003; Mizuno et al. 2004; De Villiers et al. 2006; Nagataki et al. 2006). Due to the complexity of dealing with 3D magnetic phenomena in the general relativistic regime, these studies are still in their infancy. This paper deals with the second of these questions. Zhang et al. (2003) performed fixed grid 2D special relativistic simulations of jets in the cores of massive stars. They observed a diverse phenomenology of time dependent properties in the emerging jets, including variations in the opening angle, but their resolution was not large enough to draw robust conclusions at large radii, after the jet breaks out of the star. Lazzati & Begelman (2005, hereafter LB05) showed, with an analytic model, that the star-jet interaction can be strong and can create observable consequences. They argued that even if a steady jet is injected in the stellar core, its properties at the surface of the star will be strongly time-dependent, so that even the concept of a jet opening angle becomes hard to define in a general sense. They concluded that the time-integrated jet structure would match that of the “Universal Jet” proposed by Rossi et al. (2002). Even though - as we will show in this paper - their approximations were inaccurate in places, and the jet structure they predicted is not observed in the simulations, we confirm the basic concept that the propagation of the jet through the star will strongly influence both, generating strong temporal evolution in their properties. The influence of the jet’s initial conditions has been studied recently with a set of 2D simulations (Mizuta et al. 2006). Their jets, however, are cylindrical at the base, different from any previous work and from our simulations presented here, where the outflow is set up to be conical in the core of the star.
In this paper, we present the results of a set of high-resolution 2-dimensional simulations of a jet propagating through a stellar progenitor. The jet is assumed to be already developed and mildly relativistic at $10^9$ cm inside the star, even though it is still hot and entropy-rich. The energy release is constant and the engine is left to run for a time longer than the breakout time, i.e., the time at which the jet breaks through the stellar surface. We show that the jet-star interaction creates three well-defined phases. The first one is characterized by turbulence. During this phase the jet propagates through and eventually breaks out of the star. The second phase is a transition phase, during which a heavily shocked jet flows through the opened channel. Finally, a more stable configuration is attained, with a freely flowing jet surrounded by a thin boundary layer that progressively widens its opening angle. Some of these phases can be dealt with analytically or semi-analytically with sufficient accuracy, allowing us to understand the origin of the observed behavior. The observational implications of such non-steady flows are numerous, including long dead times during the prompt emission and bumpy, complex afterglows that do not obey the rules of the simple external shock model.
This paper is organized as follows. In § 2 we describe the numerical code used. In § 3 we describe the initial conditions for both the stellar progenitor and the jet in its center while in § 4 we detail our results. In § 5 we develop an analytic modeling of the jet and its propagation, in § 6 we present afterglow calculations based on the numerical results of § 4. We finally summarize our results in § 7.
Numerical Methods
=================
The simulations presented here were performed using a modified version of the FLASH adaptive mesh refinement (AMR) code (version 2.5) in 2D cylindrical coordinates. The FLASH special relativistic hydrodynamics module utilizes piecewise-parabolic interpolation for computing fluxes at cell interfaces and uses a two-shock Riemann solver for solving the fluid equations. See Mignone et al. (2005) for a full description of the methods used in the relativistic portion of the FLASH code. An adiabatic equation of state (EoS) with a fixed adiabatic index is used. The choice of a single EoS for both the relativistic and non relativistic material is dictated by the need to keep the running times reasonable and by the fact that we will concentrate on the properties of the relativistic material. Future work, that will analyze also the behavior of the star material in detail, will adopt a more realistic EoS. Block refinement is increased if the normalized second derivative of the density or pressure is greater than $0.8$ anywhere in the block. Refinement is decreased if the normalized second derivative of the density and pressure is less than $0.2$ everywhere in the block. Appendix \[codetesting\] contains results from a number of test problems using this code.
In order to achieve higher resolution near the center of the star and close to the axis of the jet, we have modified the FLASH code so that the maximum level of refinement allowed can be varied over the simulation grid. See § \[gridsetup\] for details of the grid setup used for the simulations presented here. In order to improve the efficiency of the simulations, the FLASH code was modified so that quantities in a given cell are only computed and updated if mod$(n_{step},2^{(l_{max} - l_i)}) = 0$, where $n_{step}$ is the time step number, $l_{max}$ is the maximum level of refinement anywhere on the grid, and $l_i$ is the refinement level of a particular cell. The size of the time step used for computing new values in a cell is correspondingly increased by a factor of $2^{(l_{max} - l_i)}$. In other words, larger time steps are taken less often for cells at a lower level of refinement. This adjustment is equivalent to having a fixed Courant-Friedrichs-Levy (CFL) number at all levels of refinement, rather than a lower CFL number in less resolved cells, as is the default in FLASH. The minimum time step is determined by the maximum velocity anywhere on the grid, ensuring that the CFL condition is never violated. Additionally, in the relativistic case the maximum velocity is always $\sim c$ (1 in relativistic units), so the minimum time step is approximately constant. This improvement in efficiency is generally applicable to the entire FLASH code and not just the relativistic portion. Although some operations are still performed on all cells at all time steps, this scheme is always more efficient than the default in FLASH. For the simulations and grid setup described here, this scheme resulted in approximately a factor of 10 decrease in running time. The tests of the adaptive mesh in Appendix \[codetesting\] include this change and do not show any adverse effects.
Setup
=====
Stellar Model
-------------
For the simulations presented here, two different stellar models are used. The first is a realistic stellar model, based on model 16TI from Woosley & Heger (2006) of a Wolf-Rayet star with an initial mass of 16 M$_\sun$, metallicity of $1\%$ solar and a large angular momentum of $3.3 \times 10^{52}$ erg s$^{-1}$. The model has been evolved to core collapse, and with a final mass and radius of 13.95 M$_\sun$ and $4.077 \times 10^{10}$ cm, respectively. (see [http://www.ucolick.org/\$\\sim\$alex/GRB2/](http://www.ucolick.org/$\sim$alex/GRB2/)). Simulations using this model have names beginning with 16TI. The second model is a simple power-law model for a star with a mass of 15 M$_\sun$ and a radius of $10^{11}$ cm. The density profile is modeled as a power law $\rho \propto r^{-2.5}$ and the pressure is set such that $p = p_0
\rho^{4/3}$, where $p_0$ is computed based on $p = 1.8 \times
10^{18}$ erg cm$^{-3}$ at $r = 10^{10}$ cm. This value of pressure at $10^{10}$ cm is reasonable based on numerically modeled values for stars of similar size and mass (Heger et al. 2005). Since the stellar pressure is small compared to the jet and cocoon pressure, the actual value of the stellar pressure has little impact on the simulations. All simulations use an ultra-relativistic equation of state with $\Gamma = 4/3$, so setting $p = p_0 \rho^{4/3}$ ensures that pressure is always small compared to $\rho\,c^2$ inside the star. For both models, the density and pressure exterior to the star are set to $10^{-9}$ g cm$^{-3}$ and $9 \times 10^7$ erg cm$^{-3}$, respectively. The mass exterior to the star does not have a significant impact on the dynamics of the simulations presented here, and the exterior density and pressure are set to small, non-zero values for numerical reasons. A CFL number of $0.4$ is used for all simulations.
The stars in our simulations are not stable objects since gravity is not included. However, the timescale for the star to expand is large compared to the length of our simulations. After 50 seconds, the end time of our simulations, the material from the edge of the power-law star has moved outward by $8\times10^9$ cm, which is smaller than the distance scale for any realistic drop-off in density at the edge of the star.
Grid Setup \[gridsetup\]
------------------------
All simulations were run on an identical grid setup. The total grid size is $2.56 \times 10^{11}$ cm by $2.56 \times 10^{11}$ cm. The maximum allowed resolution varies from $7.8125 \times 10^6$ cm (13 levels of refinement) near the jet input to $2.5 \times 10^8$ cm (8 levels of refinement) far from the coordinate axis outside the star. Figure \[gridsetupfig\] shows the maximum allowed refinement in different regions. With this setup, the grid has a resolution of $6.25 \times 10^7$ cm at $10^{11}$ cm, the surface of the star for the power-law model. This corresponds to an angular resolution of $0.0358\degr$ at this radius. This is a significant improvement over Zhang et al. (2003), who used a grid in spherical coordinates with a resolution of $0.25\degr$, and is comparable to Zhang et al. (2004), who had a resolution of $10^8$ cm on a cylindrical grid at $10^{11}$ cm. The variable maximum resolution allows the lower boundary of the grid to be set at $10^9$ cm above the equator of the star. Zhang et al. (2004) used a fixed grid inside the star, and their lower boundary was placed at $10^{10}$ cm above the stellar equator. Zhang et al. (2003) was able to place the lower boundary at a radius of $2 \times 10^8$ cm in radial coordinates, but at the cost of increasingly poor resolution across the jet at large radii. Similarly, Aloy et al. (2000) could place the inner boundary at $2\times10^7$ cm but the resolution was severely degraded at the star surface.
Jet Parameters and Boundary Conditions
--------------------------------------
The jet injection is modeled as a boundary condition on the lower edge of the grid at $10^9$ cm. The opening angle and Lorentz factor of the incoming jet are varied between simulations, but are constant at all times in each run. The terminal Lorentz factor, or Lorentz factor at infinity, $\gamma_\infty$, is defined as the Lorentz factor that the material would achieve if all its internal energy were converted to kinetic energy. It is calculated as $\gamma_\infty = (1 + 4p/ \rho
c^2)\gamma$, where $\gamma$ is the local bulk Lorentz factor. $\gamma_\infty$ for the jet material is set to 400 for all simulations and the luminosity of the central engine is set to $5.32
\times 10^{50}$ erg s$^{-1}$. Table \[modelparameters\] lists the parameters used for each simulation. Outside the jet injection region, where the boundary values are fixed, the boundary conditions on the lower edge of the grid are reflective, which is appropriate assuming that symmetric jets emerge along both axes of the star. The outer boundaries of the grid use outflow boundary conditions and the symmetry axis is reflecting. Each simulation is run for 50 seconds, giving an energy input of $2.66 \times 10^{52}$ ergs per jet, or a total energy of $5.32 \times 10^{52}$ ergs assuming symmetric jets. These total energies are comparable to those assumed in previous works (see Zhang et al. 2003 for a discussion).
Results
=======
The early evolution of each of our models is generally similar to that found in previous simulations (Zhang et al. 2003, 2004). The jet ram pressure generates a bow shock that propagates sub-relativistically through the star. Cold and dense stellar material is pushed to the sides, partly mixing with shocked jet material to create a hot cocoon that surrounds the jet. This allows the younger jet material to propagate relativistically and unpolluted through an evacuated channel along the symmetry axis. Once this material reaches the stellar surface, it accelerates to its final Lorentz factor and can, once it has become optically thin, generate the observed GRB spectra.
Figure \[timesequence\] shows a time sequence of the evolution of model t10g5. The three upper panels are very similar to figures obtained by other groups with previous simulations, even though our AMR code can capture finer details. The main novelty of our work is shown in the three bottom panels, where the jet is still powered several tens of second after breakout and is evolving inside the star. As we will explain in more detail in the following sections, three phases can be identified. Initially the jet is confined, and hot turbulent material is stored in a cocoon (first two panels). When the jet head reaches the surface, the cocoon is released as a wide angle outflow (third panel). Immediately afterward, a heavily shocked jet flows outside the star (third and fourth panels). Eventually, a more stable configuration emerges (fifth panel) in which the jet is internally free-flowing, and is bounded by a shear layer at the contact discontinuity with the star. Figure \[rhopg\] shows a snapshot of simulation t10g2 at 30 seconds as an example of the structure present shortly after breakout.
The three phases are identified through the behavior of the on-axis energy flow, as shown in Fig. \[fig:phasedef\]. The transition between the precursor and the shocked jet is defined as the moment at which the energy flow along the axis becomes continuous, albeit variable. The transition between the shocked and unshocked jet is defined as the time at which the on-axis energy flow drops and becomes steady, without prominent variations. Table \[breakout\] lists the times at which the cocoon, shocked jet, and unshocked jet reach the initial surface of the star. From these data we can see some expected trends as different parameters are varied. Models with an initial Lorentz factor of 2 (t10g2 and 16TIg2) have longer breakout times than corresponding models with an initial Lorentz factor of 5 (t10g5 and 16TIg5). This is to be expected because a lower Lorentz factor at the same energy implies less momentum in the direction of propagation and a higher pressure (similar conclusions in a different geometry were obtained by Mizuta et al. 2006). This means that the jet will be less tightly collimated and will propagate more slowly.
Model t5g2, with a $5\degr$ injection opening angle, has a slightly earlier cocoon breakout time than the corresponding model with a $10\degr$ opening angle (t10g2). A smaller opening angle means that the same amount of energy and momentum is initially spread over a smaller area, making the jet more penetrating. However, as the jet is being collimated, the initial opening angle is only important at the beginning of the simulation, hence the overall difference between the two simulations is small. The shocked jet breakout time is earlier for model t10g2 by 0.8s, and the unshocked jet breakout is earlier for model t5g2 by 2s. These differences appear to depend more on the details of turbulence near the jet head than on the initial opening angle of the jet.
Simulations using the realistic stellar model (16TIg5 and 16TIg2) have earlier breakout times than simulations using the power-law stellar model and identical jet (t10g5 and t10g2, respectively). This is because the realistic stellar model is more compact, giving the star a smaller radius and higher density. The higher density leads to a more tightly collimated jet, which is therefore more penetrating. As a result, the jet is able to travel through about the same amount of total mass in a shorter time. It should be noted, however, that the average speed of the bow shock is smaller in the denser, realistic, stellar progenitor.
We also ran a simulation where a wide jet ($\theta_0=20\degr$) is injected at the center of the star. Model t20g5 appears to be morphologically different from the other simulations. It has the latest cocoon breakout time of any of the simulations, and the jet material along the propagation axis does not reach the surface of the star until the end of the simulation. Before this, relativistic material is deflected around a dense wedge of stellar material and can escape the star at large angles, but it does not appear that this would produce a classical gamma ray burst. This is in part a 2D effect. In three dimensions, the jet could be deflected sideways to move around the wedge of material, but in our 2D simulations it is forced to be axially symmetric and therefore stalls (see Zhang et al. 2004). Figure \[t20g5compare\] compares models t10g5 and t20g5. The reason why model t20g5 is different is that the initial opening angle is too large to produce a tightly collimated jet. The pressure of the incoming jet falls as $\theta^{-2}_0$, meaning that a wider jet is less penetrating. At 20 degrees, model t20g5 is not penetrating enough to pierce the star in the manner necessary to create a GRB, despite the large energy input of $5.32 \times 10^{52}$ ergs. This simulation may represent a different class of jet-powered supernova or failed gamma ray burst, but it will not be considered in latter portions of this paper that compare our simulations to classical GRB properties. Analogous conclusions were drawn from previous simulations (MacFadyen et al. 2001; Zhang et al. 2003) and for 3D precessing jets (Zhang et al. 2004). It should be remembered, however, that assuming a wide jet at $10^9$ cm in the star may be unphysical. Even if a wide jet is launched at the black-hole scale, recollimation is likely to take place at radii $10^8-10^9$ cm (see model JA in Zhang et al. 2003). As a consequence, an initial condition with a narrow, entropy rich, jet (model t5g2) is more realistic than a wide fast outflow (model t20g5) at the inner boundary of our simulations.
Angular Distribution of Energy {#angulardistributionofenergy}
------------------------------
A snapshot file containing all the simulation data is produced every 1/15th of a second in simulation time. The energy flux is determined as a function of angle and time by taking a simulation snapshot and finding all the points that will cross a fixed radius in the next 1/15th of a second. The energy at each point is spread equally over an angle of $\pm1/\gamma_\infty$ from the direction of motion of the fluid at that point. This approximately accounts for the hydrodynamic spreading (or sideways expansion) of that element of the jet and for the relativistic beaming of radiation eventually emitted by that material. The energy is then placed into angular bins. The total energy in each angular bin can then be calculated by adding the energy from all points, and correcting for geometrical effects to give the energy flux in each bin. The energy in material above a given terminal Lorentz factor (Lorentz factor at infinity) can be found by excluding points with material below that $\gamma_\infty$. The results from each snapshot can be added over time to find the total energy seen at a fixed radius at different angles for the entire length of the simulation, or for shorter time intervals. With this data, we can examine the amount of energy crossing a fixed radius as a function of time, angle, and minimum Lorentz factor. Bear in mind that, given the definition of the angle adopted, the energy distribution may be modified by hydrodynamic interactions during the propagation to larger radii[^1]. For the results presented here, data from our simulations have been taken at $2.4 \times 10^{11}$ cm and placed into 45 angular bins arranged in the following manner: 14 bins spaced every $0.25\degr$ with centers ranging from $0.125\degr$ to $3.375\degr$, 17 bins spaced every $1.0\degr$ with centers ranging from $4.0\degr$ to $20.0\degr$, and 14 bins spaced every $5.0\degr$ with centers ranging from $23.0\degr$ to $88.0\degr$.
### Energy vs. Angle
In order to determine the observational characteristics of a gamma ray burst produced from our simulations, we examine the amount of energy directed toward observers at different angles. Figure \[total\_energy\_angle\_16TIg5\] shows the total energy crossing a radius of $2.4 \times 10^{11}$ cm during the simulation. Minimum terminal Lorentz factors of 1.01, 2, 10, 50, and 200 have been used for the different lines in the figure. The energy has been converted to units of isotropic equivalent energy available at different angles. The observed isotropic energy would then be the available energy times the efficiency of converting this energy to gamma rays. Different minimum Lorentz factors are presented because the efficiency of gamma ray production is likely to vary with Lorentz factor.
For each simulation, the total isotropic equivalent energy $dE/d\Omega$ is constant or slowly decreasing from the jet axis out to some cutoff. Inside the cutoff, energy goes as $\sim$constant to $\theta^{-1}$. Beyond this, the energy decreases rapidly. In some cases there is a significant contribution from precursor energy at large angle, but when this is removed (Fig. \[no\_pre\_energy\_angle\_16TIg5\]), the cutoff in energy becomes even sharper. For large $\gamma_\infty$, this decrease in energy can be fit by an exponential or a steep power law ($dE/d\Omega \sim
\theta^{-4}$ or more). If low-energy particles are included ($1.01
\le \gamma_\infty \le 2$) then energy falls off as $\sim \theta^{-2}$ at large angles. These particles should not contribute to the prompt burst, as they do not have a Lorentz factor large enough to produce gamma rays, but they can produce X-rays and can be important to the afterglow energetics. This differs from the Zhang et al. (2004) finding that the energy is distributed as $dE/d\Omega \sim
\theta^{-3}$. The difference is due to a combination of several factors. First, since their simulations are performed on a fixed grid, the resolution is poorer at the stellar surface compared to our AMR simulations. Second, Zhang et al. (2004) consider a small range of angles for their $\theta^{-3}$ fit ($3\degr<\theta<15\degr$). As a matter of fact, some of our models (especially 16TIg2 and t10g5) could be fit by $\theta^{-3}$ power-laws in small angle intervals. Finally, their simulation domain inside the star is only from $10^{10}$ cm to $8.8\times10^{10}$ cm, a volume that may be too small to allow the development of turbulence that would result in mixing of the jet material with the stellar material. From our simulations and the comparison with previous work, it can be concluded that the energy distribution with angle is not a robust property of the collapsar, but depends on the stellar structure and the jet injection parameters.
As discussed above, the passage of jet material at a fixed radius consists of two phases: the passage of the shocked and unshocked jet. Figure \[shock\_unshock\_energy\_angle\_16TIg5\] shows the total energy from each of these phases in model 16TIg5 for $\gamma_\infty \ge 50$ at a radius of $1.2 \times 10^{11}$ cm. The shocked phase of the jet has an approximately uniform energy distribution with a sharp cutoff at the edge of the jet. During the unshocked phase, however, mass and energy are concentrated in the shocked boundary layer that surrounds the unshocked core of the jet. This means the unshocked jet does not contribute much energy near the jet axis. However, at angles outside where the shocked phase of the jet is visible, the unshocked phase is the dominant energy source. In addition, while the energy in the shocked jet phase is fixed by the stellar properties, the unshocked jet can extend in time and become more and more prominent. Only self-consistent simulations of the jet feeding, launching, and propagation can pin down any absolute normalization between the energies of the two phases. At any angle, the unshocked phase only contributes significant energy while the thin shocked boundary layer covers that angle. As the opening angle of the jet is increasing during this phase, an observer will only see the boundary layer for a relatively short time. Increasing the duration of energy injection will therefore increase the final opening angle of the jet, but it will not significantly increase the isotropic equivalent energy seen by observers already within the jet.
### Phases of Jet Evolution
Figure \[energy\_time\_angle\_16TIg5\] shows logarithmic energy (brightness) contours as a function of time (vertical axis) and angle (horizontal axis) for model 16TIg5. Different panels show different minimum $\gamma_\infty$ cutoffs. The figure shows that there are three phases in the energy flux seen at a fixed radius. First there is a precursor phase, preceding significant on-axis energy flux, followed by a shocked jet phase, and an unshocked jet phase.
The precursor phase can be divided in two subsequent events. First a thin, nearly isotropic shell of mildly relativistic material, visible in the lower energy bands of Fig. \[energy\_time\_angle\_16TIg5\]. This phase is associated to the shock preceding the jet expanding into a density gradient and carries very little energy. For this reason it has observational consequences only if the other phases of the GRB are not visible due to viewing angle constraints (see § \[secpr\]). This initial precursor is followed by a mixture of jet and stellar material which is peaked at several degrees off axis with less energy on axis. This is the material that accumulates around the jet in the form of a cocoon (Ramirez-Ruiz et al. 2002) during the initial phases when the jet is confined inside the star. This material generally has a lower Lorentz factor than the jet and is most clearly visible in the lower energy plots. A further discussion of this material is given in § \[precursors\].
After the cocoon material, the shocked jet emerges. Differently from the cocoon material, the shocked jet has a large Lorentz factor at the stellar surface. During the shocked phase, jet material between the jet head and reverse shock is passing by. The shocked portion of the jet has a highly structured interior, including turbulent structures and internal shocks. Although this makes the energy flux and width of the jet variable during this period, there is no clear trend with time in the jet properties.
Finally, the jet settles into a quasi-stationary configuration, which we call the unshocked phase. The reverse shock has passed the radius we are looking at, and the jet consists of unshocked material surrounded by a narrow boundary layer of shocked material. During this phase the jet does not have the variable structure seen in the precursor and shocked phases, but the width of the jet clearly increases with time (see § \[openingangle\]). Figure \[fig:frexp\] shows the Lorentz factor and pressure along the jet axis of model 16TIg5 at the time at which the shocked-unshocked jet boundary crosses the star surface. The figure confirms that the core of the unshocked jet, inside the boundary layer, is free streaming. Dashed lines show how pressure and Lorentz factor behave in a free-streaming, pressure dominated jet; the simulation result is in excellent agreement with this behavior. Note that the Lorentz factor deviates from the asymptotic solution at large $\gamma$. This is due to the fact that at such highly relativistic speeds, the pressure-dominated approximation does not hold any more. The comparison of Fig. \[fig:frexp\] with previous results (e.g. Fig. 2 in Aloy et al. 2000, Fig. 4, 5, and 6 in Zhang et al. 2003, or Fig. 6 and 7 in Mizuta et al. 2006) reveals much sharper features in our results. This is likely due to the increased resolution of the AMR code with respect to fixed grid codes.
It is interesting to speculate on the temporal properties of the three phases. It is usually assumed that the light curve of the GRB prompt emission is due to internal shocks driven by Lorentz factor inhomogeneities in the flow. These inhomogeneities are supposed to be imprinted by the central engine. The propagation of the outflow through the star, however, is likely to modify these structures, erasing some and amplifying others. Which ones are erased and which amplified will depend on the phase of the jet evolution. The precursor material is made of jet and stellar material that has been completely reshuffled and shocked in the bow shock and turbulent eddies surrounding the jets. As a consequence, the ejection history of the central engine has been forgotten. A similar conclusion holds for the shocked jet, even though some trace of the engine properties may be retained. In the unshocked jet, however, and especially in its freely expanding core, all variability imprinted by the engine will be frozen and advected to the radiative phase. The properties of the inner engine will therefore be more clearly seen in the tail of the GRB emission. Unluckily, this is the faintest phase.
### Opening Angle {#openingangle}
As noted from previous simulations (Zhang et al. 2003) and discussed theoretically in LB05, the opening angle is not a constant property of the outflow emerging from the star, nor is it an easily measurable quantity. We define the opening angle as follows. We first select all points that will cross a fixed radius within 1/15th of a second (the same points used to find energy flux) and with a minimum value of $\gamma_\infty$. We then compute the angles associated with all these points and define the jet opening angle as the largest. In this case, we define the angle as the geometrical angle of the point with respect to the origin and polar axis of the coordinates, since adopting the velocity vector angle introduces substantial noise (see footnote 1).
Figure \[opening\_angle\_16TIg5\] shows opening angle vs. time for different minimum values of the terminal Lorentz factor ($\gamma_\infty$) at a radius of $1.2 \times 10^{11}$ cm in our simulations. This figure emphasizes again the three phases of jet evolution. As material first reaches this radius, the opening angle is very wide for all but the highest values of $\gamma_\infty$. This is due to the precursor material discussed in § \[precursors\]. Following this is the shocked portion of the jet, characterized by a fairly constant opening angle and no consistent evolution with time. Using different values of $\gamma_\infty$ gives widely different opening angles, ranging, for example, from $\sim 2\degr$ for $\gamma_\infty = 200$ to $\sim 8\degr$ for $\gamma_\infty = 2$ in model 16TIg5. This is because material at the edge of the jet is partially mixed with stellar material, lowering its terminal Lorentz factor. After the shocked portion of the jet has passed, the jet consists of an unshocked core with a shocked boundary layer along the edge. As the unshocked jet passes, the opening angle of the jet is slowly but consistently increasing. The rate of increase is always less than linear and can be well fit by a logarithmic increase. The rate of increase is obviously much slower than the exponential increase predicted in LB05 but is in reasonable agreement with the semi-analytic results presented in this paper (§ \[analytic\]). The thickness of the boundary layer, measured as the difference between the opening angles for $\gamma_{min} = 2$ and $\gamma_{min} =
200$, decreases with time. For model 16TIg5, the thickness goes from $\sim 6\degr$ in the shocked phase down to $\sim 4\degr$ by the end of the simulation (see Fig. \[opening\_angle\_16TIg5\]).
Although the intrinsic opening angle of the jet is as small as $\theta_0 = 5\degr$, at late times the jet will be over-pressured at the base and expand to a larger opening angle. After cocoon breakout, the pressure in the cocoon decreases exponentially, while the incoming jet pressure remains fixed. When the cocoon pressure drops below the incoming jet pressure, the jet will expand near its base by an angle of up to $1/\gamma_0$, where $\gamma_0$ is the Lorentz factor of the incoming jet. For $\gamma_0 = 5$ this is about $11\degr$. Therefore, a jet with $\theta_0 = 5\degr$ and $\gamma_0 = 5$ can have a maximum opening angle of about $16\degr$. Pressure in the unshocked jet drops as $r^{-4}$, so at large radii the jet is still being collimated by the cocoon pressure. In all simulations presented here, the opening angle of the jet at large radii is always less than the maximum opening angle for that simulation.
Precursors
----------
As the jet propagates through the dense material of the star, a high density wedge of stellar material develops at the head of the jet. The jet material does not penetrate this wedge, but instead moves to the sides. Eventually this material will curl back on itself, creating large vortexes in advance of the narrowly collimated jet (Fig. \[timesequence\]). As the jet continues to propagate, the vortexes will eventually detach from the jet and be swept backward, relative to jet propagation, in a phenomenon known as vortex shedding (Scheck et al. 2002; Mizuta et al. 2004). New vortexes will then develop at the head of the jet in a repeating cycle. However, when the cocoon of material surrounding the jet breaks out of the star and into the low density material surrounding it, the cocoon material is released and these vortexes are no longer swept backward. Whatever material is being shed from the head of the jet at this time is then free to expand ballistically. The result is a significant amount of relativistic material escaping over a large angle at close to the breakout time. Even though the details of the process require a 3D simulation for a deep investigation, the formation of a cocoon of high pressure material is unavoidable (see Ramirez-Ruiz et al. 2002 for discussion of cocoon properties). Because the opening angle of the jet increases slowly, for off-axis observers there can be a long delay between when the cocoon material is seen and when the jet is seen. This vortex material could therefore be responsible for precursor events seen 10s of seconds before the main gamma ray burst (Lazzati 2005; Lazzati et al. 2005). In our simulations, lasting 50 seconds, delays of up to $\sim20$ seconds were seen between the precursor material and jet material at certain angles (Fig. \[pre\_angle\_16TIg5\]). At larger angles, the jet never comes into view, but the precursor is still visible. This could account for the observed soft, low-energy gamma ray bursts such as GRB980425 (associated with SN 1998bw, Galama et al. 1998). Whether this faint GRB had a relativistic jet associated with it is still a matter of lively debate (Waxman 2004).
### Precursor Energetics
To estimate the total energy contained in the precursor, we first find the isotropic energy vs. angle of the precursor at $2.4 \times
10^{11}$ cm, as described above (§ \[angulardistributionofenergy\]). This energy can then be added over angle to give the total energy in the precursor. The mode and full width at one tenth of the maximum of the energy-weighted direction of motion can also be calculated to give an idea of the angle over which the precursor is visible. All those quantities are given in Tab. \[precursorenergetics\] for $\gamma_\infty>10$ and $\gamma_\infty>50$.
For the five simulations, the total precursor energy with $\gamma_\infty \ge 10$ ranges from about $3 \times 10^{50}$ to $5
\times 10^{51}$ ergs, with $45\%$ to $75\%$ of that energy carried by material with $\gamma_\infty \ge 50$. This means that $\ga1\%$ of the total input energy of our simulation, $2.66 \times 10^{52}$ ergs per jet, ends up as relativistic material in the precursors. This material is spread over a wide opening angle ($\sim 40\degr$), so the isotropic equivalent energy of the precursor emission should be within an order of magnitude of the total precursor energy. Models with a later breakout time typically have more energy in the precursor material, which is expected since a longer propagation time allows more relativistic material to be shed from the jet and because simulations with a lower initial Lorentz factor have later breakout times and have less momentum in the jet, making it easier to deflect.
### Precursor Energy vs. Angle
Figure \[pre\_energy\_angle\_16TIg5\] shows isotropic equivalent energy of the precursor vs. angle for our simulations. The isotropic equivalent precursor energy can be up to $2 \times 10^{52}$ ergs, $\sim 2\%$ of the $10^{54}$ ergs typically seen in the core of the jet. Note in Fig. \[pre\_energy\_angle\_16TIg5\] that near the axis there is usually less precursor emission. This is because vortex material is being deflected away from the head of the jet. This is in part a 2D effect. In 3D, the jet would be able to wobble and the precursor material would not have to be rotationally symmetric. This should increase the amount of precursor material on axis. However, the qualitative effects would be the same. Material would still be shed from the head of the jet, creating a region of relativistic material spread over a wide opening angle near the head of the jet.
The precursor energies found in our simulations are sufficient to account for observed precursor emissions, which typically produce $\sim$ 1/1000th as many photons as the complete GRB (Lazzati 2005). As shown in Fig. \[pre\_energy\_angle\_16TIg5\], the isotropic equivalent precursor energy typically peaks at $10^{52}$ ergs. The absolute energies of observed precursors are not known because the redshifts of GRBs with precursors have not been measured. However, it is possible that precursors are only detected in relatively nearby bursts, which would on average be intrinsically fainter than GRBs in general. This could further reduce the amount of energy needed to produce the observed precursor emission, in the case that precursors have less energy available than our simulations show or are less efficient at producing gamma rays.
### Precursor Lorentz Factors
Table \[precursorenergetics\] also contains the ratios of precursor energies for material with $\gamma_\infty \ge 50$ to $\gamma_\infty
\ge 10$. As highly relativistic material ($\gamma \sim 100$) is needed to produce high-energy gamma rays, this ratio should reflect the spectral hardness of gamma ray emission from the precursor material. The precursor material is less relativistic than the jet material because some stellar material has been mixed into it. Whereas the jet material typically has $\ga 80\%$ of its energy in material with $\gamma_\infty \ge 50$, the precursor material has $\sim
60\%$ of its energy in material with $\gamma_\infty \ge 50$. A larger percentage of energy in mildly relativistic material could produce a softer spectrum of emitted radiation, as observed in GRB precursors (Lazzati 2005) and low-energy GRBs (Galama et al. 1998).
### Isotropic Precursors {#secpr}
When the cocoon breaks out of the star, before the cocoon energy of is released, there is a thin shell of mildly relativistic material formed at the leading edge of the expanding material. This shell provides a second possible source of precursor emission. The shell is nearly isotropic and is visible in Fig. \[energy\_time\_angle\_16TIg5\]. Note that this shell is not well resolved in the simulations presented here. The isotropic equivalent energy in this shell is $\sim 10^{47}$ to $\sim 10^{49}$ ergs, far lower than the energy of a typical GRB. It is therefore unlikely that radiation from this material would be detected along with a classical GRB. The shell material also has a low terminal Lorentz factor, typically around 5, and therefore may not be able to produce gamma ray photons. However, this material could still produce X-ray photons and may be observed as an X-ray flash by observers far off-axis, who see emission from neither the jet nor the vortex precursor material.
Between the jet material, vortex material and isotropic shell material, observers at different angles could see the same event as a classical GRB without a precursor, a classical GRB with a precursor, a low-energy GRB, or an X-ray flash. Figure \[pre\_angle\_16TIg5\] shows plots of energy flux vs. time for 4 different angles to illustrate different types of observed events.
At a viewing angle of $1.125\degr$, a large energy flux begins to arrive about $15$ seconds after the start of the simulation and last for $18$ seconds. After this there is a small amount of energy still arriving (from the unshocked jet) until the end of the simulation. At $5\degr$, the precursor is visible as a spike of emission beginning at $\sim14$s and lasting for about $1$s. This is followed by $\sim15$ seconds with low energy flux. Emission then begins again as the edge jet comes into view. After $44$s the emission reaches a lower constant level as the unshocked portion of the jet comes into view. At $7\degr$, the precursor is again visible starting at $14$s, but the jet does not come into view until $\sim20$ seconds later. At $12\degr$, the precursor is still seen at $14$s, but the jet is never visible.
Effects of Resolution
---------------------
In order to test the effects of resolution on our simulations, we have carried out a version of 16TIg5 at half the resolution of our other simulations. In the low-resolution version of model 16TIg5, there is not a distinct precursor phase preceding the shocked jet, as can be seen in Fig. \[energy\_time\_angle\_16TIg5lowres\]. This appears to be because the resolution of this model is not sufficient for vorticity to develop at the head of the jet. The thin isotropic shell is, however, seen in the low resolution model. In the first three seconds after reaching $2.4 \times 10^{11}$ cm, the jet is wider at lower energies than at later times and is narrower at high energies. Although this structure is not a separate precursor as seen in the high resolution simulations, it could still give rise to precursor emissions seen off-axis before the jet comes into view.
Despite a lack of distinct vortex structures, the evolution of the shocked and unshocked phases are very similar to the high resolution model. Because material at the head of the jet has not been deflected, the shocked portion of the jet is larger in the low resolution model. In other words, the reverse shock has propagated farther into the jet at $2.4\times10^{11}$ cm in the low resolution model. This would be expected as material is not being shed from the jet as in the high resolution model.
The breakout times of the cocoon and shocked jet are about 3 seconds earlier in the low resolution simulation, and unshocked jets have nearly identical breakout times (Tab. \[breakout\]). Figure \[total\_energy\_angle\_16TIg5lowres\] compares the total energy vs. angle at the two resolution and shows that they are very similar. Figure \[opeining\_angle\_16TIg5lowres\] compares the opening angle vs. time of the two resolutions. The low resolution plot shows a wider opening angle, particularly at the lower Lorentz factor cutoffs, but this is expected due to the lower resolution. Other than this, the time evolution of the opening angle is nearly identical in the two simulations.
Analytic Modeling and Interpretation {#analytic}
====================================
More insight into the results of the numerical work can be achieved if the dominant processes that shape the jet evolution and its interaction with the star can be singled out. To this aim, we developed an analytical description of the jet-cocoon-star interaction. We find that the phase during which the jet is confined inside the star (before the breakout) and the unshocked jet phase can be reasonably well approximated with a semi-analytic treatment. This allows us to compute breakout times, precursor energetics, the amount of energy given to the star and the late time evolution of the jet properties.
In LB05 we explored the dynamics of the jet-cocoon interaction under the *monolithic jet* assumption, in which the jet is assumed to be uniform across its section. In addition, the jet is assumed to satisfy Bernoulli conditions, i.e., no significant dissipation by shocks. As a consequence, the pressure exerted by the jet on the cocoon material is only the internal pressure (which is relativistically invariant for a perpendicular Lorentz boost), $$p_j=\frac{L_j}{4\,\Sigma_j\,c\,\gamma_j^2},
\label{eq:pjet}$$ where $L_j$ is the jet luminosity, $\Sigma_j$ is the jet cross section and $\gamma_j$ is the jet Lorentz factor. Under this approximation, LB05 found that the jet reaches the stellar surface very narrow and spreads exponentially afterwards.
The FLASH numerical simulations we present show that the jet does not have a uniform distribution of pressure and density but rather develops a boundary layer structure (see Fig. \[timesequence\]). The core of the jet is freely streaming out to the point at which it collides with the boundary layer, which in turn flows parallel to the jet-cocoon boundary. In addition to the jet internal pressure, we have therefore the ram pressure due to the deflection of the free streaming jet by the boundary layer. Figure \[fig:sketch\] shows a sketch of the jet geometry that we consider.
Simple geometry allows us to derive the pressure balance equation $$p_{\rm{cocoon}}=p_j+4\,p_j\,\gamma_j^2\,\sin\left[
{\rm{atan}}\left(\frac{dz}{dr_\perp}\right)-
{\rm{atan}}\left(\frac{z}{r_\perp}\right)
\right],
\label{eq:bal}$$ where $z$ is the coordinate along the jet and $r_\perp$ is the perpendicular size of the jet. Eq. \[eq:bal\] can be simplified in the approximation of a narrow ($z\gg{}r_\perp$) relativistic jet (for which the ram pressure is much larger than the internal pressure) yielding a differential equation of the form $$\frac{dr_\perp}{dz}=\frac{r_\perp}{z}-r_\perp^2\,K,
\label{eq:approx}$$ where $K=\pi\,c\,p_{\rm{cocoon}}/L_j$ is a constant related to the ratio of the ram to cocoon pressures. Eq. \[eq:approx\] has an analytic solution of the form $$r_\perp=\frac{2\,z}{K\,z^2+C},
\label{eq:solu}$$ where $C$ is a constant of integration. Eq. \[eq:solu\] can be rewritten in a clearer form as $$\theta_j=\frac{2\,\theta_0}{2+K\,\theta_0\,(z^2-z_0^2)}.
\label{eq:theta}$$ In this form is easy to see how the opening angle of the jet is initially constant, but decreases as the jet propagates. For very large values of $z$ the jet tends to close on itself. This equation will be used in the following to study the time dependence of the jet opening angle. We now concentrate on the parameter $K$. The missing piece of information to derive it is the cocoon pressure.
To compute the cocoon pressure we develop the approximations of Begelman & Cioffi (1989) and the jet propagation description of Matzner (2003). The cocoon evolution is governed by the first principle of thermodynamics which, for a relativistic temperature cocoon, reads $$d\rho_{\rm{cocoon}}=\frac{dQ-4/3\rho_{\rm{cocoon}}dV_{\rm{cocoon}}}
{V_{\rm{cocoon}}},
\label{eq:thermo}$$ where the evolution of the cocoon volume is computed as $$\frac{dV_{\rm{cocoon}}}{dt} =
2\pi\,\int_{z_0}^{z_h}r_{\rm{cocoon},\perp}\,v_{\rm{sh}}\,dz.
\label{eq:vol}$$ Here $v_{\rm{sh}}=\sqrt{\rho_{\rm{cocoon}}/3\rho_\star}$ is the cocoon expansion velocity (Begelman & Cioffi 1989; $\rho_\star$ is the matter density of the star), $r_{\rm{cocoon},\perp}$ is the transverse size of the cocoon, $z_0$ is the location of the base of the jet and $z_h$ is the position of the head of the jet.
The energy input into the cocoon, $dQ$, is calculated differently in the two phases: when the whole jet is inside the star (the cocoon is bounded) and when the jet has broken out (the cocoon is unbounded). In the first phase, the energy input is from the dissipation of the jet energy. In the second phase, energy is lost through a channel at the stellar surface, that we assume to have a surface area equal to half the jet cross section[^2]. We obtain $$dQ=\left\{
\begin{array}{cc}
L_j\,(1-\beta_h) & {\rm jet~in~star}\\
-\rho_{\rm{cocoon}}\,\Sigma_j/2\,c_s\,dt & {\rm jet~outside~the~star}
\end{array}\right.,
\label{eq:dq}$$ where $\beta_h$ is the speed of the head of the jet in units of $c$.
Equations \[eq:pjet\], \[eq:bal\], \[eq:thermo\], \[eq:vol\], and \[eq:dq\] constitute a solvable system of equations. To check the validity of the assumptions we have computed the solution of the equations and run simulations for the same progenitor and engine parameters.
Figure \[fig:jet\] shows a comparison of the simulation results (model 10g5) with the semi-analytic predictions. During the confined jet phase, before the breakout, the motion of the head of the jet is reproduced with reasonable accuracy. The energy stored in the cocoon at the time of breakout and the energy given to the stellar material (potentially powering a supernova explosion) are accurate within $10\%$, while the breakout time is reproduced within 20%. Comparison with other simulations (16TIg5, t10g2 and t5g2) show that the semi-analytic results are accurate to within $20\%$ with respect to the numerical treatment. The analytic treatment tends to be more accurate for jets injected with higher values of $\gamma\theta_0$, i.e., jets that have lost causal contact. This is not surprising since our analytic treatment does not account for the spreading of the jet due to internal motions, which is relevant in the $\gamma<\theta_0^{-1}$ case.
Afterglows
==========
Besides light curves of the prompt emission (Fig. \[pre\_angle\_16TIg5\]), we can compute afterglow light curves based on the energy distributions obtained from these simulations. To this aim, we adopt the afterglow code of Rossi et al. (2004). We input into the code the energies from model 16TIg5 and t10g5 with a lower-limit on the Lorentz factor $\gamma_\infty=5$ and we assume a constant density of the ISM with $n=10$ cm$^{-3}$. The flow is supposed to propagate from the low radii of our outer boundary to the external shock radius without re-adjusting the energy distribution and no sideways expansion of the external shock is assumed. We also simplify the computation by assuming a Lorentz factor $\gamma=400$ for the whole fireball. This simplification is due to the fact that it is not possible to input, for a given angle, material with different Lorentz factors, as obtained in the simulations. The effect of this approximation should be to slightly modify the shape of the light curve around the peak of the afterglow emission.
Results of the afterglow calculation are shown in Fig. \[fig:after\]. Light curves have been computed for different observing off-axis angles $\theta_0=0$, 2, 4, 8, 16, and 32 degrees and for two simulations: model 16TIg5 and model t10g5. The jet properties are the same in the two models, but the progenitor stars are different. In each panel we plot also the afterglow from a standard top-hat jet for comparison. The behavior of the two simulations is quite different. Model 16TI has a very flat energy distribution in its center. For this reason, the inner light curves resemble very well those of a top hat jet. Only the on-axis light curve deviates due to its high energy. While the slopes are different from those of the top-hat afterglow, the spectra are the same. This implies that the use of the so-called afterglow closure relations[^3] (Price et al. 2002) cannot be blindly applied to afterglow with the angular energy distributions derived from these simulations.
Owing to its more centrally condensed distribution, model t10g5 has afterglows for small off-axis angles that differ from each other and from the top hat example. Qualitatively, the curves are however similar to one another, with an early shallow decay followed by a sharper decay when the jet reaches causal contact. In both models 16TI and t10g5, curves at large off-axis angles show a plateau or even a bump several hours to weeks after the explosion. This is due to the fact that the radiation from the brighter jet core enters into the line of sight at late times. Comparison of the two lower panels of Fig. \[fig:after\] teaches us how the progenitor structure is important not only for the prompt GRB emission, but also for the ensuing afterglow radiation.
Summary and Conclusions
=======================
We have presented high resolution 2D simulations of the propagation of light relativistic jets inside the cores of massive stars. We use an adaptive mesh code (FLASH) that allows us to study the behavior of the jet-star interaction over a long timescale and a wide spatial range. Different stellar progenitors as well as initial conditions of the jet are explored. Thanks to the high resolution and large spatial and temporal domain of our simulations we can confirm and study in more detail jet features discussed in previous work (MacFadyen & Woosley 1999; Aloy 2000; MacFadyen et al. 2001; Zhang et al. 2003, 2004; Mizuta et al. 2006), as well as identify new phenomenologies.
The main conclusion of this paper is that, even if the central engine is stationary, the jet that propagates out of the progenitor star is characterized by three phases, all of which display significant variability. The first phase is a wide angle release of mildly to moderately relativistic material, which we call the precursor. This phase is due to the release of the turbulent shocked material that accumulates around the jet due to vortex shedding before it breaks out of the star (Ramirez-Ruiz et al. 2002; Zhang et al. 2003). This initial phase is followed by a shocked jet phase. In this second phase the jet material that flows out of the star has been heavily shocked. The energy flow is highly variable in this phase, and no temporal trend in the properties can be identified. During this phase the jet is most highly collimated. Finally, as the pressure of the cocoon decreases, the jet settles into a stable configuration with a freely expanding core surrounded by a shocked shear layer at the boundary with the cocoon material. In this phase the energy flow is almost constant, and the jet opening angle increases logarithmically with time.
This temporal evolution of the jet is also associated with the angular distribution of energy, and therefore determines what different observers see from different directions. The precursor is characterized by a wide opening angle and can be seen from most directions. The shocked jet phase is the most concentrated and can be seen only by observers within several degrees of the axis. As a consequence, such observers will see a very bright event. Observers who lie a few degrees outside the shocked jet cone will see the precursor, followed by a dead time of several tens of seconds. Eventually, when the jet opens to contain their line of sight, they will see a second phase of emission. These observers will therefore measure dead-times much longer than any timescale of the inner engine, as found in several BATSE light curves (Lazzati 2005).
The overall angular distribution of energy is complex and does not seem to follow any simple correlation with the jet or progenitor properties. In some cases, the jet is characterized by a flat core with a sudden cutoff, very similar to a top-hat jet (16TIg5), while in other cases the distribution is more centrally peaked. It seems, however, that our high resolution simulations were not able to reproduce previous analytic or numerical results like the $\theta^{-2}$ universal jet of LB05 or the $\theta^{-3}$ distribution found by Zhang et al. (2003). Such behaviors are observed only in a limited range of angles from the jet axis. We computed afterglow light curves from our energy distributions showing that differences in the afterglow phase can result from the different properties of the progenitor and/or of the jet in the core of the star. We also developed further the analytic model of LB05, refining some approximations to obtain a model that can reproduce such basic features of the simulations as the propagation of the bow shock inside the star and the precursor energy, as well as the qualitative evolution of the jet opening angle.
It is easy to find paths along which this work can be developed further. Higher resolution and dimensionality will certainly be worth exploring (Zhang et al. 2004), as well as the consequences of a more accurate equation of state capable of describing both relativistic and non-relativistic material. Including a non-relativistic wind from the accretion disk is also worthwhile, as this seems to be a ubiquitous outcome of jet-launching simulations (Proga et al. 2003; de Villiers et al. 2006). This wind component may alter the cocoon properties, which are so important in defining the behavior of the precursor and shocked jet phases. Simulations also suggest that the jet luminosity should be highly variable if not intermittent. Preliminary studies (Aloy et al. 2000) suggest that this may enhance the propagation of the jet through the star, but higher resolution simulations are required. An interesting hypothesis is that the three phases of jet propagation will respond differently to an intermittent engine, with the precursor virtually unaffected while the unshocked jet should retain most of the engine variability.
The software used in this work was in part developed by the DOE-supported ASC/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. We thank Alex Heger for providing us with the tabulated properties of his stellar models, Andrew MacFadyen for useful discussions and advice on the testing of the relativistic FLASH code and Miguel Aloy for useful discussions. This work was supported by NSF grant AST-0307502, NASA Astrophysical Theory Grant NNG06GI06G, and Swift Guest Investigator Program NNX06AB69G.
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[lllll]{}
16TIg5 & Realistic & $5.32 \times 10^{50}$ & $10\degr$ & $5$\
16TIg2 & Realistic & $5.32 \times 10^{50}$ & $10\degr$ & $2$\
t10g5 & Power Law & $5.32 \times 10^{50}$ & $10\degr$ & $5$\
t10g2 & Power Law & $5.32 \times 10^{50}$ & $10\degr$ & $2$\
t5g2 & Power Law & $5.32 \times 10^{50}$ & $5\degr$ & $2$\
t20g5 & Power Law & $5.32 \times 10^{50}$ & $20\degr$ & $5$\
[llll]{}
16TIg5 & $7.53$ & $7.73$ & $23.4$\
16TIg2 & $9.66$ & $12.0$ & $25.0$\
t10g5 & $13.1$ & $14.0$ & $29.0$\
t10g2 & $24.0$ & $31.0$ & $48.0$\
t5g2 & $21.2$ & $31.8$ & $46.0$\
16TIg5lowres & $4.93$ & $5.00$ & $23.9$\
[llllllll]{}
16TIg5 & $3.467\times10^{50}$ & $4\degr$ & $10\degr$ & $2.470\times10^{50}$ & $4\degr$ & $9\degr$ & $0.712$\
16TIg2 & $4.563\times10^{51}$ & $15\degr$ & $34\degr$ & $2.613\times10^{51}$ & $15\degr$ & $14\degr$ & $0.573$\
t10g5 & $7.948\times10^{50}$ & $15\degr$ & $54\degr$ & $4.602\times10^{50}$ & $15\degr$ & $14\degr$ & $0.579$\
t10g2 & $1.181\times10^{51}$ & $23\degr$ & $34\degr$ & $5.281\times10^{50}$ & $23\degr$ & $24\degr$ & $0.447$\
t5g2 & $2.560\times10^{51}$ & $7\degr$ & $39\degr$ & $1.898\times10^{51}$ & $7\degr$ & $39\degr$ & $0.741$\
Code Testing \[codetesting\]
============================
In order to evaluate the accuracy of the FLASH relativistic hydrodynamics module, we have carried out a series of tests and compared the results to analytic solutions where possible. These results have also been compared to the weighted essentially non-oscillatory (WENO) scheme used by Zhang & MacFadyen (2006) in their Relativistic Adaptive Mesh code (RAM hereafter). All test problems were run using FLASH’s Riemann-type solver along with a piecewise-parabolic reconstruction scheme, identical to that used for the simulations presented in this paper. Although this scheme is formally less accurate and more diffusive than that used in RAM, our testing shows that the errors produced by FLASH are similar to those produced by RAM.
1D Riemann Problem 1
--------------------
For the 1D Riemann problem, there is a discontinuity between two states at $x=0.5$ on a grid ranging from $x=0$ to $1$. The values of pressure, density, perpendicular ($v_x$) and transverse ($v_y$) velocity can be different for the left and right states. For the first Riemann problem considered, the left state has a pressure of $13.33$ and density of $10$ while the right state has a pressure of $10^{-8}$ and density of $1$. The $x$ and $y$ velocity in both states is initially $0$. For this problem an adiabatic index of $\Gamma=5/3$ and CFL number of $0.5$ are used. As time progresses, a mildly relativistic shock wave forms traveling to the right and a rarefaction wave travels to the left. The result at $t=0.4$ is then compared to the analytic solution to the identical problem (Pons et al. 2000), and the L1 error calculated. The L1 error is defined as $L1 = \Sigma _j \Delta x_j |u_j - u(x_j)|$ where $x_j$ is the coordinate of grid point $j$ and $u(x_j)$ and $u_j$ are the analytic and numerical values of proper density at grid point $j$ and $\Delta x_j$ is the grid spacing. This is identical to the definition of L1 error in RAM. Figure \[riemann1pic\] shows a comparison between the numerical and analytic solutions at a resolution of 400 grid points. Table \[riemann1table\] lists the L1 errors and convergence rates for resolutions from 100 to 3200 grid points for this model, and compares it with the error and convergence rate from RAM. Convergence rate is defines as $[\ln(L1_{i/2}/L1_i)]/[\ln{2}]$ where $L1_i$ is the $L1$ error at resolution $i$.
For Riemann type problems, the L1 error is dominated by the few points near the shock front. This limits the convergence rate of any code to about 1. It also makes the L1 error sensitive to the particular slope limiter being used. Therefore, small differences in the L1 error should not be taken to indicate a difference in the fundamental accuracy achievable by the two codes.
1D Riemann Problem 2
--------------------
For the second Riemann test, pressure is set to $1000$ in the left state and $10^{-2}$ in the right state. Density in both states in initially $1$. The velocities in both states are again set to $0$ with an adiabatic index of $5/3$ and CFL number of $0.5$. These conditions give rise to higher velocities than Riemann problem 1 and a narrower shock. Figure \[riemann2pic\] shows a comparison between the numerical and analytic solutions at a resolution of 400 grid points at $t=0.4$. Table \[riemann2table\] lists the L1 errors and convergence rates for resolutions from 100 to 3200 grid points for this model, and compares it with the error and convergence rate from RAM.
1D Riemann Problem 3
--------------------
For the third standard Riemann test problem, a situation is set up in which a strong reverse shock develops. Pressure is set to $1$ in the left state and $10$ in the right state. Density in both states in initially $1$. Velocity in the $x$ direction is set to $0.9$ in the left state and $0$ in the right state. Transverse velocity ($v_y$) is $0$ in both states. An ultra-relativistic adiabatic index ($\Gamma=4/3$) is used in this test along with a CFL number of $0.5$. Figure \[riemann3pic\] shows a comparison between the numerical and analytic solutions at a resolution of 400 grid points at $t=0.4$. Oscillations in the post-shock density of $\sim1\%$ are visible. Table \[riemann3table\] lists the L1 errors and convergence rates for resolutions from 100 to 3200 grid points for this model, and compares it with the error and convergence rate from RAM.
1D Riemann Problem 4: “Easy” Shear Velocity Test
------------------------------------------------
Analytical solutions can also be obtained for a 1D Riemann problem with a transverse (shear) velocity (Pons et al. 2000). For the “easy” version of this problem, the setup is identical to that of Riemann problem 2 above, but with a shear velocity of $v_y = 0.99$ in the right state and $v_y = 0$ in the left state. Figure \[riemann4pic\] shows a comparison between the numerical and analytic solutions at a resolution of 400 grid points at $t=0.4$. Table \[riemann4table\] lists the L1 errors and convergence rates for resolutions from 100 to 3200 grid points for this model, and compares it with the error and convergence rate from RAM.
1D Riemann Problem 5: “Hard” Shear Velocity Test
------------------------------------------------
For the “hard” shear velocity test, the setup is again identical to that of Riemann problem 2 above, but with a shear velocity of $v_y =
0.9$ in both the left and right states. This test is more difficult for numerical codes due to the shear velocity in the high-pressure left state. As the numerical solution poorly matches the analytic solution at low resolution, this test is run on higher resolution grids. Figure \[riemann5pic\] shows a comparison between the numerical and analytic solutions at a resolution of 400 grid points at $t=0.6$. Table \[riemann5table\] lists the L1 errors and convergence rates for resolutions from 100 to 6400 grid points for this model, and compares it with the error and convergence rate from RAM. It should be noted that for the last 3 Riemann problems the absolute error using FLASH is significantly small than for the F-WENO scheme in RAM. In particular, comparing figure \[riemann5pic\] to figure 9 in RAM it is apparent that FLASH produces a better fit to the analytic solution at low resolution. This is likely due to the different slope limiters being used by the two codes rather than fundamental limitations of the relativistic solvers they employ.
1D Isentropic Flow
------------------
For all the Riemann problems, the error is dominated by the region near the shock, which gives approximately first order convergence regardless of the scheme used. To examine the behavior of the code in smooth flow regions, we evaluate an isentropic smooth flow set up in a uniform reference state. The initial density for this problem is given by
$$\label{eqnrho0}
\rho_0 (x) = \rho_{ref} (1+\alpha f(x)),$$
where $\rho_{ref}$ is the density of the reference state and
$$f(x) = \left \{ \begin{array}{lll}
((x/L)^2 - 1)^4 & : & |x|<L \\
0 & : & $otherwise,$ \end{array} \right.$$
$\alpha$ and $L$ are the amplitude and width of the pulse. Pressure is given by the adiabatic equation of state $p = K\rho^\Gamma$, where $K$ is a constant. The initial velocity in the reference state ($|x|
> L$) is $0$. The initial velocity inside the pulse is found by assuming that one of the two Riemann invariants is constant,
$$J_- = \frac{1}{2} \ln(\frac{1+v}{1-v}) - \frac{1}{\sqrt{\Gamma-1}}
\ln(\frac{\sqrt{\Gamma-1}+c_s}{\sqrt{\Gamma-1}-c_s}) = const$$
where
$$\label{cs}
c_s = \sqrt{\Gamma\frac{p}{\rho+\frac{\Gamma}{\Gamma-1}p}}$$
is the sound speed. Solving for $v$ gives
$$v=\frac{e^{2(J_- + \frac{1}{\sqrt{\Gamma-1}}
\ln(\frac{\sqrt{\Gamma-1}+c_s}{\sqrt{\Gamma-1}-c_s})} - 1}
{e^{2(J_- + \frac{1}{\sqrt{\Gamma-1}}
\ln(\frac{\sqrt{\Gamma-1}+c_s}{\sqrt{\Gamma-1}-c_s})} + 1}$$
where $J_-$ is a constant calculated from the reference state. For the 1D isentropic test, our computational domain extends from $x=-0.35$ to $x=1.0$. The reference state is set to $\rho_{ref}=1$, $v_{ref} = 0$, $p_{ref}=100$, $K=100$, with the pulse amplitude $\alpha=1.0$ and pulse length $L=0.3$. An adiabatic index of $\Gamma=5/3$ and a CFL number of $0.5$ are used. The test is run until $t=0.8$, which is before a shock develops in the flow. An analytic solution for the density at this time can be found by characteristic analysis. A comparison of numerical and analytic results for density at $t=0.8$ is shown in figure \[isen1Dpic\]. L1 errors and convergence rates for resolutions from 80 to 5120 grid points are shown in Tab. \[isen1Dtable\]. This table shows that FLASH has a convergence rate of 2 for smooth flows, as expected for the solver being used (formally 2nd order accurate in time and space). The F-WENO scheme used in RAM has a convergence rate of $\sim3$, also as expected (formally 5th order accurate in space, 3rd order accurate in time). RAM is clearly more accurate in smooth flow regions, although both codes converge toward the correct solution.
2D Isentropic Flow
------------------
The isentropic flow problem can also be used to test the behavior of the FLASH code in a two dimensional situation. For this test, the computational region goes from $0.0 \le x \le 3.75$ and $0.0 \le y \le
5.0$ in 2D Cartesian coordinates. The boundary conditions of the grid are periodic. Periodic isentropic waves with a spacing of 3.0 are place in the grid such that ${\bf k}$, the normal vector perpendicular to the wave front, is ${\bf k}=(4/5,3/5)$. Note that the spatial periods in the $x$ and $y$ direction are 3.75 and 5.0, so that periodic boundary conditions are appropriate. The initial density profile is given by $\rho_0(d)$ (eqn. \[eqnrho0\]) where $d
=$mod$({\bf k}\cdot{\bf r}+S/2,S)-S/2$ and ${\bf r}=(x,y)$. The reference state is set to $\rho_{ref}=1$, $v_{ref} = 0$, $p_{ref}=100$, $K=100$, with the pulse amplitude $\alpha=1.0$ and pulse length $L=0.9$. An adiabatic index of $\Gamma=5/3$ and a CFL number of $0.5$ are used. The test is run until $t=2.4$. Table \[isen2Dtable\] shows the total L1 errors and convergence rates for grid resolutions from $48\times64$ to $768\times1024$ grid points. The convergence rate for FLASH is again about 2, as expected.
The above results confirm that the relativistic FLASH module used here converges to the correct solution for problems with known analytic results and has a comparable level of accuracy to other available relativistic hydro codes.
2D Riemann Problem
------------------
The Riemann problem can be extended to two dimensions by setting up a square grid in which each quarter has different initial conditions. Although there is no analytic solution to this problem, it has been well studied as a test of relativistic codes (Del Zanna & Bucciantini 2002; Lucas-Serrano et al. 2004; Zhang & MacFadyen 2006). The computational domain extends from $0.0 \le x \le 1.0$ and $0.0 \le y
\le 1.0$ in 2D Cartesian coordinates. The initial conditions for the four quarters of the grid are
$$\begin{array}{llll}
(\rho,v_x,v_y,p)=(0.1,0,0,0.01) & $for$ & 0.5 \le x \le 1.0 & 0.5 \le y \le 1.0 \\
(\rho,v_x,v_y,p)=(0.1,0.99,0,1.) & $for$ & 0.0 \le x \le 0.5 & 0.5 \le y \le 1.0 \\
(\rho,v_x,v_y,p)=(0.5,0,0,1.) & $for$ & 0.0 \le x \le 0.5 & 0.0 \le y \le 0.5 \\
(\rho,v_x,v_y,p)=(0.1,0,0.99,1.) & $for$ & 0.5 \le x \le 1.0 & 0.0 \le y \le 0.5
\end{array}$$
An adiabatic index of $\Gamma=5/3$ is used with a CFL number of $0.2$ on a grid of $512\times512$ grid points. Figure \[riemann2Dpic\] shows logarithmically spaced density contours of the test at $t=0.4$. Our test correctly reproduces the two curved shock fronts and sharp density spike in the upper right portion of the grid seen in other codes. The low density flow moving toward the lower left is also seen, although it appears more turbulent than in other codes. Symmetry is not perfectly preserved across the diagonal from lower left to upper right. This is because FLASH uses operator splitting rather than a Runge-Kutta integration scheme such as that used in RAM. However, this loss of symmetry is not important as our code is being used to examine jets in 2D cylindrical coordinates where there is no symmetry to preserve.
Isentropic Pulse
----------------
To examine what happens when material crosses a change in grid refinement, we set up a 2D isentropic pulse in density only that is advected across changes in refinement and eventually returned to its original location. For this problem the computational domain ranges from $0.0 \le x \le 0.9$ and $0.0 \le y \le 0.9$ in 2D Cartesian coordinates with periodic boundaries. The resolution of the mesh varies from $\Delta x=0.0225$ at the edges to $\Delta x=0.0028125$ in the innermost portion of the grid (see figure \[isenpulsepic\] for block structure). The structure of the grid is fixed and does not change as the density pulse moves. The pulse in initially centered at $x=0.45$, $y=0.45$ and is set by eqn. \[eqnrho0\] with $\alpha=10$ and $L=0.2$. Velocity everywhere is set to $v_x=0.72$ and $v_y=0.54$, for a total velocity of $v=0.9$. Pressure everywhere is $p=1$, the adiabatic index is $\Gamma=5/3$ and the CFL number is $0.5$. The test is run until $t=10$, at which time the pulse is again centered at the center of the computational domain. As can be seen in figure \[isenpulsepic\], the density at the center of the pulse has been flattened and the shape has become somewhat more square due to changes in refinement as the pulse has advected. However, there are no spurious waves and the size of the pulse has not changed significantly. At $t=10$, no fluctuations in pressure are detectable. A comparison with figure 10 in RAM for an identical test shows that while their pulse has increased in width by $\sim35\%$, there has been almost no increase in width in our test, indicating that, in this case, FLASH has less numerical diffusion.
Emery Step
----------
The Emery step problem (Emery 1968; Woodward & Colella 1984) consists of a wind flowing into a sharp vertical step in a wind tunnel with reflective upper and lower boundaries. The step causes a reverse shock to propagate into the wind. The shock will eventually collide with the upper boundary and reflect, giving rise to a Mach stem which is initially nearly vertical. By the end of the test a portion of the reflected shock has again reflected, this time off the lower boundary. For this test, a computational domain ranging from $0.0 \le x \le 3.0$ and $0.0 \le y \le 1.0$ in 2D Cartesian coordinates is used with a step of height $0.2$ beginning at $x=0.6$. Initially, the grid is filled with gas of $\rho=1.4$ and $v=0.999$ with an adiabatic index of $\Gamma=1.4$ moving with a Newtonian Mach number of $3.0$. Using eqn. \[cs\] gives a pressure of $p=0.1534$. Upper and lower boundaries, as well as the face of the step, are reflective. The left boundary is a constant inflow with these initial parameters. The right boundary uses outflow boundary conditions. A CFL number of $0.5$ is used. The test is run to a time of $t=4$. Figure \[wind1pic\] shows contours of density for the results carried out on a uniform grid of $240\times80$ grid points. Our results have very little noise in the downstream region and along the top of the step, as compared to the U-PPM results in RAM, which employ a similar solver to FLASH (see RAM figure 7). Our results appear more comparable to the F-WENO method in RAM.
This test has also been carried out on an adaptive mesh grid with 5 levels of refinement, for a maximum resolution of $3840 \times 1280$ grid points. Other than the resolution, the setup is identical to that of the uniform grid. As this problem involves mostly smooth flows, it provides a good test of the ability of FLASH to selectively refine and de-refine while still capturing shocks and discontinuities. Logarithmically spaced density contours of the results at $t=4$ are shown in the lower frame of figure \[wind1pic\]. The density and pressure are plotted in figure \[wind2pic\] along with the velocity field and block structure of the mesh. The figure is comparable to figure 15 from RAM, which shows the same results for an identical setup and maximum resolution. Both codes concentrate refinement around the shocks that develop and the contact discontinuity that originates from the bottom of the Mach stem. Using our code, Kelvin-Helmholtz instabilities are clearly seen along this contact discontinuity. This instability does not develop using RAM, which may indicate that FLASH is less diffusive.
Double Mach Reflection
----------------------
The double Mach reflection test presented here follows the same setup as in RAM. This test consists of a computational domain ranging from $0.0 \le x \le 4.0$ and $0.0 \le y \le 1.0$ in 2D Cartesian coordinates. A shock is placed on the grid moving down and to the right at a 60 degree angle to the x axis. The lower boundary is reflecting for $x > 1/6$. At the initial time, the shock is just making contact with the reflecting portion of the lower boundary. The lower boundary for $x \le 1/6$ is set to the post-shock conditions, as is the left boundary. The upper boundary is set to either the pre- or post-shock conditions depending on the time of the simulation. The right boundary is always set to the pre-shock conditions for the test considered here. The unshocked conditions are $\rho_0=1.4$, $p_0=0.0025$, and $v_0=0.0$ with an adiabatic index of $\Gamma=1.4$ in both the pre- and post-shock state. The shock is moving with a classical Mach number of $M=v_s/c_s$ of $10$ where $v_s$ is the shock speed and $c_s$ is the sound speed of the unshocked gas. The relativistic shock jump conditions can then be used to determine the post-shock conditions and the shock speed. Using equation \[cs\], we find that $v_s=10 \times c_s=0.4984$ in the observer’s frame. In the shock frame, this is the velocity of the incoming unshocked gas. Solving the relativistic shock jump conditions yields a post-shock state of $\rho_1=8.564$, $p_1=0.3808$, $v_1=0.09358$ in the shock frame. In the observer frame this transforms to $v_1=0.4247$. The speed of the leading edge of the shock is the relativistic sum of the shock speed, $v_s$, and the sound speed of the shocked gas, which is $c_{s1}=0.2321$, giving a total speed of $0.5978$ for the leading edge of the shock. This is the velocity used to determine if a point on the upper boundary is in the pre- or post-shock state. A CFL number of $0.5$ is used. The test is run to a time of $t=4$.
Figure \[doublemachpic\] shows density contours of this test run on a uniform grid of $512\times128$ grid points and an adaptive mesh grid with the same maximum resolution. The contours of the two plots are nearly identical whether or not adaptive mesh is used. Contours produced using FLASH do not appear to be as smooth as those from RAM (see RAM figure 16), particularly to the left of the vertical shock at $x = 2.7$. This may indicate that RAM produces more accurate results for this test.
Spherical Implosion in Cylindrical Coordinates
----------------------------------------------
In order to test the behavior of the FLASH code in cylindrical coordinates we have carried out tests with spherical implosions and explosions. The spherical implosion problem consists of a spherically symmetric flow converging on a single point. This test is carried out on a computational domain of $0.0 \le r \le 1.0$ and $0.0 \le z \le
1.0$ in 2D cylindrical coordinates. Initially, $\rho_0=1$ and $p=0$ everywhere and the material is flowing toward the origin at a fixed speed $v$, and the adiabatic index is $\Gamma=4/3$. The $r=0$ and $z=0$ boundaries are reflecting and the $r=1$ and $z=1$ boundaries are set to the analytic solution for a spherically converging flow for the time in the simulation (Martí et al. 1997). The analytic solution used at the boundaries is
$$\rho(r,t)=\rho_0 \left ( 1+\frac{|v|t}{r} \right )^2$$
where $r$ is the spherical radius, $t$ is the simulation time, and $v$ is the inflow velocity. Pressure at the boundaries is always $0$ and $v$ is fixed. The converging flow forms a spherical region of shocked, stationary gas which increases in radius with time according to
$$R_s=\frac{(\Gamma-1)\gamma|v|}{\gamma+1} t$$
where $\gamma$ is the Lorentz factor of the inflowing gas. The density in the post-shock state is given by
$$\rho_s=\rho(R_s,t) \left ( \frac{\gamma\Gamma+1}{\Gamma-1} \right )$$
This test was run for inflow velocities of $v=0.9$, $v=0.999$, and $v=0.99999$ to a time of $t=2$. All test were run with a CFL number of $0.2$. For these three tests, the average error in density in the post-shock gas is $3.16\%$, $1.91\%$, and $8.37\%$, respectively. Decreasing the CFL number will decrease the error, but as this is generally a hard test problem, these errors are acceptable.
Spherical Explosion in Cylindrical Coordinates
----------------------------------------------
For the spherical explosion test, the computational domain ranges from $0.0 \le r \le 1.0$ and $0.0 \le z \le 1.0$ in 2D cylindrical coordinates. Inside a radius of $R=0.4$ from the origin there is initially a gas with $\rho=1$, $p=1000$, and $v=0$ while outside this radius the gas has $\rho=1$, $p=1$, and $v=0$. The adiabatic index is $\Gamma=5/3$ in both states. This pressure discontinuity gives rise to a spherical shock traveling outward and a rarefaction wave traveling inward. Our test was carried out on a adaptive mesh grid with 4 levels of refinement for a maximum resolution of $320\times320$ grid points. The test was run with a CFL number of $0.2$ to a time of $t=0.4$. The $r=0$ and $z=0$ boundaries are reflecting and the $r=1$ and $z=1$ boundaries allow free outflow. Figure \[blastpic\] shows the density vs. (spherical) radius for points along the line $r=z$. Our results are consistent with results from other codes. The shock front is resolved by $\sim4$ grid points.
[ccccc]{}
$100$ & $0.132$ & & $0.131$ &\
$200$ & $0.0696$ & $0.92$ & $0.0725$ & $0.85$\
$400$ & $0.0357$ & $0.96$ & $0.0332$ & $1.1$\
$800$ & $0.0179$ & $1.0$ & $0.0208$ & $0.67$\
$1600$ & $0.00852$ & $1.1$ & $0.0100$ & $1.1$\
$3200$ & $0.00432$ & $0.98$ & $0.00507$ & $0.98$\
[ccccc]{}
100 & 0.206 & & 0.210 &\
200 & 0.148 & 0.48 & 0.142 & 0.56\
400 & 0.0832 & 0.83 & 0.0929 & 0.61\
800 & 0.0461 & 0.85 & 0.0554 & 0.75\
1600 & 0.0249 & 0.89 & 0.0254 & 1.1\
3200 & 0.0130 & 0.94 & 0.0151 & 0.75\
[ccccc]{}
100 & 0.0587 & & 0.0997 &\
200 & 0.0347 & 0.76 & 0.0629 & 0.67\
400 & 0.0214 & 0.70 & 0.0301 & 1.1\
800 & 0.0133 & 0.69 & 0.0169 & 0.83\
1600 & 0.00845 & 0.65 & 0.00948 & 0.83\
3200 & 0.00329 & 1.36 & 0.00524 & 0.86\
[ccccc]{}
100 & 0.627 & & 0.758 &\
200 & 0.337 & 0.90 & 0.392 & 0.95\
400 & 0.172 & 0.97 & 0.231 & 0.76\
800 & 0.0843 & 1.0 & 0.118 & 0.97\
1600 & 0.0441 & 0.93 & 0.0658 & 0.84\
3200 & 0.0232 & 0.93 & 0.0344 & 0.94\
[ccccc]{}
100 & 0.512 & & &\
200 & 0.464 & 0.14 & &\
400 & 0.325 & 0.51 & 0.521 &\
800 & 0.217 & 0.58 & 0.363 & 0.52\
1600 & 0.133 & 0.71 & 0.233 & 0.64\
3200 & 0.0833 & 0.68 & 0.126 & 0.89\
6400 & 0.0534 & 0.64 & 0.0649 & 0.96\
[ccccc]{}
$80$ & $5.48e-3$ & & $2.07e-3$ &\
$160$ & $1.55e-3$ & $1.8$ & $1.10e-4$ & $4.2$\
$320$ & $3.99e-4$ & $2.0$ & $1.70e-5$ & $2.7$\
$640$ & $1.00e-4$ & $2.0$ & $1.47e-6$ & $3.5$\
$1280$ & $2.50e-4$ & $2.0$ & $1.58e-7$ & $3.2$\
$2560$ & $5.35e-6$ & $2.2$ & $1.91e-8$ & $3.1$\
$5120$ & $1.56e-6$ & $1.8$ & $2.37e-9$ & $3.0$\
[ccccc]{}
$48\times64$ & $8.12e-3$ & & $7.35e-2$ &\
$96\times128$ & $2.23e-3$ & $1.9$ & $4.43e-3$ & $4.1$\
$192\times256$ & $5.87e-4$ & $1.9$ & $8.04e-4$ & $2.5$\
$384\times512$ & $1.48e-4$ & $2.0$ & $9.62e-5$ & $3.1$\
$768\times1024$ & $3.61e-5$ & $2.0$ & $1.12e-5$ & $3.1$\
[^1]: An alternative definition of the angle is provided by the positional angle of the point with respect to the origin of the coordinates. This definition is also prone to change with the expansion of the flow if the velocity and position vectors are not aligned.
[^2]: The factor of two is calibrated through simulations. Note that this approximation is different from that of LB05, who assumed a constant aperture of the cocoon.
[^3]: Closure relations are simple equations that associate a temporal decay of the afterglow to a given spectral slope and to a given distribution of the ambient medium into which the external shock runs.
|
---
abstract: 'We show how to extend the concept of heat capacity to nonequilibrium systems. The main idea is to consider the excess heat released by an already dissipative system when slowly changing the environment temperature. We take the framework of Markov jump processes to embed the specific physics of small driven systems and we demonstrate that heat capacities can be consistently defined in the quasistatic limit. Away from thermal equilibrium, an additional term appears to the usual energy–temperature response at constant volume, explicitly in terms of the excess work. In linear order around an equilibrium dynamics that extra term is an energy–driving response and it is entirely determined from local detailed balance. Examples illustrate how the steady heat capacity can become negative when far from equilibrium.'
author:
- 'E. Boksenbojm'
- 'C. Maes'
- 'K. Netočný'
- 'J. Pešek'
title: Heat capacity in nonequilibrium steady states
---
Introduction
============
The study of thermophysical properties of materials has played a major role in the development of thermodynamics and physics in general. A key issue is to understand how the system responds to variations in external control fields via heat exchange with its environment. The discussion simplifies for reversible thermal processes that are slow enough and pass through a sequence of equilibrium states. The heat exchange along such a process is determined by the way the system accommodates to the modified external conditions and relaxes to the new equilibrium state. Restricting to processes parameterized by temperature while other parameters (like volume, pressure etc.) are fixed, leads to the notion of heat capacity as a primary quantifier of the heat exchange. Their determination and characterization has proven very relevant in a great variety of domains ranging from industrial applications, over the study of phase transitions to fundamental tests for understanding the relation between mechanics and thermodynamics. Not surprisingly they were also key objects of study in the beginnings of quantum theory, in the further development of solid state physics and in the thermodynamics of new materials.
There is no well-established nonequilibrium theory. So far, the study of nonequilibrium heat capacities and related quantities has been mostly restricted to transient systems. There, internal relaxation is slow compared to the time-dependent control fields, with glassy systems as a paradigmatic example. A standard approach to transient systems involves frequency-dependent heat capacities as analyzed in several theoretical and experimental studies [@gard; @sta]. In contrast, the present letter considers systems that are well relaxed but under stationary nonequilibrium conditions. The study of heat capacities for such systems is largely unexplored. Such studies would include the thermal conditions of active matter; it would ask for heat capacities of bodies in which life processes take place, and it would seem to require nonequilibrium extensions of thermodynamic potentials. These questions are probably too difficult and too broad to answer at once, but nevertheless they motivate us in the initial set-up and in our modeling. A central issue is then whether and how the steady nonequilibrium functioning produces substantial deviations from the equilibrium heat capacity, not because the system has not fully relaxed but because of a totally different physics altogether.\
Preliminary calculations of steady state heat capacities within the framework of linear irreversible thermodynamics to explain certain involved conduction calorimetry experiments have been reported since almost twenty years [@cera]. We follow here more closely the ideas that were developed more recently in [@seki; @house; @stt; @ha], which is sometimes referred to as steady state thermodynamics.
To be specific, we mostly stick to a discrete set-up with a driven stochastic dynamics that is consistent with the presumed microscopic reversibility via the principle of *local* detailed balance and which covers a wide range of physically relevant nonequilibrium processes. As an example, we discuss at the end a model of driven diffusion in one and two dimensions, that naturally fits our formalism via the continuous (diffusion) limit of discrete approximations.\
Nonequilibrium model
====================
We have in mind small thermodynamically-open systems on which mechanical work is performed and which are coupled to an environment represented by a single heat bath. A crucial physical hypothesis is that the external forces are not fully conservative so that the system is always dissipative. Our aim is to analyze to what extent the heat exchanged with the bath while slowly changing its temperature can be represented by a well defined heat capacity.
To be specific, we consider Markovian dynamics with discrete states $x, y,\ldots$ representing distinct (mesoscopic) configurations of the system. It is a stochastic process with trajectories over a time-interval $[0,\tau]$ written as $$\label{eq: trajectory}
[x_t] = (x^0 \stackrel{t^1}{\rightarrow} x^1 \stackrel{t^2}{\rightarrow} \ldots \stackrel{t^n}{\rightarrow} x^n)\,,\quad
0 < t^1 < \ldots < t^n < \tau$$ each specified by a sequence of random jumps between states. Each state $x$ is given an energy $E(x)$ representing all conservative forces acting on the system. The non-conservative forces need to be introduced via the amount of work $F(x,y) = -F(y,x)$ they perform on the system when it jumps from state $x$ to $y$. Here we mostly assume that the energy function $E(x)$ and the non-conservative work function $F(x,y)$ are constant in time and we concentrate on the thermodynamic process corresponding to (slow) changes in the bath temperature $T(t)$. (For $F=0$ this would lead to the heat capacity at constant volume.)
Energy conservation on the level of a single trajectory $[x_t]$, $\,0 \leq t \leq \tau,$ can be written as $$\label{first-law}
E(x_\tau) - E(x_0) = W_F([x_t]) + Q([x_t])$$ Here the change of energy is decomposed into the work of the non-conservative forces, $W_F$, and the heat $Q$ flowing into the system. The work of conservative forces is zero by our assumption that $E(x)$ does not explicitly depend on time. The work of non-conservative forces is $$W_F([x_t]) = \sum_{j=1}^n F(x^{j-1},x^j)$$ with the sum over all jump times in the trajectory . Since $F$ does not derive from a potential, the work $W_F$ remains a non-trivial trajectory-dependent function which is $\tau-$extensive for typical paths; the same being true for the heat $Q$ for which the balance relation serves as a definition.\
The dynamics is determined by transition rates $\la^\be(x,y)$ that are time-dependent through their explicit dependence on the inverse temperature $\beta = 1/T$ (setting the Boltzmann constant to unity), $$\label{psip}
\la^\be(x,y) = \psi^\be(x,y)\,
e^{\frac{\be}{2}\,[E(x) - E(y) + F(x,y)]}$$ By the condition of local detailed balance (expressing thermal equilibrium in the coupled heat bath, see [@KLS]), the symmetry condition $\psi^\be(x,y) = \psi^\be (y,x)$ has to be always satisfied. If we now vary the temperature in time, the time-dependent distribution $\rho_t(x)$ solves the Master equation $$\label{mastereq}
\dot\rho_t(x) =
\sum_{y} [\rho_t(y) \lambda^{\be(t)}(y,x) - \rho_t(x) \lambda^{\be(t)}(x,y)]$$ We assume that for a fixed inverse temperature $\be$ the stationary distribution $\bar\rho^\be$ is unique and approached exponentially fast with relaxation time $\tau_R$; the latter provides a reference time-scale to delineate the quasistatic regime. Expectations with respect to $\bar\rho^\be$ will be denoted by $\langle \cdot \rangle^\be$.
An essential feature of our model is that its stationary regime is fundamentally different from equilibrium: despite the local detailed balance, one has $\bar\rho^\be(x)\,\la^\be(x,y) \neq \bar\rho^\be(y)\,\la^\be(y,x)$ unless $F$ derives from a potential. In particular, the system exhibits steady dissipation, the rate of which is given by the (positive) mean stationary work (or equally heat) per unit time $\langle w^\beta \rangle^\be$, the expectation value of $w^{\beta}(x) = \sum_y \la^\be(x,y)\,F(x,y)$ which is the expected power of the non-conservative forces when the system is in state $x$. Note that we allow the transition rates to depend on time only via their temperature-dependence — this condition will simplify the construction of the quasistatic limit.\
Steady heat capacity
====================
We come to our main question: Under what conditions and in what sense can *some* averaged heat $\langle Q \rangle$ along a process corresponding to $T(t)$, $0 \leq t \leq \tau$, be given the form $\int C_F\,{\textrm{d}}T$, with $C_F$ an appropriate heat capacity? This can only be true provided that $\langle Q \rangle$ is ‘geometric’ in the sense that it only depends on the values of temperature and not on how fast $T$ changes in time. Such a property is known to hold for currents in the quasistatic limit of infinitely slow process [@pump], irrespective of being in or out of equilibrium. However, the essential difference between the equilibrium and the nonequilibrium cases is that in the latter there are non-zero stationary (sometimes called ‘house-keeping’) currents which are due to the intrinsic dissipation of the nonequilibrium steady states. These non-geometric currents need to be regularized away to separate the *excess* currents that are to be seen as a natural extension of the equilibrium energy changes.
Next we explain in detail how this can be applied to construct the steady heat capacity in a consistent way. We more closely follow the formalism of Ref. [@ha] using the terms ‘house-keeping heat’ and the ‘excess heat’ for the stationary and the geometric components, respectively.\
On a somewhat intuitive level, the thermodynamic process induced by changing $\be(t)$, $0 \leq t \leq \tau$, can be considered to be quasistatic provided that the whole time interval can be suitably discretized, $\tau = N \Delta\tau$, so that (1) $\De\tau \gg \tau_R$ (relaxation time), and (2) $|\Delta \be| /\be \ll 1$ over all elementary time-intervals. Whenever $\tau \gg \tau_R$, such a discretization is possible and we can see the whole process as essentially consisting of a sequence of $N$ sudden and small temperature changes $\De\be = O(1/N)$, each one followed by relaxation to the new steady conditions.
Within the $k-$th time interval $[\tau_{k-1}, \tau_k] = [k-1,k]\,\De \tau$, the system can be thought to relax to the steady state distribution $\bar\rho^{\be(\tau_k)}$, starting at time $\tau_{k-1}$ from the steady distribution $\bar\rho^{\be(\tau_{k-1})}$ reached in the previous interval. Up to leading order, the initial distribution can be given in terms of the final steady distribution as $$\label{qs}
\bar\rho^{\be(\tau_{k-1})} = \bar\rho^{\be(\tau_k)} -
\frac{\partial\bar\rho^\be}{\partial \be}\,[\be(\tau_k) - \be(\tau_{k-1})] +
O(N^{-2})$$ while the expected work of the non-conservative forces within the relaxation process equals $$\label{kw}
\Delta_k W_F = \int_{\tau_{k-1}}^{\tau_k}
\sum_x \rho_t(x)\,w^{\be(t)}(x)\,{\textrm{d}}t$$ where $w^{\be(t)}(x) = \sum_y \la^{\be(t)}(x,y)\,F(x,y)$ is the expected power at time $t$ provided the system is in state $x$. Approximating $w^{\be(t)}$ within the entire interval $[\tau_{k-1},\tau_k]$ by $w^{\be(\tau_k)}$, we rewrite up to corrections $O(N^{-2})$ in the form $$\Delta_k W_F =
\Delta \tau \sum_x \bar\rho^{\be(\tau_k)}(x) \,w^{\be(\tau_k)}(x)
+ \int_{\tau_{k-1}}^{\tau_k} \sum_x \left[ \rho_t(x) - \bar\rho^{\be(\tau_k)}(x)\right] \,w^{\be(\tau_k)}(x)\,{\textrm{d}}t$$ The first term on the right-hand side of this equation is the house-keeping part of the work. It corresponds to the expected work if the system would be in its stationary state at every instant of time. The other term corresponds to the excess work. By assumption, $\De\tau \gg \tau_R$ and hence the system does reach the stationary state $\bar\rho^{\be(t_{k})}$, which means that we can as well take the upper limit of the integral to be $+\infty$. Formally solving the Master equation to find $\rho_t$ and after some standard manipulation, we obtain the total quasistatic work of non-conservative forces by summing over $k$: $$\label{work-quasistatic}
\langle W_F \rangle = \int_0^\tau \bigl\langle w^{\be(t)} \bigr\rangle^{\be(t)}\,{\textrm{d}}t
+\int \Bigl\langle \frac{\partial}{\partial\be}\, V^{\be} \Bigr\rangle^\be\, {\textrm{d}}\be
+ O\bigl( \frac{\tau_R}{\tau} \bigr)$$ with $$\langle w^\beta\rangle^\beta = \frac{1}{2} \sum_{x,y}
F(x,y)\,[\bar\rho^\be(x)\,\la^\be(x,y) - \bar\rho^\be(y)\,\la^\be(y,x)]$$ the steady rate of dissipation, given in the standard ‘force times current’ form. The first term in is therefore the steady state (or ‘house-keeping’) component. The second term on the other hand relates to the transient (or ‘excess’) component where we have introduced $$\label{work-relaxation}
V^{\be}(x) = \int_0^\infty [\langle w^\be(x_t)\rangle_{x_0=x}
- \langle w^{\be} \rangle^\beta ]\,{\textrm{d}}t$$ The state function $V^{\be}(x)$ is to be understood as the transient part of the mean dissipated work along the complete relaxation path started from state $x$. The function $\langle w^\be(x_t)\rangle_{x_0=x}$ yields the expected power at time $t$ given that the system was started in state $x$ at time zero. Note that $\langle w^\be(x_t)\rangle_{x_0=x} \simeq \langle w^{\be} \rangle^\beta$ for times $t\gg \tau_R$, and $\langle V^\be \rangle^\be = 0$.
In the same quasistatic regime where the system essentially passes through a succession of steady states, the expected change in energy is $\langle E(x_\tau) - E(x_0) \rangle = \int \frac{\partial}{\partial \be}\,
\langle E \rangle^\be\,{\textrm{d}}\be$. Hence, from the First Law , $\langle Q \rangle = -\int_0^\tau \langle w^{\be(t)} \rangle^{\be(t)}\,{\textrm{d}}t
+ \langle Q \rangle^{ex} + O(\tau_R / \tau)$ with the excess heat $\langle Q \rangle^{ex} = \int C_F\,{\textrm{d}}(1/\be)$ given in terms of the generalized heat capacity $$\label{heat-capacity}
C_F = -\be^2\frac{\partial}{\partial\be}\,\langle E \rangle^\be
+\be^2\Bigl\langle \frac{\partial}{\partial \be} \,V^{\be} \Bigr\rangle^\beta$$ This is our main result. The first term resembles the familiar equilibrium expression for the heat capacity at constant volume (and/or other external parameters) but now under the nonequilibrium steady state. The second term is novel and it originates from the fact that even keeping all the external parameters and forces fixed and merely changing the temperature, there is an extra non-zero work done. Part of the energy which is added to the system can be used to change the stationary currents, reminiscent of the more familiar Mayer relation between the heat capacities at constant volume and pressure. In general, the function $V^\be(x)$ non-trivially couples both variables $\be$ and $x$ so that, without further conditions, the heat capacity cannot be written as the temperature derivative of some generalized thermodynamic potential such as in the construction of (equilibrium) enthalpy.
The above derivation of formula can easily be turned into a rigorous argument [@kaji]: Using the quasistatic (or ‘adiabatic’) scaling of the time-dependent protocol, $T(t) \mapsto T(\ve t)$, $\tau \mapsto \ve^{-1}\tau$, both the work of non-conservative forces and the heat can be systematically expanded in powers of $\ve$. In this framework the house-keeping part is recognized as a linearly diverging term of order $\ve^{-1}$ and the non-quasistatic corrections are $O(\ve)$. The excess work/heat are the finite (or ‘renormalized’) parts of both by construction diverging quantities. It is precisely in this sense that they can be considered as well-defined.\
Experimental access
===================
It is crucial for the consistency of our construction that $C_F$ is defined through *excess* heat that was proven to be geometric, i.e., fully determined by the steady state properties. In principle, both the steady rate of dissipation $\langle w^\be \rangle^\beta$ and the transient work functions $V^\be(x)$ along relaxation paths can be obtained by measurement, independently of measuring the heat capacity from the quasistatic heat exchange. In this way, a specific prediction is given concerning the mutual relation between the results of *a priori* different types of experiment.
Clearly, the experimental accessibility of the generalized capacity strongly depends on whether the excess heat in the decomposition $\langle Q \rangle = -\int_0^\tau \langle w^{\be(t)} \rangle^{\be(t)}\,{\textrm{d}}t
+ \langle Q \rangle^{ex} + O(\tau_R / \tau)$ can be distinguished from the house-keeping (diverging) component $-\int_0^\tau \langle w^{\be(t)} \rangle^{\be(t)}\,{\textrm{d}}t$. A natural possibility comes from the different symmetry properties of the contributions: under the protocol reversal $\be(t) \mapsto \be(\tau - t)$, the house-keeping part is symmetric whereas the excess part is antisymmetric; the residual non-quasistatic corrections have no definite protocol-reversal behavior. Hence, the excess heat along any path can in principle be extracted by repeatedly traveling the same temperature-path back and forth and counting in only differences in the heat exchanged or the work done. At the same time, the temperature changes need to be slow enough to avoid non-quasistatic residuals. Estimating the experimental errors with such a procedure probably remains a challenging but very relevant and physically interesting problem.\
Naturally, the same questions as discussed here in the context of stochastic systems can be addressed for macroscopic bodies under nonequilibrium conditions. A particular experimental setup has already been proposed in Ref. [@cera]: The authors there employ conduction calorimetry techniques to study the heat produced by a ferroelectric sample heated by an applied high-frequency AC-current. Changing the environment temperature results in modifications of the outgoing heat current that can be directly measured in real time by an imposed thermopile. It is argued that the method is subtle enough to distinguish the steady heat currents from their excess components which, after some time-integration, yield the heat capacity by definition. In this context, the present letter provides a general microscopic (or, more precisely, mesoscopic) theory for such a type of experiments on dissipative systems.\
Linear nonequilibrium correction
================================
Some progress can be made in a close-to-equilibrium regime where we can control the steady-state properties of the system by a systematic expansion in the magnitude of the nonequilibrium driving. In the pioneering work of McLennan [@mac], he found leading nonequilibrium corrections to the canonical distribution in terms of entropy changes; see also [@KN; @mat] for recent extensions. Here we use the formulation and results of Ref. [@jmp].
For small non-conservative forces $F$, the stationary distribution $\bar\rho^\be$ can be well approximated by the McLennan ensemble, $$\label{mclennan}
\bar\rho^\be(x) \simeq \frac{1}{Z^\be}\,\exp\,[-\be E(x) - \be V^{\be}(x)]$$ in which the correction term exactly coincides with the dissipated work along relaxation paths . This formula can be justified by scaling the driving forces as $F(x,y) \mapsto {\varepsilon}F(x,y)$ and expanding in powers of ${\varepsilon}$; the McLennan distribution is proven correct up to order ${\varepsilon}$.
Up to linear order in $F$ and by the construction of $V^\be$, the nonequilibrium term in can be written in the form of equilibrium time-correlations between the energy and the power of the non-conservative forces : $$C_F \simeq -\be^2\frac{\partial}{\partial\be}\,\langle E \rangle^\be -
\be^2 \int_0^\infty \langle E(x_0)\,w^\be(x_t) \rangle^\beta_{\text{eq}}\,{\textrm{d}}t$$ Here the expectation $\langle \cdot \rangle^\be_{\text{eq}}$ is under the equilibrium distribution $\bar\rho^\be_{\text{eq}}(x) = \exp\,[-\be E(x)] / Z^\be$ and we have used that $\langle w^\be \rangle^\beta_{\text{eq}} = 0$. Finally, combining with the McLennan formula , we finally obtain the relation $$\label{thre}
C_F \simeq -\be^2 \frac{\partial}{\partial\be}\,\langle E \rangle^\be -
\be\,(\langle E \rangle^{\be} - \langle E \rangle^{\be}_{\text{eq}})$$ always correct up to linear order in the nonequilibrium driving. Hence the close-to-equilibrium heat capacity consists of two linear-response contributions: (1) the (equilibrium-like) energy–temperature response and (2) the energy–driving response, which can be further rewritten in terms of an equilibrium correlation function like in the Green-Kubo relation. While the quasistatic heat capacity on the left-hand side of derives from a thermodynamic process, the two response-functions on the right-hand side are by definition steady-state properties of the system. All three quantities in are independently measurable, at least *in principle*. Note there is no dependence on the symmetric part $\psi^\be$ in the transition rates , which is at the origin of the remarkable simplification in the close-to-equilibrium regime.
Remark that this linear order theory is only meaningful when the dynamics breaks the driving-reversal symmetry $F \mapsto -F$ (simultaneously for all transitions $x \leftrightarrow y$). Under this symmetry, the linear nonequilibrium corrections to the heat capacity vanish, $C_{{\varepsilon}F} = C_0 + O({\varepsilon}^2)$, due to the absence of the $O({\varepsilon})$ corrections in both the mean energy $\langle E \rangle^\be$ and the transient work function $V^\be$. The higher-order corrections can be obtained by a systematic expansion in powers of the parameter ${\varepsilon}$ adopted to control the distance from equilibrium [@mat; @kaji].\
The general non-perturbative formula and its close-to-equilibrium approximation for driving-reversal asymmetric systems constitute the main results of this letter. In the next section we give specific examples that go beyond the scope of the simple linear theory and on which we demonstrate some peculiar features of the steady heat capacity .\
Example: driven diffusion
=========================
As a trial nonequilibrium system we consider the case of independent colloids driven in a toroidal trap, which is experimentally feasible [@cil]. The particle motion can be modeled by the overdamped driven diffusion on a circle of unit length, $$\label{1d}
\dot{x}_t = F - E'(x_t) + \sqrt{2T(t)}\,\xi_t$$ ($\xi_t$ is standard white noise.) The driving force $F$ is constant and, to be specific, we take the potential landscape $E(x) = \sin (2\pi x)$.
The steady heat capacity $C_F$ is depicted in Fig. \[fig: CF\]; we have evaluated numerically exactly for a discrete-space approximation of the dynamics . For large temperatures the nonequilibrium correction to the steady heat capacity becomes dominated by the energy-temperature response (the first term in ) and $C_F(T)$ asymptotically approaches the equilibrium curve $C_0(T) = 1 /(2 T^2) + O(T^{-3})$, for arbitrary forcing $F$. On the other hand, at lower temperatures the nonequilibrium correction becomes relevant and we see a qualitative change of behavior across the value $F^* = 2\pi$, which we associate with the crossover between the limiting fixed point and the limiting cycle in the zero-temperature (deterministic) solution of .
Our model demonstrates that $C_F$ can obtain negative values when far from equilibrium. Although similar observations concerning negative heat capacities have been made before for systems non-weakly coupled to finite reservoirs [@hanggi], here the physical origin is different and the effect emerges due to the nonequilibrium nature of our system; see more below.
We also calculate the steady heat capacity at constant steady power, $C_W$, as defined via the excess heat along the quasistatic curve $(T,F)$ on which the steady power $\langle w^\beta \rangle^\be$ remains constant. The general relation between both heat capacities is readily found to be $$\label{mayer}
C_W = C_F
+ \be^2 \frac{\partial\langle w^\beta \rangle^\be}{\partial \be}\,
\Bigl( \frac{\partial\langle w^\beta \rangle^\be}{\partial F} \Bigr)^{-1}
\Bigl[ \frac{\partial \langle U \rangle^\be}{\partial F} -
\Bigl\langle \frac{\partial}{\partial F} \,V^{\be} \Bigr\rangle^\be
\Bigr]$$ with $W$ and $F$ related by the condition $\langle w^\beta \rangle^\be = W$. For our diffusion model , the heat capacity $C_W$ as a function of temperature is depicted in Fig. \[fig: CW\].\
To get a better understanding of how the steady heat capacity depends on the dissipative properties of the system, we further consider the two-dimensional modification of the model , $$\dot{X}_t = \vec F(X_t) - \nabla E(X_t) + \sqrt{2T(t)}\,\vec\xi_t\,,\quad
X = (x,y)$$ with the spherically symmetric potential $E(X) = \frac{\la}{2}\,r^2$, $\la > 0$ and driven by the purely rotational field $\vec F(X) = \ka\,r^\al\,\vec e_\th$ with some $\al > -1$; the standard polar coordinates $r$ and $\th$ being used here. The conservative and the non-conservative fields are mutually orthogonal, $\vec F \cdot \nabla E = 0$, and the stationary density is insensitive to the nonequilibrium driving, $\bar\rho^\be = \exp(-\be E)/Z^\be$, i.e., the same as if the system were in equilibrium. Hence, the first term in the steady heat capacity equals unity by equipartition. However, different steady states are distinguished by their mean dissipative power that equals $\langle w^\beta \rangle^\be = \Ga(\al + 1)\,\ka^2\, (2 T / \la)^\al$. The nonequilibrium correction term in $C_F$ can also be calculated analytically to yield the formula $$C_F = 1 + \frac{1}{2\la}\,\frac{\partial \langle w^\beta \rangle^\be}{\partial T}$$ This simple relation between the steady heat capacity and the mean power is not to be expected in general. Nevertheless, the relation makes it very clear that the steady heat capacity depends on how the dissipation, and not just the energy, depends on temperature. In our model the increase of temperature makes the steady states less localized around the origin and depending on whether $\al > 0$ or $-1 < \al < 0$, this corresponds to a higher, respectively lower amount of dissipation as quantified by the mean power $\langle w^\beta \rangle^\be$. As a result, the nonequilibrium correction to the heat capacity obtains the same positive, respectively negative sign. This suggests that negative steady heat capacities may generally emerge for far-from-equilibrium systems when their steady dissipation decreases sufficiently strongly with temperature — details are left to further studies. We conclude our short analysis of this model by noting the equality $C_W = C_F$ due to the $F-$independence of the stationary density $\bar\rho^\be$, [*cf.*]{} formula .\
Conclusion and open questions
=============================
We have analyzed a meaningful and consistent generalization of heat capacity to nonequilibrium systems. By applying and adapting the previously developed framework of slow transformations of nonequilibrium steady states, we have derived the basic properties of the heat capacity defined from the quasistatic heat. This construction makes physical sense because the finite excess part of the heat exchange is well-defined and geometric. In formula a general non-perturbative expression for the steady heat capacity is given in terms of the (standard) energy–temperature response but modified with a new correction intimately related to the relaxation properties of the dissipative effects — the new term derives from the transient work of the driving forces along relaxation paths.
We have demonstrated via simple examples that the steady heat capacity can take negative values as well. It has been argued that this phenomenon has to do with a specific temperature-dependence of dissipative characteristics far from equilibrium. The details of this proposal need to be further analyzed. Another relevant question is a detailed analysis of the steady heat capacity at low temperatures, in particular in regimes where the reference zero-temperature dynamical system is fundamentally different from the one in equilibrium. We expect the steady heat capacity and related nonequilibrium response functions to reveal important information about the presence of nonequilibrium phase transitions in the system [@marro]. We have also found more specific expressions for the heat capacity of close-to-equilibrium systems breaking the driving-reversal symmetry. In that case the nonequilibrium contribution to the heat capacity is directly related to the equilibrium linear response to switching on a (weak) nonequilibrium driving, see formula . Equivalently, it can be given in terms of equilibrium time-correlations resembling the Green-Kubo or fluctuation-dissipation relations.
To conclude, remark that presently the inertial degrees of freedom (the particles’ momenta) have been considered ‘fast’ with respect to ‘slow’ spatial configurations, in the usual sense of time-scale separation. By this assumption, the distribution of momenta is always Maxwellian and the contribution to the total steady heat capacity follows the equipartition theorem as $k_B /2$ per momentum degree of freedom, in the exact same way as in equilibrium. This restriction is not essential and the momenta degrees of freedom with more general stationary distributions can easily be included in the theory.
We are grateful to Bram Wynants for initial work. E.B. and C.M. acknowledge financial support from the FWO project G.0422.09N. K.N. acknowledges the support from the Grant Agency of the Czech Republic, Grant no. 202/08/0361. J.P. benefits from the Grant no. 51410 (the Grant Agency of Charles University) and from the project SVV-263301 (Charles University).
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abstract: 'It is argued that the massive gauge field theory without the Higgs mechanism can well be set up on the gauge-invariance principle based on the viewpoint that a massive gauge field must be viewed as a constrained system and the Lorentz condition, as a constraint, must be introduced from the beginning and imposed on the Yang-Mills Lagrangian. The quantum theory for the massive gauge fieldis may perfectly be established by the quantization performed in the Hamiltonian or the Lagrangian path-integral formalism by means of the Lagrange undetermined multiplier method and shows good renormalizability and unitarity.'
address: |
Department of Physics , Jilin University, Changchun 130023, People’s\
Republic of China
author:
- 'Jun-Chen Su'
title: Massive gauge field theory without Higgs mechanism
---
It is the prevailing viewpoint that the massive gauge field theory can not be set up without introducing the Higgs mechanism. The first obstacle is the gauge-non-invariance of the mass term in the massive Yang-Mills Lagrangian for a massive gauge field. On the contrary, we present an argument to show that the conventional viewpoint is not true \[1\]. In fact, a certain massive gauge field theory can be well established on the basis of gauge-invariance principle without recourse to the Higgs mechanism. The basic ideas are stated in the following.
\(1) A massive gauge field must be viewed as a constrained system. In the previous attempt of building up the massive non-Abelian gauge field theory, the massive Yang-Mills Lagrangian density written below was chosen to be the starting point \[2,3\]. $${\cal L}=-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu }A_\mu ^a
\eqnum{1}$$ where $A_\mu ^a$ are the vector potentials for the non-Abelian massive gauge fields, $$F_{\mu \nu }^a=\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a+gf^{abc}A_\mu
^bA_\nu ^c \eqnum{2}$$ are the field strengths and $m$ is the mass of gauge bosons. The first term in the Lagrangian is the ordinary Yang-Mills Lagrangian which is gauge-invariant under a whole Lie group and used to determine the form of interactions among the gauge fields themselves. The second term in the Lagrangian is the mass term which is not gauge-invariant and only affects the kinematic property of the fields. The above Lagrangian itself was ever considered to give a complete description of the massive gauge field dynamics. This consideration is not correct because the Lagrangian is not only not gauge-invariant, but also contains redundant unphysical degrees of freedom. As one knows, a full vector potential $A^\mu (x)$ can be split into two Lorentz-covariant parts: the transverse vector potential $A_T^\mu (x)$ and the longitudinal vector potential $A_L^\mu (x)$, $A^\mu (x)=A_T^\mu
(x)+A_L^\mu (x)$. The transverse vector potential $A_T^{a\mu }(x)$ contains three independent spatial components which is sufficient to represent the polarization states of a massive vector boson. Whereas, the longitudinal vector potential $A_L^{a\mu }$ appears to be a redundant unphysical variable which must be constrained by introducing the Lorentz condition $$\varphi ^a\equiv \partial ^\mu A_\mu ^a=0 \eqnum{3}$$ whose solution is $A_L^{a\mu }=0$. With this solution, the massive Yang-Mills Lagrangian may be expressed in terms of the independent dynamical variables $A_T^{a\mu }(x)$$${\cal L}=-\frac 14F_T^{a\mu \nu }F_{T\mu \nu }^a+\frac 12m^2A_T^{a\mu
}A_{T\mu }^a \eqnum{4}$$ which gives a complete description of the massive gauge field dynamics. If we want to represent the dynamics in the whole space of the full vector potential as described by the massive Yang-Mills Lagrangian in Eq.(1), the massive gauge field must be treated as a constrained system. In this case, the Lorentz condition in Eq.(3), as a constraint, is necessary to be introduced from the onset and imposed on the Lagrangian in Eq.(1) so as to guarantee the redundant degrees of freedom to be eliminated from the Lagrangian.
\(2) The gauge-invariance should generally be required for the action written in the physical subspace. Usually, the gauge-invariance is required to the Lagrangian. From the dynamical viewpoint, as we know, the action is of more essential significance than the Lagrangian. This is why in Mechanics and Field Theory, to investigate the dynamical and symmetric properties of a system, one always starts from the action for the system. Similarly, when we examine the gauge-symmetric property of a field system, in more general, we should also see whether the action for the system is gauge-invariant or not. In particular, for a constrained system such as the massive gauge field, we should see whether or not the action represented by the independent dynamical variables is gauge-invariant. This point of view is familiar to us in the mechanics for constrained systems. Certainly, in some special cases, the Lagrangian itself is gauge-invariant in the physical subspace so that the gauge-invariance of the action is ensured. This situation happens for the massless gauge fields and the massive Abelian gauge field. For the non-Abelian gauge fields, the infinitesimal gauge transformation usually is given by \[3\] $$\delta A_\mu ^a=D_\mu ^{ab}\theta ^b \eqnum{5}$$ where $$D_\mu ^{ab}=\delta ^{ab}\partial _\mu -gf^{abc}A_\mu ^c \eqnum{6}$$ This gauge transformation is different from the Abelian one in that in the physical subspace defined by the Lorentz condition, i.e., spanned by the transverse vector potential $A_T^{a\mu }$, the fields still undergo nontrivial gauge transformations $$\delta A_{T\mu }^a=D_{T\mu }^{ab}\theta ^b \eqnum{7}$$ where $$D_{T\mu }^{ab}=\delta ^{ab}\partial _\mu -gf^{abc}A_{T\mu }^c \eqnum{8}$$ Therefore, the mass term in the massive Yang-Mills Lagrangian written in Eq.(8) is not gauge-invariant. But, the action given by this Lagrangian is invariant with respect to the gauge transformation shown in Eqs.(7) and (8). In fact, noticing the identity: $f^{abc}A_T^{a\mu }A_{T\mu }^b=0$ and the transversity condition (an identity): $\partial ^\mu A_{T\mu }^a=0,$ it is easy to see $$\delta S=-m^2\int d^4x\theta ^a\partial ^\mu A_{T\mu }^a=0 \eqnum{9}$$ This shows that the dynamics of massive non-Abelian gauge fields is gauge-invariant. Alternatively, the gauge-invariance may also be seen from the action given by the Lagrangian in Eq.(1) which is constrained by the Lorentz condition in Eq.(3). Under the gauge transformation written in Eqs.(5) and (6), noticing the identity $f^{abc}A^{a\mu }A_\mu ^b=0$ and the Lorentz condition, it can be found that $$\delta S=-m^2\int d^4x\theta ^a\partial ^\mu A_\mu ^a=0 \eqnum{10}$$ This suggests that the massive non-Abelian gauge field theory may also be set up on the basis of gauge-invariance principle.
\(3) Only infinitesimal gauge transformations need to be considered in the physical subspace. In examining the gauge invariance of the action for the massive non-Abelian gauge fields, we confine ourself to consider the infinitesimal gauge transformation only. The reason for this arises from the fact that the Lorentz condition limits the gauge transformation only to take place in the vicinity of the unity of the gauge group. In other words, the residual gauge degrees of freedom existing in the physical subspace are characterized by the infinitesimal gauge transformations. This fact was pointed out in the pioneering article by Faddeev and Popov for the quantization of massless non-Abelian gauge fields \[4\]. Usually, the dynamics of massless gauge fields is described by the Yang-Mills Lagrangian. It is well-known that the Yang-Mills Lagrangian itself is not quantizable, namely, it can not be used to construct a convergent generating functional of Green’s functions even though it is gauge-invariant with respect to the whole gauge group. This is because the Yang-Mills Lagrangian contains redundant unphysical degrees of freedom and hence is not complete for describing the massless gauge field dynamics unless a suitable constraint such as the Lorentz condition is introduced to eliminate the unphysical degrees of freedom. In the article by Faddeev and Popov, the Lorentz condition is introduced through the following identity \[4\] $$\Delta [A]\int D(g)\delta [\partial ^\mu A_\mu ^g]=1 \eqnum{11}$$ which is inserted into the functional integral representing the vacuum-to vacuum transition amplitude. After doing this , the authors said that ” We must know $\Delta [A]$ is only for the transverse fields and in this case all contributions to the last integral are given in the neighborhood of the unity element of the group”. The delta-functional in Eq.(16) implies $%
\partial ^\mu (A^g)_\mu ^a=\partial ^\mu A_\mu ^a=0$ which represents the gauge-invariance of the Lorentz condition because the Lorentz condition is required to hold for all the field variables including the ones before and after gauge transformations. In the physical subspace where only the transverse fields are allowed to appear, only infinitesimal gauge transformations around the unity element are permitted and needed to be considered in the course of Faddeev-Popov’s quantization even though the Yang-Mills Lagrangian used is invariant under the whole gauge group. Obviously, there are no reasons of considering the gauge-transformation property of the fields in the region beyond the physical subspace because the fields do not exist in that region. By this point, it can be understood why in the ordinary quantum gauge field theories such as the standard model, the BRST-transformations are all taken to be infinitesimal.
According to the general procedure, the Lorentz condition (3) may be incorporated into the Lagrangian (1) by the Lagrange undetermined multiplier method to give a generalized Lagrangian. In the first order formalism, this Lagrangian is written as \[5\] $${\cal L}=\frac 14F^{a\mu \nu }F_{\mu \nu }^a-\frac 12F^{a\mu \nu }(\partial
_\mu A_\nu ^a-\partial _\nu A_\mu ^a+gf^{abc}A_\mu ^bA_\nu ^c)+\frac 12%
m^2A^{a\mu }A_\mu ^a+\lambda ^a\partial ^\mu A_\mu ^a \eqnum{12}$$ where $A_\mu ^a$ and $F_{\mu \nu }^a$ are now treated as the mutually independent variables and $\lambda ^a$ are chosen to represent the Lagrange multipliers. Using the canonically conjugate variables defined by $$\Pi _\mu ^a(x)=\frac{\partial {\cal L}}{\partial \dot A^{a\mu }}=F_{\mu
0}^a+\lambda ^a\delta _{\mu 0}={\cal \{}
\begin{tabular}{l}
$F_{k0}^a=E_k^a,$ if $\mu =k=1,2,3;$ \\
$\lambda ^a=-E_0^a,$ if $\mu =0,$%
\end{tabular}
\eqnum{13}$$ the Lagrangian in Eq.(18) may be rewritten in the canonical form $${\cal L}=E^{a\mu }\dot A_\mu ^a+A_0^aC^a-E_0^a\varphi ^a-{\cal H} \eqnum{14}$$ where $$C^a=\partial ^\mu E_\mu ^a+gf^{abc}A_k^bE^{ck}+m^2A_0^a \eqnum{15}$$ $${\cal H}=\frac 12(E_k^a)^2+\frac 14(F_{ij}^a)^2+\frac 12%
m^2[(A_0^a)^2+(A_k^a)^2] \eqnum{16}$$ here $E_\mu ^a=(E_0^a,E_k^a)$ is a Lorentz vector, ${\cal H}$ is the Hamiltonian density in which $F_{ij}^a$ are defined in Eq.(2). In the above, the four-dimensional and the spatial indices are respectively denoted by the Greek and Latin letters. From Eq.(14), it is clearly seen that the second and third terms are given respectively by incorporating the constraint condition
$$C^a=0 \eqnum{17}$$
where $C^a$ was represented in Eq.(15) and the Lorentz condition in Eq.(3) into the Lagrangian.
Now, let us first perform the quantization of the massive non-Abelian gauge fields in the Hamiltonian path-integral formalism \[5\]. In accordance with the general procedure of the quantization, we should first write the generating functional of Green’s functions via the independent canonical variables which are now chosen to be the transverse parts of the vectors $%
A_\mu ^a$ and $E_\mu ^a$ $$Z[J]=\frac 1N\int D(A_T^{a\mu },E_T^{a\mu })exp\{i\int d^4x[E_T^{a\mu }\dot A%
_{T\mu }^a-{\cal H}^{*}(A_T^{a\mu },E_T^{a\mu })+J_T^{a\mu }A_{T\mu }^a]\}
\eqnum{18}$$ where ${\cal H}^{*}(A_T^{a\mu },E_T^{a\mu })$ is the Hamiltonian which is obtained from the Hamiltonian in Eq.(16) by replacing the constrained variables $A_L^{a\mu }$ and $E_L^{a\mu }$ with the solutions of equations (3) and (17) $${\cal H}^{*}(A_T^{a\mu },E_T^{a\mu })={\cal H}(A^{a\mu },E^{a\mu })\mid
_{\varphi ^a=0,C^a=0} \eqnum{19}$$ As mentioned before, Eq.(3) leads to $A_L^{a\mu }=0$. Noticing this solution and the decomposition $E^{a\mu }(x)=E_T^{a\mu }+E_L^{a\mu }(x)$, when setting $$E_L^{a\mu }(x)=\partial _x^\mu Q^a(x) \eqnum{21}$$ where $Q^a(x)$ is a scalar function, one may get from Eq.(17) an equation obeyed by the scalar function $Q^a(x)$ $$K^{ab}(x)Q^b(x)=W^a(x) \eqnum{22}$$ where $$K^{ab}(x)=\delta ^{ab}\Box _x-gf^{abc}A_T^{c\mu }(x)\partial _\mu ^x
\eqnum{23}$$ and $$W^a(x)=gf^{abc}E_T^{b\mu }(x)A_{T\mu }^c(x)-m^2A_T^{a0}(x) \eqnum{24}$$ With the aid of the Green’s function $G^{ab}(x-y)$ (the ghost particle propagator) which satisfies the following equation $$K^{ac}(x)G^{cb}(x-y)=\delta ^{ab}\delta ^4(x-y) \eqnum{25}$$ one may find the solution to the equation (22) as follows $$Q^a(x)=\int d^4yG^{ab}(x-y)W^b(y) \eqnum{26}$$ From the expressions given in Eqs.(21) and (26), one can see that the $%
E_L^{a\mu }(x)$ is a complicated functional of the variables $A_T^{a\mu }$ and $E_T^{a\mu }$ so that the Hamiltonian ${\cal H}^{*}(A_T^{a\mu
},E_T^{a\mu })$ is of much more complicated functional structure which is not convenient for constructing the diagram technique in the perturbation theory. Therefore, it is better to express the generating functional in Eq.(18) in terms of the variables $A_\mu ^a$ and $E_\mu ^a$. For this purpose, it is necessary to insert the following delta-functional into Eq.(18) \[5\] $$\delta [A_L^{a\mu }]\delta [E_L^{a\mu }-E_L^{a\mu }(A_T^{a\mu },E_T^{a\mu
})]=detM\delta [C^a]\delta [\varphi ^a] \eqnum{27}$$ where $M$ is the matrix whose elements are defined by the Poisson bracket $$\begin{tabular}{l}
$M^{ab}(x,y)=\{C^a(x),\varphi ^b(y)\}=\int d^4x\{\frac{\delta C^a}{\delta
A_\mu ^a(x)}\frac{\delta \varphi ^b}{\delta E^{a\mu }(x)}-\frac{\delta C^a}{%
\delta E_\mu ^a(x)}\frac{\delta \varphi ^b}{\delta A^{a\mu }(x)}\}$ \\
$=D_\mu ^{ab}(x)\partial _x^\mu \delta ^4(x-y)$%
\end{tabular}
\eqnum{28}$$ The relation in Eq.(27) is easily derived from equations (3) and (17) by applying the property of delta-functional. Upon inserting Eq.(27) into Eq.(18) and utilizing the Fourier representation of the delta-functional $$\delta [C^a]=\int D(\eta ^a/2\pi )e^{i\int d^4x\eta ^aC^a} \eqnum{29}$$ we have $$\begin{tabular}{l}
$Z[J]=\frac 1N\int D(A_\mu ^a,E_\mu ^a,\eta ^a)detM\delta [\partial ^\mu
A_\mu ^a]\exp \{i\int d^4x[E^{a\mu }\dot A_\mu ^a$ \\
$+\eta ^aC^a-{\cal H}(A^{a\mu },E^{a\mu })+J^{a\mu }A_\mu ^a]\}$%
\end{tabular}
\eqnum{30}$$ In the above exponential, there is a $E_0^a$-related term $E_0^a(\partial
_0A_0^a-\partial _0\eta ^a)$ which permits us to perform the integration over $E_0^a$, giving a delta-functional $$\delta [\partial _0A_0^a-\partial _0\eta ^a]=det|\partial _0|^{-1}\delta
[A_0^a-\eta ^a] \eqnum{31}$$ The determinant $det|\partial _0|^{-1}$, as a constant, may be put in the normalization constant $N$ and the delta-functional $\delta [A_0^a-\eta ^a]$ will disappear when the integration over $\eta ^a$ is carried out. The integral over $E_k^a$ is of Gaussian-type and hence easily calculated. After these manipulations, we arrive at $$\begin{tabular}{l}
$Z[J]=\frac 1N\int D(A_\mu ^a)detM\delta [\partial ^\mu A_\mu ^a]exp\{i\int
d^4x[-\frac 14F^{a\mu \nu }F_{\mu \nu }^a$ \\
$+\frac 12m^2A^{a\mu }A_\mu ^a+J^{a\mu }A_\mu ^a]\}$%
\end{tabular}
\eqnum{32}$$ When employing the familiar expression \[4\] $$detM=\int D(\bar C^a,C^a)e^{i\int d^4xd^4y\bar C^a(x)M^{ab}(x,y)C^b(y)}
\eqnum{33}$$ where $\bar C^a(x)$ and $C^a(x)$ are the mutually conjugate ghost field variables and the following limit for the Fresnel functional $$\delta [\partial ^\mu A_\mu ^a]=\lim_{\alpha \to 0}C[\alpha ]e^{-\frac i{%
2\alpha }\int d^4x(\partial ^\mu A_\mu ^a)^2} \eqnum{34}$$ where $C[\alpha ]\sim \prod_x(\frac i{2\pi \alpha })^{1/2}$ and supplementing the external source terms for the ghost fields, the generating functional in Eq.(32) is finally given in the form $$Z[J,\overline{\xi },\xi ]=\frac 1N\int D(A_\mu ^a,\bar C^a,C^a)exp\{i\int
d^4x[{\cal L}_{eff}+J^{a\mu }A_\mu ^a+\overline{\xi }^aC^a+\bar C^a\xi ^a]\}
\eqnum{35}$$ where $${\cal L}_{eff}=-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2-\partial ^\mu \bar C%
^aD_\mu ^{ab}C^b \eqnum{36}$$ which is the effective Lagrangian for the quantized massive non-Abelian gauge field in which the third and fourth terms are the so-called gauge-fixing term and the ghost term respectively. In Eq.(36), the limit $%
\alpha \to 0$ is implied. Certainly, the theory may be given in arbitrary gauges $(\alpha \ne 0)$. In this case, as will be seen soon later, the ghost particle will acquire a spurious mass $\mu =\sqrt{\alpha }m$.
To confirm the result of the quantization given above, let us turn to quantize the massive non-Abelian gauge fields in the Lagrangian path-integral formalism. For later convenience, the massive Yang-Mills Lagrangian in Eq.(1) and the Lorentz constraint condition in Eq.(3) are respectively generalized to the following forms $${\cal L}_\lambda =-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a-\frac 12\alpha (\lambda ^a)^2 \eqnum{37}$$ and $$\partial ^\mu A_\mu ^a+\alpha \lambda ^a=0 \eqnum{38}$$ where $\lambda ^a(x)$ are the extra functions which will be identified with the Lagrange multipliers and $\alpha $ is an arbitrary constant playing the role of gauge parameter. According to the general procedure for constrained systems, Eq.(38) may be incorporated into Eq.(37) by the Lagrange undetermined multiplier method, giving a generalized Lagrangian $${\cal L}_\lambda =-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a+\lambda ^a\partial ^\mu A_\mu ^a+\frac 12\alpha (\lambda ^a)^2
\eqnum{39}$$ This Lagrangian is obviously not gauge-invariant. However, for building up a correct gauge field theory, it is necessary to require the action given by the Lagrangian in Eq.(39) to be invariant with respect to the gauge transformations shown in Eqs.(5) and (6) so as to guarantee the dynamics of the gauge field to be gauge-invariant. By this requirement, noticing the identity $f^{abc}A^{a\mu }A_\mu ^b=0$ and applying the constraint condition in Eq.(38), we have $$\delta S_\lambda =-\frac 1\alpha \int d^4x\partial ^\nu A_\nu ^a(x)\partial
^\mu ({\cal D}_\mu ^{ab}(x)\theta ^b(x))=0 \eqnum{40}$$ where $${\cal D}_\mu ^{ab}(x)=\delta ^{ab}\frac{\mu ^2}{\Box _x}\partial _\mu
^x+D_\mu ^{ab}(x) \eqnum{41}$$ in which $\mu ^2=\alpha m^2$ and $D_\mu ^{ab}(x)$ was defined in Eq.(6). From Eq.(38), we see $\frac 1\alpha \partial ^\nu A_\nu ^a=-\lambda ^a\ne 0$. Therefore, to ensure the action to be gauge-invariant, the following constraint condition on the gauge group is necessary to be required $$\partial _x^\mu ({\cal D}_\mu ^{ab}(x)\theta ^b(x))=0 \eqnum{42}$$ These are the coupled equations satisfied by the parametric functions $%
\theta ^a(x)$ of the gauge group. Since the Jacobian of the equations is not singular, $detM\ne 0$ where $M$ is the matrix whose elements are $$\begin{tabular}{l}
$M^{ab}(x,y)=\frac{\delta (\partial _x^\mu {\cal D}_\mu ^{ac}(x)\theta ^c(x))%
}{\delta \theta ^b(y)}\mid _{\theta =0}$ \\
$=\delta ^{ab}(\Box _x+\mu ^2)\delta ^4(x-y)-gf^{abc}\partial _x^\mu (A_\mu
^c(x)\delta ^4(x-y))$%
\end{tabular}
\eqnum{43}$$ the above equations are solvable and would give a set of solutions which express the functions $\theta ^a(x)$ as functionals of the vector potentials $A_\mu ^a(x)$. The constraint conditions in Eq.(42) may also be incorporated into the Lagrangian in Eq.(39) by the Lagrange undetermined multiplier method. In doing this, it is convenient, as usually done, to introduce ghost field variables $C^a(x)$ in such a fashion \[3-5\]: $\theta ^a(x)=\varsigma
C^a(x)$ where $\varsigma $ is an infinitesimal Grassmann’s number. In accordance with this relation, the constraint condition in Eq.(42) can be rewritten as $$\partial ^\mu ({\cal D}_\mu ^{ab}C^b)=0 \eqnum{44}$$ which usually is called ghost equation. When this constraint condition is incorporated into the Lagrangian in Eq.(39) by the Lagrange multiplier method, we obtain a more generalized Lagrangian as follows $${\cal L}_\lambda =-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a+\lambda ^a\partial ^\mu A_\mu ^a+\frac 12\alpha (\lambda ^a)^2+%
\bar C^a\partial ^\mu ({\cal D}_\mu ^{ab}C^b) \eqnum{45}$$ where $\bar C^a(x)$, acting as Lagrange multipliers, are the new scalar variables conjugate to the ghost variables $C^a(x)$.
At present, we are ready to formulate the quantization of the massive gauge field in the Lagrangian path-integral formalism. As we learn from the Lagrange multiplier method, the dynamical and constrained variables as well as the Lagrange multipliers in Eq.(45) can all be treated as free ones, varying arbitrarily. Therefore, we are allowed to use this kind of Lagrangian to construct the generating functional of Green’s functions $$\begin{tabular}{l}
$Z[J^{a\mu },\overline{\xi }^a,\xi ^a]=\frac 1N\int D(A_\mu ^a,\bar C%
^a,C^a,\lambda ^a)\exp \{i\int d^4x[{\cal L}_\lambda (x)$ \\
$+J^{a\mu }(x)A_\mu ^a(x)+\overline{\xi }^a(x)C^a(x)+\overline{C}^a(x)\xi
^a(x)]\}$%
\end{tabular}
\eqnum{46}$$ where $D(A_\mu ^a,\cdots ,\lambda ^a)$ denotes the functional integration measure, $J_\mu ^a$, $\overline{\xi }^a$ and $\xi ^a$ are the external sources coupled to the gauge and ghost fields and $N$ is the normalization constant. Looking at the expression of the Lagrangian in Eq.(46), it is seen that the integral over $\lambda ^a(x)$ is of Gaussian-type. Upon completing the calculation of this integral, we finally arrive at $$\begin{tabular}{l}
$Z[J^{a\mu },\overline{\xi }^a,\xi ^a]=\frac 1N\int D(A_\mu ^a,\bar C%
^a,C^a,)\exp \{i\int d^4x[{\cal L}_{eff}(x)$ \\
$+J^{a\mu }(x)A_\mu ^a(x)+\overline{\xi }^a(x)C^a(x)+\overline{C}^a(x)\xi
^a(x)]\}$%
\end{tabular}
\eqnum{47}$$ where $${\cal L}_{eff}=-\frac 14F^{a\mu \nu }F_{\mu \nu }^a+\frac 12m^2A^{a\mu
}A_\mu ^a-\frac 1{2\alpha }(\partial ^\mu A_\mu ^a)^2-\partial ^\mu \bar C^a%
{\cal D}_\mu ^{ab}C^b \eqnum{48}$$ is the effective Lagrangian given in the general gauges. In the Landau gauge ($\alpha \rightarrow 0$), The Lagrangian in Eq.(48) just gives the result in Eq.(36). It has been proved \[1\] that the above quantization carried out by means of the Lagrange multiplier method is equivalent to the Faddeev-Popov approach of quantization \[4\].
There are three points we would like to emphasize: (1) In the quantization by the Lagrange multiplier method, the gauge-invariance is always to be required even in the arbitrary gauge. Moreover, it has been found that the action given by the Lagrangian in Eq.(48) is invariant under a kind of BRST-transformations \[6\]. Thus, the quantum non-Abelian gauge field theory is set up from beginning to end on the firm basis of gauge-invariance. (2) In the Lagrangian path-integral formalism, as shown before, the quantized result is derived by utilizing the infinitesimal gauge transformations. This result is identical to that obtained by the quantization in the Hamiltonian path-integral formalism. In the latter quantization, we only need to calculate the classical Poisson brackets without concerning any gauge transformation. This fact reveals that to get a correct quantum theory in the Lagrangian path-integral formalism, the infinitesimal gauge transformations are only necessary to be taken into account and thereby confirms the fact that in the physical subspace restricted by the Lorentz condition, only the infinitesimal gauge transformations are possible to exist. (3) From the generating functional shown in Eqs.(47) and (48), one may derive the gauge boson propagator as follows $$iD_{\mu \nu }^{ab}(k)=-i\delta ^{ab}\{\frac{g_{\mu \nu }-k_\mu k_\nu /k^2}{%
k^2-m^2+i\varepsilon }+\frac{\alpha k_\mu k_\nu /k^2}{k^2-\mu
^2+i\varepsilon }\} \eqnum{50}$$ which is of good renormalizable behavior. In the zero-mass limit, this propagator with the massive Yang-Mills Lagrangian and the generating functional together all go over to the results given in the massless gauge field theory, different from the quantum theory established previously from the massive Yang-Mills Lagrangian alone without any constraint \[7-10\]. For the previous theory, there occurs a severe contradiction in the zero-mass limit that the massive Yang-Mills Lagrangian in Eq.(1) is converted to the massless one, but, the propagator is not and of a singular behavior. In particular, the previous theory was shown to be nonrenormalizable \[3, 7-9\] because the unphysical longitudinal fields and residual gauge degrees of freedom are not excluded from the theory.
Up to the present, we limit ourself to discuss the gauge fields themselves without concerning fermion fields. For the gauge fields, in order to guarantee the mass term in the action to be gauge-invariant, the masses of all gauge bosons are taken to be the same. If fermions are included, Obviously, the QCD with massive gluons fulfils this requirement because all the gluons can be considered to have the same mass. Such a QCD, as has been proved, is not only renormalizable, but also unitary \[6\]. The renormalizability and unitarity are warranted by the fact that the unphysical degrees of freedom in the theory have been removed by the constraint conditions in Eqs.(38) and (45). The gauge-fixing term and the ghost term in Eq.(49) just play the role of counteracting the unphysical degrees of freedom contained in the massive Yang-Mills Lagrangian as verived by the perturbative calculations \[6\].
References
==========
\[1\] J. C. Su, Nuovo Cimento 117B (2002) 203.
\[2\] C. N. Yang and R. L. Mills, Phys. Rev. 96 (1954) 191.
\[3\] C. Itzykson and F-B, Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).
\[4\] L. D. Faddeev and V. N. Popov, Phys. Lett. B25 (1967) 29.
\[5\] L. D. Faddeev, Theor. Math. Phys., 1 (1970) 1.
\[6\] J. C. Su, hep.th/9805192; 9805193; 9805194.
\[7\] H. Umezawa and S. Kamefuchi, Nucl. Phys. 23 (1961) 399.
\[8\] A. Salam, Nucl. Phys. 18 (1960) 681; Phys. Rev. 127 (1962) 331.
\[9\] D. G. Boulware, Ann. Phys. 56 (1970) 140.
\[10\] P. Senjanovic, Ann. Phys. (N.Y.) 100 (1976) 227.
\[11\] C. Grosse-Knetter, Phys. Rev. D48 (1993) 2854.
\[12\] N. Banerjee, R. Banerjee and G. Subir, Ann. Phys. (N. Y.) 241 (1995) 237.
|
---
author:
- 'C.A. Narayan, K. Saha, & C.J. Jog'
date: 'Received; accepted'
title: Constraints on the halo density profile using HI flaring in the outer Galaxy
---
Introduction
============
It is a well-known observational fact that the atomic hydrogen layer flares in the outer Galaxy (Kulkarni, Heiles & Blitz 1982; Knapp 1987; Wouterloot et al. 1990; Diplas & Savage 1991; Merrifield 1992; Nakanishi & Sofue 2003). HI flaring is also noticed in many external galaxies seen edge-on (Brinks & Burton 1984; Olling 1996a; Matthews & Wood 2003). A possible cause for flaring could be that the total gravitational force acting perpendicular to the disk plane decreases with radius while the velocity dispersion of HI is observed to be nearly constant (Lewis 1984). The contribution to the total perpendicular gravitational force comes mainly from the stellar disk, gas and the dark matter. While the stellar disk dominates within the optical region of a normal disk galaxy, the outer region is dominated by the dark matter. The disk and halo seem to dominate in different regions of a galaxy because of their different density distributions. Although both decrease from the center, the disk density decreases rapidly whereas the halo density decreases much more slowly, so that the halo extends to several times the size of the optical disk. Thus the halo plays a major role in determining the vertical disk structure beyond the optical region. This makes the outer Galactic HI layer sensitive to the mass and the density profile of the halo, and hence it can be used as a diagnostic to study the halo properties.
The first attempt towards studying the halo parameters using HI layer was made by Olling (1995), who has developed a model where the observed thickness of the HI layer can be used to predict the shape of the halo. This method has been used to show that the halos of NGC 4244 (Olling 1996b) and NGC 891 (Becqueart & Combes 1997) are highly flattened with their axis ratios in the range of 0.2 - 0.4. Olling & Merrifield (2000) use the same method to find an axis ratio of 0.8 for the halo of our Galaxy. However, on spanning a larger parameter space and considering more factors which affect the HI scaleheight, they find the halo to be closer to spherical for any R$_{\circ} >$ 7 kpc where R$_{\circ}$ is the distance between sun and the Galactic center (Olling & Merrifield 2001). Another measurement of the shape the halo of our Galaxy comes from an entirely different method. Using the kinematics of the tidal streams of stars surrounding the Galaxy, called the Sagittarius stream, Ibata et al. (2001) show that the Galactic dark halo is almost spherical. They conclusively rule out axis ratios $<$ 0.7 for our Galaxy. In general, there is a lot of interest now in determining the shape of the dark matter halo in a galaxy (see e.g., Natarajan 2002, and the references therein).
The focus of this paper is to determine the halo parameters which provide the best fit (in the least square sense) to the observed HI scaleheight. The halo parameters we intend to explore are its central mass density $\rho_{\circ}$, core radius R$_c$, the shape (or axis ratio) $q$ and the power law index of the density profile $p$. We calculate the HI scaleheight, numerically, based on our self-consistent model for the Galaxy (Narayan & Jog 2002) and compare the results with the observations. The gravitational force due to the dark matter halo is incorporated in the model as an external force acting on the disk. The effect of the disk gravity on the halo is thus neglected in this model so that the halo is taken to be rigid or non-responsive. The HI scaleheight is first obtained using the simplest profile for the halo - a spherical screened isothermal density profile. Subsequently, the shape of this halo is varied. Next we study the effects of including a non-isothermal halo by varying the power law index of its density profile. We find that the best fit to the data is obtained for a spherical halo whose density falls more rapidly ($p$=1.5-2) than that of an isothermal halo ($p$=1).
The formulation of the problem is described in Section 2 while the method of calculation and parameters used in the model are discussed in Section 3. Section 4 is devoted to the results of this paper, and in Sections 5 and 6, we present discussion and conclusions.
Formulation of the problem
==========================
Vertical equilibrium
--------------------
We treat the Galactic disk to consist of three major components: the stars, the interstellar atomic hydrogen gas HI, and the interstellar molecular hydrogen gas H$_2$, which are coupled gravitationally. We assume that the three components are axisymmetric and are in hydrostatic equilibrium in the $z$ direction. We use the galactic cylindrical co-ordinates $R, \phi, z$. Then the disk dynamics under the force field due to the rigid halo can be described by the Poisson equation, and the force equation along the normal to the plane (the equation of pressure equilibrium) for each component. The Poisson equation for the axisymmetric galactic system in cylindrical geometry is given as:
$$\frac{\partial^2 \Phi_{tot}}{\partial z^2} + \frac{1}{R}
\frac{\partial}{\partial R} (R \frac{\partial \Phi_{tot}}{\partial R}) \:
= \: 4 \pi G \: (\sum_{i=1}^3 \rho_i + \rho_h )
\eqno(1)$$
where $\rho_i$ with $i$ = 1 to 3 denotes the mass density for each disk component, $\rho_h$ denotes the same for the halo, and $\Phi_{total}$ denotes the potential due to the disk and the halo. For a disk with a flat rotation curve, the radial term in the above equation is identically equal to zero at the mid-plane ($z$ = 0). The rotation curve of our Galaxy is not observed to be strictly flat (Merrifield 1992; Honma & Sofue 1997). But quantitatively, we find that the radial term contributes to less than 1 $\%$ change in the HI scaleheight. Hence, this term can be neglected and equation (1) reduces to:
$$\frac{\partial^2 \Phi_{tot}}{\partial z^2} \: = \:
4 \pi G \: (\sum_{i=1}^3 \rho_i + \rho_h ) \eqno(2)$$
The equation for pressure equilibrium in the vertical direction for each component is given by (e.g. Rohlfs 1977):
$$\frac{\partial}{\partial z}(\rho_i \overline{v^2_z}_i) +
\rho_i \frac{\partial \Phi_{tot}}{\partial z} \: = \: 0 \eqno(3)$$
On combining the above two equations, we get the equation for the vertical equilibrium of each component in the disk under the field of the halo to be:
$$\frac{\partial}{\partial z} \left [ \frac {\overline{(v^2_z)}_i} {\rho_i}
\frac{\partial \rho_i}{\partial z} \right]
= -4\pi G \: ( \sum_{i=1}^3 \rho_i + \rho_h ) \eqno(4)$$
Here, we assume that the vertical velocity dispersion is independent of distance from the disk plane, that is, the disk is taken to be isothermal. Note that the above treatment [*does not*]{} assume the disk to be thin. Under the thin disk approximation, the disk contribution to the radial term in equation (1) would drop out and equation (4) would reduce to :
$$\frac{\partial}{\partial z} \left [ \frac {\overline{(v^2_z)}_i} {\rho_i}
\frac{\partial \rho_i}{\partial z} \right] = -4\pi G \: ( \sum_{i=1}^3 \rho_i ) + \left . \frac{\partial K_z}{\partial z} \right |_{halo} \eqno(5)$$
where K$_z$ is the vertical force due to the halo. The above equation was used to calculate the stellar and gas scaleheights in the inner region of the Galaxy in our earlier paper (Narayan & Jog 2002). We find that treating the HI disk to be thin in the present case would overestimate the HI scaleheight by as much as $10\%$. Hence, we have used the general approach for a thick disk for HI gas (eq.\[4\]) in the present paper.
Dark matter halo
----------------
We assume a four-parameter halo model as described by the following density profile (de Zeeuw & Pfenniger 1988, Becquaert & Combes 1997)
$$\rho(R,z) \: = \: \frac{\rho_{\circ}(q)} {\left(1 + \frac{m^2}{R^2_c(q)}\right)^p} \eqno(6)$$
where $\rho_{\circ}$ is the central mass density of the halo, $R_c$(q) is the core radius, $q$ is the axis ratio and $p$ is the index. Here by definition, $ m^2 = R^2 + {z^2}/{q^2} $ represents the surfaces of concentric ellipsoids. Note that $q$ = 1 would give rise to a spherical halo, while $q = c/a <1$ gives an oblate halo and $q = c/a >1$ describes a prolate halo, where $a$ is the axis in the disk plane and $c$ is that along the vertical direction.
By varying $p$ one can generate different halo density profiles. $p$=1 gives a screened isothermal halo. The mass density here goes as r$^{-2}$ at large radii (r$\gg$R$_c$). This leads to the mass within a spheroid $M(r) \propto r$ and a flat rotation curve. For $p$=1.5, $\rho \propto r^{-3}$ at large r just like the NFW halo (Navarro et al. 1996) and this gives $M(r) \propto
log(r)$. i.e., the total mass goes over to infinity (similar to the case of $p$=1) but more gradually. This family of halos gives rise to a falling rotation curve for r$\gg$R$_c$. $p$=2 halos have their mass density falling much faster ($\propto
r^{-4}$ at large r). $M(r)$ saturates to a finite value, unlike the other two cases. The rotation curve falls faster than the former class.
### Halo shape : isothermal halo
In the above equation $p$ = 1 gives a screened isothermal halo. Varying $q$ in this profile would give an axisymmetric ’pseudo-isothermal’ halo of a different shape. When the shape of a halo of fixed mass changes to either prolate or oblate, its central density, $\rho_{\circ}$, and the core radius, R$_c$, are bound to change in order to conserve the mass. Therefore to know the exact density profile for any ellipsoid, we need to first calculate the $\rho_{\circ}$ and $R_c$ as functions of $q$. We find that $\rho_{\circ}(q)$ and $R_c(q)$ are related to their spherical counterparts by the following relations:
$$\rho_{\circ}(q) = \rho_{\circ}(1) \frac{1}{q}\left ( \frac{e}{sin^{-1}e} \right)^3$$ $$R_c(q) = R_c(1) \left ( \frac{sin^{-1}e}{e} \right) \eqno (7)$$
These are obtained by imposing the following two constraints : the mass within a thin spheroidal shell (Binney & Tremaine 1987, pg 54) and the terminal velocity of the halo should be independent of $q$. Here $e$ is the eccentricity, and $e=\sqrt{1 - q^2}$ for the oblate case ($q<1$) ; and $e=\sqrt{1 - (1/q^2)}$ for the prolate case ($q > 1$). Figures (1a) and (1b) show the plots for $\rho_o(q)$ and $R_c(q)$ respectively as a function of oblateness and figures (1c) and (1d) give the corresponding plots for prolate shapes.
Comparison with previous work
-----------------------------
We note that relations for $\rho_o(q)$ and $R_c(q)$ for the oblate case have been previously obtained by Olling (1995) (see figure 2 of his paper) but by using a different method. The rotation velocity at core radius and the terminal velocity are the two constraints used to derive the relations for $\rho_o(q)$ and $R_c(q)$ in his work. This method also yields similar results but the method is cumbersome and the relations are approximations.
In this respect, we find that our method is advantageous in the following ways : first, it gives simple and accurate relations for $R_c(q)$ and $\rho_o(q)$, and second, these relations can be used for the oblate as well for the prolate cases along with the appropriate relations between $e$ and $q$ as given above, whereas Olling considered only oblate halos.
The ’global approach’ originally proposed by Olling (1995) and subsequently used by Olling & Merrifield (2000, 2001), to calculate the HI scaleheight is a general one and can be used even if the stellar disk is truncated before the HI layer ends. The method used in this paper is the so-called ’local approach’ (see Sect. 2.1), where the density obtained as a solution is based on the local graviational potential. This method, which was previously used by Spitzer (1942) and Bahcall (1984) to get classic results for the vertical disk distribution, is adopted in our work too because of its simplicity. The drawback of this method is that it does not yield self-consistent results (vertical density distribution and scaleheight of HI in this case) for truncated stellar disks. In our work (as in the works of Spitzer and Bahcall), the assumed density distributions of all the disk components and the halo are continuous. Hence the results are very close approximations of the exact self-consistent solutions (obtained by using the global approach) and are therefore valid.
Calculations and input parameters
=================================
Equation (4) represents the three coupled equations for the three disk components (stars, HI, and H$_2$) which are to be solved for the corresponding density distributions. The vertical density distribution for each component responding to the total potential of the disk and the halo, is solved for numerically as an initial value problem, using the fourth order Runge-Kutta method of integration (Press et al 1994). The details of this procedure are presented in our earlier paper (Narayan & Jog 2002). At any radius R, the HWHM of the vertical density distribution is defined as the scaleheight. Repetition of the calculation at regular intervals of the radius gives us the ’model’ scaleheight curve.
The input parameters for the model for each disk component are its surface density and vertical velocity dispersion. The surface densities for HI and H$_2$ are taken from the observations of Wouterloot et al. (1990). The stellar disk surface density is assumed to fall exponentially with distance from the center. The stellar disk mass and the surface density at any radius can be calculated using the following measured/inferred quantities : the stellar surface density at solar region $\Sigma_{\odot}$, the disk scalelength $R_d$ and the distance of sun from the center R$_{\circ}$. We use $\Sigma_{\odot}$ = 45 M$_{\odot} pc^{-2}$ which is consistent with 48 $\pm$ 9 M$_{\odot} pc^{-2}$ obtained by Kuijken & Gilmore (1991) and 52 $\pm$ 13 M$_{\odot} pc^{-2}$ obtained by Flynn & Fuchs (1994) for the total surface density, after the gas density is subtracted. We use the IAU recommended value for R$_{\circ}$ (=8.5 kpc) and $R_d$ is equal to 3.2 kpc (Mera et al. 1998) in accordance with the recent determinations of smaller disk scalelength for our Galaxy.
The stellar vertical dispersion is derived from the observations of radial dispersion by Lewis & Freeman (1989) and the assumption that the ratio of the vertical to radial velocity dispersion is equal to 0.45 at all radii in the Galaxy, equal to its observed value in the solar neighbourhood as obtained from the analysis of the Hipparcos data (Dehnen & Binney 1998, Mignard 2000). The vertical velocity dispersion for H$_2$ is taken to be 5 km s$^{-1}$ (Stark 1984, Clemens 1985).
HI velocity dispersion
----------------------
The HI velocity dispersion is observed to be almost constant with radius and is about 9$\pm$1 km s$^{-1}$ (Spitzer 1978; Malhotra 1995) in the inner Galaxy (upto solar circle). Beyond the solar circle, however, the dispersion is not yet measured. A study of 200 external galaxies (Lewis 1984) shows that the observed dispersion has a very narrow range, about 8$\pm$1 km s$^{-1}$, consistent with observations of our Galaxy. Sicking’s (1997) work shows that in two external galaxies, dispersion decreases slowly upto the outer edge of the HI layer. In a number of other galaxies, the velocity dispersion decreases and then saturates to a constant value of 7$\pm$1 km s$^{-1}$ (Shostak & van der Kruit 1984; Dickey 1996; Kamphuis 1993). This decrease in dispersion is perhaps due to the lesser number density of supernovae in the outer region (McKee & Ostriker 1977) . A major part of the observed dispersion is non-thermal in origin and the supernovae could be the major source for this whereas the thermal contribution comes upto just about 1 km s$^{-1}$ (Spitzer 1978).
In this work, the dispersion is taken to be 9 km s$^{-1}$ at 9 kpc, consistent with Malhotra (1995). Between 9-20 kpc the dispersion is allowed to decrease linearly with a slope of -0.2 km s$^{-1}$kpc$^{-1}$, as it is found to give the best fit to the data. Beyond 20 kpc, which is about twice the optical disk size, it is kept fixed at 7 km s$^{-1}$, equivalent to that observed in external galaxies. This is perhaps justified in the absence of direct observations in the outer Galaxy.
Results
=======
In this section we calculate the HI scaleheight versus radius in the outer Galaxy using the above parameters for the disk components and different density profiles for the dark matter halo. We then compare our results with observed HI scaleheight data. Of the various observations of HI scaleheight in the outer Galaxy by different authors, we find that results of Knapp (1987), Wouterloot et al (1990) and Merrifield (1992) are consistent with each other. Of these, Wouterloot et al give scaleheight values upto a very large distance (nearly 3R$_{\circ}$) with closest sampling (bin size of about 250 pc). Therefore we use their data to fit with the results from our model. Unfortunately, the error bars associated with these data points are not given. So, we compute least square (which is equivalent to $\chi^2$ with unit error bars) of the model-generated curve in order to measure its goodness of fit. Merrifield (1992) has pointed out that Wouterloot’s data has to be corrected for beam-size effects but we note that the correction on the derived HI scaleheight is so small (fig 4 of Merrifield 1992) that it can be ignored in our study.
The first choice of dark matter profile used here is that proposed as a part of the complete mass model of the Galaxy by Mera et al. (1998) based on microlensing observations. It is a simple screened isothermal spherical halo ($p$ = 1) with $R_c$ = 5 kpc, the central density $\rho_0$= 0.035 M$_{\odot} pc^{-3}$ and the terminal rotation velocity = 220 km $s^{-1}$. Their stellar disk parameters are $\Sigma_{\odot}$ = 45 M$_{\odot} pc^{-2}$, $R_d$ = 3.2 kpc and R$_{\circ}$ = 8.5 kpc. Figure 2 (solid line) shows that the HI scaleheight obtained using this profile matches well with observations upto about 20 kpc. But, beyond this region the calculated values fall much below the observed points, thereby giving a poor fit (a high value of least square).
Shape of the halo
-----------------
Next, we consider whether halo of any different axisymmetric shape can improve the fit with the data. In doing so the remaining halo parameters and the disk parameters are kept unchanged. Such halos, especially those of oblate shape have gained popularity as discussed in Section 1.
Figure 2 shows the resulting scaleheight curves from our model for a range of values for $q$, the axis ratio. In addition to the oblate halos that are extensively studied in the literature, we also consider halos that are prolate shaped. This makes the study of the effect of halo shape on the HI flaring complete. Also, there is some evidence in recent literature in favour of prolate-shaped halos. For example, Ideta et al. (2000) find that prolate halo helps sustain warps better. Figure 2 shows that the scaleheight upto 16 kpc radius remains almost unaffected by the change in halo shape and the effect of shape is prominent only beyond 20 kpc. Also note that both the oblate and the prolate cases reduce to that of the spherical halo under limiting conditions (i.e., for q = 1). A definite trend is observed as the shape is varied from oblate to prolate. The oblate halo tends to reduce the scaleheight and this reduction increases with flattening. The effect is exactly opposite for the prolate-shaped halo. This is because the mid-plane halo density increases on compressing the halo (oblate shape) and decreases on elongating it (prolate shape). This results in a higher constraining force due to the oblate shape, and vice versa for the prolate shape.
It is clearly seen that the results from the pseudo-isothermal models (p=1;q$\neq$1) do not fit the observations well irrespective of the choice of shape and the axis ratio. The least square for the oblate case increases with increasing oblateness. Though the curves generated by prolate halos also do not give a good fit, the least square is lower than that for the oblate case. The overall trend shown by the prolate curves matches with observations better but they are still far from giving a good fit to the data.
Halo density profiles
---------------------
Since the variation in the shape of the halo does not lead to a good agreement between the model and the observations, we consider change in the density profiles as characterised by the index $p$ (eq.\[6\]). As $p$ is varied between 1, 1.5 and 2, the shape of the halos is kept spherical for the sake of simplicity. For each value of $p$, a realistic range of $\rho_o$ and $R_c$ is chosen to form a grid of ($\rho_o$,$R_c$) pairs. The core density $\rho_o$ is varied between 0.001-0.1 $M_{\odot}pc^{-3}$ in steps of 0.002 $M_{\odot}pc^{-3}$ and $R_c$ is varied between 4-15 kpc in steps of 100 pc. The total (disk+halo) circular speed at the solar point is calculated corresponding to each of these grid points. The HI scaleheight is calculated for only those grid points which give circular speed in the range determined by the relation between Galactic constants, $\Theta_{\circ}$ = (27$\pm$2.5)$R_{\circ}$ km s$^{-1}$ (Kerr & Lynden-Bell 1986, Reid et al. 1999). This is the first constraint used to narrow down the number of possibilities for the best fit halo model. The second constraint is that the least square value of the model scaleheight curve should be minimum. We find that just a small subset ($\sim$10) of the entire set of grid points ($\sim$5000) gives minimum values of the least square. The final constraint imposed on this subset is that the total disk+halo rotation curves generated by these halos should show the main trends seen in the observed rotation curve (Merrifield 1992; Honma & Sofue 1997) such as the rise beyond the solar point and fall beyond 2$R_{\circ}$. The above procedure is repeated for other values of index $p$.
For the specific choice of $R_{\circ}$ = 8.5 kpc ($R_d$ = 3.2 kpc; $\Sigma_{\odot}$ = 45$M_{\odot}pc^{-2}$), the $\Theta_{\circ}$ is expected to be in the range 230$\pm$21 km s$^{-1}$. The least square value is found to be minimum for $p$ = 2 and $\rho_o$, $R_c$ are in the range of 0.035-0.093 $M_{\odot}pc^{-3}$ and 7-9.5 kpc respectively. These halos span the entire range of allowed $\Theta_{\circ}$ ie, between 210-250 km s$^{-1}$. Figure 3 shows the case of $\rho_o$ = 0.035 $M_{\odot}pc^{-3}$ and $R_c$ = 9.4 kpc. As the central density increases (and $R_c$ decreases), the $\Theta_{\circ}$ increases progressively. In each case, the total rotation curve rises from the solar point, reaches a peak value and then falls beyond. But the rise becomes more gradual, the peak shifts towards centre and the fall becomes steeper as the central density increases (for the higher end of $\Theta_{\circ}$), thus growing more and more inconsistent with the observed rotation curve. Therefore we find that halos with $\rho_o$ = 0.035-0.06 $M_{\odot}pc^{-3}$ and $R_c$ = 8-9.5 kpc , which give $\Theta_{\circ}$ = 210 to 230 km s$^{-1}$ give a good fit to the HI data as well as produce realistic rotation curves.
The choice of $p$=1.5 for the halo profile also gives a reasonable fit to the data points, though not as good as for $p$=2. The fall in the rotation curve generated by $p$=1.5 is so gradual that it almost appears to be constant in the region of interest. This is however certainly within the error bars of the observed rotation curve (Honma & Sofue 1997). The least square values for the $p$=1 halos are so large (for physically meaningful $\rho_o$, $R_c$) that they do not pass the very first constraint used to obtain the best fit halos.
Thus our model shows a preference for $1.5 < p \leq 2$ in the density profile for the halo (eq.\[6\]), corresponding to a density fall-off proportional to r$^{-\beta}$ with $3 <\beta \leq 4$, at very large radii ($R\gg R_c$). For $p$=2, the density falls to $10^{-4}$ of the central density at just 10 core radii ($\sim$100kpc). In comparison, the fractional fall-off for isothermal case ($p=1$) is only $10^{-2}$ at the same distance. This steeper density fall can give rise to the rapid observed flaring between 20-24 kpc. See Fig. 4a for a comparison between $p$=1 and $p$=2 density profiles. Note that at large radii ($R\gg R_c$), $p$=1.5 goes over to the popular NFW profile for the dark halo.
The p=2 halo gives a finite mass on integrating upto infinity. For example, the best fit halo has a total mass of 2.8$\times
10^{11}$ M$_{\odot}$ of which about 90% lies within 100kpc. In contrast, the mass of an isothermal halo is linearly proportional to the distance of integration and therefore becomes infinite at infinity (see Fig. 4b). Since most of the mass of p=2 halos are confined to within a few hundreds of kpc (or a few decades of core radii), these can be regarded as naturally truncated halos or ’finite sized’ halos. This concept has very important consequences in the cosmological scenario.
In Figure 4c, we plot the rotation curves corresponding to $p=1$ and $2$ cases. This illustrates the difference between these two model cases especially at large radii, and is in a form that is directly amenable to future observational checks. In the case of $p=1$, the rotation curve becomes flat at large radii in keeping with the linearly rising mass of the halo (compare with Fig. 4b); whereas in the case of $p=2$, the rotation curve begins to drop beyond $\sim$ 12 kpc.
Comparison with other deductions
--------------------------------
Observationally, the vast database of SDSS has allowed researchers to check the large-scale behaviour of the dark matter halos. Prada et al. (2003) find that the radial density profile $\propto r^{-3}$ as predicted by most modern cosmological models, is consistent with the observed velocity dispersion of satellites of isolated galaxies. This corresponds to $p$=1.5 in our model. Fischer et al. (2000) and McKay et al. (2002), on the other hand find that the halo mass density falls off as r$^{-4}$ at very large radii (corresponding to $p$=2 in our model) i.e., for r $\gg$ 260 h$^{-1}$ kpc, which is the minimum size limit for an isolated galaxy. This limit corresponds to about 400kpc within which 95% of mass is confined for our best fit halos. These support the overall trend in the results from our work. However, we need to be careful in comparing our results with the above SDSS studies which are on isolated galaxies while our Galaxy has at least one massive close neighbour.
The faster than isothermal fall-off is also supported by many numerical simulation studies in the literature. In their simulations on formation of dark halos, Avila-Reese et al. (1999) find that most halos tend to have density profiles $\propto r^{-\beta}$ where $\beta$ falls in the range 2.5 - 3.8, in the outer region. Bournaud et al. (2003) find from their simulations, that dark halos should extend to at least ten times further than their stellar disks, in order to be able to explain the formation of tidal dwarf galaxies. We note that our best fit halo certainly extends much beyond the corresponding limit of $\sim$ 120kpc. Further support for the choice of $p=2$ comes from studies which show that the halo core radius is comparable to the optical/Holmberg radius for a galaxy (Salucci 2001). As $p$ increases, the best fit value for R$_c$ also increases. Hence $p$=2 halos give the largest value for R$_c$ (8-10 kpc) which is closest to the Homberg radius estimated for our Galaxy (e.g., Binney & Merrifield 1998).
In the present work, the halos that provide a reasonable fit to the observed flaring have $p$ in the range of 1.5-2 but were all spherical in shape. This result is consistent with recent findings on the shape of dark halo of our Galaxy (Ibata et al. 2001; Olling & Merrifield 2001). These recent studies indicate that the shape is nearly spherical (see Sect. 1). Thus we find that the results for both density profile and shape ($p$ and $q$) for the dark halo of our Galaxy as deduced from the HI flaring, are consistent with recent observational evidence and theoretical works.
Discussion
==========
[**(1) Asymmetry in the Galaxy :**]{} A very crucial assumption in our model is that the Galactic disk and the halo are axisymmetric. This is done for simplicity, as was also done by previous authors (Olling & Merrifield 2000, 2001). We note, however, that the outer Galaxy shows asymmetry or lopsidedness. There is observational evidence for kinematical lopsidedness where the cut-off in the fourth and the first quadrants differs by 25 km s$^{-1}$ (Burton 1988), and also for spatial lopsidedness where the measurable column density extends much farther out in the north than in the south (upto 4 R$_{\odot}$ and 2.2 R$_{\odot}$ respectively)- see Merrifield (2002), also see Nakanishi & Sofue (2003). Thus the analysis and the conclusions from our paper are largely based on the northern data and that is the limitation in the analysis of the present paper.
[**(2) Galactic constants :**]{} We have built the stellar disk model based on the IAU-recommended values for the galactic constants - R$_{\circ}$ = 8.5 kpc and $\Theta_{\circ}$ = 220 km s$^{-1}$. It would definitely be interesting and also worthwhile to know how the results for the halo density profile would vary with the assumed galactic constants. For example, Olling & Merrifield (2001) find the effect of varying these constants on the inferred axis ratio of the halo. Unfortunately, all the observational inputs for our model, like the HI and H$_2$ surface densities, HI scaleheight and the stellar velocity dispersion, are based on the IAU-recommended galactic constants and rescaling them for different values of the constants is beyond the scope of this paper.
[**(3) Self-gravity of the gas :**]{} It is interesting to check the change brought about by excluding the HI self-gravity on its vertical scaleheight. The difference is not negligible, it is seen to be about 10-20$\%$ within the optical disk (R$<$ 4R$_d$) and also beyond R $>$ 6R$_d$. This could be because these regions are dominated by the stellar disk and by the dark matter respectively. For the region where 4 $<$ R/R$_d$ $<$ 6, the difference is substantial ($\sim$50$\%$) suggesting that in this range, the gas gravity is crucial in negotiating the hydrostatic equilibrium for the HI layer. Thus neglecting it may lead to a serious overestimate of the HI scaleheight in general at all radii in the outer Galaxy (as already cautioned by Olling 1995) and particularly in the intermediate range of radii.
Conclusions
===========
We calculate the HI scaleheight in the outer Galaxy using a Galactic disk model taking the dark matter halo also into account. In the outer Galaxy, the dark matter halo is the key component that decides the scaleheight of HI, hence we calculate the radial variation in the HI scaleheight as a function of the shape and density profile of the halo. Based on the method of least squares we show that neither oblate nor prolate-shaped isothermal halos can provide a good fit to the observations.
Instead, the best agreement with data is provided by a spherical halo and a density profile that is $\propto r^{-\beta}$ with $3<\beta \leq4$ in the peripheral parts of the Galaxy ($1.5<p \leq 2$). In such a halo, the density falls off stepper than the-often-used isothermal halo. The rotation curves produced by these best-fit halos are also in good agreement with the observed one. This result seems to be in good agreement with the recent trend seen in the literature on the numerical simulations of halo formation, as well as the halo density profiles deduced from the SDSS data.
[**Acknowledgements**]{}
We would like to thank the referee, Albert Bosma, for his critical comments and the many suggestions, which have vastly improved the quality of this paper. We also thank Anish Roshi and Rekhesh Mohan for discussions on HI observations and their analysis. K.S. would like to thank the CSIR-UGC, India for a Senior research fellowship.
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[**Figure 1.**]{} This plot shows the variation of halo parameters as a function of its axis ratio (see eq.\[7\]). Figures 1(a) and (b) show the variation in the central density and the core radius, with oblateness (q = minor axis/major axis). Figures 1 (c) and (d) show the same as a function of prolateness (=minor axis/major axis). This dependence on axis ratio arises from keeping the mass within a thin spheroidal shell of the halo and the terminal velocity of the halo invariant of $q$.
[**Figure 2.**]{} The calculated HI scaleheight is shown as a function of the galactocentric radius in the outer region of the Galaxy. The shape of an initially spherical isothermal halo ($\rho_{\circ}$ = 0.035 M$_{\odot}$ pc$^{-3}$ ; R$_c$ = 5 kpc) is changed keeping its mass constant. This plot shows the results from our model for different axis ratios. The solid line is due to spherical shape ($q = c/a $= 1); the dashed lines are due to oblate halos ($c/a$ = 0.8, 0.6, 0.4) and the dotted lines are due to prolate halos ($a/c$ = 0.8, 0.6, 0.4). The points show the observed values. Note that neither the oblate nor the prolate-shaped halos are clearly favoured by the data.
[**Figure 3.**]{} The best fit (based on the least square value) for the observed HI scaleheight is given by a halo with $p$ = 2 where $p$ is the index in the halo density profile (solid line). For comparison, results for a typical isothermal halo ($p$ = 1) is also shown (dashed line).
[**Figure 4a.**]{} A log-normal plot of the halo mass density (in units of M$_{\odot}$pc$^{-3}$) as a function of the galactocentric radius, for the best-fit halo (p=2) and the isothermal halo (p=1). Although the two begin to differ around the optical edge of the stellar disk itself, the effect on HI scaleheight becomes noticeable only after 20 kpc (see Fig. 3). The corresponding p=1.5 profile (which is equivalent to the NFW density profile at large radii) falls between the two shown profiles but is closer to the p=2 profile.
[**Figure 4b.**]{} The behaviour of M(R) (mass contained within a spheroid of radius R) as a function of R, is shown here for the two kinds of halos. For an isothermal halo, the mass tends to infinity whereas for p=2, it tends converges to a finite value as R increases. The mass tends to infinity for p=1.5 as well, but rather gradually compared to that of p=1.
[**Figure 4c.**]{} Rotation curves for the cases p=1 and p=2. For p=1, the rotation curve becomes flat at large radii whereas for p=2, the curve begings to fall beyond a radius of $\sim$ 12 kpc.
|
---
abstract: 'We study fairness in collaborative-filtering recommender systems, which are sensitive to discrimination that exists in historical data. Biased data can lead collaborative filtering methods to make unfair predictions against minority groups of users. We identify the insufficiency of existing fairness metrics and propose four new metrics that address different forms of unfairness. These fairness metrics can be optimized by adding fairness terms to the learning objective. Experiments on synthetic and real data show that our new metrics can better measure fairness than the baseline, and that the fairness objectives effectively help reduce unfairness.'
author:
- Sirui Yao
- Bert Huang
bibliography:
- 'yao-nips17.bib'
title: New Fairness Metrics for Recommendation that Embrace Differences
---
|
---
abstract: 'This expository paper features a few highlights of Richard Stanley’s extensive work in Ehrhart theory, the study of integer-point enumeration in rational polyhedra. We include results from the recent literature building on Stanley’s work, as well as several open problems.'
address: |
Department of Mathematics\
San Francisco State University\
San Francisco, CA 94132\
U.S.A.
author:
- Matthias Beck
bibliography:
- 'bib.bib'
date: 13 September 2015
title: 'Stanley’s Major Contributions to Ehrhart Theory'
---
[^1]
Introduction
============
This expository paper features a few highlights of Richard Stanley’s extensive work in Ehrhart theory, with some pointers to the recent literature and open problems. Pre-Stanley times saw two major results in this area: In 1962, Eugène Ehrhart established the following fundamental theorem for a [*lattice polytope*]{}, i.e., the convex hull of finitely many integer points in $\R^d$.
\[thm:ehrhart\] If $\P \subset \R^d$ is a lattice polytope and $n \in \Z_{ >0 }$ then $${\operatorname{ehr}}_\P(n) := \# \left( n \P \cap \Z^d \right)$$ evaluates to a polynomial in $n$ (the [*Ehrhart polynomial*]{} of $\P$). Equivalently, the accompanying generating function (the [*Ehrhart series*]{} of $\P$) evaluates to a rational function: $${\operatorname{Ehr}}_\P(x) := 1 + \sum_{ n>0 } {\operatorname{ehr}}_\P(n) \, x^n = \frac{ \h_\P(x) }{ (1-x)^{ \dim(\P)+1 } }$$ for some polynomial $\h_\P(x)$ of degree at most $\dim(\P)$, the [*Ehrhart $h$-vector*]{} of $\P$.[^2]
We remark that the step from an Ehrhart polynomial to its rational generating function is a mere change of variables: the coefficients of $\h_\P(x)$ express ${\operatorname{ehr}}_\P(n)$ in the binomial-coefficient basis $\binom n k$, $\binom {n+1} k$, …, $\binom {n+k} k$, where $k = \dim(\P)$.
In 1971, I.G. Macdonald proved the following *reciprocity theorem*, which had been conjectured (and proved for several special cases) by Ehrhart.
\[thm:macdon\] The evaluation of the Ehrhart polynomial of $\P$ at a negative integer yields $${\operatorname{ehr}}_\P(-n) = (-1)^{ \dim(\P) } {\operatorname{ehr}}_{ \P^\circ } (n) \, ,$$ where $\P^\circ$ denotes the (relative) interior of $\P$. Equivalently, we have the following identity of rational functions: $${\operatorname{Ehr}}_\P(\tfrac 1 x) = (-1)^{ \dim(\P) + 1 } {\operatorname{Ehr}}_{ \P^\circ } (x) \, ,$$ where ${\operatorname{Ehr}}_{ \P^\circ } (x) := \sum_{ n>0 } {\operatorname{ehr}}_{ \P^\circ }(n) \, x^n$.
Stanley made several fundamental contributions to Ehrhart theory, starting in the 1970s. This paper attempts to highlight some of them, roughly in historical order. The starting point, in Section \[sec:anand\], is Stanley’s proof of the Anand–Dumir–Gupta conjecture. Section \[sec:reciprocity\] features a reciprocity theorem of Stanley that generalizes Theorem \[thm:macdon\], and Section \[sec:inequ\] contains Stanley’s inequalities on Ehrhart $h$-vectors. Throughout the paper we mention open problems, and Section \[sec:beyond\] gives some recent results in Ehrhart theory building on Stanley’s work.
“I am grateful to K. Baclawski for calling my attention to the work of Eugène Ehrhart...” {#sec:anand}
=========================================================================================
The starting point for Stanley’s work in Ehrhart theory is arguably his proof of the *Anand–Dumir–Gupta conjecture* [@ananddumirgupta]; this proof appeared in a 1973 paper from which the quote of the section header is taken [@stanleymagic p. 631]. (One could, in fact, argue that *$P$-partitions*, which were introduced in Stanley’s Ph.D. thesis [@stanleythesis] and are featured in Gessel’s article in this volume, already show an Ehrhart-theoretic flavor, but this geometric realization came slightly later in Stanley’s work.) The conjecture concerns the counting function $H_n(r)$, the number of $(n \times n)$-matrices with nonnegative integer entries that sum to $r$ in every row and column; these matrices are often referred to as *semimagic squares*. The function $H_n(r)$ goes back to MacMahon [@macmahon], who computed the first nontrivial case, $H_3 (r) = \binom {r+5} 5 - \binom {r+2} 5$. Anand, Dumir, and Gupta conjectured that
- $H_n(r)$ is a polynomial in $r$ for any fixed $n$,
- this polynomial has roots at $-1, -2, \dots, -n+1$, and
- it satisfies the symmetry relation $H_n(-r) = (-1)^{ n-1 } H_n(r-n)$.
Stanley showed that one could use what is now called the *Elliott–MacMahon algorithm* (going back to [@elliott] and [@macmahon]; see also [@andrewspa1] and its dozen follow-up papers) in connection with the *Hilbert syzygy theorem* (see, e.g., [@eisenbudbook]) to prove the Anand–Dumir–Gupta conjecture.[^3] Stanley realized that $H_n(r)$ is closely related to the geometry of the *Birkhoff–von Neumann polytope* $\B_n$, the set of all doubly-stochastic $(n \times n)$-matrices. In modern language, $H_n(r)$ equals the Ehrhart polynomial of $\B_n$, and because $\B_n$ is *integrally closed* (essentially by the Birkhoff–von Neumann theorem that the extreme points of $\B_n$ are precisely the permutation matrices; see, e.g., [@brunsgubeladzektheory] for more on integrally closed polytopes), this Ehrhart series equals the *Hilbert series* of the semigroup algebra generated by the integer point in $\B_n \times \{ 1 \}$, graded by the last coordinate. Stanley proved that this Hilbert series has the following properties, which are a direct translation of the Anand–Dumir–Gupta conjecture into the language of generating functions:
\[thm:stanleysemimagic\] For each $n$ there exists a palindromic polynomial $h_n(x)$ of degree $n^2 - 3n + 2$ such that $$1 + \sum_{ r>0 } H_n(r) \, x^r = \frac{ h_n(x) }{ (1-x)^{ n^2 - 2n + 2 } } \, .$$
One could skip the detour through commutative algebra and directly realize this rational generating function as the Ehrhart series of $\B_n$, with the needed properties to affirm the Anand–Dumir–Gupta conjecture. Nevertheless, the algebraic detour is worth taking; aside from its inherent elegance, it allowed Stanley to realize that the monomials $z_1^{ m_1 } z_2^{ m_2 } \cdots
z_{n^2}^{ m_{n^2} }$, where $(m_1, m_2, \dots, m_{ n^2 }) \in \Z_{ \ge 0 }^{ n^2 }$ ranges over all semimagic squares, generate a *Cohen–Macaulay algebra*, and this implies, following Hochster’s work [@hochster]:
\[thm:stanleysemimagicintegral\] The polynomial $h_n(x)$ in Theorem \[thm:stanleysemimagic\] has nonnegative coefficients.
Stanley conjectured that the coefficients of $h_n(x)$ in Theorem \[thm:stanleysemimagic\] are also *unimodal* (the coefficients increase up to some point and then decrease), which was proved by Athanasiadis some three decades later [@athanasiadismagic]; we will say more about this in Section \[sec:beyond\].
Another problem connected to Theorem \[thm:stanleysemimagic\] (mentioned by Stanley but certainly older than his work) is still wide open, namely, the quest for the volume of $\B_n$, which equals $h_n(1)$ after a suitable normalization. This volume is known only for $n \le 10$, though there has been recent progress, e.g., in terms of asymptotic and combinatorial formulas [@beckpixton; @canfieldmckay; @deloeraliuyoshida].
It is worth noting that Stanley proved versions of Theorems \[thm:stanleysemimagic\] and \[thm:stanleysemimagicintegral\] for the more general *magic labellings* of graphs (and semimagic squares correspond to such labellings for a complete bipartite graph $K_{ nn }$). In this more general context, the associated counting functions become *quasipolynomials* of period 2 (see, e.g., [@stanleyec1 Chapter 4] for more about quasipolynomials), foreshadowing in some sense Zaslavsky’s work on enumerative properties of *signed graphs*: Stanley’s magic labellings are essentially flows on all-negative signed graphs [@nnz; @zaslavskyorientationsignedgraphs].
Stanley’s work on the Anand–Dumir–Gupta conjecture was not just the starting point of his contributions to Ehrhart theory. As he mentions in [@stanleyhowupperbound], it opened the door to what are now called *Stanley–Reisner rings* and the use of Cohen–Macaulay algebras in geometric combinatorics, famously leading to Stanley’s proof of the *upper bound conjecture* for spheres [@stanleyupperbound]. Stanley’s appreciation for the polynomials $H_n(r)$ is also evident in his writings: they are prominently featured in his influential books [@stanleycombcommalg; @stanleyec1]. We close this section by mentioning that a highly readable account of commutative-algebra concepts behind the Anand–Dumir–Gupta conjecture can be found in [@brunssemimagic].
Reciprocity {#sec:reciprocity}
===========
Theorem \[thm:macdon\] is an example of a *combinatorial reciprocity theorem*: we get interesting information out of a counting function when we evaluate it at a *negative* integer (and so, a priori the counting function does not make sense at this number). We remark that the last part of the Anand–Dumir–Gupta conjecture follows from Theorem \[thm:macdon\] applied to the Birkhoff–von Neumann polytope $\B_n$, as one can easily show that ${\operatorname{ehr}}_{ \B_n^\circ } (r) = H_n(r-n)$.
Stanley had discovered reciprocity theorems for $P$-partitions and order polynomials in his thesis [@stanleythesis], and so it was natural for him to realize Theorem \[thm:macdon\] as a special case of a wider phenomenon. A [*rational cone*]{} is a set of the form $\left\{ \x \in \R^d : \, \A \, \x \le \0 \right\}$ for some integer matrix $\A$. It is not hard to see that the multivariate generating function $$\sigma_\K \left( z_1, z_2, \dots, z_d \right) := \sum_{ \left( m_1, m_2, \dots, m_d \right) \in \K \cap \Z^d }
z_1^{ m_1 } z_2^{ m_2 } \cdots z_d^{ m_d }$$ evaluates to a rational function in $z_1, z_2, \dots, z_d$ if $\K$ is a rational cone (see, e.g., [@ccd Chapter 3]). Stanley proved that this rational function satisfies a reciprocity theorem.
\[thm:stanleyrec\] If $\K$ is a rational cone then $$\sigma_\K \left( \tfrac{ 1 }{ z_1 } , \tfrac{ 1 }{ z_2 } , \dots, \tfrac{ 1 }{ z_d } \right) = (-1)^{\dim(\K)} \, \sigma_{
\K^\circ } \left( z_1, z_2, \dots, z_d \right) .$$
A simple proof of Theorem \[thm:stanleyrec\], based on the ideas of [@bsstanleyirrational], can be found in [@ccd Chapter 4]. True to the theme of this survey, our description of Theorem \[thm:stanleyrec\] is geometric, while Stanley’s viewpoint presented in [@stanleyreciprocity] is based on integral solutions of a system of integral linear equations; in this language, the reciprocity is between *nonnegative* and *positive* solutions. (It is not hard to see that both viewpoints are equivalent.) This language also connects once more to the Elliott–MacMahon algorithm mentioned in Section \[sec:anand\], and Stanley uses this connection in [@stanleyreciprocity] to present his *monster reciprocity theorem*, which (a bit oversimplified) can be thought of an affine version of Theorem \[thm:stanleyrec\]. It has been recently revitalized by Xin [@xinmonsterrec].
We finish this section with a sketch how Theorem \[thm:macdon\] follows as a corollary of Theorem \[thm:stanleyrec\]. Given a lattice polytope $\P \subset \R^d$, we consider its [*homogenization*]{} $${\operatorname{cone}}(\P) := \sum_{ \v \text{ vertex of } \P } \R_{ \ge 0 } (\v, 1)$$ by lifting the vertices of $\P$ into $\R^{ d+1 }$ to the hyperplane $x_{ d+1 } = 1$ and taking the nonnegative span of this lifted version of $\P$. Thus (by the *Minkowski–Weyl theorem*—see, e.g., [@ziegler Lecture 1]) ${\operatorname{cone}}(\P)$ is a rational cone, and because ${\operatorname{cone}}(\P) \cap \{ \x \in \R^d : \, x_{ d+1 } = n \}$ is identical to $n \P$ (embedded in $\{ \x \in \R^d : \, x_{ d+1 } = n \}$), $${\operatorname{Ehr}}_\P(x) = \sigma_{ {\operatorname{cone}}(\P) } (1, 1, \dots, 1, x) \, .$$ Theorem \[thm:macdon\] follows now by specializing all but one variable in Theorem \[thm:stanleyrec\].
Ehrhart inequalities {#sec:inequ}
====================
Theorem \[thm:stanleysemimagicintegral\] generalizes to all lattice polytopes, and this is arguably Stanley’s most important contribution to the intrinsic study of Ehrhart polynomials.
\[thm:nonneg\] For any lattice polytope $\P$, the Ehrhart $h$-vector $\h_\P(x)$ has nonnegative coefficients.
Even though Stanley explicitly states this result first in [@stanleydecomp] (and gives a proof using a shelling triangulation argument), he attributes Theorem \[thm:nonneg\] to [@stanleymagic Proposition 4.5]—the magic-graph-labelling version of Theorem \[thm:stanleysemimagicintegral\]. Theorem \[thm:nonneg\] can be viewed as a starting point for the problem of classifying Ehrhart polynomials/Ehrhart $h$-vectors. This problem is wide open, already in dimension 3. (In dimension 2, it is essentially solved by Pick’s theorem [@pick] and an inequality of Scott [@scott].)
As part of his study of monotonicity of $h$-vectors of Cohen–Macaulay complexes, Stanley deduced the following refinement of Theorem \[thm:nonneg\].
\[thm:monoton\] If $\P \subseteq \Q$ are lattice polytopes then $\h_\P(x) \le \h_\Q(x)$ (component-wise).
Theorem \[thm:nonneg\] can be realized as a corollary of Theorem \[thm:monoton\] by choosing $\P$ to be a *unimodular simplex* (i.e., a $d$-dimensional lattice polytope of volume $\frac 1 {d!}$, which comes with the Ehrhart $h$-vector $\h_\P(x) = 1$).
One can show that Theorems \[thm:nonneg\] and \[thm:monoton\] also hold for *rational* $d$-polytopes $\P$ and $\Q$ (i.e., polytopes whose vertices have rational coordinates) if their Ehrhart series are written in the form $$\frac{ \h_{\P/\Q} (x) }{ (1-x^p)^{ d+1 } }$$ for some $p \in \Z_{ >0 }$ such that $p\P$ and $p\Q$ are lattice polytopes. (The accompanying Ehrhart counting functions for $\P$ and $\Q$ are then quasipolynomials, and $p$ is a period of them.) The arguably simplest proofs of (rational versions of) Theorems \[thm:nonneg\] and \[thm:monoton\] are given in [@bsstanleyirrational].
A natural question is whether there are any natural *upper* bounds complementing Theorem \[thm:nonneg\]. Of course, the volume (and therefore $\h_\P(1)$, the sum of the Ehrhart-$h$ coefficients) of a lattice polytope $\P$ can be arbitrarily large, but one can ask for upper bounds given certain data.
The volume of a lattice polytope $\P$ (and therefore also the coefficients of $\h_\P(x)$) is bounded by a number that depends only on the degree and the leading coefficient of $\h_\P(x)$.
This result was conjectured by Batyrev [@batyrevhvector] and improves a classic theorem of Lagarias and Ziegler [@lagariasziegler] that the volume of a lattice polytope is bounded by a number depending only on its dimension and the number of its interior lattice points, if the latter is positive.
Theorem \[thm:nonneg\] can be extended in a direction different from that of Theorem \[thm:monoton\], namely, one can establish inequalities *among* the coefficients of an Ehrhart $h$-vector. Stanley derived one set of such inequalities as a corollary of his study of Hilbert functions of semistandard graded Cohen–Macaulay domains [@stanleyinequ]; thus the following result holds in a more general situation. The [*degree*]{} of a lattice polytope $\P$ is the degree of its Ehrhart $h$-vector $\h_\P(x)$.
\[thm:stanlineq\] If $\P$ is a $d$-dimensional lattice polytope of degree $s$ then its Ehrhart $h$-vector $\h_\P(x) = \h_s x^s + \h_{ s-1 } x^{ s-1 } + \dots + \h_0$ satisfies $$\h_0 + \h_1 + \dots + \h_j \le \h_s + \h_{ s-1 } + \dots + \h_{ s-j } \quad \text{ for } \quad 0 \le j \le d.$$
(In the above theorem and below, we define $\h_j = 0$ whenever $j < 0$ or $j$ is larger than the degree of $\P$.) Theorem \[thm:stanlineq\] complements inequalities discovered by Hibi around the same time [@hibiehrhartineq; @hibilowerbound]; they were more recently improved by Stapledon. Together with Theorem \[thm:stanlineq\] and the trivial inequality $\h_1 \ge \h_d$ (which follows from the facts $\h_1 = \# (\P \cap \Z^d) - d - 1$ and $\h_d = \# (\P^\circ \cap \Z^d)$), the following result gives the state of the art in terms of linear constraints for the Ehrhart coefficients that can be easily written down in general.
\[thm:stapledonineq\] If $\P$ is a $d$-dimensional lattice polytope of degree $s$ and codegree $l := d+1-s$, then its Ehrhart $h$-vector $\h_\P(x) = \h_s x^s + \h_{ s-1 } x^{ s-1 } + \dots + \h_0$ satisfies $$\begin{aligned}
\h_2 + \h_3 + \dots + \h_{ j+1 } \ge \h_{ d-1 } + \h_{ d-2 } + \dots + \h_{ d-j } \quad &\text{ for } \quad 0
\le j \le \lfloor \tfrac d 2 \rfloor - 1, \\
\h_{ 2-l } + \h_{ 3-l } + \dots + \h_1 \le \h_j + \h_{ j-1 } + \dots + \h_{ j-l+1 } \quad &\text{ for } \quad 2 \le j \le d - 1. \end{aligned}$$
We will say more about this theorem in Section \[sec:beyond\] and finish this section with the remark that a unimodular $d$-simplex satisfies each of the inequalities of Theorems \[thm:stanlineq\] and \[thm:stapledonineq\] with equality.
Stanley & beyond {#sec:beyond}
================
The Ehrhart $h$-vector is philosophically close to the $h$-vector of a simplicial complex. This statement can be made much more precise, as we will show in this final section, in which we give a flavor of recent results that build on Stanley’s work in Ehrhart theory.
The starting point can once more be found in Stanley’s papers; the following result follows essentially from the definition $$h_T(z) := \sum_{ k=-1 }^{ d } f_k \, z^{ k+1} \, (1-z)^{ d-k }$$ of the [*$h$-vector*]{} of a given triangulation $T$ of a $d$-dimensional polytope (here $f_k$ denotes the number of $k$-simplices in $T$) and the fact that a unimodular simplex has a trivial Ehrhart $h$-vector (a triangulation is [*unimodular*]{} if all of its simplices are). Nevertheless, the following identity is an important base case for structural properties of Ehrhart $h$-vectors to come.
\[thm:unimodularehrharth\] If $\P$ is a lattice polytope that admits a unimodular triangulation $T$ then $${\operatorname{Ehr}}_\P(z) = \frac{ h_T(z) }{ (1-z)^{ \dim(\P) + 1 } } \, .$$ In words, the Ehrhart $\h$-vector of $\P$ is given by the $h$-vector of $T$.
The hope is now to use properties of the $h$-vector of a triangulation to say something about Ehrhart $h$-vectors; for example, if $T$ is the cone over a boundary triangulation of $\P$ then $h_T(z)$ satisfies the *Dehn–Sommerville equations* (see, e.g., [@deloerarambausantos] for more about triangulations). Unfortunately, not all lattice polytopes admit unimodular triangulations in dimension $\ge 3$—in fact, most do not—and so Theorem \[thm:unimodularehrharth\] needs some tweaking before we can apply it to general lattice polytopes. This tweaking, due to Betke and McMullen [@betkemcmullen], has two main ingredients: the [*link*]{} of a simplex $\Delta$ in a triangulation $T$ $${\operatorname{link}}(\Delta) := \left\{ \Omega \in T : \, \Omega \cap \Delta = {\varnothing}, \ \Omega \subseteq \Phi \text{ for some } \Phi \in T \text{ with } \Delta \subseteq \Phi
\right\} ,$$ and its *box polynomial* $$B_\Delta(x) := \sum_{ \m \in \Pi(\Delta) \cap \Z^{ d+1 } } x^{ {\operatorname{height}}(\m) }$$ where we define $$\Pi(\Delta) := \left\{ \sum_{ \v \text{ vertex of } \Delta } \!\!\!\!\lambda_\v (\v, 1) : \, 0 < \lambda_\v < 1 \right\}$$ and ${\operatorname{height}}(\m)$ denotes the last coordinate of $\m$. (Geometrically, $\Pi(\Delta)$ is the open fundamental parallelepiped of ${\operatorname{cone}}(\Delta)$.) For the empty simplex ${\varnothing}$ of a triangulation, we set $B_{\varnothing}(x) = 1$.
\[thm:betkemcmullen\] Fix a triangulation $T$ of the lattice polytope $\P$. Then $$\h_\P(x) = \sum_{ \Delta \in T } h_{ {\operatorname{link}}(\Delta) } (x) \, B_\Delta(x) \, .$$
If a simplex $\Delta \in T$ is unimodular then $B_\Delta(x) = 0$, unless $\Delta = {\varnothing}$. Thus, if $T$ is unimodular then the sum in Theorem \[thm:betkemcmullen\] collapses to $h_{ {\operatorname{link}}({\varnothing}) } (x) \, B_{\varnothing}(x) = h_T(x)$, and so Theorem \[thm:unimodularehrharth\] is a corollary to Theorem \[thm:betkemcmullen\]. (This argument also shows, in general, that $\h_\P(x) \ge h_T(x)$ component-wise.) Furthermore, since all ingredients for the sum in Theorem \[thm:betkemcmullen\] are nonnegative, this gives another (and the first combinatorial) proof of Theorem \[thm:nonneg\].
Theorem \[thm:betkemcmullen\] was greatly extended by Stanley in (and served as some motivation to) his work on local $h$-vectors of subdivisions [@stanleylocalhvectors]; see Athanasiadis’ contribution to this volume. Payne gave a different, multivariate generalization of Theorem \[thm:betkemcmullen\] in [@payneehrharttriang].
Theorem \[thm:betkemcmullen\] has a powerful consequence when $\P$ has an interior lattice point; this consequence was fully realized only by Stapledon [@stapledondelta] who extended it to general lattice polytopes—Theorem \[thm:stapledonab\] below. Namely, if a lattice polytope $\P$ has an interior lattice point, it admits a regular triangulation that is a cone (at this point) over a boundary triangulation. This has the charming effect that each $h_{ {\operatorname{link}}(\Delta) } (x)$ appearing in Theorem \[thm:betkemcmullen\] is palindromic (due to the afore-mentioned Dehn–Sommerville equations). Since the box polynomials are palindromic and both kinds of polynomials have nonnegative coefficients, a little massaging of the identity in Theorem \[thm:betkemcmullen\] gives:
\[cor:betkemcmullen\] Suppose $\P$ is a $d$-dimensional lattice polytope that contains an interior lattice point. Then there exist unique polynomials $a(x)$ and $b(x)$ with nonnegative coefficients such that $$\h_\P(x) = a(x) + x \, b(x) \, ,$$ $a(x) = x^d \, a(\frac 1 x)$, and $b(x) = x^{ d-1 } \, b(\frac 1 x)$.
The identities for $a(x)$ and $b(x)$ say that $a(x)$ and $b(x)$ are palindromic polynomials; the degree of $a(x)$ is necessarily $d$, while the degree of $b(x)$ is $d-1$ or smaller; in fact, $b(x)$ can be zero—this happens if and only if $\P$ is the translate of a *reflexive* polytope (i.e., a lattice polytope whose dual is also a lattice polytope), due to Theorem \[thm:hibi\] below. Stapledon recently introduced a weighted variant of the Ehrhart $\h$-vector which is *always* palindromic, motivated by motivic integration and the cohomology of certain toric varieties [@stapledonweightedehrart]. One can easily recover $\h_\P(x)$ from this weighted Ehrhart $\h$-vector, but one can also deduce the palindromy of both $a(x)$ and $b(x)$ as coming from the same source (and this perspective has some serious geometric applications).
The statements that $a(x)$ and $b(x)$ in Corollary \[cor:betkemcmullen\] have nonnegative coefficients are straightforward translations of Hibi’s and Stanley’s inequalities on Ehrhart $h$-vectors mentioned above (right after and in Theorem \[thm:stanlineq\]), in the case that the dimension and the degree of $\P$ are equal (which is equivalent to $\P$ containing an interior lattice point). The full generality of Theorem \[thm:stanlineq\] as well as Theorem \[thm:stapledonineq\] follow from the following generalization of Corollary \[cor:betkemcmullen\].
\[thm:stapledonab\] Suppose $\P$ is a $d$-dimensional lattice polytope of degree $s$ and codegree $l = d+1-s$. Then there exist unique polynomials $a(x)$ and $b(x)$ with nonnegative coefficients such that $$\left( 1 + x + \dots + x^{ l-1 } \right) \h_\P(x) = a(x) + x^l \, b(x) \, ,$$ $a(x) = x^d \, a(\frac 1 x)$, $b(x) = x^{ d-l } \, b(\frac 1 x)$, and, writing $a(x) = a_d x^d + a_{ d-1 } x^{ d-1 } + \dots + a_0$, $$1 = a_0 \le a_1 \le a_j \quad \text{ for } \quad 2 \le j \le d-1.$$
Stapledon has recently improved this theorem further, giving infinitely many classes of linear inequalities among Ehrhart-$h$ coefficients [@stapledonadditive]. This exciting new line of research involves additional techniques from additive number theory.
One can, on the other hand, ask which classes of polytopes satisfy more special sets of equalities or inequalities. Arguably the most natural such equalities/inequalities are those expressing palindromy and unimodality.
Lattice polytopes with palindromic Ehrhart $h$-vectors are completely classified by the following theorem, which first explicitly surfaced in Hibi’s work on reflexive polytopes but can be traced back to Stanley’s work on Hilbert functions of Gorenstein rings.
\[thm:hibi\] If $\P$ is a lattice polytope of degree $s$ and codegree $l = d+1-s$, then its Ehrhart $h$-vector is palindromic if and only if $l \P$ is a translate of a reflexive polytope.
The following result is a start towards a unimodality classification; it was proved by Athanasiadis [@athanasiadishstareulerian] and independently by Hibi and Stanley (unpublished).
\[thm:athan\] If the $d$-dimensional lattice polytope $\P$ admits a regular unimodular triangulation, then $$\h_{ \lfloor \frac{ d+1 }{ 2 } \rfloor } \ge \dots \ge \h_{ d-1 } \ge \h_d$$ and $$\h_j \le \binom{ \h_1 + j - 1 }{ j } \quad \text{ for } \quad 0 \le j \le d.$$
Stapledon’s work in [@stapledondelta] implies further that if the *boundary* of $\P$ admits a regular unimodular triangulation, then $$\h_{ j+1 } \ge \h_{ d-j } \quad \text{ for } \quad 0 \le j \le \lfloor \tfrac d 2 \rfloor - 1$$ (which was also proved in [@athanasiadishstareulerian] under the stronger assumption that $\P$ admits a regular unimodular triangulation) and $$\h_0 + \dots + \h_{ j+1 } \le \h_d + \dots + \h_{ d-j } + \binom{ \h_1 - \h_d + j + 1 }{ j+1 } \quad \text{ for } \quad 0 \le j \le \lfloor \tfrac d 2 \rfloor - 1.$$ Naturally, if in addition to the conditions in Theorem \[thm:athan\], $\P$ has degree $d$ and $\h_\P(x)$ is palindromic, then $\h_\P(x)$ is unimodal. The proof of Theorem \[thm:athan\] starts with Theorem \[thm:unimodularehrharth\] and then shows that the $h$-vector of the unimodular triangulation satisfies the stated inequalities. Athanasiadis’ approach was inspired by work of Reiner and Welker on order polytopes of graded posets and a connection between the *Charney–Davis* and *Neggers–Stanley conjectures* [@reinerwelker] and can be taken further: Athanasiadis used similar methods to prove Stanley’s conjecture mentioned in Section \[sec:anand\] that the Ehrhart $h$-vector of the Birkhoff–von Neumann polytope is unimodal [@athanasiadismagic]. Bruns and Römer generalized this to any Gorenstein polytope that admits a regular unimodular triangulation [@brunsroemer].
Going into a somewhat different direction, Schepers and van Langenhoven recently proved that lattice parallelepipeds have a unimodal Ehrhart $h$-vector [@schepersvanlangenhoven]. The conjecture that any integrally closed polytope (of which both parallelepipeds and the Birkhoff–von Neumann polytope are examples) has a unimodal Ehrhart $h$-vector remains open, even for integrally closed reflexive polytopes (though recent work of Braun and Davis give some pointers of what could be tried here [@braundavis]); this is closely related to a conjecture of Stanley that every standard graded Cohen–Macaulay domain has a unimodal $h$-vector [@stanleylogconcave].
[^1]: We thank Ben Braun, Martin Henk, Richard Stanley, and Alan Stapledon for helpful discussions and suggestions. This work was partially supported by the U. S. National Science Foundation (DMS-1162638).
[^2]: The Ehrhart $h$-vector is also known by the names of *$\h$-vector/polynomial* and *$\delta$-vector/polynomial*.
[^3]: Stanley’s comment in [@stanleyhowupperbound p. 6] is amusing in this context: “I had taken a course in graduate school on commutative algebra that I did not find very interesting. It did not cover the Hilbert syzygy theorem. I had to learn quite a bit of commutative algebra from scratch in order to understand the work of Hilbert.”
|
---
abstract: 'We present a quantum theory of light based on quantum cellular automata (QCA). This approach allows us to have a thorough quantum theory of free electrodynamics encompassing an hypothetical discrete Planck scale. The theory is particularly relevant because it provides predictions at the macroscopic scale that can be experimentally tested. We show how, in the limit of small wave-vector $\bk$, the free Maxwell’s equations emerge from two Weyl QCAs derived from informational principles in Ref. [@d2013derivation]. Within this framework the photon is introduced as a composite particle made of a pair of correlated massless Fermions, and the usual Bosonic statistics is recovered in the low photon density limit. We derive the main phenomenological features of the theory, consisting in dispersive propagation in vacuum, the occurrence of a small longitudinal polarization, and a saturation effect originated by the Fermionic nature of the photon. We then discuss whether these effects can be experimentally tested, and observe that only the dispersive effects are accessible with current technology, from observations of arrival times of pulses originated at cosmological distances.'
author:
- Alessandro
- Giacomo Mauro
- Paolo
bibliography:
- 'bibliography.bib'
title: Quantum Cellular Automaton Theory of Light
---
Introduction
============
The Quantum Cellular Automaton (QCA) is the quantum version of the popular cellular automaton of von Neumann [@neumann1966theory]. It describes the finite evolution of a discrete set of quantum systems, each one interacting with a finite number of neighbors via the unitary transformation of a single step evolution. The idea of a quantum version of a cellular automaton was already contained in the early work of Feynman [@feynman1982simulating], and later has been object of investigation in the quantum-information community [@schumacher2004reversible; @arrighi2011unitarity; @gross2012index], with special enphasis on the so-called Quantum Walks (QW) which decribes the one particle sector of QCA’s with evolution linear in a quantum field [@grossing1988quantum; @succi1993lattice; @meyer1996quantum; @bialynicki1994weyl; @ambainis2001one].
The interest in QCAs is motivated by their potential applications in several fields, like the statistical mechanics of lattice systems and the quantum computation with microtraps [@cirac2000scalable] and with optical lattices [@bloch2004quantum]. Moreover, Quantum Walks have been used in the design of new quantum algorithms with a computational speed-up [@childs2003exponential; @farhi2007quantum].
Recently, the idea that QCA could be used to describe a more fundamental discrete Plank scale dynamics from which the usual Quantum Field Theory emerges [@darianopla; @BDTqcaI; @d2013derivation], is gathering increasing attention [@farrelly2014causal; @arrighi2013dirac]. The proposal of modeling Planck scale physics with a classical automaton on a discrete background first appeared in the work of ’t Hooft [@t1990quantization], and Quantum Walks were considered for the simulation of Lorentz-covariant differential equations in Refs. [@succi1993lattice; @bialynicki1994weyl; @meyer1996quantum; @PhysRevA.73.054302; @Yepez:2006p4406].
Up to now, most of the interest was focused on the emergence of the Dirac equation for a free Fermionic field. The choice of cosidering Fermions as the elementary physical systems is motivated by the idea that the amount of information that can be stored in a finite volume must be finite, as also suggested by black hole physics [@bekenstein1973black; @hawking1975particle]. However, the question whether a Fermionic QCA could recover the dynamics of a Bosonic field was never addressed before. Here we will see how free electrodynamics emerges from two Weyl QCAs [@d2013derivation] with Fermionic fields. The dynamical equations resulting in the limit of small wavevector $\bk$ are the Maxwell’s equations. However, for high value of $\bk$ the discreteness of the Planck scale manifests itself, producing deviations from Maxwell. Most notably, the QCA dynamics introduces a $\bk$-dependent speed of light, a feature that was already considered in some approaches to quantum gravity, and that could be in principle experimentally detected in astrophysical observations [@ellis1992string; @lukierski1995classical; @Quantidischooft1996; @amelino1998tests; @amelino2001testable; @amelino2001planck; @PhysRevLett.88.190403; @PhysRevLett.96.051301; @ellis2013probes].
In the present approach the photon turns out to be a composite particle made of a pair of correlated massless Fermions. This scenario closely resembles the neutrino theory of light of De Broglie [@de1934nouvelle; @jordan1935neutrinotheorie; @kronig1936relativistically; @perkins1972statistics; @perkins2002quasibosons] which suggested that the photon could be composed of a neutrino-antineutrino pair bound by some interaction. The failure of the neutrino theory of light was determined by the fact that a composite particle cannot obey the exact Bosonic commutation relations [@pryce1938neutrino]. However, as it was shown in Ref. [@perkins2002quasibosons], the non-Bosonic terms introduce negligible contribution at ordinary energy densities. In our case, as a consequence of the composite nature of the photon, we have that the number of photons that can occupy a single mode is bounded. However, as we will see, a saturation effect originated by the Fermionic nature of the photon is far beyond the current laser technology.
In Section \[sec:quant-cell-autom\], after recalling some basic notions about the QCA,we review the Weyl automaton of Ref. [@d2013derivation]. In Section \[s:Maxw\] we build a set of Fermionic bilinear operators, which in Sect. \[sec:recov-maxw-dynam\] are proved to evolve according to the Maxwell equations. In Section \[sec:photons-as-composite\] we will show that the polarization operators introduced in Sect. \[sec:recov-maxw-dynam\] can be considered as Bosonic operators in a low energy density regime. As a spin-off of this analysis we found a result that completes the proof, given in Ref. [@PhysRevLett.104.070402], that the amount of entanglement quantifies whether pairs of Fermions can be considered as independent Bosons. Section \[sec:phen-analys\] presents the phenomenological consequences of the present QCA theory, the most relevant one being the the appearence of a $\bk$-dependent speed of light. In the same section we discuss possible experimental tests of such $\bk$-dependence in the astrophysical domain, and we compare our result with those from Quantum Gravity literature [@ellis1992string; @lukierski1995classical; @Quantidischooft1996; @amelino1998tests; @amelino2001testable; @amelino2001planck; @PhysRevLett.88.190403; @PhysRevLett.96.051301; @ellis2013probes]. We conclude with Section \[sec:conclusions\] where we review the main results and discuss future developments.
The Weyl automaton: a review {#sec:quant-cell-autom}
============================
The basic ingredient of the Maxwell automaton is Weyl’s, whis has been derived in Ref. [@d2013derivation] from first principles. Here, we will briefly review the construction for completeness.
A QCA represents the evolution of a numerable set $G$ of cells $g\in G$, each one containing an array of Fermionic local modes. The evolution occurs in discrete identical steps, and in each one every cell interacts with a the others. The Weyl automaton is derived from the following principles: unitarity, linearity, locality, homogeneity, transitivity, and isotropy. Unitarity means just that each step is a unitary evolution. Linearity means that the unitary evolution is linear in the field. Locality means that at each step every cell interacts with a finite number of others. We call cells interacting in one step [*neighbors*]{}. The neighboring notion also naturally defines a graph over the automaton, with $g$ as vertices and the neighboring couples as edges. Homogeneity means both that all steps are the same, all cells are identical systems, and the set of interactions with neigbours is the same for each cell, hence also the number of neigbours, and the dimension of the cell field array, which we will denote by $s>0$. We will denote by $A$ the matrix representing the linear unitary step. Transitivity means that every two cells are connected by a path of neighbours. Isotropy means that the neighboring relation is symmetric, and there exists a group of automorphisms for the graph for which the automaton itself is covariant. Homogeneity, transitivity, and isotropy together imply that $G$ is a group, and the graph is a Cayley graph $\Gamma(G,S_+)$ where $G=\<S_+|R\>$ is a presentation of $G$ with generator set $S_+$ and relator set $R$. The set of neighboring cells is then given by $S:=S_+\cup S_-$ where $S_-$ is the set of the inverse generators. Linearity, locality, and homogeneity imply that each step can be described in terms of transition matrices $A_h\in\rm{M}(\mathbb{C},s)$ for each $h\in S$, and then the step is described mathematically as follows $$\begin{aligned}
\label{eq:automagraph}
\psi_g(t+1) = \sum_{h\in S}A_h\psi_{hg}(t)\end{aligned}$$ where $\psi_g(t)$ is the $s$-array of field operators at $g$ at step $t$. Therefore, upon denoting by $T_g$ $g\in G$ the unitary representation of $G$ on $\ell^2(G)$, $T_g|f\>:=|gf\>$, for $f\in G$, $A$ is a unitary operator on $\ell^2(G)\otimes\mathbb{C}^s$ of the form $$\begin{aligned}
A:=\sum_{h\in S}T_h\otimes A_h.
\label{eq:walk}\end{aligned}$$ Covariance of the isotropy property means precisely that the group $L$ of automorphisms of the graph is a transitive permutation group of $S_+$, and there exists a (generally projective) unitary representation $U_l$ $l\in L$ of $L$ such that $$\begin{aligned}
A=\sum_{h\in S}T_{lh}\otimes U_l A_{h}U_l^\dag,\qquad \forall l\in L.
\label{eq:iso}\end{aligned}$$
In Ref. [@d2013derivation] attention was restricted to group $G$ quasi-isometrically embeddable in an Euclidean space, which is then [*virtually Abelian*]{} [@Cornulier07], namely it has an Abelian subgroup $G'\subset G$ of finite index, namely with a finite number of cosets. Then it can be shown the automaton is equivalent to another one with group $G'$ and dimension $s'$ multiple of $s$. We further assume that the representation of the isotropy group $L$ induced by the embedding is orthogonal, which implies that the graph neighborhood is embedded in a sphere. We call such a property [*orthogonal isotropy*]{}.
For $s=1$ the automaton is trivial, namely $A=I$. For $s=2$ and for Euclidean space $\mathbb R^3$ one has $G=\mathbb Z^3$, and the Cayley graphs satisfying orthogonal isotropy are the Bravais lattices. The only lattice that has a nontrivial set of transition matrices giving a unitary automaton is the BCC lattice. We will label the group element as vectors $\bx\in\mathbb{Z}^3$, and use the customary additive notation for the group composition, whereas the unitary representation of $\mathbb{Z}^3$ is expressed as follows $$T_{\bvec z}|\bvec x\>=|\bvec z+\bvec x\>.$$ Being the group Abelian, we can Fourier transform, and the operator $A$ can be easily block-diagonalized in the $\bk$ representation as follows $$\begin{aligned}
\label{eq:weylautomata}
A = \int_B\operatorname d^3 \! \bk \, |{\bk}\>\< {\bk}| \otimes A_{\bk}\end{aligned}$$ with $A_\bk:=\sum_{\bvec h\in S}\bvec k\,e^{-i\bvec k\cdot\bvec h}A_\bh$ unitary for every $\bk\in
B$, and the vectors $|{\bk}\>$ given by $$|\bk\>:=\frac1{\sqrt{2\pi}^3}\sum_{\bx\in G}e^{i\bk\cdot\bx}|\bx\>,$$ is a Dirac-notation for the direct integral over $\bk$, and the domain $B$ is the first Brillouin zone of the BCC. There are only two QCAs, with unitary matrices $$\label{eq:weylautomata2}
A^{\pm}_{\bk} := d^{\pm}_{\bk} I+\tilde{\bn}^{\pm}_{\bk}\cdot\boldsymbol{\sigma}
=\exp[-i\bvec{n}^{\pm}_{\bk} \cdot \boldsymbol{\sigma}],$$ where $$\begin{aligned}
&\tilde{\bn}^{\pm}_{\bk} :=
\begin{pmatrix}
s_x c_y c_z \mp c_x s_y s_z\\
\mp c_x s_y c_z - s_x c_y s_z\\
c_x c_y s_z \mp s_x s_y c_z
\end{pmatrix}\!\!,\,
{\bn}^{\pm}_{\bk}:=\frac{\lambda^{\pm}_{\bk}\tilde{\bn}^{\pm}_{\bk}}{\sin\lambda^{\pm}_{\bk}},\nonumber\\
&\d^{\pm}_{\bk} := (c_x c_y c_z \pm s_x s_y s_z ),\;
\lambda^{\pm}_{\bk}:=\arccos(d^{\pm}_{\bk}),\nonumber\end{aligned}$$ and $$c_\alpha := \cos({k}_\alpha/\sqrt{3}),\;s_\alpha:= \sin({k}_\alpha/\sqrt{3}),\;\alpha = x,y,z.\nonumber$$ The matrices $A_{\bk}^\pm$ in Eq. describe the evolution of a two-component Fermionic field, $$\begin{aligned}
\label{eq:automa1}
{\psi} ({\bk},t+1) =
A_{\bk}^\pm {\psi} ({\bk},t),
\quad
{\psi} ({\bk},t) : =
\begin{pmatrix}
{\psi}_R ({\bk},t)\\
{\psi}_L ({\bk},t)
\end{pmatrix}.
\end{aligned}$$ The adimensional framework of the automaton corresponds to measure everything in Planck units. In such a case the limit $|{\bk}|\ll 1$ corresponds to the relativistic limit, where on has $$\bn^{\pm}({\bk})\sim\tfrac{{\bk}}{\sqrt{3}},\quad A^{\pm}_{\bk}\sim\exp[-i\tfrac{{\bk}}{\sqrt{3}} \cdot\boldsymbol{\sigma}],$$ corresponding to the Weyl’s evolution, with $\tfrac{{\bk}}{\sqrt{3}}$ playing the role of momentum.
The Maxwell automaton {#s:Maxw}
=====================
In order to build the Maxwell dynamics, we need to consider two different Weyl QCAs the first one acting on a Fermionic field $\psi(\bk)$ by matrix $A_\bk$ as in Eq. (\[eq:automa1\]), and the second one acting on the field $\varphi(\bk)$ by the complex conjugate matrix $A_\bk^*=\sigma_y
A_\bk\sigma_y$, i.e. $$\begin{aligned}
\label{eq:automa2}
{\varphi} (\bk,t+1) = A_\bk^*{\varphi} (\bk,t), \quad
{\varphi} (\bk,t) =
\begin{pmatrix}
{\varphi}_R (\bk,t)\\
{\varphi}_L (\bk,t)
\end{pmatrix}.\end{aligned}$$ The matrix $A_\bk$ can be either one of the Weyl matrices $A^\pm_{\bk}$, and the whole derivation is independent of the choice.
The Fermionic fields ${\varphi}$ and ${\psi}$ are independent and obey the following anti-commutation relations $$\begin{aligned}
&[\psi_i(\bk),\psi_j(\bk') ]_+ =
[\varphi_i(\bk),\varphi_j(\bk') ]_+ =\nonumber\\
&[\varphi_i(\bk),\psi_j(\bk') ]_+=
[\varphi_i(\bk),\psi^\dagger_j(\bk') ]_+
=0
\nonumber \\
&[\psi_i(\bk),\psi^\dagger_j(\bk') ]_+ =
[\varphi_i(\bk),\varphi^\dagger_j(\bk') ]_+
=
\delta_B(\bk-\bk')\delta_{i,j} \nonumber \\
&i,j = R,L \qquad \bk,\bk' \in B,
\label{eq:commutationrel}\end{aligned}$$ where $\delta_B(\bk)$ is the 3d Dirac’s comb delta-distribution (which repeats periodically with $\mathbb R^3$ tasselated into Brillouin zones).
Given now two arbitrary fields ${\eta}(\bk)$ and ${\theta}(\bk)$ we define the following bilinear function $$\begin{aligned}
\label{eq:gimu}
\!\! G_f^{\mu}(\eta,\theta,\bk) := \!\!
\int\!\!\frac{\d\bq}{(2\pi)^3} f_{\bk}(\bq)
{{\eta}}^T
\left(\tfrac{\bk}{2}-\bq\right)
\sigma^{\mu}
{\theta}
\left(\tfrac{\bk}{2}+\bq\right) \end{aligned}$$ where $\sigma^0:= I $, $\sigma^1:= \sigma^x$, $\sigma^2:= \sigma^y $, $\sigma^3:=\sigma^z$ and $\int\frac{\d\bq}{(2\pi)^3} |f_{\bk}(\bq)|^2 =1, \forall \bk$. In the following we will also treat the vector part $\boldsymbol\sigma:=(\sigma^1,\sigma^2,\sigma^3)$ of the four-vector $\sigma^\mu$ separately. This allows us to define the following operators $$\begin{aligned}
\label{eq:bilinears}
%\begin{split}
F^{\mu}(\bk) :=G_f^\mu(\varphi,\psi,\bk)
% =
%\int
%{\!\! \d \! m_{\bq}(\bk) \,}
%{{\psi}}^T
%\left(\tfrac{\bk}{2}-\bq\right)
%\sigma^{\mu}
%{\varphi}
% \left(\tfrac{\bk}{2}+\bq\right) \\
%\end{split}\end{aligned}$$
In the following sections we study the evolution of the bilinear functions $F^{\mu}(\bk)$ and their commutation relations and show that, in the relativistic limit and for small particle densities the quantum Maxwell equations are recovered for both choices of $A_\bk=A^\pm_\bk$.
The Maxwell dynamics {#sec:recov-maxw-dynam}
====================
In the following we will use the short notations $$\label{eq:notaz}
[Z\eta](\bk):=Z_{\bk}\eta(\bk),\quad [ZW]_\bk:=Z_{\bk}W_\bk,$$ for $\eta$ a field and $Z$ and $W$ matrices. If the fields $\psi$ and $\varphi$ evolve according to Eqs. and , then the evolution of the bilinear functions $F^{\mu}(\bk)$ introduced in Eq. obeys the following equation $$\begin{aligned}
\label{eq:exactevolution}
&F^{\mu}(\bk,t) = G_f^\mu([{A^*}^t\varphi],[A^t\psi],\bk),\end{aligned}$$ where we used the notation in (\[eq:notaz\]). Now, let us define $$\begin{aligned}
&\tilde F^\mu(\bk,t):= G_f^\mu([{U^{\bk,t}}^*\varphi],[U^{\bk,t}\psi],\bk), \nonumber\\
&U^{\bk,t}_\bq :=A^{-t}_{\tfrac{\bk}2} A^t_\bq,
\label{eq:defU}\end{aligned}$$ where we remind that $[{U^{\bk,t}}^*\varphi](\bq):={U_\bq^{\bk,t}}^*\varphi(\bq)$. Clearly, one has $[A^t\eta]=[A_{\frac{\bk}{2}}^tU^{\bk,t}\eta]$. We now need the identity $$\begin{aligned}
&\exp (-\tfrac{i}{2}\bvec{v}\cdot \boldsymbol{\sigma})
\boldsymbol{\sigma} \exp (\tfrac{i}{2}\bvec{v} \cdot \boldsymbol{\sigma}) =
\Exp(-i\bvec{v} \cdot \bvec{J}) \boldsymbol{\sigma},\nonumber\\
&\exp (-\tfrac{i}{2}\bvec{v}\cdot \boldsymbol{\sigma})
\sigma^0 \exp (\tfrac{i}{2}\bvec{v} \cdot \boldsymbol{\sigma}) =\sigma^0,\end{aligned}$$ where the matrix $\Exp(-i\bvec{v}\cdot\bvec{J})$ acts on $\boldsymbol{\sigma}$ regarded as a vector, and $\bvec J=(J_x, J_y,J_z)$ is the vector of angular momentum operators. We can then recast Eq. in terms of the following functions $$\begin{aligned}
\bvec{F}(\bk,t) &:=
(
F^{1}(\bk,t), F^{2}(\bk,t), F^{3}(\bk,t)
) ^T, \label{eq:ftilde}\end{aligned}$$ and $\tilde{\bvec{F}}(\bk,t)$ similarly defined, obtaining $$\begin{aligned}
\label{eq:evolutionwithrotation}
& F^{0}(\bk,t) =
\tilde{F}^{0}(\bk,t), \nonumber\\
& \bvec{F}(\bk,t) =
\Exp\left(-2i {\bvec{n}}_{\tfrac{\bk}{2}} \cdot
\bvec{J}t\right)
\tilde{\bvec{F}}(\bk,t). \end{aligned}$$ If we assume that $$\begin{aligned}
\label{eq:approxf}
\int_{|\bq| \geq \bar{q}(\bk)}\frac{\d\bq}{(2\pi)^3} |f_{\bk}(\bq)|^2
\ll 1 \quad\mbox{for}\ \bar{q}(\bk) \ll |\bk|, \end{aligned}$$ by taking the Taylor expansion of ${\bvec{n}}_{\tfrac{\bk}{2}+\bq}$ with respect to $\bq$ we can make the approximation $$\begin{aligned}
{U}^{\bk,t}_{\tfrac{\bk}2\pm\bq}&\simeq \exp\left(i
{\bn}_{\tfrac{\bk}{2}} \cdot \boldsymbol{\sigma}t\right)
\exp\left[-i\left({\bn}_{\tfrac{\bk}{2}}
\pm\bvec{l}_{\bk,\bq}\right) \cdot
\boldsymbol{\sigma}t\right] \nonumber \\
&
\simeq
\exp \left(\pm
i c_{\bk,\bq}
\frac{{\bvec{n}}_{\frac{\bk}{2}}}{|{\bvec{n}}_{\frac{\bk}{2}}|} \cdot \boldsymbol{\sigma}
t\right)+ O \big( \tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|}
\big) \label{eq:approxU0}
,\end{aligned}$$ where $\bvec{l}_{\bk, \bq} := J_{{\bvec{n}}}\left(\frac{\bk}{2}\right)\bq $ and $J_{{\bvec{n}}}\left(\frac{\bk}{2}\right)$ denotes the Jacobian matrix of the function $\bvec{n}_\bk$ evaluated at $\frac{\bk}{2}$ and $c_{\bk,\bq} :=
\frac{{\bvec{n}}_{\frac{\bk}{2}}}{|{\bvec{n}}_{\frac{\bk}{2}}|} \cdot
\bvec{l}_{\bk, \bq}$ (the proof of Eq. \[eq:approxU0\] is given in Appendix \[sec:proof-eq.-eqref\]). By introducing the transverse field operators $$\begin{aligned}
\label{eq:transverse}
\begin{split}
\tilde{\bvec{F}}_T(\bk,t) :=\tilde{\bvec{F}}(\bk,t)-
\left(\frac{\bvec{n}_{\frac{\bk}{2}}}{|\bvec{n}_{\frac{\bk}{2}}|} \cdot
\tilde{\bvec{F}}(\bk,t) \right)
\frac{\bvec{n}_{\frac{\bk}{2}}}{|\bvec{n}_{\frac{\bk}{2}}|} \\
\bvec{F}_T(\bk,t) := \bvec{F}(\bk,t) -
\left(\frac{\bvec{n}_{\frac{\bk}{2}}}{|\bvec{n}_{\frac{\bk}{2}}|} \cdot
{\bvec{F}}(\bk,t) \right)
\frac{\bvec{n}_{\frac{\bk}{2}}}{|\bvec{n}_{\frac{\bk}{2}}|}.
\end{split}\end{aligned}$$ and using Eq. into Eq. we get (see Appendix \[sec:proof-eq.-eqref-1\]) $$\begin{aligned}
\label{eq:transverse2}
\begin{split}
\tilde{\bvec{F}}_T(\bk,t) =
{\bvec{F}}_T(\bk)
+
O \big( \tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big).
\end{split}\end{aligned}$$ Finally, combining Eq. with Eq. we obtain a closed expression for the time evolution of the operator ${\bvec{F}_T}(\bk)$, $$\begin{aligned}
\begin{split}
\label{eq:maxwell}
\bvec{F}_T(\bk,t) =
\exp\left[\left(2 \bvec{n}_{\tfrac{\bk}{2}} \cdot
\bvec{J}\right)t\right]
{\bvec{F}_T}(\bk) +\Lambda(\bk,t),
\end{split}\end{aligned}$$ where $\|\Lambda(\bk,t)\|= O \big( \tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big)$. Taking the time derivative in Eq. and reminding the definition we obtain $$\begin{aligned}
\begin{split} \label{eq:maxwell2}
&\partial_t\bvec{F}_T(\bk,t) =
2\bvec{n}_{\frac{\bk}{2}} \times \bvec{F}_T(\bk,t)+
\partial_t \Lambda(\bk,t)\\
&2\bvec{n}_{\frac{\bk}{2}} \cdot \bvec{F}_T(\bk,t) = 0,
\end{split}\end{aligned}$$ where $\|\partial_t \Lambda(\bk,t)\|=O \big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big)$ (see Appendix \[sec:proof-eq.-eqref-1\]).
Let now $\bvec{E}$ and $\bvec{B}$ be two Hermitian operators defined by the relation $$\begin{aligned}
\label{eq:electric and magnetic field}
&\bvec{E}:=|{\bn}_{\tfrac\bk2}|(\bvec{F}_T+\bvec{F}_T^\dag),\quad\bvec{B}:=i|{\bn}_{\tfrac\bk2}|(\bvec{F}_T^\dag-\bvec{F}_T),\nonumber\\
&2|{\bn}_{\tfrac\bk2}|\bvec{F}_T=\bvec{E} + i \bvec{B}.\end{aligned}$$ We now show that in the limit of small wavevectors $\bk$ and by interpreting $ \bvec{E}$ and $ \bvec{B} $ as the electric and magnetic field the usual vacuum Maxwell’s equations can be recovered. For $|\bk| \ll 1$ one has $2\bvec{n}_{\frac{\bk}{2}} \simeq \bk/\sqrt3$, and Eq. becomes $$\begin{aligned}
\begin{split} \label{eq:maxwell3}
&\partial_t\bvec{F}_T(\bk,t) =
\frac{\bk}{\sqrt3} \times \bvec{F}_T(\bk,t)
\\
&\bk\cdot \bvec{F}_T(\bk,t) = 0
\end{split} \; .\end{aligned}$$ As in Ref. [@d2013derivation], we recover physical dimensions from the previous adimensional equations using Planck units, taking $c:=l_P/t_P$, time measured in Planck times $t\to t*t_P$, and lengths measured in Planck lenghts as $x\to x*\sqrt{3}l_P$, the $\sqrt{3}l_P$ corresponding to the distance between neighboring cells. Then Eq. becomes $$\begin{aligned}
\begin{split} \label{eq:maxwellposition}
&\partial_t\bvec{F}_T(\bvec{x},t) =
-ic\nabla\times \bvec{F}_T(\bvec{x},t)
\\
&\nabla \cdot \bvec{F}_T(\bvec{x},t) = 0
\end{split} \end{aligned}$$ which in terms of $\bvec{E}$ and $\bvec{B}$ become the vacuum Maxwell’s equations $$\begin{aligned}
\label{eq:maxwellstandard}
\begin{array}{lcl}
\nabla \cdot \bvec{E}=0 & &\nabla \cdot \bvec{B} =0\\
\partial_t \bvec{E} = c\nabla \times \bvec{B} && \partial_t \bvec{B} = -c\nabla \times \bvec{E} \;\;.
\end{array}\end{aligned}$$ Introducing the polarization vectors $\bvec{u}_\bk^1$ and $\bvec{u}_\bk^2$ satisfying $$\bvec{u}^i_\bk \cdot \bn_{\bk} =\bvec u^1_\bk\cdot\bvec u^2_\bk= 0,\ |\bvec u^i_\bk|=1,\
(\bvec u^1_\bk\times\bvec u^2_\bk)\cdot\bn_\bk>0,$$ we can now interpret the following operators $$\begin{aligned}
\gamma^i(\bk) &:= \bvec{u}^i_\bk\cdot\bvec{F}(\bk,0),\quad i=1,2,
\label{eq:polarization}\end{aligned}$$ as the two polarization operators of the field. In the light of this analysis, one can conclude that the automaton discrete evolution leads to modified Maxwell’s equations in the form of Eqs. , with the electromagnetic field rotating around $\bn_{\tfrac{\bk}{2}}$ instead of $\bk$. Moreover, since in this framework the photon is a composite particle, the internal dynamics of the consitutent Fermions is responsible for an additional term $O \big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big)$. As a consequence of this distorsion, one can immediately see that the electric and magnetic fields are no longer exactly transverse to the wave vector but we have the appearence of a longitudinal component of the polarization (see Fig. \[fig:elmwave\]). In Section \[sec:phen-analys\] we discuss the new phenomenology that emerges from Eqs. .
![(colors online) A rectilinear polarized electromagnetic wave. We notice that the polarization plane (in green) is sligtly tilted with respect the plane orthogonal to $\bk$ (in gray).[]{data-label="fig:elmwave"}](graph_wave_QCATOL-beta-c.pdf){width="8cm"}
Photons as composite Bosons {#sec:photons-as-composite}
===========================
In the previous section we proved that the operators defined in Eq. dynamically evolve according to the free Maxwell’s equation. However, in order to interpret $\bvec{E}(\bk)$ and $\bvec{B}(\bk)$ as the electric and magnetic fields we need to show that they obey the correct commutation relation. The aim of this paragraph is to show that, in a regime of low energy density, the polarization operators defined in Eq. actually behave as independent Bosonic modes.
In order to avoid the technicalities of the continuum we now suppose to confine the system in finite volume $\mathcal{V}$. The finiteness of the volume introduces a discretization of the momentum space and the operators $\psi(\bk)$, $\varphi(\bk)$, obey Eq. where the periodic Dirac delta is replaced by the Kronecker delta. All the integrals over the Brillouin zone are then replaced by sums, and the polarization operators of Eq. become $$\gamma^i(\bk) :=
\sum_{\bq}
f_{\bk}(\bq)
{\varphi}^T \left(\tfrac{\bk}{2}-\bq\right)
(
\bvec{u}^i_{\tfrac{\bk}2}
\cdot
\boldsymbol{\sigma}
)
{\psi} \left(\tfrac{\bk}{2}+\bq\right).$$ These operators can be simply expressed in terms of the functions $\gamma_{\alpha,\beta}(\bvec{k})$ defined as follows $$\begin{aligned}
\gamma_{\alpha,\beta}(\bvec{k}) := \sum_{\bvec{q}}
{f}_{\bk}(\bvec{q})
\varphi_\alpha
\left(\tfrac{\bk}{2}-\bq\right)
\psi_\beta
\left(\tfrac{\bk}{2}+\bq\right),
\nonumber \\
\alpha, \beta = R,L.
\label{eq:basicobjects}\end{aligned}$$ Since the polarisation operators $\gamma^i(\bk)$ are linear combinations of $\gamma_{\alpha,\beta}(\bk)$, it is useful to compute the commutation relations of the latter. We have $$\begin{aligned}
\label{eq:basiccommutation}
&[\gamma_{\alpha,\beta}(\bvec{k}) ,\gamma_{\alpha',\beta'}(\bvec{k}') ]_{-} = 0,\nonumber\\
&[\gamma_{\alpha,\beta}(\bvec{k})
,\gamma^\dagger_{\alpha',\beta'}(\bvec{k}') ]_{-} =
\delta_{\alpha,\alpha'} \delta_{\beta,\beta'}
\delta_{\bvec{k},\bvec{k}'} -\Delta_{\alpha,\alpha',\beta,\beta',\bk,\bk'},\nonumber\\
&\Delta_{\alpha,\alpha',\beta,\beta',\bk,\bk'}:=\left( \delta_{\alpha,\alpha'}H^+_{\psi, \beta', \beta,\bvec{k}',\bvec{k}}+ \delta_{\beta,\beta'}H^-_{\varphi, \alpha', \alpha,\bvec{k}',\bvec{k}}\right), \nonumber\\
&H^\pm_{\eta, \alpha', \alpha,\bvec{k}',\bvec{k}} := \sum_{\bvec{q}}{f}_{\bk}(\bvec{q}) {f}_{\bk'}^*(\tfrac{\bvec{k}' - \bvec{k}}2+\bvec{q})\nonumber\\
&\qquad\times\eta_{\alpha'}^\dagger \left(\tfrac{ 2\bvec{k}' -
\bvec{k}}2\pm\bvec{q}\right) \eta_{\alpha} \left(
\tfrac{\bvec{k}}2\pm{\bvec{q}}\right).
%&\Delta_{\psi, \alpha', \alpha,\bvec{k}',\bvec{k}}: = \sum_{\bvec{q}}
% {f}_{\bk}(\bvec{q}){f}_{\bk'}^*(2(\bvec{k}' - \bvec{k})+\bvec{q})
% \\
% &\qquad\psi_{\alpha'}^\dagger \left( 2\bvec{k}' -
% \bvec{k}+\tfrac{\bvec{q}}{2}\right)
%\psi_{\alpha} \left( \bvec{k}+\tfrac{\bvec{q}}{2}\right).\end{aligned}$$ Then the operators $\gamma_{\alpha,\beta}$ fail to be Bosonic annihilation operators because of the apperance of the operator $\Delta_{\alpha,\alpha', \beta, \beta',\bvec{k},\bvec{k}'}$ in the commutation relation . However, if we restrict to the subset $\mathcal{S}$ of states such that $\Tr[\rho H^-_{\varphi, \beta', \beta,\bvec{k}',\bvec{k}}] \simeq 0$ and $\Tr[\rho H^+_{\psi, \alpha', \alpha,\bvec{k}',\bvec{k}}] \simeq 0$ for all $\rho \in
\mathcal{S}$, we could make the approximation $[\gamma_{\alpha,\beta}(\bvec{k}) ,\gamma^\dagger_{\alpha',\beta'}(\bvec{k}')
]_{-} \simeq
\delta_{\alpha,\alpha'} \delta_{\beta,\beta'}
\delta_{\bvec{k},\bvec{k}'} $. If we consider the modulus of the expectation value of the operators $H^\pm_{\eta, \beta', \beta,\bvec{k}',\bvec{k}} $ we have $$\begin{aligned}
\label{eq:boundfordelta}
&|\< H^\pm_{\eta, \beta', \beta,\bvec{k}',\bvec{k}} \>|
% \left| \left\< \sum_{\bvec{q}}
% {f}_{\bk}(\bvec{q}){f}_{\bk'}^*(2(\bvec{k} - \bvec{k}')+\bvec{q})
% \phi_{\beta'}^\dagger \left( 2\bvec{k}' -
% \bvec{k}-\tfrac{\bvec{q}}{2}\right)
%\phi_{\beta} \left( \bvec{k}-\tfrac{\bvec{q}}{2}\right )
%\right \> \right | \leq\\
\leq\sum_{\bvec{q}}
\left| {f}_{\bk}(\bvec{q}) \right |
\left| {f}_{\bk'}^*(\tfrac{\bvec{k}' - \bvec{k}}2+\bvec{q}) \right |\nonumber\\
&\quad\times
\left|\left\< \eta_{\beta'}^\dagger \left( \tfrac{2\bvec{k}' -
\bvec{k}}2\pm\bvec{q}\right)
\eta_{\beta} \left(\tfrac{ \bvec{k}}2\pm\bvec{q}\right )
\right \> \right | \leq\nonumber\\
%\sum_{\bvec{q}}
% \left|{f}_{\bk}(\bvec{q}) \right |
% \left|{f}_{\bk'}^*(2(\bvec{k} - \bvec{k}')+\bvec{q}) \right |
%\sqrt{
%\left\<
%\phi_{\beta'}^\dagger \left( 2\bvec{k}' -
% \bvec{k}-\tfrac{\bvec{q}}{2}\right)
% \phi_{\beta'} \left( 2\bvec{k}' -
% \bvec{k}-\tfrac{\bvec{q}}{2}\right)
%\right \>
%\left\<
%\phi^\dagger_{\beta} \left( \bvec{k}-\tfrac{\bvec{q}}{2}\right )
%\phi_{\beta} \left( \bvec{k}-\tfrac{\bvec{q}}{2}\right )
%\right \>
%}\leq\\
&\qquad\sqrt{
\<
\Gamma^\pm_{\eta,\beta,\bvec{k}}
\>
\<
\Gamma^\pm_{\eta,\beta',\bvec{k}'}
\>
},\\
&\Gamma^\pm_{\eta,\beta,\bvec{k}} =
\sum_{\bvec{q}}
\left|{f}_{\bk}(\bvec{q}) \right |^2
\eta^\dagger_{\beta} \left(\tfrac{ \bvec{k}}2\pm\bvec{q}\right )
\eta_{\beta} \left(\tfrac{ \bvec{k}}2\pm\bvec{q}\right ),\end{aligned}$$ where we repeatedly applied the Schwartz inequality.
The operators $\Gamma^-_{\varphi,\beta,\bvec{k}}$ and $\Gamma^+_{\psi, \alpha,\bvec{k}}$ can be interpreted as number operators “shaped” by the probability distribution $|{f}_{\bk}(\bvec{q})|^2$. If we suppose $| {f}_\bk(\bvec{q})|^2$ to be a constant function over a region $\Omega_\bk$ which contains $N_\bk$ modes, i.e. $|{f}_\bk(\bvec{q})|^2 = \tfrac{1}{N_\bk}$ if $\bvec{q}\in \Omega_\bk$ and $|{f}_\bk(\bvec{q})|^2 = 0 $ if $\bvec{q} \not\in \Omega_\bk$, we have $$\begin{aligned}
\left\<
\Gamma^+_{\psi, \alpha,\bvec{k}}
\right\> =
\frac{1}{N_\bk}\sum_{\bvec{q}\in\Omega_\bk}
\left\<
\psi^\dagger_{\alpha} \left( \tfrac{\bvec{k}}2+\bvec{q}\right )
\psi_{\alpha} \left(\tfrac{ \bvec{k}}2+\bvec{q}\right )
\right\> = \frac{M_{\psi,\alpha,\bvec{k}}}{N_\bk}\end{aligned}$$ where we denoted with $M_{\psi,\alpha,\bvec{k}}$ the number of $\psi_{\alpha}$ Fermions in the region $\Omega_k $ (clearly the same result applies to $\Gamma^-_{\varphi, \beta,\bvec{k}}$). Then, if we consider states $\rho$ such that $M_{\xi,\chi,\bvec{k}}/ N_\bk \leq
\varepsilon$ for all $\xi_{\chi}$ and $\bvec{k}$ and for $\varepsilon \ll 1$ we can safely assume $[\gamma_{\alpha,\beta}(\bvec{k})
,\gamma^\dagger_{\alpha',\beta'}(\bvec{k}') ]_{-} =
\delta_{\alpha,\alpha'} \delta_{\beta,\beta'}
\delta_{\bvec{k},\bvec{k}'} $ in Eq. which after an easy calculation gives $$\begin{aligned}
\label{eq:commutationpolarization}
[\gamma^i (\bvec{k}),{\gamma^j}^\dag (\bvec{k}')]_- =
\delta_{i,j} \delta_{\bvec{k},\bvec{k}'}\quad i = 0,1,2,3.\end{aligned}$$ In Eq. , besides the previously defined transverse polarizations $\gamma^1 (\bvec{k}) $ and $\gamma^2 (\bvec{k}) $, we considered also the “longitudinal” polarization operator $\gamma^3 (\bvec{k}) := \sum_{\bvec{q}}
f_\bk(\bvec{q})
{\varphi}^T
\left( \tfrac{\bvec{k}}2 -\bvec{q} \right)
(
\bvec{e}_{\tfrac{\bk}2} \cdot
\boldsymbol{\sigma}
)
{\psi}
\left(\tfrac{
\bvec{k}}2 +\bvec{q}
\right)$, where $\bvec e_{\bk}:=\bn_\bk/|\bn_\bk|$, and the “timelike” polarization operator $\gamma^0 (\bvec{k}) := \sum_{\bvec{q}}
{f}_\bk(\bvec{q})
{\varphi}^T
\left( \tfrac{ \bvec{k}}2 - \bvec{q} \right)
I
{\psi}
\left(
\tfrac{\bvec{k}}2+\bvec{q}
\right)$.
This result tells us that, as far as we restrict ourselves to states in $\mathcal{S}$ we are allowed to interpret the operators $\gamma^i(\bvec{k})$ as $4$ independent Bosonic field modes and then to interpret $\bvec{E}$ and $\bvec{B}$ defined in Eq. as the electric and the magnetic field operators. This fact together with the evolution given by Eq. proves that we realized a consistent model of quantum electrodynamics in which the photons are composite particles made by correlated Fermions whose evolution is described by a cellular automaton.
Composite Bosons and entanglement {#sec:comp-bosons-entangl}
---------------------------------
The results that we had in this section are in agreement with the recent works [@combescot2001new; @rombouts2002maximum; @avancini2003compositeness; @combescot2003n] which studied the conditions under which a pair of Fermionic fields can be considered as a Boson. In Refs. [@PhysRevA.71.034306; @PhysRevLett.104.070402] it was shown that a sufficient condition is that the two Fermionic fields $\psi,\phi$ are sufficiently entangled. More precisely, for a composite Boson $c := \sum_{i} f(i) \psi_i \phi_i $, $\sum_{i} |f(i)|^2=1$ one has $$[c,c^\dag] = 1- (\Gamma_\psi + \Gamma_{\phi}),$$ where $$\Gamma_\psi = \sum_{i} |f(i)|^2 \psi^\dag_i \psi_i,\quad\Gamma_\phi = \sum_{i} |f(i)|^2 \phi^\dag_i \phi_i,$$ and in Ref. [@PhysRevLett.104.070402] it was shown that the following bound holds $$\forall N \geq 1,\quad NP \geq \<{N}|\Gamma_\psi |{N} \>\geq P,$$ and the same holds for $\Gamma_\phi$, where $P = \sum_{i=1}^N |f(i)|^4$ is the purity of the reduced state of a single particle and $|{N}\> = \tfrac{1}{\sqrt{N!}} \chi_N(c^\dag)^N |{0}\>$ ($\chi_N$ is a normalization constant). From this result, the authors of Ref. [@PhysRevLett.104.070402] concluded that, as far as $P,NP \approx 0$, $c$ and $c^\dag$ can be safely considered as a Bosonic annihilation/creation pair. Our criterion, which restricts the state $\rho$ to satisfy $\Tr[\rho
\Gamma_\psi],\Tr[\rho\Gamma_\phi] \leq \varepsilon$ in this simplified scenario, gives the criterion in Refs. [@PhysRevA.71.034306; @PhysRevLett.104.070402] for $\rho=|N\>\<N|$. Moreover it is interesting to show that the technique applied in the derivation of Eq. can be used to answer an open question raised in Ref. [@PhysRevLett.104.070402]. The conjecture is that, given two different composite Bosons $c_1 = \sum_{i} f_1(i) \psi_i \phi_i$ and $c_2 = \sum_{i} f_2(i) \psi_i \phi_i$ such that $ \sum_{i} f_1(i) f_2(i)^* =0$, the commutation relation $[c_1,c_2^\dag ]$ should vanish as the two purities $P_1$ and $P_2$ ($P_a = \sum_{i=1}^N
|f_a(i)|^4$) decrease. Since $[c_1,c_2^\dag ] = - \sum_i f_1(i) f_2(i)^* (\psi_i^\dag \psi_i +
\phi_i^\dag \phi_i )$ we have $$|\< [c_1,c_2^\dag ]\>| \leq \sum_x \sqrt{\<\Gamma^{(1)}_x \>
\<\Gamma^{(2)}_x \> },$$ by the same reasoning that we followed in the derivation of Eq. . Combining this last inequality with the condition $ \< N |\Gamma^{(i)}_x | N\> \leq NP$ we have $|\<N| [c_1,c_2^\dag ]|N\>| \leq 2NP $ which proves the conjecture.
Phenomenological analysis {#sec:phen-analys}
=========================
We now investigate the new phenomenology predicted from the modified Maxwell equations and the modified commutation relations , with a particular focus on practically testable effects.
Let us first have a closer look at the dynamics described by Eq. . If $\bvec{u}_+$ and $\bvec{u}_-$ are the two eigenvectors of the matrix $\Exp [( 2 \bvec{n}_{\frac{\bk}{2}}
\cdot \bvec{J} ) t ]$, corresponding to eigenvalues $e^{\mp i2 |\bvec{n}_{\frac{\bk}{2}}|t} $, Eq. can be written as $$\begin{aligned}
\label{eq:maxwellsolved}
\bvec{F}_T(\bk,t) =
e^{-i2|\bvec{n}_{\frac{\bk}{2}}|t} \gamma_+(\bk) \bvec{u}_+
+
e^{i2|\bvec{n}_{\frac{\bk}{2}}|t} \gamma_-(\bk) \bvec{u}_-\end{aligned}$$ where the corresponding polarization operators $\gamma_\pm(\bk)$ are defined according to Eq. . According to Eq. the angular frequency of the electromagnetic waves is given by the modified dispersion relation $$\begin{aligned}
\label{eq:modifieddisprelmax}
\omega(\bk) = 2 | \bvec{n}_{\tfrac{\bk}{2}} | .\end{aligned}$$ The usual relation $\omega(\bk) = | \bk | $ is recovered in the $| \bk | \ll 1$ regime. The speed of light is the group velocity of the electromagnetic waves, i.e. the gradient of the dispersion relation. The major consequence of Eq. is that the speed of light depends on the value of $\bk$, as for Maxwell’s equations in a dispersive medium.
The phenomenon of a $\bk$-dependent speed of light was already analyzed in the in the context of quantum gravity where many authors considered the hypothesis that the existence of an invariant length (the Planck scale) could manifest itself in terms of modified dispersion relations [@ellis1992string; @lukierski1995classical; @Quantidischooft1996; @amelino2001testable; @PhysRevLett.88.190403]. In these models the $\bk$-dependent speed of light $c(\bk)$, at the leading order in $k :=| \bk |$, is expanded as $c(\bk) \approx 1 \pm \xi k^{\alpha}$, where $\xi $ is a numerical factor of order $1$, while $\alpha$ is an integer. This is exactly what happens in our framework, where the intrinsic discreteness of the quantum cellular automata $A^\pm$ leads to the dispersion relation of Eq. from which the following $\bk$-dependent speed of light $$\begin{aligned}
\label{eq:freqdepsol}
c^\mp(\bk) \approx 1 \pm 3\frac{k_x k_y k_z}{|\bk|^2} \approx
1 \pm \tfrac{1}{\sqrt{3}}k,\end{aligned}$$ can be obtained by computing the modulus of the group velocity and power expanding in $\bk$ with the assumption $ k_x = k_y = k_z = \tfrac{1}{\sqrt{3}} k $, ($k = |\bk|$). It is interesting to observe that depending on the automaton $A^{+}(\bk)$ of $A^{-}(\bk)$ in Eq. we obtain corrections to the speed of light with opposite sign. Moreover the correction is not isotropic and can be superluminal, though uniformly bounded for all $\bk$ as shown for the Weyl automaton in Ref. [@d2013derivation].
Models leading to modified dispersion relations recently received attention because they allow one to derive falsifiable predictions of the Plank scale hypothesis. These can be experimentally tested in the astrophysical domain, where the tiny corrections to the usual relativistic dynamics can be magnified by the huge time of flight. For example, observations of the arrival times of pulses originated at cosmological distances, like in some $\gamma$-ray bursts[@amelino1998tests; @abdo2009limit; @vasileiou2013constraints; @amelino2009prospects], are now approaching a sufficient sensitivity to detect corrections to the relativistic dispersion relation of the same order as in Eq. .
![(colors online) The graphics shows the vector $2\bvec{n}_{\tfrac{\bk}{2}}$ (in green), which is orthogonal to the polarization plane, the wavevector $\bk$ (in red) and the group velocity $\nabla \omega (\bk)$ (in blue) as function of $\bk$ for the value $|\bk|= 0.8$ and different directions. Notice that the three vectors are not parallel and the angles between them depend on $\bk$. Such anisotropic behavior can be traced back to the anisotropy of the dispersion relation of the Weyl automaton.[]{data-label="fig:relvectors"}](graph_vectors_QCATOL-rc2.pdf){width="8cm"}
A second distinguishing feature of Eq. is that the polarization plane is neither orthogonal to the wavevector, nor to the group velocity, which means that the electromagnetic waves are no longer exactly transverse (see Figs. \[fig:elmwave\] and \[fig:relvectors\]). However the angle $\theta$ between the polarization plane and the plane orthogonal to $\bk$ or $\nabla\omega(\bk)$ is of the order $\theta \approx 2k$, which gives $10^{-15}\mathrm{rad}$ for a $\gamma$-ray wavelength, a precision which is not reachable by the present technology. Since for a fixed $\bk$ the polarization plane is constant, exploiting greater distances and longer times does not help in magnifying this deviation from the usual electromagnetic theory.
Finally, the third phenomenological consequence of our modelling is that, since the photon is described as a composite Boson, deviations from the usual Bosonic statistics are in order. As we proved in Section \[sec:photons-as-composite\], the choice of the function ${f}_\bk(\bvec{q})$ determines the regime where the composite photon can be approximately treated as a Boson. However, independently on the details of function ${f}_\bk(\bvec{q})$ one can easily see that a Fermionic saturation of the Boson is not visible, e.g. for the most powerful laser [@dunne2007high] one has a approximately an Avogadro number of photons in $10^{-15}$cm${}^3$, whereas in the same volume on has around $10^{90}$ Fermionic modes.
Another test for the composite nature of photons is provided by the prediction of deviations from the Planck’s distribution in Blackbody radiation experiments. A similar analysis was carried out in Ref. [@perkins2002quasibosons], where the author showed that the predicted deviation from Planck’s law is less than one part over $10^{-8}$, well beyond the sensitivity of present day experiments.
Conclusions {#sec:conclusions}
===========
In this paper we derive a complete theoretical framework of the free quantum radiation field at the Planck scale, based on a quantum Weyl automaton derived from first principles in Ref. [@d2013derivation]. Differently from previous arguments based just on discreteness of geometry, the present approach provides fully quantum theoretical treatment that allows for precise observational predictions which involve electromagnetic radiation, e. g. about deep-space astrophysical sources. Within the present framework the electromagnetic field emerges from two correlated massless Fermionic fields whose evolution is given by the Weyl automaton. Then the electric and magnetic field are described in terms of bilinear operators of the two constituent Fermionic fields. This framework recalls the so-called “neutrino theory of light” considered in Refs. [@de1934nouvelle; @jordan1935neutrinotheorie; @kronig1936relativistically; @perkins1972statistics; @perkins2002quasibosons].
The automaton evolution leads to a set of modified Maxwell’s equations whose dynamics differs from the usual one for ultra-high wavevectors. This model predicts a longitudinal component of the polarization and a $\bk$-dependent speed of light. This last effect could be observed by measuring the arrival times of light originated at cosmological distances, like in some $\gamma$-ray bursts, exploiting the huge distance scale to magnify the tiny corrective terms to the relativistic kinematics. This prediction agrees with the one presented in Ref. [@amelino1998tests] where $\gamma$-ray bursts were for the first time considered as tests for physical models with non-Lorentzian dispersion relations. Within this perspective, our quantum cellular automaton singles out a specific modified dispersion relation as emergent from a Planck-scale microscopic dynamics.
Another major feature of the proposed model, is the composite nature of the photon which leads to a modification of the Bosonic commutation relations. Because of the Fermionic structure of the photon we expect that the Pauli exclusion principle could cause a saturation effects when a critical energy density is achieved. However, an order of magnitude estimation shows that the effect is very far from being detectable with the current laser technology.
As a spin-off of the analysis of the composite nature of the photons, we proved a result that strenghten the thesis that the amount of entanglement quantifies whether a pair of Fermions can be treated as a Boson [@PhysRevA.71.034306; @PhysRevLett.104.070402]. Indeed we showed that, even in the case of several composite Bosons, the amount of entanglement for each pair is a good measure of how much the different pair of Fermions can be treated as independent Bosons. This question was proposed as an open problem in Ref. [@PhysRevLett.104.070402].
The results of this work leave a lot of room for future investigation. The major question is the study of how symmetry transformations can be represented in the model. The scenario we considered is restricted to a fixed reference frame and in order to properly recover the standard theory we should discuss how the Poincarè group acts on our physical model. This analysis could be done following the lines of Ref. [@bibeau2013doubly] where it is shown how a QCA dynamical model is compatible with a deformed relativity model [@amelino2002relativity; @PhysRevLett.88.190403] which exhibits a non-linear action of the Poincarè group.
Proof of Eq. {#sec:proof-eq.-eqref}
=============
Given two vector $\bvec{a},\bvec{a}' \in \mathbb{R}^3$, we define $$\begin{aligned}
&U = R_\bvec{-a}R_{\bvec{a}+\bvec{a}'}\\
&R_\bvec{-a} = \exp(i \bvec{a} \cdot \boldsymbol{\sigma} \, t)
\nonumber \\
&R_{\bvec{a}+\bvec{a}'}\exp ( -i \left(\bvec{a}+ \bvec{a}' \right) \cdot
\boldsymbol{\sigma} \, t ), \nonumber . \end{aligned}$$ By explicit computation $R_{\bvec{a}+\bvec{a}'}$ can be written as $$\begin{aligned}
&R_{\bvec{a}+\bvec{a}'}=\exp ( -i (\bvec{a}+ \bvec{a}' )
\cdot \boldsymbol{\sigma} \, t ) = \nonumber \\
&= \exp (-i t |\bvec{a}+ \bvec{a}' |
\bvec{e}_{\bvec{a}+\bvec{a}'} \cdot \boldsymbol{\sigma} ) = \nonumber \\
&=I \cos(|\bvec{a}+ \bvec{a}' |t) -i
\sin\left(|\left(\bvec{a}+ \bvec{a}' \right)|t\right)
\bvec{e}_{\bvec{a}+\bvec{a}'} \cdot \boldsymbol{\sigma} \label{eq:decompoR}\end{aligned}$$ where we introduced $\bvec{e}_\bvec{a} = \frac{\bvec{a}}{|\bvec{a}|} $ and $ \bvec{e}_{\bvec{a}+\bvec{a}'} = \frac{\bvec{a}+\bvec{a}'}{|\bvec{a}+\bvec{a}'|}$. For $|\bvec{a}'| \ll |\bvec{a}|$ we have $$\begin{aligned}
\begin{split}
\label{eq:approxmod}
& |\bvec{a} + \bvec{a}'| = \sqrt{|\bvec{a}|^2 + |\bvec{a}'|^2 + 2
\bvec{a}\cdot \bvec{a}'} =\\
&=|\bvec{a}| + \frac{\bvec{a}\cdot
\bvec{a}'}{|\bvec{a}|^2} + O\left( \tfrac{ |\bvec{a}'|^2}{ |\bvec{a}|^2} \right)
\end{split}
\end{aligned}$$ and $$\begin{aligned}
|\bvec{e}_{\bvec{a}+\bvec{a}'} - \bvec{e}_\bvec{a}| &=
\left|\tfrac{\bvec{a}+\bvec{a}'}{|\bvec{a}+\bvec{a}'|} -
\tfrac{\bvec{a}}{|\bvec{a}|} \right|=
\left|\tfrac{
\bvec{a} (-|\bvec{a}+\bvec{a}'|+|\bvec{a}|) +
\bvec{a}' |\bvec{a}|
}
{|\bvec{a}+\bvec{a}'||\bvec{a}|}
\right|\leq\nonumber\\
&\leq
\tfrac{|\bvec{a}'|}{|\bvec{a} + \bvec{a}'|} +
1- \tfrac{|\bvec{a}|}{|\bvec{a} + \bvec{a}'|} =
O\left(\tfrac{|\bvec{a}'|}{|\bvec{a} |}\right).
\label{eq:apprxversor}\end{aligned}$$ Then, for $|\bvec a'|\ll|\bvec a|$ we obtain $$\begin{aligned}
&R_{\bvec{a}+\bvec{a}'} =
I \cos\left(\left( |\bvec{a}| + \tfrac{\bvec{a}\cdot
\bvec{a}'}{|\bvec{a}|}\right)t\right) + \nonumber
\\ &-i \sin\left(\left( |\bvec{a}| + \tfrac{\bvec{a}\cdot \bvec{a}'}{|\bvec{a}|}\right)t\right) \bvec{e}_{\bvec{a}} \cdot \boldsymbol{\sigma} +
\Lambda'(\bvec{a},\bvec{a}') + \Theta'(\bvec{a},\bvec{a}')= \nonumber \\
& \exp\left(-i t \left( |\bvec{a}| + \tfrac{\bvec{a}\cdot
\bvec{a}'}{|\bvec{a}|}\right) \bvec{e}_\bvec{a} \cdot
\boldsymbol{\sigma}\right) +
\Lambda'(\bvec{a},\bvec{a}') + \Theta'(\bvec{a},\bvec{a}',t) \nonumber\end{aligned}$$ where $\Lambda'(\bvec{a},\bvec{a}')$ + $\Theta'(\bvec{a},\bvec{a}',t)$ are a couple of operators such that $$\begin{aligned}
|\Lambda'(\bvec{a},\bvec{a}')| =
O\left(\tfrac{|\bvec{a}'|}{|\bvec{a}|}\right), \quad
|\Theta'(\bvec{a},\bvec{a}')| =
O\left(\tfrac{|\bvec{a}'|^2}{|\bvec{a}|^2}t\right)
\nonumber\end{aligned}$$ from which we finally get $$\begin{aligned}
\label{eq:approximU}
& U = \exp\left(-it \frac{\bvec{a}\cdot
\bvec{a}'}{|\bvec{a}|}\bvec{e}_\bvec{a} \cdot \boldsymbol{\sigma}\right)
+
\Lambda(\bvec{a},\bvec{a}') + \Theta(\bvec{a},\bvec{a}',t) \nonumber
\\
&|\Lambda(\bvec{a},\bvec{a}')| =
O\left(\tfrac{|\bvec{a}'|}{|\bvec{a}|}\right), \quad
|\Theta(\bvec{a},\bvec{a}')| =
O\left(\tfrac{|\bvec{a}'|^2}{|\bvec{a}|^2}t\right) \end{aligned}$$ which leads to Eq. if we identify $\bvec{a}=\bvec{n}_\bk$, $\bvec{a}'=\bvec{l}_{\bk, \bq}$.
Proof of Eq. {#sec:proof-eq.-eqref-1}
=============
Let us introduce the vectors $\bvec u^1_\bk, \bvec u^2_\bk \in \mathbb{R}^3$ such that $$\begin{aligned}
\label{eq:vectors}
\begin{split}
\bvec u^1_\bk \cdot \bvec n_\bk=0 \quad
\bvec u^2_\bk :=\bvec{e}_\bk \times\bvec u^1_\bk \quad
\bvec{e}_\bk := {|\bvec{n}_{\tfrac{\bk}{2}}|}^{-1}{\bvec{n}_{\tfrac{\bk}{2}}}.
\end{split}\end{aligned}$$ The transverse field $ \tilde{\bvec{F}}_T(\bk,t)$ defined in Eq. can then be written in the basis $\{\bvec u^i_\bk\}$ as $$\begin{aligned}
\begin{split}
\tilde{\bvec{F}}_T(\bk,t) =
\begin{pmatrix}
\bvec u^1_{\tfrac{\bk}2}\cdot\tilde{\bvec F}^{\bvec{u}_1}(\bk,t) \\
\bvec u^2_{\tfrac{\bk}2}\cdot\tilde{\bvec F}^{\bvec{u}_2}(\bk,t)
\end{pmatrix}
\end{split}\end{aligned}$$ Reminding the definition we have $$\begin{aligned}
&\bvec u^i_{\tfrac{\bk}2}\cdot\tilde{\bvec F}(\bk,t) =
\int\!\!\frac{\d\bq}{(2\pi)^3} f_{\bk}(\bq)
{{\varphi}}^T
\left(\tfrac{\bk}{2}-\bq\right)
Q^i
{\psi}
\left(\tfrac{\bk}{2}+\bq\right) \nonumber\\
&\qquad Q^i (\bk,\bq,t):= ({U}^{\bk,t}_{\tfrac{\bk}2-\bq})^\dag
\bvec u^i_{\tfrac{\bk}2} \cdot\boldsymbol{\sigma}
{U}^{\bk,t}_{\tfrac{\bk}2+\bq}. \label{eq:operatO}\end{aligned}$$ If we insert Eq. , which can be written as $$\begin{aligned}
& {U}^{\bk,t}_{\tfrac{\bk}2\pm\bq} = R_{\pm\xi \bvec{e}} + O\big( \tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|}
\big) \\
&R_{\pm\xi \bvec{e}} := \exp( \pm i \xi \bvec{e} \cdot
\boldsymbol{\sigma}) \quad
\xi := c_{\bk,\bq} t\;,\end{aligned}$$ inside Eq. we have $$\begin{aligned}
& Q^i (\bk,\bq,t)=
R_{-\xi \bvec{e}}
\bvec u^i_{\tfrac{\bk}2} \cdot\boldsymbol{\sigma}
R_{-\xi \bvec{e}}
+ O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|}\big)= \nonumber\\
&=\bvec u^i_{\tfrac{\bk}2} \cdot\boldsymbol{\sigma} + O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big) =
Q^i (\bk,\bq,0) + O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|}\big)
\label{eq:operatorOev},\end{aligned}$$ where we used the identity $$\begin{aligned}
(\bvec{a}\cdot \boldsymbol{\sigma})
(\bvec{b} \cdot \boldsymbol{\sigma})(\bvec{a}\cdot \boldsymbol{\sigma})=-\bvec{b} \cdot \boldsymbol{\sigma}\end{aligned}$$ holding for $\bvec a\cdot\bvec b=0$, $|\bvec a|=|\bvec b|=1$, which implies $$\begin{aligned}
\exp(i \xi \bvec{e} \cdot \boldsymbol{\sigma})
\bvec u^i_{\tfrac{\bk}2}\cdot \boldsymbol{\sigma}
\exp(i \xi \bvec{e} \cdot \boldsymbol{\sigma}) =
\bvec u^i_{\tfrac{\bk}2}\cdot \boldsymbol{\sigma} \quad \forall \xi \in \mathbb{R}.\end{aligned}$$ Inserting Eq. in Eq. we have $$\begin{aligned}
\label{eq:ftildeapproxevolut}
\bvec u^i_{\tfrac{\bk}2}\cdot\tilde{\bvec F}(\bk,t) =
\bvec u^i_{\tfrac{\bk}2}\cdot\tilde{\bvec F}(\bk,0) + O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big)
\quad i =1,2 \nonumber\end{aligned}$$ which then implies $$\begin{aligned}
\tilde{\bvec{F}}_T(\bk,t) = \tilde{\bvec{F}}_T(\bk,0) + O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big) ={\bvec{F}}_T(\bk) + O\big(
\tfrac{\bar{q}(\bk)}{|\bvec{n}_{\frac{\bk}{2}}|} \big) \nonumber\end{aligned}$$
|
---
abstract: 'We show that a two-dimensional topological superconductor (TSC) can be realized in a hybrid system with a conventional $s$-wave superconductor proximity-coupled to a quantum anomalous Hall (QAH) state from the Rashba and exchange effects in single layer graphene. With very low or even zero doping near the Dirac points, i.e., two inequivalent valleys, this TSC has a Chern number as large as four, which supports four Majorana edge modes. More importantly, we show that this TSC has a robust topologically nontrivial bulk excitation gap, which can be larger or even one order of magnitude larger than the proximity-induced superconducting gap. This unique property paves a way for the application of QAH insulators as seed materials to realize robust TSCs and Majorana modes.'
author:
- 'L. Wang'
- 'M. W. Wu'
title: Topological superconductor with a large Chern number and a large bulk excitation gap in single layer graphene
---
[^1]
[^2]
INTRODUCTION
============
Majorana modes can naturally exist in topological superconductors (TSCs).[@kitaev; @alicea1; @beenakker; @hasan; @qi] The intrinsic TSC has been predicted to exist in superconducting Sr$_2$RuO$_4$ with $p$-wave paring state.[@ivanov; @mackenzie] However, this has not yet been experimentally confirmed. Recently, many efforts have been devoted to design artificial TSCs.[@fu; @sau; @sau2; @alicea2; @lutchyn; @alicea3; @halperin; @stanescu; @mourik; @yzhou; @bysun; @nadj; @hui; @dumitrescu; @jianli; @qi2; @jwang; @poyhonen; @jjhe; @tewari; @tewari2; @mdiez; @wong; @haim; @rontynen; @dutreix; @jianli2] So far, most studies focus on the effective $p$-wave superconductors in hybrid systems with conventional $s$-wave superconductors in proximity to strong topological insulators,[@fu] semiconductors with strong spin-orbit coupling (SOC), [@sau; @sau2; @alicea2; @lutchyn; @alicea3; @halperin; @stanescu; @mourik; @yzhou; @bysun; @jjhe; @tewari; @tewari2; @mdiez; @wong; @haim; @dutreix] or ferromagnetic atom chains.[@nadj; @hui; @dumitrescu; @jianli; @poyhonen; @rontynen; @jianli2] Some attention has also been paid to the conventional $s$-wave superconductors coupled to quantum anomalous Hall (QAH) insulators such as topological insulators with magnetic dopants.[@qi2; @jwang] Among all the above TSCs, multiple spatially overlapping Majorana modes, which greatly benefit the transport properties, can only coexist in one-dimensional (two-dimensional) TSCs belonging to Class BDI [@poyhonen; @haim; @dumitrescu; @hui; @jianli; @jjhe; @tewari; @tewari2; @mdiez; @wong] (D [@qi2; @jwang; @rontynen; @jianli2]) with integer topological invariant.[@schnyder] In reality, the one-dimensional TSCs in Class BDI can easily reduce to the ones indexed by Class D with zero or one Majorana mode.[@poyhonen; @haim; @dumitrescu; @hui; @tewari; @tewari2; @mdiez; @wong] As for the two-dimensional TSCs in Class D, the number of the Majorana modes or the Chern number is limited upto two.[@qi2; @jwang; @jianli2] More Majorana modes or larger Chern numbers are limited by large chemical potential (i.e., very high doping) and an overall much smaller bulk excitation gap than the proximity-induced superconducting gap.[@rontynen; @jianli2]
In this work, we show that a two-dimensional TSC can be realized in a hybrid system with a conventional $s$-wave superconductor proximity-coupled to a QAH state [@qiao] due to the Rashba SOC [@Rashba] and exchange field in single layer graphene. Interestingly, with very low or even zero doping near the Dirac points, i.e., two inequivalent valleys, the TSC from the QAH state has a Chern number reaching as large as four, hosting four Majorana edge modes. More importantly, these Majorana modes are protected by a bulk excitation gap, which can be larger or even one order of magnitude larger than the superconducting gap from the proximity effect. This is in strong contrast to the case of effective $p$-wave superconductors where the excitation gap is always smaller than the superconducting gap.[@sau; @sau2; @alicea2; @lutchyn; @alicea3; @halperin; @stanescu; @mourik; @yzhou; @bysun] As the large topologically nontrivial gap has been shown to be probably most important for applications in topological insulators,[@hasan; @qi] topological crystalline insulators,[@cniu] and QAH insulators,[@cniu; @qiao2; @gangxu] our finding, i.e., reporting a large bulk excitation gap in the TSC is crucial to the field of TSCs and Majorana modes. This paves a way to obtain robust TSCs and Majorana modes using the QAH states. We also address the experimental feasibility of the TSC from the QAH state.
This paper is organized as follows. In Sec. II, we present our model and lay out the tight-binding Hamiltonian of single layer graphene. Then, we calculate the topological invariant in Sec. III. We further presents the results on the phase diagram, Majorana edge states and bulk excitation gap in Sec. IV. Finally, we summarize and discuss in Sec. V.
MODEL AND HAMILTONIAN
=====================
The real-space tight-binding Hamiltonian of single layer graphene with the Rashba SOC, exchange field and proximity-induced $s$-wave superconductivity is given by [@qiao; @kane; @ezawa] $$\begin{aligned}
H&=&-t\sum_{\langle i,j\rangle\alpha}c^{\dagger}_{i\alpha}c_{j\alpha}
+i\lambda\sum_{\langle i,j\rangle\alpha\beta}{({{\mbox{\boldmath$\sigma$\unboldmath}}}^{\alpha\beta}\times {\bf d}_{ij})}_zc^{\dagger}_{i\alpha}c_{j\beta}\nonumber\\
&&\mbox{}-\mu\sum_{i\alpha}c^{\dagger}_{i\alpha}c_{i\alpha}
+V_z\sum_{i\alpha}c^{\dagger}_{i\alpha}\sigma_z^{\alpha\alpha}c_{i\alpha}\nonumber\\
&&\mbox{}+\Delta\sum_{i}(c^{\dagger}_{i\uparrow}c^{\dagger}_{i\downarrow}+{\rm H.c.}),\label{hamil} \end{aligned}$$ where $\langle i,j\rangle$ represents the nearest-neighboring sites and $c_{i\alpha}\ (c^{\dagger}_{i\alpha})$ annihilates (creates) an electron with spin $\alpha$ at site $i$. The first term stands for the nearest-neighbor hopping with $t=2.7\ $eV [@neto] being the hopping energy. The second term denotes the Rashba SOC with $\lambda$, ${{\mbox{\boldmath$\sigma$\unboldmath}}}$ and ${\bf d}_{ij}$ representing the coupling strength, Pauli matrices for real spins and a unit vector from site $j$ to site $i$, respectively. $\mu$ in the third term is the chemical potential. $V_z$ ($\Delta$) in the fourth (fifth) term corresponds to exchange field (superconducting gap from the proximity effect).
To start, we transform the Hamiltonian of Eq. (\[hamil\]) to the Bogoliubov-de Gennes (BdG) one in the momentum space. Specifically, $$\begin{aligned}
H=\frac{1}{2}\sum_{\bf k}\Phi^{\dagger}_{\bf k}H_{\rm BdG}({\bf k})\Phi_{\bf k}\end{aligned}$$ where $\Phi^{\dagger}_{\bf k}=(\psi^{\dagger}_{{\rm A}\uparrow}(\bf
k),\ \psi^{\dagger}_{{\rm B}\uparrow}(\bf k),\ \psi^{\dagger}_{{\rm A}\downarrow}(\bf
k),\ \psi^{\dagger}_{{\rm B}\downarrow}(\bf k),\ \psi_{{\rm A}\downarrow}(-{\bf
k}),\\ \psi_{{\rm B}\downarrow}(-{\bf k}),\ -\psi_{{\rm A}\uparrow}(-{\bf
k}),\ -\psi_{{\rm B}\uparrow}(-{\bf k}))$ with $\psi^{\dagger}_{i\alpha}({\bf k})$ creating an electron with spin $\alpha$ and momentum ${\bf k}$ counted from the momentum $\Gamma$ at sublattice $i$ ($i={\rm A}$, ${\rm B}$) and $$\begin{aligned}
H_{\rm BdG}({\bf k})=\left(\begin{array}{cc}
H_e({\bf k})-\mu & \Delta\\
\Delta & \mu-\sigma_yH_e^*(-{\bf k})\sigma_y\label{BdG}
\end{array}\right).\end{aligned}$$ $H_e({\bf k})$ represents tight-binding Hamiltonian without the $s$-wave superconductivity, which can be written as $$\begin{aligned}
H_{e}({\bf k})=\left(\begin{array}{cccc}
V_z & f({\bf k}) & 0 & h_1({\bf k})\\
f^*({\bf k}) & V_z & h_2^*({\bf k}) & 0\\
0 & h_2({\bf k}) & -V_z & f({\bf k})\\
h_1^*({\bf k}) & 0 & f^*({\bf k}) & -V_z\label{hamilnosc}
\end{array}\right)\end{aligned}$$ where $f({\bf k})=-t[(2\cos \frac{k_x}{2}\cos \frac{k_y}{2\sqrt{3}}+\cos
\frac{k_y}{\sqrt{3}})-i(2\cos \frac{k_x}{2}\sin \frac{k_y}{2\sqrt{3}}-\sin
\frac{k_y}{\sqrt{3}})]$, $h_1({\bf k})=-\lambda [(\cos \frac{k_x}{2}+\sqrt{3}\sin
\frac{k_x}{2})\sin \frac{k_y}{2\sqrt{3}}+\sin \frac{k_y}{\sqrt{3}}]-i\lambda [-\cos \frac{k_y}{\sqrt{3}}+\cos
\frac{k_y}{2\sqrt{3}}(\cos \frac{k_x}{2}+\sqrt{3}\sin \frac{k_x}{2})]$ and $h_2({\bf k})=\lambda [(\sqrt{3}\sin \frac{k_x}{2}-\cos \frac{k_x}{2})\sin
\frac{k_y}{2\sqrt{3}}-\sin \frac{k_y}{\sqrt{3}}]+i\lambda [\cos \frac{k_y}{\sqrt{3}}-\cos
\frac{k_y}{2\sqrt{3}}(\cos \frac{k_x}{2}-\sqrt{3}\sin \frac{k_x}{2})]$. Note that the lattice constant is set to be unity in the calculation for simplicity.
TOPOLOGICAL INVARIANT
=====================
Before investigating the topological properties of $H_{\rm BdG}({\bf k})$, we first identify the gap closing conditions. The gap closing of the BdG Hamiltonian $H_{\rm BdG}({\bf k})$ is equivalent to the existence of bulk zero energy states due to particle-hole symmetry. The condition for bulk zero energy states is obtained by calculating ${\rm
det}(H_{\rm BdG})=0$. We find that the gap closes at the momenta $\Gamma$ (single one), $M$ (three inequivalent ones) and $K$ (two inequivalent ones) points with the corresponding conditions given by $(\mu\pm
3t)^2=V_z^2-\Delta^2$, $(\mu\pm t)^2=V_z^2-\Delta^2$ and $\mu^2=V_z^2-\Delta^2$, respectively. It is noted that $+$ ($-$) stands for lower (higher) energy band at the momentum $\Gamma$ or $M$. The detailed calculation is shown in Appendix \[appB\]. Obviously, our system is topologically trivial in the case of $|V_z|<|\Delta|$. As for $|V_z|\ge |\Delta|$, we have ten critical chemical potentials in order, i.e., $\mu_{1,2}=3t\pm
\sqrt{V_z^2-\Delta^2}$, $\mu_{3,4}=t\pm \sqrt{V_z^2-\Delta^2}$, $\mu_{5,6}=\pm \sqrt{V_z^2-\Delta^2}$, $\mu_{7,8}=-t\pm \sqrt{V_z^2-\Delta^2}$, and $\mu_{9,10}=-3t\pm \sqrt{V_z^2-\Delta^2}$ by assuming $|V_z|,|\Delta|\ll t$, which divide the system into eleven topological regimes.
These topological regimes are characterized by the Chern number $C_1$ since $H_{\rm BdG}(\bf k)$ belongs to Class D with integer topological invariant.[@schnyder] $C_1$ can be calculated by [@ghosh] $$\begin{aligned}
C_1=\frac{1}{2\pi}\int_{\rm BZ}d^2{\bf k}f_{xy}({\bf k})\end{aligned}$$ with the Berry curvature $$\begin{aligned}
f_{xy}({\bf k})=i\sum_{m,n}(f_m-f_n){u_m^{\dagger}({\bf
k})[\partial_{k_x}H_{\rm BdG}({\bf k})]u_n({\bf k})}\nonumber\\
\mbox{}\times {u_n^{\dagger}({\bf k})[\partial_{k_y}H_{\rm BdG}({\bf k})]u_m({\bf k})}/{[E_m({\bf k})-E_n({\bf k})]^2}.\end{aligned}$$ Here, $u_m({\bf k})$ is the $m$-th eigenvector of $H_{\rm BdG}({\bf k})$ with the corresponding eigenvalue being $E_m({\bf k})$; $f_m=1\ (0)$ for occupied (empty) band. The Chern number of all topological regimes is given by $$\begin{aligned}
C_1=\left\{
\begin{array}{ll}
1\ (-1), &\mu_2<\mu<\mu_1\\
-3\ (3), &\mu_4<\mu<\mu_3\\
4\ (-4), &\mu_6<\mu<\mu_5\\
-3\ (3), &\mu_8<\mu<\mu_7\\
1\ (-1), &\mu_{10}<\mu<\mu_9\\
0, &\mbox{other regimes}
\end{array}
\right.\end{aligned}$$ when $V_z>0\ (V_z<0)$. It is seen that $|C_1|=1\ (3)$ near the momentum $\Gamma\
(M)$ point, which is consistent with the number of the zero energy states in Ref. . These Majorana modes require very large chemical potential (of the order of eV), clearly unachievable experimentally. It is noted that the study on the Majorana modes near the Dirac points is absent in Ref. . In this work, with very low or even zero doping near the Dirac points, i.e., $K$ (two inequivalent ones), we have a Chern number as large as four.
Results
=======
Phase diagram
-------------
In the following, we focus on the investigation near the Dirac points. We first study the topological phase diagram as shown in Fig. \[fig1\](a). The phase boundaries between the topological and nontopological superconductors (NTSCs) are determined by the dashed curves, i.e., $V_z^2=\mu^2+\Delta^2$. To further distinguish the TSCs (i.e., $V_z^2>\mu^2+\Delta^2$), we suppress the $s$-wave superconductivity. Without the $s$-wave superconductivity, we show the bulk energy spectrum of the low energy effective Hamiltonian near the Dirac points $H_e^{\rm eff}$ (see Appendix \[appA\]) in Fig. \[fig1\](b). When the chemical potential lies in the gap ($|\mu|<E_0$), eg., $\mu_{\rm in}$, the system behaves as a QAH state with the Chern number $|N|=2$.[@qiao] Note that $E_0$ is the absolute value of the minimum (maximum) energy of the conduction (valence) band with the formula given in Appendix \[appA\]. This QAH state in proximity to an $s$-wave superconductor becomes a TSC with the Chern number $2|N|=4$ [@qi2] (see regime I). When the chemical potential is tuned out of the gap below the upper limit $|V_z|$, eg., ${\mu_{\rm out}}$, the system is in a metallic phase with two Fermi surfaces in each valley as shown in Fig. \[fig1\](b) ($K^{\prime}$ valley is not shown here). With the $s$-wave superconductivity included, the effective paring near each of these four Fermi surfaces is equivalent to that of a $p$-wave superconductor.[@sau; @alicea2; @jianli] Each of these effective $p$-wave superconductors hosts a Majorana edge mode, which is in agreement with the Chern number near the Dirac points, i.e., $|C_1|=4$. This effective $p$-wave superconductor from metal is labeled as regime II. Similarly, the NTSCs (i.e., $V_z^2<\mu^2+\Delta^2$) can also be divided into two regimes, i.e., regime III (from the QAH state) and regime IV (from metal).
![(Color online) (a) Topological phase diagram in the ($\mu,V_z$) space with $\Delta\ne 0$ or $\Delta=0$. The dashed curves, i.e., $V^2_z=\Delta^2+\mu^2$ are the phase boundaries between the TSC and NTSC whereas the dotted ones, i.e, $\mu^2=E_0^2$ stand for the phase boundaries between the QAH state and metal. (b) Bulk energy spectrum of $H_{e}^{\rm eff}$ near the $K$ point with $k_y=0$ and $\Delta=0$. $V_z$ ($-V_z$) is the upper (lower) limit of the chemical potential in the topological nontrivial regime ($V_z^2>\mu^2+\Delta^2$). $\mu_{\rm in}$ and $\mu_{\rm out}$ stand for the chemical potential in and out of the gap, respectively. $V_z=6\ $meV and $\lambda=4\ $meV.[]{data-label="fig1"}](figww1a.eps "fig:"){width="6.cm"} ![(Color online) (a) Topological phase diagram in the ($\mu,V_z$) space with $\Delta\ne 0$ or $\Delta=0$. The dashed curves, i.e., $V^2_z=\Delta^2+\mu^2$ are the phase boundaries between the TSC and NTSC whereas the dotted ones, i.e, $\mu^2=E_0^2$ stand for the phase boundaries between the QAH state and metal. (b) Bulk energy spectrum of $H_{e}^{\rm eff}$ near the $K$ point with $k_y=0$ and $\Delta=0$. $V_z$ ($-V_z$) is the upper (lower) limit of the chemical potential in the topological nontrivial regime ($V_z^2>\mu^2+\Delta^2$). $\mu_{\rm in}$ and $\mu_{\rm out}$ stand for the chemical potential in and out of the gap, respectively. $V_z=6\ $meV and $\lambda=4\ $meV.[]{data-label="fig1"}](figww1b.eps "fig:"){width="8.5cm"}
![(Color online) (a) and (b) represent the energy spectrum of zigzag graphene ribbon with the Rashba SOC, exchange field and proximity-induced $s$-wave superconductivity near the $K$ and $K^{\prime}$ points, respectively. (c) ((d)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) $|k_x|$ near the $K$ point at one edge (i.e., $y=0$) with $v_x>0$ (only part of the ribbon is shown). A (B) refers to A (B) sublattice. The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=0$ and $\Delta=2\ $meV. []{data-label="fig2"}](figww2a.eps "fig:"){width="4.2cm"} ![(Color online) (a) and (b) represent the energy spectrum of zigzag graphene ribbon with the Rashba SOC, exchange field and proximity-induced $s$-wave superconductivity near the $K$ and $K^{\prime}$ points, respectively. (c) ((d)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) $|k_x|$ near the $K$ point at one edge (i.e., $y=0$) with $v_x>0$ (only part of the ribbon is shown). A (B) refers to A (B) sublattice. The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=0$ and $\Delta=2\ $meV. []{data-label="fig2"}](figww2b.eps "fig:"){width="4.2cm"} ![(Color online) (a) and (b) represent the energy spectrum of zigzag graphene ribbon with the Rashba SOC, exchange field and proximity-induced $s$-wave superconductivity near the $K$ and $K^{\prime}$ points, respectively. (c) ((d)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) $|k_x|$ near the $K$ point at one edge (i.e., $y=0$) with $v_x>0$ (only part of the ribbon is shown). A (B) refers to A (B) sublattice. The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=0$ and $\Delta=2\ $meV. []{data-label="fig2"}](figww2c.eps "fig:"){width="4.2cm"} ![(Color online) (a) and (b) represent the energy spectrum of zigzag graphene ribbon with the Rashba SOC, exchange field and proximity-induced $s$-wave superconductivity near the $K$ and $K^{\prime}$ points, respectively. (c) ((d)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) $|k_x|$ near the $K$ point at one edge (i.e., $y=0$) with $v_x>0$ (only part of the ribbon is shown). A (B) refers to A (B) sublattice. The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=0$ and $\Delta=2\ $meV. []{data-label="fig2"}](figww2d.eps "fig:"){width="4.2cm"}
Majorana edge states
--------------------
As the effective $p$-wave superconductors (regime II) have been widely investigated in the literature,[@sau; @sau2; @alicea2; @lutchyn; @alicea3; @halperin; @stanescu; @mourik; @yzhou; @bysun] we concentrate on the TSC from the QAH state (regime I). The Majorana edge states are studied in thick graphene ribbons. The numerical method is detailed in Appendix \[appC\]. We plot the energy spectrum of zigzag graphene ribbon near the $K$ and $K^{\prime}$ points in Figs. \[fig2\](a) and (b), respectively. We find that there exist four zero energy states in each valley. These eight states can be divided into two categories, i.e., four propagate along the same direction $+x$ ($-x$) determined by the group velocity $v_x=\frac{1}{\hbar}\frac{\partial E(k_x)}{\partial k_x}$ $>0$ ($<0$). Moreover, the four states in the same category are at the same edge, which is in agreement with the magnitude of the Chern number, i.e., $|C_1|=4$. This indicates that these eight zero energy states are topologically protected Majorana edge states. Specifically, we choose two of them at the same edge with $v_x>0$ near the $K$ point and show the real space probability amplitude of the one with smaller and larger $|k_x|$ in Figs. \[fig2\](c) and (d), respectively. Note that we separate the A and B sublattices by the blue solid and red dashed curves. It is seen that the amplitudes of both A and B sublattices in two Majorana edge states show obvious decay and oscillation. However, the penetration lengths are different between these two Majorana edge states.
![(Color online) Bulk excitation gap $E_{\rm gap}$ of the TSC from the QAH state as a function of $\Delta$. The solid curves with diamonds, crosses and plus signs correspond to the numerical results at $\mu=0$, $1\ $meV and $2\ $meV, respectively. The analytical results at $\mu=0$, $1\
$meV and $2\ $meV are separately represented by the symbols of squares, upward triangles and downward triangles. Note that for the analytical results at $\mu\ne 0$, only two limits, i.e., $\Delta\sim 0$ and $\Delta\sim \Delta_c$ are calculated. In addition, the dotted (dashed) curve corresponds to $E_{\rm gap}=\Delta$ ($E_{\rm gap}=10\Delta$). $V_z=6\ $meV and $\lambda=4\ $meV.[]{data-label="fig3"}](figww3.eps){width="8.5cm"}
Bulk excitation gap
-------------------
### Chemical potential dependence
The above Majorana edge states are protected by a bulk excitation gap of the TSC from the QAH state. With different chemical potentials chosen in the gap of a QAH system, the bulk excitation gap as a function of the proximity-induced superconducting gap is plotted in Fig. \[fig3\]. In the $\Delta=0$ limit, the system can be considered as two copies of QAH insulators as shown in Fig. \[fig1\](b) but with an energy shift of $-\mu$ ($\mu$) for the particle (hole) one. Then, the bulk excitation gap of our system is determined by these two QAH insulators, i.e., $E_{\rm
gap}=E_{0}-|\mu|$. This nonzero bulk excitation gap in the limit $\Delta=0$ strongly indicates that the bulk excitation gap can be much larger than $\Delta$ especially for small $\Delta$. It is emphasized that the nonzero excitation gap in the $\Delta=0$ limit is totally different from the case of the effective $p$-wave superconductors where the excitation gap is exactly zero in the limit $\Delta=0$.[@alicea2] At the critical point $\Delta_c=\sqrt{V_z^2-\mu^2}$, the bulk excitation gap of our system becomes zero. In between, the bulk excitation gap shows a monotonic decrease with increasing $\Delta$. We emphasize that during this process, $E_{\rm
gap}$ can be larger or even one order of magnitude larger than $\Delta$ by referring to $E_{\rm gap}=\Delta$ (dotted curve) and $E_{\rm
gap}=10\Delta$ (dashed curve). For example, $E_{\rm gap}=4.02\ $meV ($\Delta=0.3\ $meV) at $\mu=0$; $E_{\rm gap}=3.22\ $meV ($\Delta=0.3\ $meV) at $\mu=1\ $meV; $E_{\rm gap}=2.23\ $meV ($\Delta=0.2\ $meV) at $\mu=2\ $meV. This marked enlargement of the gap is in strong contrast to the effective $p$-wave superconductors where the bulk excitation gap is always smaller than the induced superconducting gap.[@sau; @sau2; @alicea2; @lutchyn; @alicea3; @halperin; @stanescu; @mourik; @yzhou; @bysun] This makes our proposal, i.e., the TSC from the QAH state, very promising for the realization of robust Majorana modes in experiments.
To have a better understanding of the behavior of the bulk excitation gap of the TSC from the QAH state, we also perform an analytic derivation. Near the Dirac points, the BdG Hamiltonian $H_{\rm BdG}({\bf k})$ in Eq. (3) can be expanded as a low energy effective one with $H_e({\bf k})$ \[see Eq. (\[hamilnosc\])\] being replaced by $H^{\rm
eff}_{e}({\bf k})$ \[see Eq. (\[effhamil\])\]. The secular equation of the eigenvalue $E$ is ${\rm det}[H_{\rm BdG}({\bf k})-EI_{8\times 8}]=0$ where $I_{8\times 8}$ is a unit matrix. After a careful calculation, we have $$\begin{aligned}
&&[\alpha_1^2-4V_z^2\alpha_3+4\alpha_1 (\lambda_R^2-\mu^2-\mu
V_z)+4\alpha_2(\mu^2-\lambda_R^2)]^2\nonumber\\
&&\mbox{}-64V_z^2\alpha_3(\lambda_R^2-\mu^2-\mu
V_z)^2+8[\alpha_1\mu-2(\mu+V_z)\nonumber\\
&&\mbox{}\times(\mu^2-\lambda_R^2)][(\mu+V_z)(\alpha_1^2-4V_z^2\alpha_3)-2\mu
\alpha_1\alpha_2]=0\nonumber\\\label{secular}\end{aligned}$$ with $\alpha_1=\alpha_2-\alpha_3+\alpha_4$, $\alpha_2=v_f^2k_x^2$, $\alpha_3=E^2$, $\alpha_4=\Delta^2-V_z^2+\mu^2$, $v_f=3t/2$ and $\lambda_R=3\lambda/2$. Note that we focus on the calculation near the $K=(4\pi/3,0)$ ($\tau=1$) and set $k_y=0$ by considering the isotropy of the low energy effective Hamiltonian. It is very difficult to obtain the eigenvalues by solving Eq. (\[secular\]) directly. Instead of the eigenvalues, we are interested in the bulk excitation gap here. Differentiating Eq. (\[secular\]) with respect to $\alpha_2$ and then employing the extreme value condition of the excitation gap (i.e., $\frac{\partial
\alpha_3}{\partial \alpha_2}=0$), we have $$\begin{aligned}
\alpha_3^3-g_2\alpha_3^2-g_1\alpha_3-g_0=0\label{diff}\end{aligned}$$ where $g_2=3(\alpha_2+\alpha_4)+2(2\lambda_R^2-\mu^2+2V_z^2)$, $g_1=-3(\alpha_2+\alpha_4)^2+4(-2\lambda_R^2-V_z^2+\mu^2)(\alpha_2+\alpha_4)+4\alpha_2(\lambda_R^2+\mu^2)-8V_z^2(\lambda_R^2+\mu^2)$ and $g_0=(\alpha_2+\alpha_4)^3-2(\mu^2-2\lambda_R^2)(\alpha_2+\alpha_4)^2+4(\alpha_2+\alpha_4)
[\alpha_2(-\lambda_R^2-\mu^2)-2\lambda_R^2(\mu^2-V_z^2-\lambda_R^2)]+8\alpha_2(\mu^4-\lambda_R^4)
+8\alpha_4\lambda_R^2(\mu^2-\lambda_R^2)$.
At $\mu=0$, Eq. (\[diff\]) can be simplified to $(\alpha_2+\alpha_4-\alpha_3)(4\alpha_3^2+q_1\alpha_3+q_2)=0$ with $q_1=-8(\alpha_2+\alpha_4)-16(\lambda_R^2+V_z^2)$ and $q_2=4(\alpha_2+\alpha_4)^2+16\lambda_R^2(\alpha_2+\alpha_4)-16\lambda_R^2\alpha_2+32\lambda_R^2V_z^2$. Since the equation $4\alpha_3^2+q_1\alpha_3+q_2=0$ is inconsistent with the gap closing condition, we only have $\alpha_2+\alpha_4-\alpha_3=\alpha_1=0$. With this condition together with Eq. (\[secular\]), one obtains the bulk excitation gap $E_{\rm gap}=E_0(1-|\Delta|/|V_z|)$, which is linearly dependent on $\Delta$ and agrees very well with the numerical results as shown in Fig. \[fig3\]. Specially, for $E_{\rm
gap}>|\Delta|$ ($E_{\rm gap}>10|\Delta|$), we have $|\Delta|<E_0|V_z|/(E_0+|V_z|)\equiv \Delta_1$ ($|\Delta|<E_0|V_z|/(E_0+10|V_z|)\equiv \Delta_2\approx 0.1E_0$). These conditions will guide the experiments to obtain robust TSCs and Majorana modes. As for the case of $\mu\ne 0$, it is very difficult for us to obtain an exact analytic solution. Only the analytical results in two limits, i.e., $|\Delta|\sim \Delta_c$ and $\Delta\sim 0$, are given. In the $|\Delta|\sim
\Delta_c$ limit, we have $E_{\rm
gap}=(\Delta_c-|\Delta|)\Delta^2_c|\mu^2-\lambda_R^2|/\sqrt{{V_z^2(\Delta_c^2+\lambda_R^2)(\mu^4+\lambda_R^2\Delta_c^2)}}$. In the limit $\Delta\sim 0$, $E_{\rm
gap}=\sqrt{(E_0-\mu)^2-\Delta^2w_2/w_1}$ with $w_1=-\lambda_R^2\mu^2V_z^2+\mu (\lambda_R^2+V_z^2)(-\lambda_R^2+\mu^2-V_z^2)E_0$ and $w_2=\mu (\lambda_R^2-\mu^2)(\lambda_R^2+V_z^2)E_0-16\lambda_R^2V_z^2(\lambda_R^4-\mu^2V_z^2-2\lambda_R^2\mu^2
+\lambda_R^2V_z^2)/(\lambda_R^2+V_z^2)$ by assuming $0<\mu<E_0$. The analytical results at $\mu\ne 0$ in both limits agree fairly well with the numerical ones as shown in Fig. 3.
![(Color online) Numerical results of bulk excitation gap $E_{\rm gap}$ of the TSC from the QAH state as a function of the proximity-induced superconducting gap $\Delta$ at $\mu=0$ (a) under different $\lambda$ with $V_z=6\ $meV and (b) under different $V_z$ with $\lambda=4\ $meV.[]{data-label="S2"}](figww5a.eps "fig:"){width="8.5cm"} ![(Color online) Numerical results of bulk excitation gap $E_{\rm gap}$ of the TSC from the QAH state as a function of the proximity-induced superconducting gap $\Delta$ at $\mu=0$ (a) under different $\lambda$ with $V_z=6\ $meV and (b) under different $V_z$ with $\lambda=4\ $meV.[]{data-label="S2"}](figww5b.eps "fig:"){width="8.5cm"}
### Rashba SOC strength and exchange field dependences
We then turn to investigate the effects of the Rashba SOC and exchange field on the bulk excitation gap of the TSC from the QAH state. In Figs. \[S2\](a) and (b), we plot the dependence of the bulk excitation gap on the proximity-induced superconducting gap at $\mu=0$ under different Rashba SOC strengths and exchange fields, respectively. It is seen that the bulk excitation gap increases with the increase of either the Rashba SOC strength or exchange field. This can be easily understood from $E_{\rm gap}=E_0(1-|\Delta|/|V_z|)$ mentioned above where $E_0$ (see Appendix \[appA\]) increases with increasing Rashba SOC strength and exchange field.
SUMMARY AND DISCUSSION
======================
In summary, we have proposed that in the presence of proximity-induced $s$-wave superconductivity, the QAH state due to the Rashba SOC and exchange field in single layer graphene can become a two-dimensional TSC. With very low or even zero doping near the Dirac points, i.e., two inequivalent valleys, we show that this TSC, which exhibits a Chern number as large as four and hosts four Majorana edge modes, has a bulk excitation gap being larger or even one order of magnitude lager than the proximity-induced superconducting gap. The unique feature is in strong contrast to the case of the effective $p$-wave superconductors where the bulk excitation gap is always smaller than the proximity-induced superconducting gap. This also applies to other QAH systems as seed materials to obtain robust TSCs and Majorana modes.
Finally, we address the experimental feasibility of the TSC from the QAH state. Single layer graphene on the (111) surface of an antiferromagnetic insulator BiFeO$_3$ can have an exchange field ($V_z=142\ $meV) and Rashba SOC ($\lambda=1.4\ $meV), realizing a QAH insulator with a gap being $2E_0=4.2\ $meV.[@qiao2] This QAH state ($|\mu|<E_0$) in proximity to a conventional $s$-wave superconductor (eg., Nb with a large superconducting gap $\Delta_{\rm Nb}=0.83\ $meV [@jwang]) becomes a TSC since the topologically nontrivial condition $\Delta^2+\mu^2<V_z^2$ is easily satisfied due to $|\mu|<E_0\ll |V_z|$ and $|\Delta|<|\Delta_{\rm Nb}|\ll |V_z|$. With $\Delta=0.5\ $meV ($\Delta_2=0.21\ $meV $<\Delta<\Delta_1=2.1\ $meV) for estimation, we have the bulk excitation gap $E_{\rm gap}=2.05\ $meV, $1.56\ $meV and $1.06\ $meV, corresponding to a temperature of $23.8\ $K, $18.1\ $K and $12.3\ $K, at $\mu=0$, $0.5\ $meV and $1\ $meV, respectively. The large excitation gap (of the order of $10\ $K) ensures that robust Majorana modes can be achieved.
This work was supported by the National Natural Science Foundation of China under Grant No. 11334014 and 61411136001, the National Basic Research Program of China under Grant No. 2012CB922002 and the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB01000000.
$H_e({\bf k})$ in Eq. (3) near the Dirac points {#appA}
===============================================
Near the Dirac points, i.e., $K=(4\pi/3,0)\ (\tau=1)$ and $K^{\prime}=(-4\pi/3,0)\ (\tau=-1)$, $H_e({\bf k})$ in Eq. (3) can be expanded as a low energy effective Hamiltonian
$$\begin{aligned}
H^{\rm eff}_{e}({\bf k})=\left(\begin{array}{cccc}
V_z & v_f(\tau k_x-ik_y) & 0 & i\lambda_R(1-\tau)\\
v_f(\tau k_x+ik_y) & V_z & -i\lambda_R(1+\tau) & 0\\
0 & i\lambda_R(1+\tau) & -V_z & v_f(\tau k_x-ik_y)\\
i\lambda_R(\tau-1) & 0 & v_f(\tau k_x+ik_y) & -V_z\label{effhamil}
\end{array}\right).\end{aligned}$$
The energy spectrum of this effective Hamiltonian is shown in Fig. 1(b). The minimum (maximum) energy of the conduction (valence) band is $E_0=|V_z\lambda_R|/\sqrt{V_z^2+\lambda_R^2}$ ($-E_0$) after a simple calculation and then the band gap is given by $2E_0$.
Gap closing condition of the BdG Hamiltonian $H_{\rm BdG}({\bf k})$ {#appB}
===================================================================
The gap of $H_{\rm BdG}({\bf k})$ closes at the momenta $\Gamma$ (single one), $M$ (three inequivalent ones) and $K$ (two inequivalent ones) points. Specifically, at the momentum $\Gamma$, the Rashba SOC vanishes \[see Eq. (\[hamilnosc\])\], which is similar to the previous studies in semiconductors.[@sau; @sau2] The gap closing condition is given by $(\mu\pm
3t)^2=V^2_z-\Delta^2$ with $+$ ($-$) representing lower (higher) energy band at $\Gamma$ after a simple calculation. As for the momentum $M$, the Rashba SOC does not cause spin splitting but lead to an energy shift for the spin degenerate bands. We take $M=(0,\frac{2\sqrt{3}\pi}{3})$ for example and $H_{\rm BdG}(M)$ \[see Eq. (3)\] reads
$$\begin{aligned}
H_{\rm BdG}(M)=\left(\begin{array}{cccccccc}
-\mu+V_z & \frac{\sqrt{3}i-1}{2}t & 0 & -\lambda(i+\sqrt{3}) & \Delta & 0 & 0 & 0\\
\frac{-\sqrt{3}i-1}{2}t & -\mu+V_z & -\lambda(\sqrt{3}-i) & 0 & 0 & \Delta & 0 & 0\\
0 & -\lambda(i+\sqrt{3}) & -\mu-V_z & \frac{\sqrt{3}i-1}{2}t & 0 & 0 & \Delta & 0\\
-\lambda(\sqrt{3}-i) & 0 & \frac{-\sqrt{3}i-1}{2}t & -\mu-V_z & 0 & 0 & 0 & \Delta\\
\Delta & 0 & 0 & 0 & \mu+V_z & -\frac{\sqrt{3}i-1}{2}t & 0 & \lambda(i+\sqrt{3})\\
0 & \Delta & 0 & 0 & \frac{\sqrt{3}i+1}{2}t & \mu+V_z & \lambda(\sqrt{3}-i) & 0\\
0 & 0 & \Delta & 0 & 0 & \lambda(i+\sqrt{3}) & \mu-V_z & -\frac{\sqrt{3}i-1}{2}t\\
0 & 0 & 0 & \Delta & \lambda(\sqrt{3}-i) & 0 & \frac{\sqrt{3}i+1}{2}t & \mu-V_z
\end{array}\right).\end{aligned}$$
Performing a unitary transformation as $\tilde{H}_{\rm
BdG}(M)=U_M^{\dagger}H_{\rm BdG}(M)U_M$ with $$\begin{aligned}
U_M=\frac{\sqrt{2}}{2}\left(\begin{array}{cccccccc}
0 & \frac{1-\sqrt{3}i}{2} & 0 & \frac{-1+\sqrt{3}i}{2} & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0\\
\frac{1-\sqrt{3}i}{2} & 0 & \frac{-1+\sqrt{3}i}{2} & 0 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1-\sqrt{3}i}{2} & 0 & \frac{-1+\sqrt{3}i}{2}\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 1\\
0 & 0 & 0 & 0 & \frac{1-\sqrt{3}i}{2} & 0 & \frac{-1+\sqrt{3}i}{2} & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 0
\end{array}\right),\end{aligned}$$ one obtains $$\begin{aligned}
\tilde{H}_{\rm BdG}(M)=\left(\begin{array}{cccccccc}
-t-\mu-V_z & 0 & 0 & -2i\lambda & \Delta & 0 & 0 & 0\\
0 & -t-\mu+V_z & -2i\lambda & 0 & 0 & \Delta & 0 & 0\\
0 & 2i\lambda & t-\mu-V_z & 0 & 0 & 0 & \Delta & 0\\
2i\lambda & 0 & 0 & t-\mu+V_z & 0 & 0 & 0 & \Delta\\
\Delta & 0 & 0 & 0 & t+\mu-V_z & 0 & 0 & 2i\lambda\\
0 & \Delta & 0 & 0 & 0 & t+\mu+V_z & 2i\lambda & 0\\
0 & 0 & \Delta & 0 & 0 & -2i\lambda & -t+\mu-V_z & 0\\
0 & 0 & 0 & \Delta & -2i\lambda & 0 & 0 & -t+\mu+V_z
\end{array}\right).\end{aligned}$$ At $\mu\sim t$, the block with the diagonal terms being $-t-\mu\mp V_z$ and $t+\mu\mp V_z$ in $\tilde{H}_{\rm BdG}(M)$ is far from gap closing whereas the gap closing is determined by the remaining one. By considering that $|\lambda|\ll t$, we use the Löwdin partition method [@lowdin; @winkler] to obtain the effective Hamiltonian for the block determining the gap closing as $$\begin{aligned}
H_{\rm eff}(M)=\left(\begin{array}{cccc}
t-\mu-V_z+\frac{2\lambda^2}{t-V_z} & 0 & \Delta & 0\\
0 & t-\mu+V_z+\frac{2\lambda^2}{t+V_z} & 0 & \Delta\\
\Delta & 0 & -t+\mu-V_z-\frac{2\lambda^2}{t+V_z} & 0\\
0 & \Delta & 0 & -t+\mu+V_z+\frac{2\lambda^2}{-t+V_z}
\end{array}\right).\end{aligned}$$ Then, the gap closing condition is $(t-\mu-V_z+\frac{2\lambda^2}{t-V_z})(-t+\mu-V_z-\frac{2\lambda^2}{t+V_z})-\Delta^2=0$ or $(t-\mu+V_z+\frac{2\lambda^2}{t+V_z})(-t+\mu+V_z-\frac{2\lambda^2}{-t+V_z})-\Delta^2=0$. As $|V_z|\ll t$, both conditions become $(t-\mu+\frac{2\lambda^2}{t})^2=V_z^2-\Delta^2$ approximately. Furthermore, by considering that $|\lambda|\ll t$, we neglect the energy shift of $2\lambda^2/{t}$ and then the gap closing condition at the momentum $M$ with $\mu\sim t$ is given by $(t-\mu)^2=V_z^2-\Delta^2$. Similarly, the gap closing condition at $M$ with $\mu\sim -t$ is $(t+\mu)^2=V_z^2-\Delta^2$ under the approximation $|\lambda|,|V_z|\ll t$.
In contrast to the momenta $\Gamma$ and $M$, the Rashba SOC at the Dirac points contributes to a finite spin splitting. Specifically, with $K=(4\pi/3,0)$, $H_{\rm BdG}(K)$ \[see Eq. (3)\] can be written as $$\begin{aligned}
H_{\rm BdG}({K})=\left(\begin{array}{cccccccc}
-\mu+V_z & 0 & 0 & 0 & \Delta & 0 & 0 & 0\\
0 & -\mu+V_z & -3i\lambda & 0 & 0 & \Delta & 0 & 0\\
0 & 3i\lambda & -\mu-V_z & 0 & 0 & 0 & \Delta & 0\\
0 & 0 & 0 & -\mu-V_z & 0 & 0 & 0 & \Delta\\
\Delta & 0 & 0 & 0 & \mu+V_z & 0 & 0 & 0\\
0 & \Delta & 0 & 0 & 0 & \mu+V_z & 3i\lambda & 0\\
0 & 0 & \Delta & 0 & 0 & -3i\lambda & \mu-V_z & 0\\
0 & 0 & 0 & \Delta & 0 & 0 & 0 & \mu-V_z\\
\end{array}\right),\end{aligned}$$ which can be divided into two independent $4\times 4$ parts, i.e., $H_1$ ($H_2$) without (with) the Rashba SOC terms. Then, we have $$\begin{aligned}
H_1=\left(\begin{array}{cccc}
-\mu+V_z & 0 & \Delta & 0\\
0 & -\mu-V_z & 0 & \Delta\\
\Delta & 0 & \mu+V_z & 0\\
0 & \Delta & 0 & \mu-V_z\\
\end{array}\right),\end{aligned}$$ which is exactly the same as the Hamiltonian of semiconductors with the Rashba SOC, magnetic field and proximity-induced $s$-wave superconductivity at the momentum $\Gamma$.[@sau; @sau2] This indicates that both have the same gap closing condition, i.e., $V^2_z=\mu^2+\Delta^2$.[@sau; @sau2] As for $H_2$ (not shown), due to the existence of the nonzero Rashba SOC terms, the gap is always opened. Therefore, the gap closing condition at $K$ is just the one in $H_1$ part. Similar analysis can be applied to $K^{\prime}$ and we obtain the same gap closing condition as $K$.
Numerical method for calculating Majorana edge states in zigzag and armchair graphene ribbons {#appC}
=============================================================================================
We investigate the Majorana edge states near the Dirac points in both zigzag and armchair graphene ribbons. We first study the case of zigzag configuration. The Hamiltonian of zigzag ribbon can be obtained from Eq. (1) by choosing a unit cell and performing a Fourier transformation along the direction parallel to the edge (assuming $x$-direction). Note that the unit cell of the zigzag ribbon is the same as the one in Ref. . Specifically,
$$\begin{aligned}
H_{\rm zigzag}&=&-t\sum_{k_x}\sum_{\langle j_1,j_2
\rangle\sigma}[1+|{\rm sgn}(x_{j_2}-x_{j_1})|e^{ik_x {\rm
sgn}(x_{j_2}-x_{j_1})}]c^{\dagger}_{k_xj_1\sigma}c_{k_xj_2\sigma}+\sum_{k_x}\sum_{j\sigma}(\sigma
V_z-\mu)c^{\dagger}_{k_xj\sigma}c_{k_xj\sigma}\nonumber\\
&&\mbox{}+\Delta\sum_{k_x}\sum_{j}(c^{\dagger}_{k_xj\uparrow}c^{\dagger}_{-k_xj\downarrow}+{\rm
H.c.})+i\lambda\sum_{k_x}\sum_{\langle j_1,j_2
\rangle\sigma\sigma^{\prime}}[(\sigma_x^{\sigma\sigma^{\prime}}d^y_{j_1j_2}-\sigma_y^{\sigma\sigma^{\prime}}d^x_{j_1j_2})
+|{\rm sgn}(x_{j_2}-x_{j_1})|e^{ik_x{\rm sgn}(x_{j_2}-x_{j_1})}\nonumber\\
&&\mbox{}\times(\sigma_x^{\sigma\sigma^{\prime}}d^y_{j_1j_2}
+\sigma_y^{\sigma\sigma^{\prime}}d^x_{j_1j_2})]c^{\dagger}_{k_xj_1\sigma}c_{k_xj_2\sigma^{\prime}}\end{aligned}$$
where $x_{j_2}-x_{j_1}$ is the relative position between $j_2$-th and $j_1$-th atoms in the unit cell along the $x$-direction and sgn stands for the sign function. By exactly diagonalizing $H_{\rm
zigzag}$, one obtains the eigenvalues and eigenstates. However, this method fails due to the computational limitations when the width of the ribbon becomes very large (eg., of the order of $10^4$ atoms in the unit cell in our calculation). Alternatively, the zigzag ribbon with the leading term, i.e., the hopping term, can be solved analytically near the Dirac points.[@neto] Near $K$ ($\tau=1$) and $K^{\prime}$ ($\tau=-1$), the eigenstates are given by $$\begin{aligned}
\Psi_{\tau k_x}^{z,\varepsilon}({\bf r})&=&Ae^{i(\tau|K|+k_x)x}\nonumber\\
&&\hspace{-1.6cm}\mbox{}\times\left(\begin{array}{c}
-v_f[(z-\tau k_x)e^{zy}+(z+\tau k_x)e^{-zy}]/\varepsilon \\
e^{zy}-e^{-zy} \\
\end{array}\right),\label{zig}\end{aligned}$$ with the eigenvalues being $\varepsilon^2=v_f^2(k_x^2-z^2)$ and $A=\sqrt{\frac{\sqrt{3}}{|2(e^{2zL}-e^{-2zL})/z-8L|}}$. $L$ is the width of the ribbon and $z$ is determined by the equation $e^{-2zL}=(k_x+\tau z)/(k_x-\tau z)$. Note that if $z_0$ is a solution of this equation, so does $-z_0$. As $\Psi_{\tau k_x}^{z_0,\varepsilon}=-\Psi_{\tau k_x}^{-z_0,\varepsilon}$, only one of these two equivalent eigenstates needs to be taken. Then, one can use these eigenstates in Eq. (\[zig\]) with additional spin and particle-hole degrees of freedom included to construct complete basis functions for $H_{\rm zigzag}$. We diagonalize the Hamiltonian matrix of $H_{\rm zigzag}$ and obtain the energy spectrum and wavefunctions as shown in Fig. 2.
![(Color online) (a) Energy spectrum of armchair graphene ribbon in the presence of the Rashba SOC, exchange field and $s$-wave superconductivity from the proximity effect. (b) ((c)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) momentum $|k_x|$ ($k_x<0$) at one edge (i.e., $y=40000$) (only part of the ribbon is shown). The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=2\ $meV and $\Delta=2\ $meV. []{data-label="S1"}](figww4a.eps "fig:"){width="6cm"}\
![(Color online) (a) Energy spectrum of armchair graphene ribbon in the presence of the Rashba SOC, exchange field and $s$-wave superconductivity from the proximity effect. (b) ((c)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) momentum $|k_x|$ ($k_x<0$) at one edge (i.e., $y=40000$) (only part of the ribbon is shown). The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=2\ $meV and $\Delta=2\ $meV. []{data-label="S1"}](figww4b.eps "fig:"){width="6cm"} ![(Color online) (a) Energy spectrum of armchair graphene ribbon in the presence of the Rashba SOC, exchange field and $s$-wave superconductivity from the proximity effect. (b) ((c)) Real space probability amplitude $|\psi|$ across the width for the Majorana edge state with a smaller (larger) momentum $|k_x|$ ($k_x<0$) at one edge (i.e., $y=40000$) (only part of the ribbon is shown). The fluctuations of $|\psi|$ at the positions far away from the edge are due to numerical error. Here, $V_z=6\ $meV, $\lambda=4\ $meV, $\mu=2\ $meV and $\Delta=2\ $meV. []{data-label="S1"}](figww4c.eps "fig:"){width="6cm"}
We turn to the case of armchair graphene ribbon with the Hamiltonian being
$$\begin{aligned}
H_{\rm armchair}&=&-t\sum_{k_x}\sum_{\langle j_1,j_2
\rangle\sigma}c^{\dagger}_{k_xj_1\sigma}c_{k_xj_2\sigma}+\sum_{k_x}\sum_{j\sigma}(\sigma
V_z-\mu)c^{\dagger}_{k_xj\sigma}c_{k_xj\sigma}+\Delta\sum_{k_x}\sum_{j}(c^{\dagger}_{k_xj\uparrow}c^{\dagger}_{-k_xj\downarrow}+{\rm
H.c.})\nonumber\\
&&\hspace{-1.5cm}\mbox{}+i\lambda\sum_{k_x}\sum_{\langle j_1,j_2
\rangle\sigma\sigma^{\prime}}(\sigma_x^{\sigma\sigma^{\prime}}d^y_{j_1j_2}-\sigma_y^{\sigma\sigma^{\prime}}d^x_{j_1j_2})
c^{\dagger}_{k_xj_1\sigma}c_{k_xj_2\sigma^{\prime}}
-t\sum_{k_x}\sum_{j^*_1j^*_2\sigma}[e^{i\sqrt{3}k_x}(\delta_{j^*_1,j^*_2}+\delta_{j^*_1+1,j^*_2})c^{\dagger}_{k_xj^*_1\sigma}c_{k_xj^*_2\sigma}+{\rm
H.c.}]\nonumber\\
&&\hspace{-1.5cm}\mbox{}+i\lambda\sum_{k_x}\sum_{j^*_1j^*_2\sigma\sigma^{\prime}}\{e^{i\sqrt{3}k_x}
[\delta_{j^*_1+1,j^*_2}(\frac{\sqrt{3}}{2}\sigma_x^{\sigma\sigma^{\prime}}-\frac{1}{2}\sigma_y^{\sigma\sigma^{\prime}})
-\delta_{j^*_1,j^*_2}(\frac{\sqrt{3}}{2}\sigma_x^{\sigma\sigma^{\prime}}+\frac{1}{2}\sigma_y^{\sigma\sigma^{\prime}})]
c^{\dagger}_{k_xj^*_1\sigma}c_{k_xj^*_2\sigma^{\prime}}+{\rm H.c.}\}\end{aligned}$$
in which $j^*_1$ ($j^*_2$) represents the $j^*_1$-th ($j^*_2$-th) atom of the first (fourth) column in the unit cell. Note that the unit cell of the armchair ribbon is the same as the one in Ref. and the edges lie along the $x$-direction. Similar to the case of the zigzag graphene ribbon, we first solve the armchair ribbon with only the hopping term analytically near the Dirac points. The eigenstates read $$\begin{aligned}
\Psi_{k_x}^{k_n,\varepsilon}({\bf r})&=&2Ae^{ik_xx}\sin[(|K|+k_n)y]\nonumber\\
&&\mbox{}\times\left(\begin{array}{c}
-v_f(k_x-ik_n)/\varepsilon \\
i \\
\end{array}\right),\end{aligned}$$ where the eigenvalues are $\varepsilon^2=v_f^2(k_x^2+k_n^2)$ with $k_n=n\pi/L-|K|$ and $A=\frac{1}{\sqrt{8L}}$. These eigenstates construct complete basis functions for $H_{\rm armchair}$ with additional spin and particle-hole degrees of freedom. By diagonalizing the Hamiltonian matrix, one obtains the energy spectrum and eigenstates of armchair graphene ribbon as shown in Fig. \[S1\]. In Fig. \[S1\](a), we find that there exist eight zero energy states, corresponding to four Majorana fermions at each edge, which is similar to the case of zigzag ribbon. We then show the real space probability amplitude of two Majorana edge states at the same edge with a smaller and larger momentum $|k_x|$ ($k_x<0$) in Figs. \[S1\](b) and (c), respectively. It is seen that both show obvious decays and oscillations but the decay lengths and oscillation periods are different.
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[^1]: wlf@mail.ustc.edu.cn
[^2]: mwwu@ustc.edu.cn
|
---
author:
- |
<span style="font-variant:small-caps;">**S. I. Shirokov,$^{1}$**</span>[^1] <span style="font-variant:small-caps;">**A. A. Raikov,$^{2}$**</span> and <span style="font-variant:small-caps;">**Yu. V. Baryshev$^{1}$**</span>\
*$^{1}$St. Petersburg State University, Universitetskii pr. 28, St. Petersburg 198504, Russia\
*$^{2}$Main (Pulkovo) Astronomical Observatory, Russian Academy of Sciences,\
*Pulkovskoe shosse d. 65 cor. 1, St. Petersburg 196140, Russia\
***
title: '**Spatial Distribution of Gamma-Ray Burst Sources**'
---
Introduction
============
According to current observational data, the sources of gamma-ray bursts (GRB) are explosions of massive supernova stars (long GRB) and mergers of neutron stars (short GRB) in distant galaxies. Thus, the spatial distribution of GRB reflects the large-scale distribution of galaxies, so that analyzing the distribution of gamma-ray bursts in space and time is an important task for the study of the evolution of the large-scale structure of the universe. The extreme luminosity of GRB makes it possible to detect their sources at large red shifts, and the available completeness of the surveys of GRB (e.g., Swift) makes it possible to use samples of GRB with known red shifts for preliminary analysis of their spatial distribution over a wide range of scales.
The standard cosmological model assumes a uniform distribution of matter in the universe, including dark matter and dark energy. Observations of the spatial distribution of visible matter (galaxies) do, however, reveal nonuniformities over scales much longer than the standard correlation length, 10 Mpc (the Sloan Great Wall, size $\sim300$ Mpc at $z\sim0.07$ \[1,2\]). In addition, the power law dependence of the conditional density of the distribution of galaxies $\Gamma(r) \propto r^{-1}$ on scales up to 100 Mpc corresponds to a fractal dimensionality $D \sim 2$ \[3-5\]. Large nonuniformities have recently been discovered in the distribution of galaxies in the SDSS/CMASS survey (the BOSS Great Wall, size $\sim400$ Mpc at $z\sim0.47$ \[6\]), as well as in the deep galactic survey COSMOS (Super Large Clusters with sizes $\sim 1000$ Mpc at $z\sim1$ \[7-9\]).
The spatial distribution of gamma-ray bursts has been analyzed in a number of papers \[10-13\]. Thus, the distribution of 244 GRB has been analyzed \[10\] as part of the Swift mission using the $\xi$-function method. The correlation length was $r_0\approx388$ h$^{-1}$ Mpc, $\gamma=1.57\pm0.65$(at the $1\sigma$ level), and the uniformity scale was $r\geq7700$ h$^{-1}$. Other approaches to the study of the correlation properties of the spatial structures are the conditional density method \[3,4\] and the pairwise distance method \[14\]. In Ref. 11 it was applied for the first time to 201 GRB with known (at that time) angular coordinates and red shift. An estimate of $D\cong2.2\div2.5$ was obtained for the fractal dimensionality. This method also makes it possible to detect close pairs and triplets of points. Thus, for example, a spatially isolated group of five GRB was detected with coordinates $23^h$ $50^m < \alpha < 0^h$ $50^m$ and $5^\circ < \beta < 25^\circ$ a redshift of $0.81 < z < 0.97$. If GRB events are regarded as indicators of the presence of matter in space, then this group is indirect evidence of a supercluster of galaxies within this range of coordinates.
A giant ring of GRB with a diameter of 1720 Mpc at red shifts of $0.78<z<0.86$ has been discovered \[12\].The probability that this structure was found randomly is $2 \times 10^{-6}$. 352 GRB have been found \[13\] to have an estimated dimensionality in terms of the $\Lambda CDM$ model of $D\approx2.3\pm0.1$. In other models, $D\approx2.5$. The latest numerical predictions in terms of the $\Lambda CDM$ model show \[15\] that at large red shifts it is already possible to observe clusterization of matter, so the detection of structures in the distribution of GRB is an important problem.
In this paper, we estimate fractal dimensionality using the conditional density method for GRB for the first time and compare this estimate with estimates derived from the pairwise distance method. As a comparison with the GRB catalog, artificial fractal and uniform catalogs are modeled. Effects owing to the sample geometry are taken into account, in particular: limits on the maximum radius sphere and cutoff of the galactic belt. The evolution of luminosity with increasing red shift is examined. For the first time these plots are compared with a uniform distribution, so it is possible to compare the efficiency of the two methods. The power law dependence over a large range of scales also shows up more clearly.
The sample
==========
The Swift Gamma-ray Burst Mission on line catalog \[16\], which has been extended in Ref. 17 and in another on line catalog \[18\], is used as the basis for the GRB catalog. Our catalog of GRB sources with known red shifts contains a total of 384 objects. Our approach makes it possible to use all the points to determine the fractal dimensionality without additional truncations and selection. The only condition for including objects in the combined catalog is the existence of angular coordinates and red shifts. Thus, of the working sample ($<8$ Gpc), 377 GRB remain, for 360 of which the luminosities have been determined. The combined catalog has been updated to June 2017.
The first rows of our working sample are shown in Figs. 1 and 2. The headings of the tables correspond to:the name of the event, galactic coordinates, the estimated time of the event, the received radiation in $10^{-7}$ erg/cm$^{-2}$ in the $15\div150$ keV band during the time $T_{90}$ of the event, the red shift, the mission (Swift), metric coordinates, distance to the GRB source, the logarithm of the flux, and the logarithm of the luminosity.
Methods
=======
All the model calculations were done using the original Fractal Dimension Estimator program, which is described elsewhere.
[.48]{}[>[=0.08]{}rrrrrc]{} name & $l$ & $b$ & $T_{90}$ & $F_{obs}$ & $z$\
151215A & 177.25358 & 8.55309 & 17.80 & 3.10 & 2.590 150423A & 9.70821 & 59.24722 & 0.22 & 0.63 & 1.394 141121A & 200.39117 & 26.85321 & 549.90 & 53.00 & 1.470 ...
\[tab:OR\]
[.48]{}[>[=0.08]{}rrrrrc]{} $X_{Mpc}$ & $Y_{Mpc}$ & $Z_{Mpc}$ & $R_{Mpc}$ & $lg S_{obs}$ & $lg L_b$\
–5862.4 & 281.2 & 882.7 & 5935.1 & –0.75 & 51.97\
2099.7 & 359.2 & 3580.2 & 4166.0 & 0.45 & 52.53\
–3605.5 & –1340.3& 1947.5 & 4311.5 & –1.01 & 51.11\
...\
\[tab:CR\]
Conditional density
-------------------
This method, which is discussed in detail in Refs. 3 and 4, essentially involves counting the number of points in spheres of different radii. The conditionality is that the center of a sphere is a point in the set. For correct operation of this method at large scales, it is necessary to take the effect of the boundary of the set into account \[19\]. The concentration inside a sphere of radius $r$ is given by
$$n(r) = \frac {1}{N_c(r)}\sum_{i=1}^{N_c(r)} \frac {N_i(r)}{V(r)},$$
where $N_c(r)$ is the number of spheres inside the sample, $N_i(r)$ is the number of points in a sphere, and $V(r)$ is the volume of a sphere. On intermediate scales this distribution has a $D-3$ power law dependence,
$$\Gamma^*(r)=\langle n(r'<r) \rangle_p$$
where $\langle ... \rangle_p$ denotes averaging over all points in the sample.
Pairwise distances
-------------------
The distribution of pairwise distances for sets with integral dimensionality is discussed in Ref. 20.
$$f(l)=Dl^{D-1}(L/2)^{-D}I_{\mu}( \frac{D+1}{2},\frac{1}{2}),$$
where $D$ is the integral dimensionality of the set, $l$ is the distance between a pair of points, $L$ is the greatest distance within the set, and $I_\mu(p,q)$ is the incomplete Bessel function with $\mu=1-l^2/L^2$. For $l<<L$ there is an asymptote
$$f(l)\sim l^{D-1},$$
which also is retained for sets with a fractional dimensionality \[21,22,14\].
![The distribution of $T_{90}$ with respect to red shift $z$ for all the GRB sources[]{data-label="fig:T90"}](fig3)
![ Observed luminosity distributions of the GRB sources in fixed distance intervals.[]{data-label="fig:Lum"}](fig4)
Comparison of the methods
-------------------------
For a convenient comparison of the results of the pairwise distances and conditional density methods, plots of the ratio of the curves for a fractal or GRB to a uniform sample are ultimately examined. In this configuration the slope of the working segment, which is determined by the condition of least squares for the deviations, is equal to $D-3$ for each of the methods in logarithmic coordinates.
For sparse sets with $N<10^3$ the distributions of the conditional density and pairwise distances are characterized by substantial random deviations from a power law dependence. To reduce the influence of this effect, the curves are averaged over a sufficient number of runs for the given set. This is done by specifying different zero points for the pseudorandom number generator. This operation is carried out for the conditional density distributions, as well as for the pairwise distance distributions.
Calculating the distance and luminosity
---------------------------------------
In the standard cosmological model the metric distances are given in terms of red shift by the formula \[4\]
$$R(z)_{Mpc}=\frac{c}{H_0} \int_0^z (\Omega^0_v+\Omega_m^0(1+z')^3-\Omega_k^0(1+z')^2)^{-1/2} dz',$$
where $H_0=70$ km/s/Mpc is the Hubble constant, $c=3\cdot10^{10}$ cm/s is the speed of light, $\Omega^0_v=0.7$, $\Omega_m^0=0.3$, and $\Omega_k^0=0$ are the cosmological parameters, and $z$ is the red shift. Then the spherical coordinates are transformed to Cartesian, since the space is Euclidean. The luminosity in terms of the $\Lambda CDM$ model is given by
$$L(z)=4\pi S_{obs} R(z)_{sm}^2 (1+z)^2,$$
where $S_{obs}$ erg/s/cm$^2$ is the GRB flux, which equals the ratio of the radiation received in the $15\div150$ keV range to the time $T_{90}$, and $z$ is the red shift.
The model catalogs
------------------
Fractal and uniform sets analogous to the GRB catalog were modeled for comparison with the actual sample. Since strong absorption of visible radiation (leading to additional selection of GRB when measuring their red shift) occurs near the galactic belt, it is necessary examine the influence of this effect on the estimated fractal dimensionality. Thus, two cases were examined. Examples of these are shown in Fig. 6. The first covers the entire celestial sphere, while the band from $-10$ to $+10$ degrees of galactic latitude is cut out in the second. All of these samples are bounded by a sphere of radius 8 Gpc.
![Distribution of the luminosities of the GRB catalog and the uniform model catalog with respect to distance.[]{data-label="fig:LvsR"}](fig5)
To determine the fractal dimensionality over the entire volume right away, it is necessary to account for the observed selection effects, e.g., the Malmquist effect, for all the points. For this, the observed profiles in spherical layers with a step size of 1 Gpc shown in Fig. 4 are taken as the model visible luminosity function. Since introducing a model selectivity with respect to luminosity affects the conditional density distribution, the ratio of the curves for fractals and GRB to the uniform curves is examined.
Results
=======
General properties of the GRB source catalog
--------------------------------------------
The radial distributions of the GRB catalog are shown in Fig. 1 (integrated distributions) and Fig. 2 (differential distributions). The distribution of the time an event is observed, $T_{90}$, with respect to red shift is shown in Fig. 3. At present, the drift with increasing red shift toward shorter event times predicted by the standard model has not yet been observed. Our result is consistent with Ref. 23. A definitive answer to this question would require a significant increase in the number of GRB at large red shifts and it would be necessary to study the dependence of $T_{90}$ on the distance to the GRB source.
GRB can serve as an indicator of galactic clusters, so it is possible look for close pairs in the spatial distribution of GRB in Table 3. Of the 18 pairs, it is possible to identify spatially distinctive structures of three and four gamma-ray bursts, as well as two pairs for which the distance between the sources is less than 100 Mpc at $z\approx0.013$ and $z\approx1.43$.
[.48]{}[>[=0.08]{}ccrrrc]{} N & Designation & $d_{Mpc}$ & $l$ & $b$ & $z$\
175 & 111005A & 79.4 & 338.33759 & 34.63886 & 0.013\
4 & 100316D & & 266.91664 & –19.78007 & 0.014\
175 & 111005A & 195.6 & 338.33759 & 34.63886 & 0.013\
158 & 060218A & & 166.86303 & –32.86884 & 0.033\
4 & 100316D & 150.1 & 266.91664 & –19.78007 & 0.014\
158 & 060218A & & 166.86303 & –32.86884 & 0.033\
158 & 060218A & 293.1 & 166.86303 & –32.86884 & 0.033\
94 & 051109B & & 100.54662 & –19.39992 & 0.080\
234 & 060505A & 264.9 & 22.09128 & –53.71345 & 0.089\
54 & 060614A & & 344.08607 & –43.94594 & 0.130\
32 & 061201A & 206.0 & 315.71506 & –38.23391 & 0.111\
54 & 060614A & & 344.08607 & –43.94594 & 0.130\
117 & 130427A & 261.1 & 206.48629 & 72.51440 & 0.340\
18 & 130603B & & 236.47527 & 68.43758 & 0.356\
13 & 110328A & 299.3 & 86.71625 & 39.42626 & 0.354\
84 & 151027A & & 90.49260 & 28.48382 & 0.380\
25 & 140903A & 276.7 & 44.40465 & 50.11996 & 0.351\
89 & 101213A & & 37.17510 & 45.89511 & 0.414\
27 & 070724A & 282.8 & 184.32601 & –73.81347 & 0.457\
315 & 091127A & & 197.38677 & –66.73665 & 0.490\
22 & 141212A & 242.7 & 155.24497 & –38.01808 & 0.596\
304 & 130215A & & 163.06996 & –39.75075 & 0.597\
104 & 150323A & 189.0 & 174.83011 & 36.29286 & 0.593\
168 & 110106B & & 172.91003 & 40.47816 & 0.618\
291 & 080916A & 236.0 & 333.57758 & –50.49415 & 0.689\
124 & 150821A & & 329.53474 & –52.37413 & 0.755\
161 & 050824A & 248.8 & 122.21283 & –40.26635 & 0.830\
77 & 080710A & & 116.98198 & –43.17503 & 0.845\
77 & 080710A & 291.4 & 116.98198 & –43.17503 & 0.845\
271 & 060912A & & 113.49717 & –41.34504 & 0.937\
206 & 160131A & 231.1 & 207.86277 & –25.13766 & 0.970\
116 & 120907A & & 208.46983 & –29.19799 & 0.970\
50 & 161108A & 269.0 & 221.80332 & 78.92167 & 1.159\
268 & 90530 & & 212.46605 & 77.98286 & 1.266\
48 & 050822X & 77.0 & 255.27955 & –54.45517 & 1.434\
203 & 050318A & & 256.44382 & –55.23286 & 1.440\
\[tab:paires\]
Fractal dimensionality
----------------------
Figure 5 is a comparison of the luminosity distribution of the GRB with the model uniform catalog after application of the selectivity function with respect to luminosity shown in Fig. 4. Figure6 illustrates the visible differences between the real sample of GRB and the model catalogs for two geometries. In the case of the truncated celestial sphere, the sample is a superposition of two hemispheres. Plots of the conditional density and mutual distances are shown in Figs. 7 and 8, respectively. The graphs of the fractals shown here are average curves over 17 runs and this is the cause of the corresponding deviations for the points in the graphs. The amplitude of the deviations in the pairwise distance method is a factor of two larger than for the conditional density.In all runs of the fractal and uniform samples to which these methods were applied directly, the number of points is roughly equal to the number of GRB. This is done by creating an excess of points in the initial run, and then by selecting them uniformly.
For a more accurate result, the excess in the observed number of GRB on small radial scales must be taken into account, as discussed in Ref. 24.
![Plots of the normalized conditional density for GRB (solid circles) and fractals $D=2.0$ (squares) and $D=2.5$ (circles) for the full celestial sphere. Unity corresponds to a uniform distribution.[]{data-label="fig:CD"}](fig7)
![ Plots of the reduced pairwise distances for GRB (solid circles) and fractals $D=2.0$ (squares) and $D=2.5$ (circles) for the truncated celestial sphere.Unity corresponds to a uniform distribution.[]{data-label="fig:PD"}](fig8)
Conclusions
===========
For model sets with $N>10^3$ points, the accuracy of the estimated fractal dimensionality reaches $\Delta D=0.06$ for the conditional density and $\Delta D=0.03$ for pairwise distances prior to introduction of the model luminosity function and prior to truncation of the galactic belt. After introducing limitations on the geometry and luminosity with the same total number of points, the accuracy of the method decreased by a factor of two. It is noteworthy that the radius of the sphere, and not the diameter, is treated as the parameter in the conditional density, so that the horizontal axis can be multiplied by a factor of two when comparing the methods. The main parameters determining the accuracy of the estimated fractal dimensionality are the number of points and the number of runs. During testing of the accuracy, it was found that the pairwise distance method is better than the conditional density method for short scales.
Thus, the pairwise distance curve reaches a power law dependence at a scale equal to twice the size of the elementary cell of the fractal structure. At the same time the conditional density begins to operate at a scale that is $4\div8$ times the size of the elementary cell, depending on the representativeness of the sample and the value of the fractal dimensionality. On large scales both methods have problems. In the conditional density, averaging is over a small number of spheres, while in the pairwise distances the distribution behaves unpredictably or is close to uniform.
For the case of a full celestial sphere, the estimated fractal dimensionality of the distribution of the GRB sources was $D=2.6\pm0.12$ at $R=1.5\div2.5$ ($d=3\div5$) Gpc for the conditional density and $D=2.6\pm0.06$ for $l=1.5\div5.5$ Gpc. In the case with a truncated galactic belt, the conditional density does not yield a unique result and its distribution does not differ from uniform. The pairwise distances have a stable power law dependence with $D=2.6\pm0.06$ and essentially do not change the interval for the linear segment $l=1.5\div5.5$. Thus, on scales of $\approx 3\div5$ Gpc, there is a power law correlation between the methods.
At distances close to $D=3$, when selection effects are taken into account the methods yield a systematically high estimate for the model sets by roughly $\Delta D=0.1$. Given this bias, it is necessary to correct the result in order to obtain a more accurate estimate. Thus, the estimated fractal dimensionality of the observed GRB distribution is $D=2.55\pm0.06$.
[99]{}
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[^1]: E-mail: arhath.sis@yandex.ru
|
---
abstract: |
Mediation analysis has been used in many disciplines to explain the mechanism or process that underlies an observed relationship between an exposure variable and an outcome variable via the inclusion of mediators. Decompositions of the total causal effect of an exposure variable into effects characterizing mediation pathways and interactions have gained an increasing amount of interest in the last decade. In this work, we develop decompositions for scenarios where the two mediators are causally sequential or non-sequential. Current developments in this area have primarily focused on either decompositions without interaction components or with interactions but assuming no causally sequential order between the mediators. We propose a new concept called natural counterfactual interaction effect that captures the two-way and three-way interactions for both scenarios that extend the two-way mediated interactions in literature. We develop a unified approach for decomposing the total effect into the effects that are due to mediation only, interaction only, both mediation and interaction, neither mediation nor interaction within the counterfactual framework. Finally, we illustrate the proposed decomposition method using a real data analysis where the two mediators are causally sequential.\
\
**Keywords**: causal inference, interaction, mediation, causally sequential mediators
---
Decomposition of the Total Effect for Two Mediators: A Natural Counterfactual Interaction Effect Framework
Xin Gao[^1]$^,$[^2], Li Li$^1$, Li Luo$^{2,}$[^3]
Introduction
============
Mediation analysis has become the technique of choice to identify and explain the mechanism that underlies an observed relationship between an exposure or treatment variable and an outcome variable via the inclusion of intermediate variables, known as mediators. Decompositions of the total effect of the exposure into effects characterizing mediation pathways and interactions help researchers understand the effects through different mechanisms and have gained much attention in literature and application in the last decade [@vmul; @vpre; @d; @s; @m; @v3; @v4; @vbook; @b]. In our motivating example, we are interested in the effects of drinking alcohol on Systolic Blood Pressure (SBP) via the mediators, Body Mass Index (BMI) and Gamma Glutamyl Transferase (GGT), and their interaction effects. Besides, the mediator BMI is previously reported to affect GGT and not vice versa, and hence the two mediators are causally sequential. Current developments in this area for scenarios considering two mediators have primarily focused on either decomposition without interaction components, or decomposition allowing interactions but assuming no causally sequential order between the mediators. Daniel [@d] and Steen et al. [@s] discussed the decompositions in a general framework with causally sequential mediators; however, their decompositions do not include interaction components. Bellavia and Valeri [@b] proposed a decomposition with components describing interactions, but they assumed these mediators are causally non-sequential.
In this work, we develop decomposition methods for the scenarios when the two mediators are causally sequential and extensive interaction effects exist where existing decomposition methods are limited. Our approach also applies to the non-sequential mediators’ scenario. We present a unified approach for decomposing the total effect into the components that are due to mediation only, interaction only, both mediation and interaction, neither mediation nor interaction within the counterfactual framework. Our decomposition methods are motivated by VanderWeele’s four-way decomposition [@v4] of the total effect with one mediator, where the interaction effects include a reference interaction effect for interaction only and a mediated interaction effect for both mediation and interaction. VanderWeele [@v4] emphasized that these interaction terms are often considered of the greatest public health importance [@Rothman1; @Rothman2; @Hosmer]. We also propose a new concept called natural counterfactual interaction effect for describing the two-way and three-way interactions in the two-mediator scenarios that extend the mediated interaction from VanderWeele’s work [@v4]. Since the causal structures are more complex with two mediators, the decompositions have multiple terms for mediation only, interaction only, and both mediation and interaction. More importantly, we find that the terms for interaction only are all identifiable at the individual level when the two mediators are causally non-sequential, but some of them are no longer identifiable when the two mediators are causally sequential. When the two mediators are casually non-sequential, our decomposition uses a different approach from what was proposed by Bellavia and Valeri [@b]. For example, their population-averaged mediated interaction effect between $A$ and $M_1$ is evaluated by controlling $M_2$ at a fixed level while our natural counterfactual interaction effect is essentially a weighted mediated interaction effect where the weights are determined by the distributions of both mediators in the population.
The rest of the paper is organized as follows: Section 2 reviews VanderWeele’s four-way decomposition; Section 3 presents decompositions of total effect for two-mediator scenarios; Section 4 lays out identification assumptions and gives the empirical formulas for computing each component in the decomposition with two causally sequential mediators; Section 5 presents our real data analysis; Section 6 concludes the paper with discussions.
Decomposition of the total effect in a single-mediator scenario
===============================================================
Counterfactual definitions
--------------------------
Consider the single-mediator scenario in Figure \[fig1\]. Counterfactual formulas give the potential value of outcome $Y$ or mediator $M$ that would have been observed if the exposure $A$ or mediator $M$ were fixed at a certain level [@vbook; @riden; @p01]. Let $Y(a)$ denote the potential value of $Y$ that would have been observed if the exposure $A$ were fixed at a constant level $a$ [@vbook]. Similarly, $M(a)$ denotes the potential value of $M$ that would have been observed if $A$ were fixed at $a$ and $Y(a,m)$ denotes the potential value of $Y$ that would have been observed if $A$ and $M$ were fixed at $a$ and $m$, respectively [@vbook]. A nested counterfactual formula $Y(a, M(a^{\ast}))$ denotes the potential value of $Y$ that would have been observed if the exposure were fixed at $a$ and the mediator $M$ were set to what would have been observed or potential value when the exposure were fixed at $a^{\ast}$ (Figure \[fig2\]) [@vbook].\
Two-way decomposition
---------------------
The total effect ($TE$) of the exposure $A$ for an individual is defined as the difference between $Y(a)$ and $Y(a^{\ast})$ [@vbook], where $a$ and $a^{\ast}$ are the treatment and reference level of the exposure $A$, respectively. The classical decomposition of the total effect has two components: natural direct effect ($NDE$) and natural indirect effect ($NIE$) [@vbook; @p01; @rsem]. $NDE$ represents the causal effect along the direct path from $A$ to $Y$ and $NIE$ represents the causal effect along the indirect path from $A$ through $M$ to $Y$. The effects are defined using the following formulas: $$\begin{aligned}
TE & = & Y(a)-Y(a^\ast)\\
& = & Y(a,M(a)) - Y(a^{\ast},M(a^\ast))\\
& = & Y(a,M(a)) - Y(a,M(a^\ast)) + Y(a,M(a^\ast)) - Y(a^{\ast},M(a^\ast)),\\
\\
NDE & = & Y(a,M(a^\ast)) - Y(a^\ast,M(a^\ast)),\\
\\
NIE & = & Y(a,M(a)) - Y(a,M(a^\ast)).\end{aligned}$$
The second equality of $TE$ follows by the composition axiom [@vbook; @a] and the third equality of $TE$ follows by subtracting and adding the same counterfactual formula $Y(a,M(a^\ast))$. $NDE$ is the difference in the potential value of outcome when $A$ goes from $a^\ast$ to $a$ and $M$ is at its potential value $M(a^\ast)$. $NIE$ is the difference in the potential value of outcome had $M$ goes from $M(a^\ast)$ to $M(a)$ while $A$ is at its treatment level $a$. In literature, $NDE$ and $NIE$ are also referred to as pure direct effect ($PDE$) [@riden] and total indirect effect ($TDE$) [@riden], respectively. Furthermore, $NDE$ also corresponds to a path-specific effect proposed by Pearl [@p01].
Four-way decomposition with interactions
----------------------------------------
VanderWeele [@v4] proposed a four-way decomposition in a single-mediator scenario where the exposure interacts with the mediator. The total effect of the exposure on the outcome is decomposed into components due to mediation only, to interaction only, to both mediation and interaction, and to neither mediation nor interaction. These four components are termed as pure indirect effect ($PIE$), reference interaction effect ($INT_{ref}(m^\ast)$), mediated interaction effect ($INT_{med}$) and controlled direct effect ($CDE(m^\ast)$), respectively, where $m^\ast$ is an arbitrarily chosen fixed reference level of the mediator $M$. At the individual level, the four components are expressed in general forms [@v4]: $$\begin{aligned}
CDE(m^\ast) & = & Y(a,m^\ast)-Y(a^\ast,m^\ast),\\
\\
INT_{ref}(m^\ast) & = & \sum_{m} [Y(a,m)-Y(a^\ast,m)-Y(a,m^\ast)+Y(a^\ast,m^\ast)]\times I(M(a^\ast)=m),\\
\\
INT_{med} & = & \sum_{m} [Y(a,m)-Y(a^\ast,m)-Y(a,m^\ast)+Y(a^\ast,m^\ast)]\\
& & \times [I(M(a)=m)-I(M(a^\ast)=m)],\\
\\
PIE & = & \sum_{m}[Y(a^\ast,m)-Y(a^\ast,m^\ast)]\times [I(M(a)=m)-I(M(a^\ast)=m)].\end{aligned}$$
The reference and mediated interaction effects can also be expressed in the form of the counterfactual formulas in our view: $$\begin{aligned}
INT_{ref}(m^\ast) & = & Y(a,M(a^\ast))- Y(a^\ast,M(a^\ast))-Y(a,m^\ast)+Y(a^\ast,m^\ast),\\
\\
INT_{med} & = & Y(a,M(a))-Y(a^\ast,M(a))-Y(a,M(a^\ast))+Y(a^\ast,M(a^\ast)). $$
$CDE$ measures the effect of $A$ had $M$ be fixed at level $m^\ast$. $INT_{ref}(m^\ast)$ measures the change in the effect of $A$ had $M$ go from $m^\ast$ to $M({a^\ast})$. If $M({a^\ast}) = m^\ast$, $INT_{ref}(m^\ast)$ for the individual considered is reduced to zero. $INT_{med}$ describes the change in the effect of $A$ had $M$ go from $M({a^\ast})$ to $M({a})$. When $A$ has no effect on the mediator, $M({a^\ast})=M({a})$, and $INT_{med}$ becomes zero. $PIE$ describes the effect of $M$ when $A$ is set at $a^\ast$ and $M$ goes from $M({a^\ast})$ to $M({a})$.
When $A$ and $M$ are both binary with the conditions $a=1$, $a^\ast=0$ and $m^\ast=0$, the counterfactual definitions of the components become [@v4]: $$\begin{aligned}
CDE(0) & = & Y(1,0)-Y(0,0),\\
\\
INT_{ref}(0) & = & [Y(1,1)-Y(1,0)-Y(0,1)+Y(0,0)]\times M(0),\\
\\
INT_{med} & = & [Y(1,1)-Y(1,0)-Y(0,1)+Y(0,0)]\times[M(1)-M(0)],\\
\\
PIE & = & [Y(0,1)-Y(0,0)]\times[M(1)-M(0)],\end{aligned}$$ where $1$ is the treatment level and $0$ is the reference level [@v4].\
Both $INT_{ref}$ and $INT_{med}$ have an additive interaction $[Y(1,1)-Y(1,0)-Y(0,1)+Y(0,0)]$ term which will be non-zero for an individual if the joint effect of having both the exposure and the mediator present differs from the sum of the effects of having only the exposure or mediator present. The additive interaction effect is generally considered of great public health importance [@Rothman1; @Rothman2; @Hosmer]. The difference between $INT_{ref}$ and $INT_{med}$ is that $INT_{ref}$ is non-zero only if the mediator is present in the absence of exposure (i.e. $M(0) = 1$) whereas $INT_{med}$ is non-zero only if the exposure has an effect on the mediator (i.e. $M(1)-M(0) \neq 0$).
Based on the counterfactual formula form of mediated interaction $INT_{med}$, we propose the natural counterfactual interaction effect and provide the following definition. The mediated interaction effect and natural counterfactual interaction effect are mathematically equivalent in the single mediator scenario, we define it from a different perspective only for building up the concepts for scenarios with two mediators in section 3.\
**Definition 1**. Natural counterfactual interaction effect of $A$ and $M$ in a single-mediator scenario: $$\begin{aligned}
NatINT_{AM} := Y(a,M(a)) - Y(a^\ast,M(a)) - Y(a,M(a^\ast)) + Y(a^{\ast},M(a^\ast)),\end{aligned}$$ where $M(a^\ast)$ and $M(a)$ denote the potential values of $M$ that would have occurred if $A$ were fixed at $a^\ast$ and $a$, respectively.
Decomposition of the total effect in two-mediator scenarios
===========================================================
When two mediators are considered, two-way interaction of the two mediators and three-way interaction of the exposure and the two mediators are likely to exist [@v4; @b; @vbook]. There may also be a causal sequence between the two mediators, i.e. there is a direct causal link between the two mediators (Figure \[fig4\]). There is limited research on how to define interactions when the two mediators are causally sequential. We aim to develop interpretable interactions concepts and decomposition approaches for the two-mediator scenarios.
Mediators causally non-sequential
---------------------------------
We first consider the scenario when the two mediators are causally non-sequential, i.e., there is no direct causal link between the two mediators, which is shown in Figure \[fig3\]. Below, we define two-way natural counterfactual interaction effects of $A$ and $M_1$, $A$ and $M_2$, $M_1$ and $M_2$, and a three-way natural counterfactual interaction effect of $A, M_1$ and $M_2$.\
\
**Definition 2**. Natural counterfactual interaction effects in a causally non-sequential two-mediator scenario: $$\begin{aligned}
NatINT_{AM_1} & := & Y\left(a,M_1(a),M_2(a^\ast)\right)-Y\left(a^\ast,M_1(a),M_2(a^\ast)\right)\\
& & - Y\left(a,M_1(a^\ast),M_2(a^\ast)\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast)\right),\\
\\
NatINT_{AM_2} & := & Y\left(a,M_1(a^\ast),M_2(a)\right)- Y\left(a^\ast,M_1(a^\ast),M_2(a)\right)\\
& & -Y\left(a,M_1(a^\ast),M_2(a^\ast)\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast)\right),\\
\\
NatINT_{M_1M_2} & := & Y\left(a^\ast,M_1(a),M_2(a)\right)-Y\left(a^\ast,M_1(a^\ast),M_2(a)\right)\\
& & -Y\left(a^\ast,M_1(a),M_2(a^\ast)\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast)\right), \\
\\
NatINT_{AM_1M_2} & := & Y\left(a,M_1(a),M_2(a)\right)-Y\left(a^\ast,M_1(a),M_2(a)\right)\\
& & - Y\left(a,M_1(a^\ast),M_2(a)\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a)\right)\\
& & -Y\left(a,M_1(a),M_2(a^\ast)\right)+Y\left(a^\ast,M_1(a),M_2(a^\ast)\right)\\
& & +Y\left(a,M_1(a^\ast),M_2(a^\ast)\right)-Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast)\right).\end{aligned}$$
$NatINT_{AM_1}$, $NatINT_{AM_2}$, and $NatINT_{AM_1M_2}$ are components that capture the effects due to both mediation and interaction with the exposure. $ NatINT_{M_1M_2}$ describes the effect due to mediation and interaction between the two mediators. When measuring the interaction between $A$ and $M_1$, $M_2$ is not fixed but takes its potential value $M_2(a^\ast)$ for each individual had the exposure been the reference level. Similarly, when measuring the interaction between $A$ and $M_2$, $M_1$ is not fixed but takes its potential value $M_1(a^\ast)$ for the individual. The three-way interaction $NatINT_{AM_1M_2}$ is similar to a three-way additive interaction. To demonstrate the similarity, we consider that $A$ is binary with the conditions $a=1$ and $a^\ast =0$; $NatINT_{AM_1M_2} $ becomes $$\begin{aligned}
&& Y\left(1,M_1(1),M_2(1)\right)-Y\left(0,M_1(1),M_2(1)\right)
- Y\left(1,M_1(0),M_2(1)\right)+Y\left(0,M_1(0),M_2(1)\right)\\
\\
&&-Y\left(1,M_1(1),M_2(0)\right)+Y\left(0,M_1(1),M_2(0)\right)
+Y\left(1,M_1(0),M_2(0)\right)-Y\left(0,M_1(0),M_2(0)\right).\end{aligned}$$
The above three-way interaction measures the change in the two-way interaction between $A$ and $M_1$ when $M_2$ goes from $M_2(0)$ to $M_2(1)$. It also measures the change in the interaction between $A$ and $M_2$ when $M_1$ goes from $M_1(0)$ to $M_1(1)$ or the change in the interaction between $M_1$ and $M_2$ when $A$ goes from $0$ to $1$.
In Appendix A, we show that the total effect can be decomposed into ten components at the individual level: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)\\
& & +INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)+ NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}\\
& & + NatINT_{M_1M_2} + PIE_{M_1} + PIE_{M_2},\end{aligned}$$ where $m_1^\ast$ and $m_2^\ast$ are fixed reference levels for $M_1$ and $M_2$, respectively, $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast),\\
\\
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & Y(a, M_1(a^\ast),m_2^\ast)-Y(a^\ast,M_1(a^\ast),m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast),\\
\\
INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast) & = & Y(a, m_1^\ast, M_2(a^\ast))-Y(a^\ast, m_1^\ast, M_2(a^\ast))-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast),\\
\\
INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast) & = & Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))-Y(a,m_1^\ast,M_2(a^\ast)) \\
&& + Y(a^\ast,m_1^\ast,M_2(a^\ast)) -Y(a,M_1(a^\ast),m_2^\ast)+Y(a^\ast,M_1(a^\ast),m_2^\ast) \\
&& + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast), \\
\\
PIE_{M_1} &= & Y(a^\ast,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)), \\
\\
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)).\end{aligned}$$
Similar to the four-way decomposition, $CDE$ denotes controlled direct effect due to neither mediation nor interaction, $INT_{ref}$’s denote reference interaction effects due to interactions only, and $PIE$’s denote pure indirect effects due to mediation only [@v4]. $NatINT_{M_1M_2}$ can be interpreted as the effect due to the mediation through both $M_1$ and $M_2$, and the interaction between $M_1$ and $M_2$. Since the interaction is not involved with the change in exposure $A$, the interpretation can be simply put as the effect due to the mediation through both $M_1$ and $M_2$ only. These ten components are displayed in Table 1 assuming that $A$, $M_1$ and $M_2$ are binary with $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$.
Bellavia and Valeri [@b] proposed a ten-component decomposition for the same directed acyclic graph in Figure \[fig3\]. We show in Appendix B that their decomposition is a special case of our proposed decomposition under the extra condition $M_1(0)=M_2(0)=0$. Their $CDE$ and $INT_{ref}$’s have corresponding terms in our decomposition but their mediated interaction effects and pure natural indirect effects are different from our natural counterfactual interactions and pure indirect effects. The top panel in Figure \[fig4\] illustrates their mediated interaction effect between $A$ and $M_1$ where $M_2$ is assigned a fixed value at $m_2^\ast=0$. The bottom panel in Figure \[fig4\] illustrates the natural counterfactual interaction effect between $A$ and $M_1$, where both $M_1$ and $M_2$ take their potential values.
Our natural counterfactual interaction effects account for the distributions of $M_1(0)$ and $M_2(0)$. If the population distribution of $M_2(0)$ has probability of $1$ taking the value $0$, the $NatINT_{AM_1}$ is consistent with the mediated interaction effect between $A$ and $M_1$ as proposed by Bellavia and Valeri. However, if the population distribution of $M_2(0)$ does not have probability of $1$ taking the value $0$, the natural counterfactual interaction effects are more suitable to describe the population average of the counterfactual interaction effects. Table 1 lists the specific decomposition components. Table 2 presents the results under the extra condition $M_1(0)=M_2(0)=0$, which are reduced to those proposed by Bellavia and Valeri [@b].
Mediators causally sequential
-----------------------------
In this section, we consider the scenario where the two mediators are causally sequential, i.e., there is a direct causal link from mediator $M_1$ to $M_{2}$ (Figure \[fig5\]). Let $M_2(a^\ast, M_{1}(a))$ be the potential value of $M_2$ if $A$ were fixed at $a^\ast$ and $M_1$ were at its potential value had $A$ been set at $a$. Similarly, $M_2(a^\ast, M_{1}(a^\ast))$ denotes the potential value of $M_2$ if $A$ were fixed at $a^\ast$ and $M_1$ were at its potential value had $A$ been set at $a^\ast$. Counterfactual values for $Y$ are expressed using nested formulas but not all of them are identifiable. For example, $Y\left(a,M_1(a),M_2(a,M_1(a^\ast))\right)$ is not identifiable since it has two distinct counterfactual values of mediator $M_1$, i.e., $M_1(a)$ and $M_1(a^\ast)$, which means $M_1$ is activated by two different values of $A$ at the same time. Avin et al. [@a] showed that such counterfactual formulas are not identifiable. We present identifiable decomposition components only for those identifiable counterfactual formulas of $Y$.\
\
**Definition 3**. Natural counterfactual interaction effects in a causally sequential two-mediator scenario: $$\begin{aligned}
NatINT_{AM_1} & := & Y\left(a,M_1(a),M_2(a^\ast,M_1(a))\right)-Y\left(a^\ast,M_1(a),M_2(a^\ast,M_1(a))\right)\\
& & - Y\left(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right),\\
\\
NatINT_{AM_2} & := & Y\left(a,M_1(a^\ast),M_2(a,M_1(a^\ast))\right)-Y\left(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast))\right)\\
& & -Y\left(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right),\\
\\
NatINT_{M_1M_2} & := & Y\left(a^\ast,M_1(a),M_2(a,M_1(a))\right)-Y\left(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast))\right)\\
& & -Y\left(a^\ast,M_1(a),M_2(a^\ast,M_1(a))\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right),\\
\\
NatINT_{AM_1M_2} & := & Y\left(a,M_1(a),M_2(a,M_1(a))\right)-Y\left(a^\ast,M_1(a),M_2(a,M_1(a))\right)\\
& & - Y\left(a,M_1(a^\ast),M_2(a,M_1(a^\ast))\right)+Y\left(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast))\right)\\
& & -Y\left(a,M_1(a),M_2(a^\ast,M_1(a))\right)+Y\left(a^\ast,M_1(a),M_2(a^\ast,M_1(a))\right)\\
& & +Y\left(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right)-Y\left(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))\right).\end{aligned}$$
These interaction terms are similar to those in Definition 2 except that $M_2$ has an extra input from $M_1$. In $NatINT_{AM_1} $, $M_2$ is neither fixed nor set at a level independent of $M_1$; rather, $M_2$ changes whenever $M_1$ changes. Therefore, $NatINT_{AM_1} $ captures the change in the total effect of $M_1$ (going from $M_1(a^\ast)$ to $M_1(a)$) on the response when $A$ goes from $a^\ast$ to $a$. In $NatINT_{M_1M_2}$, $M_2$ would still partially depend on the level of $M_1$. Hence this component describes the interaction between $M_1$ and $M_2$ had $M_2$ only change its exposure input. Similarly, the three-way interaction $NatINT_{AM_1M_2}$ can be interpreted as the change in the interaction between $A$ and $M_1$ when $M_2$ has its exposure input going from $a^\ast$ to $a$.
We show in Appendix C that the total effect can be decomposed into 9 components at the individual level: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)\\
& & + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}+ NatINT_{M_1M_2}\\
& & + PIE_{M_1} + PIE_{M_2},\end{aligned}$$ where $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast),\\
\\
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & Y(a,M_1(a^\ast),m_2^\ast)-Y(a^\ast,M_1(a^\ast),m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast),\\
\\
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a,M_1(a^\ast),m_2^\ast)\\
& & - Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),m_2^\ast),\\
\\
PIE_{M_1} &= & Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))), \\
\\
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))).\end{aligned}$$
Compared to the decomposition in Section 3.1, reference interaction effects in the above case have fewer terms. $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$ are summed into $INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ to have identifiable effects. We show the detailed proof in Appendix D. $INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$ and $M_2$ only, conditioning on the potential value of $M_1$ at the reference level $a^\ast$. Because of the direct causal link between the two mediators, $M_2$ possesses two types of mediation, $M_2(1,1)-M_2(0,1)\neq 0$ and $M_2(1,0)-M_2(0,0)\neq 0$. They collectively contribute to $NatINT_{AM_1M_2}$ and $NatINT_{M_1M_2}$ (Appendix C). These nine components and their interpretations are shown in Table 3 for the special case when $A$, $M_1$ and $M_2$ are binary with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$.\
Identification assumptions and empirical formulas
=================================================
The decompositions for one- and two-mediator scenarios thus far have been primarily conceptual. The individual-level effects in the decompositions cannot be identified from data, but under certain assumptions on confounding the population-averages of those components can be identified from data [@v3].
Identification assumptions
--------------------------
We first consider a single-mediator scenario. Four identification assumptions are required [@vcon], which are listed below as ($A^\prime 1$) – ($A^\prime 4$): $$\begin{aligned}
& & Y(a,m) \perp A|C \hspace{2.4cm} (A^\prime 1)\\
& & Y(a,m) \perp M|\{A,C\} \hspace{1.5cm} (A^\prime 2)\\
& & M(a) \perp A|C \hspace{2.8cm} (A^\prime 3)\\
& & Y(a,m) \perp M(a^\ast)|C, \hspace{1.5cm} (A^\prime 4)\end{aligned}$$ where $C$ is a set of covariates. The assumptions above state that given a covariate set $C$ or $\{A,C\}$, there exist no unmeasured variables confounding the association between exposure $A$ and outcome $Y$ ($A^\prime 1$); no unmeasured variables confounding the association between mediator $M$ and outcome $Y$ ($A^\prime 2$) and no unmeasured variables confounding the association between exposure $A$ and mediator $M$ ($A^\prime 3$) [@vbook]. ($A^\prime 4$) is a strong assumption and a few researchers published their works on this topic [@v4; @s; @ralt]. It could be interpreted as there exist no variables that are causal descendants of exposure $A$, and in the meantime, that confound the association between mediator $M$ and outcome $Y$ [@s; @p01].\
The analogs of ($A^\prime 1$) – ($A^\prime 4$) for a directed acyclic graph with two sequential mediators can be found by first considering $M_1$ and $M_2$ as a set [@s]. Namely, we have four corresponding identification assumptions ($A1$) – ($A4$): $$\begin{aligned}
& & Y(a,m_1,m_2) \perp A|C \hspace{5.1cm} (A1)\\
& & Y(a,m_1,m_2) \perp \{M_1,M_2\}|\{A,C\} \hspace{2.9cm} (A2)\\
& & \{M_1(a),M_2(a,m_1)\} \perp A|C \hspace{3.85cm} (A3)\\
& & Y(a,m_1,m_2) \perp \{M_1(a^\ast),M_2(a^\ast,m_1)\}|C. \hspace{1.55cm} (A4)\\\end{aligned}$$
Similarly, the assumptions above state that given a covariate set $C$ or $\{A,C\}$, there exists no unmeasured variables confounding the association between exposure $A$ and outcome $Y$ ($A1$), no unmeasured variables confounding the association between the mediator set $\{M_1,M_2\}$ and outcome $Y$ ($A2$), no unmeasured variables confounding the association between exposure $A$ and the mediator set $\{M_1,M_2\}$ ($A3$) and no unmeasured variables that are causal descendants of exposure $A$, and in the meantime, that confound the association between the mediator set $\{M_1,M_2\}$ and outcome $Y$ ($A4$) [@s; @vcon].
In order to account for the confounding between $M_1$ and $M_2$, two more assumptions are required: $$\begin{aligned}
& & M_2(a,m_1) \perp M_1|\{A,C\} \hspace{2.2cm} (A5)\\
& & M_2(a,m_1) \perp M_1(a^\ast)|C, \hspace{2.2cm} (A6)\end{aligned}$$ where ($A5$) and ($A6$) state, respectively, that there exists no unmeasured variables confounding the association between $M_1$ and $M_2$ given $\{A,C\}$, and no unmeasured variables that are causal descendants of exposure $A$, and in the meantime, are confounding the association between $M_1$ and $M_2$ [@s].
Steen et al [@s] presented comprehensive identification assumptions for the causal structures with multiple mediators and pointed out that weaker identification assumptions than ($A1$) – ($A6$) can be considered under certain decompositions.
Empirical formulas
------------------
Suppose a set of covariates $C$ satisfies the assumptions on confounding for a decomposition. We can obtain the expected value of each component in the decomposition using the iterated conditional expectation rule. We focus on the scenario with two causally sequential mediators. Suppose $M_1$ and $M_2$ are categorical and let $p_{am_1m_2}=E[Y|A=a,M_1=m_1,M_2=m_2, C = c]$. The following formulas can be obtained: $$\begin{aligned}
E\left[CDE(m_1^\ast,m_2^\ast)\right]& = & p_{am_1^\ast m_2^\ast}-p_{a^\ast m_1^\ast m_2^\ast}\\
\\
E[INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)] & = & \sum_{m_1}(p_{am_1 m_2^\ast}-p_{a m_1^\ast m_2^\ast}-p_{a^\ast m_1 m_2^\ast}+p_{a^\ast m_1^\ast m_2^\ast})\\
& & \times Pr(M_1=m_1|a^\ast, c)\\
\\
E[INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)] & = & \sum_{m_2}\sum_{m_1}(p_{am_1 m_2}-p_{a m_1 m_2^\ast}-p_{a^\ast m_1 m_2}+p_{a^\ast m_1 m_2^\ast})\\
& & \times Pr(M_1=m_1|a^\ast, c)\\
& & \times Pr(M_2=m_2|a^\ast,m_1, c)\\
\\
E[NatINT_{AM_1}] & = & \sum_{m_2}\sum_{m_1}(p_{am_1 m_2}-p_{a^\ast m_1 m_2})\\
& & \times Pr(M_2=m_2|a^\ast,m_1, c)\\
& & \times [Pr(M_1=m_1|a, c)-Pr(M_1=m_1|a^\ast, c)]\\
\\
E[NatINT_{AM_2}] & = & \sum_{m_2}\sum_{m_1}(p_{am_1 m_2}-p_{a^\ast m_1 m_2})\\
& & \times Pr(M_1=m_1|a^\ast, c)\\
& & \times [Pr(M_2=m_2|a,m_1, c)-Pr(M_2=m_2|a^\ast,m_1, c)]\\
\\
E[NatINT_{AM_1M_2}] & = & \sum_{m_2}\sum_{m_1}(p_{am_1 m_2}-p_{a^\ast m_1 m_2})\\
& & \times [Pr(M_1=m_1|a, c)-Pr(M_1=m_1|a^\ast, c)]\\
& & \times [Pr(M_2=m_2|a,M_1=m_1, c)-Pr(M_2=m_2|a^\ast,M_1=m_1, c)]\\
\\
E[NatINT_{M_1M_2}] & = & \sum_{m_2}\sum_{m_1}p_{a^\ast m_1 m_2}\\
& & \times [Pr(M_1=m_1|a, c)-Pr(M_1=m_1|a^\ast, c)]\\
& & \times [Pr(M_2=m_2|a,m_1, c)-Pr(M_2=m_2|a^\ast,m_1, c)]\\
\\
E[PIE_{M_1}] & = & \sum_{m_2}\sum_{m_1}p_{a^\ast m_1 m_2}\\
& & \times Pr(M_2=m_2|a^\ast,m_1, c)\\
& & \times [Pr(M_1=m_1|a, c)-Pr(M_1=m_1|a^\ast, c)]\\
\\
E[PIE_{M_2}] & = & \sum_{m_2}\sum_{m_1}p_{a^\ast m_1 m_2}\\
& & \times Pr(M_1=m_1|a^\ast, c)\\
& & \times [Pr(M_2=m_2|a,m_1, c)-Pr(M_2=m_2|a^\ast,m_1, c)].\end{aligned}$$
When $M_1$ and $M_2$ are continuous, empirical formulas can be obtained by replacing the sums by integrations and the conditional probabilities by conditional densities.
Relations to linear models
--------------------------
Suppose $Y$, $M_1$, and $M_2$ are continuous. For the scenario with two causally sequential mediators, we assume that the following regression models for $Y$, $M_1$, and $M_2$ are specified: $$\begin{aligned}
E[Y|A,M_1,M_2,C] & = & \theta_0 + \theta_1A + \theta_2M_1 + \theta_3M_2 + \theta_4AM_1 + \theta_5AM_2 + \theta_6M_1M_2\\
& & + \theta_7AM_1M_2 + \theta_8^\prime C\\
E[M_2|A,M_1,C] & = & \beta_0 + \beta_1A + \beta_2M_1 + \beta_3AM_1 + \beta_4^\prime C\\
E[M_1|A,C] & = & \gamma_0 + \gamma_1A + \gamma_2^\prime C, \end{aligned}$$ the results on the effect components are given in Appendix E.
For the scenario with two causally non-sequential mediators, assume that a set of covariates $C$ satisfies the identification assumptions for the decomposition and assume that the following regression models for $Y$, $M_1$, and $M_2$ are specified: $$\begin{aligned}
E[Y|A,M_1,M_2,C] & = & \theta_0 + \theta_1A + \theta_2M_1 + \theta_3M_2 + \theta_4AM_1 + \theta_5AM_2 + \theta_6M_1M_2\\
& & + \theta_7AM_1M_2 + \theta_8^\prime C\\
E[M_2|A,C] & = & \beta_0 + \beta_1A + \beta_4^\prime C\\
E[M_1|A,C] & = & \gamma_0 + \gamma_1A + \gamma_2^\prime C, \end{aligned}$$ the results can be obtained as a special case of those derived from the scenario with two causally sequential mediators by setting parameters $\beta_2$ and $\beta_3$ to zero.\
Illustration with real data
===========================
To illustrate the concept of natural counterfactual interaction effect and the decomposition methods, we used the 2015-2016 data from the National Health and Nutrition Examination Survey on the hazard of drinking alcohol as a contribution to the abnormal pattern in mortality [@d; @l]. The dataset was downloaded from [http://www.cdc.gov/nhanes]{}. Exposure $A$ is alcohol drinking, mediator $M_1$ is Body Mass Index (BMI), mediator $M_2$ is the log-transformed Gamma Glutamyl Transferase (GGT), and outcome $Y$ is Systolic Blood Pressure (SBP). Sex and Age are considered a sufficient set satisfying the assumption on confounding. In addition, BMI is known to affect GGT. The hypothetical causal diagram is shown in Figure \[fig6\].
Log transformation was performed for $M_2$ due to the skewness of the data. The fixed reference levels of $M_1$ and $\log(M_2)$ were chosen at their corresponding mean levels where $m_1^\ast=29.5$ and ${\log{(m_2)}}^\ast=3.05$. Three linear models were fit for $Y$, $\log(M_2)$ and $M_1$. The 95% confidence intervals were obtained by using a bootstrap method [@v].
Table \[tab::data1\] presents the decomposition of total effect conditional on males and the mean level of age at $48.3$. The controlled direct effect is $0.238$ (95% C.I. = $-0.969$ to $1.429$); the reference interaction effect between $A$ and $M_1$ is $-0.059$ ($-0.203$ to $0.039$); the sum of two reference interaction effect is $-0.115$ ($-0.516$ to $0.219$); the natural counterfactual interaction effect between $A$ and $M_1$ is $-0.018$ ($-0.125$ to $0.056$); the natural counterfactual interaction effect between $A$ and $\log(M_2)$ is $-0.026$ ($-0.194$ to $0.095$); the natural counterfactual interaction effect among $A$, $M_1$ and $\log(M_2)$ is $0.000386$ ($-0.0059$ to $0.0082$); the natural counterfactual interaction effect between $M_1$ and $\log(M_2)$ is $0.000873$ ($-0.0094$ to $0.0123$); the pure direct effect is $0.0636$ ($-1.226$ to $1.317$); the pure indirect effect through $M_1$ is $-0.0409$ ($-0.206$ to $0.109$); the pure indirect effect through $\log(M_2)$ is $0.143$ ($0.00803$ to $0.363$); the total effect is $0.123$ ($-1.178$ to $1.396$). The results of the decomposition of the total effect conditional on females and the mean level of age are shown in Table \[tab::data2\]. It can be seen that the pure indirect effect through $\log(M_2)$ is the only significant effect contributing to the outcome for both females and males.
Conclusion
==========
In this work, we develop decompositions for scenarios where the two mediators are causally sequential or non-sequential. We propose a unified approach for decomposing the total effect into components that are due to mediation only, interaction only, both mediation and interaction, and neither mediation nor interaction within the counterfactual framework. The decomposition was implemented via a new concept called natural counterfactual interaction effect that we proposed to describe the two-way and three-way interactions for both scenarios that extend the two-way mediated interactions in existing literature. To estimate the components of our proposed decompositions, we lay out the identification assumptions. We also derive the formulas when the response is assumed to be continuous with a linear model.
We believe that our proposed new concept of natural counterfactual interaction effects and the decomposition methods for the causal framework with two sequential or non-sequential mediators provide a powerful tool to decipher the refined path effects while appropriately account for the interaction effects among the exposure and mediators. The counterfactual interaction effects evaluate the interaction terms that involve mediators by treating them at the natural levels. There is a gap in existing research of decomposing total effect into mediation and interaction effects for the scenario of two sequential mediators, and our proposed methods have the potential to fill in the gap. The proposed work provides the foundation to generalize into decomposition of total effect for more complicated causal structures involving more than two sequential mediators, which we will explore in the future work.
**Effect** **Definition** **Interpretation**
--------------------------------- --------------------------------------------------------------- ------------------------------------------------------------------------
$ CDE(0,0)$ $Y(1,0,0)-Y(0,0,0)$ Due to neither mediation nor interaction
$INT_{ref\mbox{-}AM_1}(0,0)$ $[Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times M_1(0)$ Due to the interaction between $A$ and $M_1$ only
$INT_{ref\mbox{-}AM_2}(0,0) $ $[Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]\times M_2(0)$ Due to the interaction between $A$ and $M_2$ only
$INT_{ref\mbox{-}AM_1M_2}(0,0)$ $[Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)$ Due to the interaction between $A$, $M_1$ and $M_2$ only
$-Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]$
$\times M_1(0)\times M_2(0)$
$NatINT_{AM_1}$ $\sum_{m_2}[Y(1,1,m_2)I(M_2(0)=m_2)-Y(0,1,m_2)I(M_2(0)=m_2)$ Due to the mediation through $M_1$ and the interaction
$- Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]$ between $A$ and $M_1$ conditioning on the potential value
$\times[M_1(1)-M_1(0)]$ of $M_2$ with the fixed reference level $a^\ast=0$
$NatINT_{AM_2}$ $\sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)$ Due to the mediation through $M_2$ and the interaction
$- Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]$ between $A$ and $M_2$ conditioning on the potential value
$\times[M_2(1)-M_2(0)]$ of $M_1$ with the fixed reference level $a^\ast=0$
$ NatINT_{AM_1M_2}$ $[Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)$ Due to the mediation through both $M_1$ and $M_2$ and the
$- Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]$ interaction between $A$, $M_1$ and $M_2$
$ \times[M_1(1)-M_1(0)]\times[M_2(1)-M_2(0)]$
$NatINT_{M_1M_2}$ $[Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]$ Due to the mediation through both $M_1$ and $M_2$ only
$\times[M_1(1)-M_1(0)]\times[M_2(1)-M_2(0)]$
$PIE_{M_1}$ $\sum_{m_2}[Y(0,1,m_2)I(M_2(0)=m_2)-Y(0,0,m_2)I(M_2(0)=m_2)]$ Due to the mediation through $M_1$ only conditioning on
$\times[M_1(1)-M_1(0)]$ the potential value of $M_2$ with the fixed reference level $a^\ast=0$
$PIE_{M_2}$ $\sum_{m_1}[Y(0,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)]$ Due to the mediation through $M_2$ only conditioning on
$\times[M_2(1)-M_2(0)]$ the potential value of $M_1$ with the fixed reference level $a^\ast=0$
: Decomposition of the Total Effect in a Two Non-sequential Mediators Scenario When $A$, $M_1$ and $M_2$ are Binary with $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$
**Effect** **Definition** **Interpretation**
--------------------- -------------------------------------------- -----------------------------------------------------------
$NatINT_{AM_1}$ $[Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]$ Due to the mediation through $M_1$ and the interaction
$\times[M_1(1)-M_1(0)]$ between $A$ and $M_1$ conditioning on $M_2(0)=0$
$NatINT_{AM_2}$ $[Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]$ Due to the mediation through $M_2$ and the interaction
$\times[M_2(1)-M_2(0)]$ between $A$ and $M_2$ conditioning on $M_1(0)=0$
$ NatINT_{AM_1M_2}$ $[Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)$ Due to the mediation through both $M_1$ and $M_2$ and the
$- Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]$ interaction between $A$, $M_1$ and $M_2$ conditioning on
$ \times[M_1(1)M_2(1)-M_1(0)M_2(0)]$ $M_1(0)=M_2(0)=0$
$NatINT_{M_1M_2}$ $[Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]$ Due to the mediation through both $M_1$ and $M_2$ only
$\times [M_1(1)M_2(1)-M_1(0)M_2(0)]$ conditioning on $M_1(0)=M_2(0)=0$
$PIE_{M_1}$ $[Y(0,1,0)-Y(0,0,0)]\times[M_1(1)-M_1(0)]$ Due to the mediation through $M_1$ only conditioning on
$M_2(0)=0$
$PIE_{M_2}$ $[Y(0,0,1)-Y(0,0,0)]\times[M_2(1)-M_2(0)]$ Due to the mediation through $M_2$ only conditioning on
$M_1(0)=0$
: Proposed Interaction and Pure Indirect Effects for Non-Sequential Two Mediators Scenario with Binary $A$, $M_1$ and $M_2$ under the Extra Conditions $M_1(0)=M_2(0)=0$
Component Estimate 95% C.I.
------------------------------------------------------------- ------------ ------------------
$ CDE(m_1^\ast,\log(m_2)^\ast)$ $0.238$ $-0.969, 1.429$
$INT_{ref\mbox{-}AM_1}(m_1^\ast,\log(m_2)^\ast)$ $-0.059$ $ -0.203, 0.039$
$INT_{ref\mbox{-}A\log(M_2)+AM_1\log(M_2)}(\log(m_2)^\ast)$ $-0.115$ $-0.516,0.219$
$NatINT_{AM_1}$ $-0.018$ $-0.125,0.056$
$NatINT_{A\log(M_2)}$ $-0.026$ $-0.194,0.095$
$NatINT_{AM_1\log(M_2)}$ $0.000386$ $-0.0059,0.0082$
$NatINT_{M_1\log(M_2)}$ $0.000873$ $-0.0094,0.0123$
$PDE$ $0.0636$ $-1.226,1.317$
$PIE_{M_1}$ $-0.0409$ $-0.206,0.109$
$PIE_{\log(M_2)}$ $0.143$ $0.00803,0.363$
$TE$ $0.123$ $-1.178,1.396$
: Illustration with Real Data: Decomposition of Total Effect Conditional on Males and the Mean Age. \[tab::data1\]
Component Estimate 95% C.I.
------------------------------------------------------------- ------------ --------------------
$ CDE(m_1^\ast,\log(m_2)^\ast)$ $0.238$ $-0.969, 1.429$
$INT_{ref\mbox{-}AM_1}(m_1^\ast,\log(m_2)^\ast)$ $0.087$ $ -0.0359, 0.263$
$INT_{ref\mbox{-}A\log(M_2)+AM_1\log(M_2)}(\log(m_2)^\ast)$ $0.0658$ $-0.395,0.533$
$NatINT_{AM_1}$ $-0.0207$ $-0.135,0.060$
$NatINT_{A\log(M_2)}$ $-0.0286$ $-0.206,0.0896$
$NatINT_{AM_1\log(M_2)}$ $0.000377$ $-0.00586,0.00863$
$NatINT_{M_1\log(M_2)}$ $0.000860$ $-0.00936,0.0117$
$PDE$ $0.391$ $-0.828,1.581$
$PIE_{M_1}$ $-0.0448$ $-0.219,0.114$
$PIE_{\log(M_2)}$ $0.137$ $0.00752,0.353$
$TE$ $0.435$ $-0.788,1.629$
: Illustration with Real Data: Decomposition of Total Effect Conditional on Females and the Mean Age. \[tab::data2\]
\[0.5\][![Directed acyclic graph of a single-mediator scenario.[]{data-label="fig1"}](New_Figures/FIGURE_1_n.pdf "fig:")]{}
\[0.5\][![Nested counterfactual formula $Y(a, M_1(a^\ast))$.[]{data-label="fig2"}](New_Figures/FIGURE_2_n.pdf "fig:")]{}
\[0.5\][![Directed acyclic graph with two non-sequential mediators.[]{data-label="fig3"}](New_Figures/FIGURE_3_n.pdf "fig:")]{}
\[0.5\][![A comparison between the mediated interaction effect and the natural counterfactual interaction effect between $A$ and $M_1$ in a non-sequential two-mediator scenario.[]{data-label="fig4"}](New_Figures/FIGURE_4_n.pdf "fig:")]{}
\[0.5\][![Directed acyclic graph with two sequential mediators.[]{data-label="fig5"}](New_Figures/FIGURE_5_n.pdf "fig:")]{}
\[0.5\][![The directed acyclic graph for the study on hazard of drinking alcohol.[]{data-label="fig6"}](New_Figures/FIGURE_6_n.pdf "fig:")]{}
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially supported by UNM Comprehensive Cancer Center Support Grant NCI P30CA118100, the Biostatistics shared resource and UNM METALS Superfund Research Center (1P42ES025589).
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Appendix A. Decomposition of total effect with the notion of natural counterfactual interaction effect in a non-sequential two-mediator scenario and the corresponding interpretations {#appendix-a.-decomposition-of-total-effect-with-the-notion-of-natural-counterfactual-interaction-effect-in-a-non-sequential-two-mediator-scenario-and-the-corresponding-interpretations .unnumbered}
======================================================================================================================================================================================
Suppose we have a directed acyclic graph as shown in Figure \[fig3\]. We show in the following that the total effect can be decomposed into the following 10 components at the individual level: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)\\
& & +INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)+ NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}\\
& & + NatINT_{M_1M_2} + PIE_{M_1} + PIE_{M_2},\end{aligned}$$ where the natural counterfactual interaction effects are listed in Definition 2. We also give the corresponding interpretation for each component.\
*Proof*:
We first decompose the total effect into total direct effect ($TDE$) [@riden], seminatural indirect effect through $M_1$ ($SIE_{M_1}$) [@p14] and pure indirect effect (path-specific effect) through $M_2$ ($PIE_{M_2}$) [@riden; @p01]. $$\begin{aligned}
TE & = & Y(a)-Y(a^\ast)\\
\\
& = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & +Y(a^\ast,M_1(a),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & +Y(a^\ast,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)),\end{aligned}$$ where the second equality follows the composition axiom [@vbook; @a] and the third equality follows by adding and subtracting the same counterfactual formulas.
The formulas of $TDE$, $SIE_{M_1}$ and $PIE_{M_2}$ are presented as follows: $$\begin{aligned}
TDE & = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
\\
SIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
\\
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)),\end{aligned}$$ where $TE = TDE+SIE_{M_1}+PIE_{M_2}$.
We focus on $TDE$ in the next step and decompose it into natural counterfactual interaction effects and pure direct effect ($PDE$) [@riden; @p01] by subtracting $PDE$ from $TDE$, where $PDE$ satisfies the definition of a path-specific effect [@p01] and equals the following contrast of two counterfactual formulas: $$\begin{aligned}
PDE & = & Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)).\end{aligned}$$
We have the following results: $$\begin{aligned}
TDE-PDE & = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
& & +Y(a^\ast,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
& & +Y(a^\ast,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & +Y(a^\ast,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a,M_1(a^\ast),M_2(a^\ast))\\
& & +Y(a,M_1(a^\ast),M_2(a))-Y(a,M_1(a^\ast),M_2(a))\\
& & +Y(a,M_1(a),M_2(a^\ast))-Y(a,M_1(a),M_2(a^\ast))\\
\\
& = & Y(a,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & - Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
& & + Y(a,M_1(a^\ast),M_2(a))- Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
& & + Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & - Y(a,M_1(a^\ast),M_2(a))+Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a),M_2(a^\ast))+Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)),\end{aligned}$$ where the second equality follows by adding and subtracting the same counterfactual formulas, and the third equality follows by rearranging all the terms.
Therefore, we have the following formulas satisfying Definition 2: $$\begin{aligned}
NatINT_{AM_1} & = & Y(a,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & - Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
NatINT_{AM_2} & = & Y(a,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
NatINT_{AM_1M_2} & = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & - Y(a,M_1(a^\ast),M_2(a))+Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a),M_2(a^\ast))+Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast)).\end{aligned}$$
Accordingly, $TDE$ can be decomposed into the following components: $$\begin{aligned}
TDE &=& PDE + NatINT_{AM_1}+NatINT_{AM_2}+NatINT_{AM_1M_2}.\end{aligned}$$
We next focus on $PDE$ (path-specific effect) and decompose it into $CDE$ and reference interaction effects [@v4; @b]: $$\begin{aligned}
PDE & = & Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)\\
& & +Y(a^\ast,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)\\
& & +Y(a^\ast,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)\\
& & +Y(a,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2)+Y(a,m_1,m_2^\ast)-Y(a,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_1}[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
& & + \sum_{m_2}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast).\end{aligned}$$
According to the derivation above, the following formulas can be obtained: $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & \sum_{m_1}[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
\\
INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_2(a^\ast)=m_2)\\
\\
INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2).\end{aligned}$$
With a little mathematical derivation, $INT_{ref\mbox{-}AM_1}$, $INT_{ref\mbox{-}AM_2}$ and $INT_{ref\mbox{-}AM_1M_2}$ can be expressed in the form of the counterfactual formula: $$\begin{aligned}
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & Y(a,M_1(a^\ast),m_2^\ast)-Y(a^\ast,M_1(a^\ast),m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,M_2(a^\ast))-Y(a^\ast,m_1^\ast,M_2(a^\ast))-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast) & = & Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
& & -Y(a,m_1^\ast,M_2(a^\ast))+Y(a^\ast,m_1^\ast,M_2(a^\ast))\\
& & -Y(a,M_1(a^\ast),m_2^\ast)+Y(a^\ast,M_1(a^\ast),m_2^\ast)\\
& & +Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast).\end{aligned}$$
Therefore, $PDE$ can be decomposed into the following components: $$\begin{aligned}
PDE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast).\end{aligned}$$
We next focus on $SIE_{M_1}$ and try to decompose it into $PIE_{M_1}$ and $NatINT_{M_1M_2}$ by subtracting $PIE_{M_1}$ from $SIE_{M_1}$: $$\begin{aligned}
SIE_{M_1}-PIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a^\ast,M_1(a),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & NatINT_{M_1M_2},\end{aligned}$$ where $NatINT_{M_1M_2}$ satisfies Definition 2.
Therefore, $SIE_{M_1}$ can be decomposed into the following components: $$\begin{aligned}
SIE_{M_1} & = & PIE_{M_1}+ NatINT_{M_1M_2}.\end{aligned}$$
Combining all the derivations above, we have the decomposition of total effect as follows: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)\\
& & +INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)+ NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}\\
& & + NatINT_{M_1M_2} + PIE_{M_1} + PIE_{M_2}.\end{aligned}$$
We next present the interpretation for each component assuming binary $A$, $M_1$ and $M_2$ with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ for illustration purpose. While other interpretations were proposed in the literature [@v4; @b], our work represent a different and more flexible interpretation from the perspective of population averages which accounts for the distribution of the mediators in the causal structure.
controlled direct effect {#controlled-direct-effect .unnumbered}
------------------------
With the specified conditions, the controlled direct effect can be written as: $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
\Rightarrow \quad\quad CDE(0,0) & = & Y(1,0,0)-Y(0,0,0).\end{aligned}$$
$CDE(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to neither mediation nor interaction.
reference interaction effects {#reference-interaction-effects .unnumbered}
-----------------------------
With the specified conditions, the reference interaction effect between $A$ and $M_1$ can be written as: $$\begin{aligned}
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & \sum_{m_1}[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
\\
\Rightarrow INT_{ref\mbox{-}AM_1}(0,0) & = & \sum_{m_1}[Y(1,m_1,0)-Y(0,m_1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=m_1)\\
\\
& = & [Y(1,0,0)-Y(0,0,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=0)\\
& & + [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=1)\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=1)\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times M_1(0).\end{aligned}$$
$INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$ and $M_1$ only.\
The reference interaction effect between $A$ and $M_2$ can be written as: $$\begin{aligned}
INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_2(a^\ast)=m_2)\\
\\
\Rightarrow INT_{ref\mbox{-}AM_2}(0,0) & = & \sum_{m_2}[Y(1,0,m_2)-Y(0,0,m_2)-Y(1,0,0)+Y(0,0,0)]\times I(M_2(0)=m_2)\\
\\
& = & [Y(1,0,0)-Y(0,0,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_2(0)=0)\\
& & + [Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]\times I(M_2(0)=1)\\
\\
& = & [Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]\times I(M_2(0)=1)\\
\\
& = & [Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]\times M_2(0).\end{aligned}$$
$INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$ and $M_2$ only.\
The reference interaction effect between $A$, $M_1$ and $M_2$ can be written as: $$\begin{aligned}
INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast)=m_2)\\
\\
\Rightarrow INT_{ref\mbox{-}AM_1M_2}(0,0) & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)-Y(0,m_1,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & -Y(1,m_1,0)+Y(0,m_1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times I(M_1(0)=m_1)\times I(M_2(0)=m_2)\\
\\
& = & \sum_{m_2}[Y(1,0,m_2)-Y(0,0,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & -Y(1,0,0)+Y(0,0,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times I(M_1(0)=0)\times I(M_2(0)=m_2)\\
& & + \sum_{m_2}[Y(1,1,m_2)-Y(0,1,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & -Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0)=m_2)\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)\\
& & -Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0)=0)\\
& & + [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & -Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0)=1)\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & -Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times M_1(0)\times M_2(0).\end{aligned}$$
$INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$, $M_1$ and $M_2$ only.
natural counterfactual interaction effects {#natural-counterfactual-interaction-effects .unnumbered}
------------------------------------------
The natural counterfactual interaction effect between $A$ and $M_1$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_1} & = & Y(a,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & - Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_2(a^\ast)=m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_2(a^\ast)=m_2)-Y(a^\ast,m_1,m_2)I(M_2(a^\ast)=m_2)]\\
& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_2(a^\ast)=m_2)-Y(a^\ast,m_1,m_2)I(M_2(a^\ast)=m_2)\\
& & - Y(a,m_1^\ast,m_2)I(M_2(a^\ast)=m_2)+Y(a^\ast,m_1^\ast,m_2)I(M_2(a^\ast)=m_2)]\\
& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)],\end{aligned}$$ where the sixth equation follows by adding two extra terms which do not change the value of $NatINT_{AM_1}$.\
With the specified conditions, $NatINT_{AM_1}$ can be written as: $$\begin{aligned}
NatINT_{AM_1} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)I(M_2(0)=m_2)-Y(0,m_1,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
\\
& = & \sum_{m_2}[Y(1,0,m_2)I(M_2(0)=m_2)-Y(0,0,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & + \sum_{m_2}[Y(1,1,m_2)I(M_2(0)=m_2)-Y(0,1,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
\\
& = & \sum_{m_2}[Y(1,1,m_2)I(M_2(0)=m_2)-Y(0,1,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
\\
& = & \sum_{m_2}[Y(1,1,m_2)I(M_2(0)=m_2)-Y(0,1,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[M_1(1)-M_1(0)],\end{aligned}$$ where the indicator function $I(M_2(0)=m_2)$ indicates that $M_2$ is at its potential value $M_2(0)$ which may vary with respect to different individuals.\
$NatINT_{AM_1}$ can be interpreted as the effect due to the mediation through $M_1$ and the interaction between $A$ and $M_1$ conditioning on the potential value of $M_2$ with the fixed reference level $a^\ast$.\
The natural counterfactual interaction effect between $A$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_2} & = & Y(a,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)-Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)-Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)\\
& & - Y(a,m_1,m_2^\ast)I(M_1(a^\ast)=m_1) + Y(a^\ast,m_1,m_2^\ast)I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)],\end{aligned}$$ where the sixth equation follows by adding two extra terms which do not change the value of $NatINT_{AM_2}$.\
With the specified conditions, $NatINT_{AM_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_2} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)I(M_1(0)=m_1)-Y(0,m_1,m_2)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,0)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=0)-I(M_2(0)=0)]\\
& & + \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[M_2(1)-M_2(0)],\end{aligned}$$ where the indicator function $I(M_1(0)=m_1)$ indicates that $M_1$ is at its potential value $M_1(0)$ which may vary with respect to different individuals.\
$NatINT_{AM_2}$ can be interpreted as the effect due to the mediation through $M_2$ and the interaction between $A$ and $M_2$ conditioning on the potential value of $M_1$ with the fixed reference level $a^\ast$.\
The natural counterfactual interaction effect between $A$, $M_1$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_1M_2} & = & Y(a,M_1(a),M_2(a))-Y(a^\ast,M_1(a),M_2(a))\\
& & - Y(a,M_1(a^\ast),M_2(a))+Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a,M_1(a),M_2(a^\ast))+Y(a^\ast,M_1(a),M_2(a^\ast))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)][I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & - Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)],\end{aligned}$$ where the fifth equation follows by adding six extra terms which do not change the value of $NatINT_{AM_1M_2}$.\
With the specified conditions, $NatINT_{AM_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_1M_2} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)-Y(0,m_1,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & - Y(1,m_1,0)+Y(0,m_1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & \sum_{m_2}[Y(1,0,m_2)-Y(0,0,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & - Y(1,0,0)+Y(0,0,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
& & + \sum_{m_2}[Y(1,1,m_2)-Y(0,1,m_2)-Y(1,0,m_2)+Y(0,0,m_2)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=0)-I(M_2(0)=0)]\\
& + & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1)-M_2(0)],\end{aligned}$$
$NatINT_{AM_1M_2}$ can be interpreted as the effect due to the mediation through both $M_1$ and $M_2$, and the interaction between $A$, $M_1$ and $M_2$.\
The natural counterfactual interaction effect between $M_1$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{M_1M_2} & = & Y(a^\ast,M_1(a),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a))\\
& & -Y(a^\ast,M_1(a),M_2(a^\ast))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a^\ast,m_1,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a^\ast,m_1,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)],\end{aligned}$$ where the fourth equation follows by adding three extra terms which do not change the value of $NatINT_{M_1M_2}$.\
With the specified conditions, $NatINT_{M_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{M_1M_2} & = & \sum_{m_2}\sum_{m_1}[Y(0,m_1,m_2)-Y(0,0,m_2)-Y(0,m_1,0)+Y(0,0,0)]\\& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & \sum_{m_2}[Y(0,0,m_2)-Y(0,0,m_2)-Y(0,0,0)+Y(0,0,0)]\\& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
& & + \sum_{m_2}[Y(0,1,m_2)-Y(0,0,m_2)-Y(0,1,0)+Y(0,0,0)]\\& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & [Y(0,1,0)-Y(0,0,0)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=0)-I(M_2(0)=0)]\\
& & + [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1)-M_2(0)].\end{aligned}$$
$NatINT_{M_1M_2}$ can be interpreted as the effect due to the mediation through both $M_1$ and $M_2$, and the interaction between $M_1$ and $M_2$. Since the interaction is not involved with the change in exposure $A$, the interpretation can be simply put as the effect due to the mediation through both $M_1$ and $M_2$ only.
pure indirect effects {#pure-indirect-effects .unnumbered}
---------------------
The pure indirect effect through $M_1$ can be rewritten as: $$\begin{aligned}
PIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a^\ast))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]I(M_2(a^\ast)=m_2).\end{aligned}$$
With the specified conditions, $PIE_{M_1}$ can be written as: $$\begin{aligned}
PIE_{M_1} & = & \sum_{m_2}\sum_{m_1}Y(0,m_1,m_2)[I(M_1(1)=m_1)-I(M_1(0)=m_1)]I(M_2(0)=m_2)\\
\\
& = & \sum_{m_2}Y(0,0,m_2)[I(M_1(1)=0)-I(M_1(0)=0)]I(M_2(0)=m_2)\\
& & + \sum_{m_2}Y(0,1,m_2)[I(M_1(1)=1)-I(M_1(0)=1)]I(M_2(0)=m_2)\\
\\
& = & - \sum_{m_2}Y(0,0,m_2)[I(M_1(1)=1)-I(M_1(0)=1)]I(M_2(0)=m_2)\\
& & + \sum_{m_2}Y(0,1,m_2)[I(M_1(1)=1)-I(M_1(0)=1)]I(M_2(0)=m_2)\\
\\
& = & \sum_{m_2}[Y(0,1,m_2)-Y(0,0,m_2)][I(M_1(1)=1)-I(M_1(0)=1)]I(M_2(0)=m_2)\\
\\
& = & \sum_{m_2}[Y(0,1,m_2)I(M_2(0)=m_2)-Y(0,0,m_2)I(M_2(0)=m_2)][M_1(1)-M_1(0)],\end{aligned}$$ where the third equation follows by the facts that $I(M_1(1)=0)=1-I(M_1(1)=1)$ and $I(M_1(0)=0)=1-I(M_1(0)=1)$ and the indicator function, $I(M_2(0)=m_2)$, indicates that $M_2$ is at its potential value $M_2(0)$ which may vary with respect to different individuals.\
$PIE_{M_1}$ can be interpreted as the effect due to the mediation through $M_1$ only, conditioning on the potential value of $M_2$ with the fixed reference level $a^\ast$.\
The pure indirect effect through $M_2$ can be rewritten as: $$\begin{aligned}
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)[I(M_2(a)=m_2)-I(M_2(a^\ast)=m_2)].\end{aligned}$$
With the specified conditions, $PIE_{M_2}$ can be written as: $$\begin{aligned}
PIE_{M_2} & = & \sum_{m_2}\sum_{m_1}Y(0,m_1,m_2)I(M_1(0)=m_1)[I(M_2(1)=m_2)-I(M_2(0)=m_2)]\\
\\
& = & \sum_{m_1}Y(0,m_1,0)I(M_1(0)=m_1)[I(M_2(1)=0)-I(M_2(0)=0)]\\
& & + \sum_{m_1}Y(0,m_1,1)I(M_1(0)=m_1)[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & - \sum_{m_1}Y(0,m_1,0)I(M_1(0)=m_1)[I(M_2(1)=1)-I(M_2(0)=1)]\\
& & + \sum_{m_1}Y(0,m_1,1)I(M_1(0)=m_1)[I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & \sum_{m_1}[Y(0,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)][I(M_2(1)=1)-I(M_2(0)=1)]\\
\\
& = & \sum_{m_1}[Y(0,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)][M_2(1)-M_2(0)],\end{aligned}$$ where the third equation follows by the facts that $I(M_2(1)=0)=1-I(M_2(1)=1)$ and $I(M_2(0)=0)=1-I(M_2(0)=1)$ and the indicator function, $I(M_1(0)=m_1)$, indicates that $M_1$ is at its potential value $M_1(0)$ which may vary with respect to different individuals.\
$PIE_{M_2}$ can be interpreted as the effect due to the mediation through $M_2$ only, conditioning on the potential value of $M_1$ with the fixed reference level $a^\ast$.\
Appendix B. The mediated interaction effects in a non-sequential two-mediator scenario {#appendix-b.-the-mediated-interaction-effects-in-a-non-sequential-two-mediator-scenario .unnumbered}
======================================================================================
Suppose we have a directed acyclic graph as shown in Figure \[fig3\]. We show that the mediated interaction effects proposed by Bellavia and Valeri [@b], if they exist (not equal to zero), are equivalent to the natural counterfactual interaction effects when $A$, $M_1$ and $M_2$ are binary with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$, $m_2^\ast=0$ and $M_1(0)=M_2(0)=0$.
*Proof*:
mediated interaction effect between $A$ and $M_1$ {#mediated-interaction-effect-between-a-and-m_1 .unnumbered}
-------------------------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $NatINT_{AM_1}$ can be written as: $$\begin{aligned}
NatINT_{AM_1} & = & \sum_{m_2}[Y(1,1,m_2)I(M_2(0)=m_2)-Y(0,1,m_2)I(M_2(0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0)=m_2)+Y(0,0,m_2)I(M_2(0)=m_2)]\\
& & \times[M_1(1)-M_1(0)].\end{aligned}$$
If we apply the condition $M_2(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
NatINT_{AM_1} & = & [Y(1,1,0)I(M_2(0)=0)-Y(0,1,0)I(M_2(0)=0)\\
& & - Y(1,0,0)I(M_2(0)=0)+Y(0,0,0)I(M_2(0)=0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & + [Y(1,1,1)I(M_2(0)=1)-Y(0,1,1)I(M_2(0)=1)\\
& & - Y(1,0,1)I(M_2(0)=1)+Y(0,0,1)I(M_2(0)=1)]\\
& & \times[M_1(1)-M_1(0)]\\
& = & [Y(1,1,0)-Y(0,1,0) - Y(1,0,0)+Y(0,0,0)]\times[M_1(1)-M_1(0)],\end{aligned}$$ where the second equality follows by the condition $M_2(0)=0$. This expression is identical to the mediated interaction effect between $A$ and $M_1$ proposed by Bellavia and Valeri [@b].\
mediated interaction effect between $A$ and $M_2$ {#mediated-interaction-effect-between-a-and-m_2 .unnumbered}
-------------------------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $NatINT_{AM_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_2} & = & \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[M_2(1)-M_2(0)].\end{aligned}$$
If we apply the condition $M_1(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
NatINT_{AM_2} & = & [Y(1,0,1)I(M_1(0)=0)-Y(0,0,1)I(M_1(0)=0)\\
& & - Y(1,0,0)I(M_1(0)=0)+Y(0,0,0)I(M_1(0)=0)]\\
& & \times[M_2(1)-M_2(0)]\\
& & + [Y(1,1,1)I(M_1(0)=1)-Y(0,1,1)I(M_1(0)=1)\\
& & - Y(1,1,0)I(M_1(0)=1)+Y(0,1,0)I(M_1(0)=1)]\\
& & \times[M_2(1)-M_2(0)]\\
\\
& = & [Y(1,0,1)-Y(0,0,1) - Y(1,0,0) + Y(0,0,0)
\times[M_2(1)-M_2(0)],\end{aligned}$$ where the second equality follows by the condition $M_1(0)=0$. This expression is identical to the mediated interaction effect between $A$ and $M_2$ proposed by Bellavia and Valeri [@b].
mediated interaction effect between $A$, $M_1$ and $M_2$ {#mediated-interaction-effect-between-a-m_1-and-m_2 .unnumbered}
--------------------------------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $NatINT_{AM_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_1M_2} & = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1)-M_2(0)].\end{aligned}$$
If we apply the conditions $M_1(0)=0$ and $M_2(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
NatINT_{AM_1M_2} & = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[M_1(1)-0]\\
& & \times[M_2(1)-0]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times M_1(1)\times M_2(1)\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times [M_1(1)M_2(1)-0]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)-Y(1,0,1)+Y(0,0,1)\\
& & - Y(1,1,0)+Y(0,1,0)+Y(1,0,0)-Y(0,0,0)]\\
& & \times [M_1(1)M_2(1)-M_1(0)M_2(0)],\end{aligned}$$ where the last equality is identical to the mediated interaction effect between $A$, $M_1$ and $M_2$ proposed by Bellavia and Valeri [@b].
pure natural indirect effect between $M_1$ and $M_2$ ($PNIE_{M_1M_2}$) {#pure-natural-indirect-effect-between-m_1-and-m_2-pnie_m_1m_2 .unnumbered}
----------------------------------------------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $NatINT_{M_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{M_1M_2} & = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1)-M_2(0)].\end{aligned}$$
If we apply the conditions $M_1(0)=0$ and $M_2(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
NatINT_{M_1M_2} & = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times[M_1(1)-0]\\
& & \times[M_2(1)-0]\\
\\
& = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times M_1(1) \times M_2(1)\\
\\
& = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [M_1(1)M_2(1)-0]\\
\\
& = & [Y(0,1,1)-Y(0,0,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [M_1(1)M_2(1)-M_1(0)M_2(0)],\end{aligned}$$ where the last equality is identical to the pure natural indirect effect between $M_1$ and $M_2$ ($PNIE_{M_1M_2}$) proposed by Bellavia and Valeri [@b].
pure indirect effect through $M_1$ {#pure-indirect-effect-through-m_1 .unnumbered}
----------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $PIE_{M_1}$ can be written as: $$\begin{aligned}
PIE_{M_1} & = & \sum_{m_2}[Y(0,1,m_2)I(M_2(0)=m_2)-Y(0,0,m_2)I(M_2(0)=m_2)][M_1(1)-M_1(0)].\end{aligned}$$
If we apply the condition $M_2(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
PIE_{M_1} & = & [Y(0,1,0)-Y(0,0,0)][M_1(1)-M_1(0)],\end{aligned}$$ where the equality is identical to the pure natural indirect effect through $M_1$ ($PNIE_{M_1}$) proposed by Bellavia and Valeri [@b].
pure indirect effect through $M_2$ {#pure-indirect-effect-through-m_2 .unnumbered}
----------------------------------
From Appendix A, we know that with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ the $PIE_{M_2}$ can be written as: $$\begin{aligned}
PIE_{M_2} & = & \sum_{m_1}[Y(0,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)][M_2(1)-M_2(0)].\end{aligned}$$
If we apply the condition $M_1(0)=0$, the equation can be simplified to the following expression: $$\begin{aligned}
PIE_{M_2} & = & [Y(0,0,1)-Y(0,0,0)][M_2(1)-M_2(0)],\end{aligned}$$ where the equality is identical to the pure natural indirect effect through $M_2$ ($PNIE_{M_2}$) proposed by Bellavia and Valeri [@b].
graphical comparison between the mediated interaction effect and the natural counterfactual interaction effect between $A$ and $M_1$ {#graphical-comparison-between-the-mediated-interaction-effect-and-the-natural-counterfactual-interaction-effect-between-a-and-m_1 .unnumbered}
------------------------------------------------------------------------------------------------------------------------------------
With the conditions $a=1$ and $a^\ast=0$, the natural counterfactual interaction effect can be written as: $$\begin{aligned}
NatINT_{AM_1} & = & Y(1,M_1(1),M_2(0))-Y(0,M_1(1),M_2(0))\\
& & - Y(1,M_1(0),M_2(0))+Y(0,M_1(0),M_2(0)),\end{aligned}$$ which is illustrated in Figure \[fig4\] B.\
If we apply the condition $M_2(0)=0$, the natural counterfactual interaction effect will be simplified to the mediated interaction effect between $A$ and $M_1$: $$\begin{aligned}
NatINT_{AM_1} & = & Y(1,M_1(1),0)-Y(0,M_1(1),0))\\
& & - Y(1,M_1(0),0)+Y(0,M_1(0),0),\end{aligned}$$ which is illustrated in Figure \[fig4\] A.
Appendix C. Decomposition of total effect in a sequential two-mediator scenario {#appendix-c.-decomposition-of-total-effect-in-a-sequential-two-mediator-scenario .unnumbered}
===============================================================================
Suppose we have a directed acyclic graph as shown in Figure \[fig5\]. We show that the total effect can be decomposed into the following 9 components at the individual level: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)\\
& & + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}+ NatINT_{M_1M_2}\\
& & + PIE_{M_1} + PIE_{M_2},\end{aligned}$$ where all the natural counterfactual interaction effects are listed in Definition 3. We also give the corresponding interpretation for each component.\
*Proof*:
We first decompose the total effect into total direct effect ($TDE$) [@riden], seminatural indirect effect through $M_1$ ($SIE_{M_1}$) [@p14] and pure indirect effect (path-specific effect) through $M_2$ ($PIE_{M_2}$) [@riden; @p01]. $$\begin{aligned}
TE & = & Y(a)-Y(a^\ast)\\
\\
& = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & + Y(a^\ast,M_1(a),M_2(a,M_1(a))) - Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & + Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))),\end{aligned}$$ where the second equality follows by the composition axiom [@vbook; @a] and the third equality follows by adding and subtracting the same identifiable counterfactual formulas.
The formulas of $TDE$, $SIE_{M_1}$ and $PIE_{M_2}$ are presented below: $$\begin{aligned}
TDE & = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
\\
SIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a,M_1(a))) - Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
\\
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))),\end{aligned}$$ where $TE = TDE + SIE_{M_1}+PIE_{M_2}$.
We next focus on $TDE$ and decompose it into natural counterfactual interaction effects and pure direct effect ($PDE$) [@riden; @p01] by subtracting $PDE$ from $TDE$, where $PDE$ satisfies the definition of a path-specific effect [@p01] and equals the following difference of two identifiable counterfactual formulas: $$\begin{aligned}
PDE & = & Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))).\end{aligned}$$
We have the following results: $$\begin{aligned}
TDE-PDE & = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
& & +Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
& & +Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & +Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
& & +Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & +Y(a,M_1(a),M_2(a^\ast,M_1(a)))-Y(a,M_1(a),M_2(a^\ast,M_1(a)))\\
\\
& = & Y(a,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
& & +Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
& & +Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a),M_2(a^\ast,M_1(a)))+Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\end{aligned}$$ where the second equality follows by adding and subtracting the same identifiable counterfactual formulas and the third equality follows by rearranging all the terms to satisfy the definition of the natural counterfactual interaction effects.
Therefore, we have the following formulas satisfying Definition 3: $$\begin{aligned}
NatINT_{AM_1} & = & Y(a,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
NatINT_{AM_2} & = & Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
NatINT_{AM_1M_2} & = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & -Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a),M_2(a^\ast,M_1(a)))+Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast))).\end{aligned}$$
Accordingly, $TDE$ can be decomposed into the following components: $$\begin{aligned}
TDE = PDE + NatINT_{AM_1} + NatINT_{AM_2} + NatINT_{AM_1M_2}.\end{aligned}$$
We next focus on $PDE$ (path-specific effect) and decompose it into $CDE$ and reference interaction effects [@b; @v4]:
$$\begin{aligned}
PDE & = & Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1} Y(a,m_1,m_2)\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & -\sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = &\sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = &\sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & +\sum_{m_2}\sum_{m_1}[Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = &\sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & +Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)\\
& & +Y(a^\ast,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1,m_2^\ast)-Y(a,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & +Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
& = & \sum_{m_1} [Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
& & + \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast).\end{aligned}$$
According to the derivation above, the following formulas can be obtained: $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & \sum_{m_1} [Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
\\
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2).\end{aligned}$$
With a little mathematical derivation, $INT_{ref\mbox{-}AM_1}$ and $INT_{ref\mbox{-}AM_2+AM_1M_2}$ can be expressed in the form of the counterfactual formula: $$\begin{aligned}
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & Y(a,M_1(a^\ast),m_2^\ast)-Y(a^\ast,M_1(a^\ast),m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a,M_1(a^\ast),m_2^\ast)\\
& & - Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),m_2^\ast)\end{aligned}$$
It is worth noting that $INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ cannot be separated into $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$ since both of the two terms are non-identifiable, which will be discussed in details in Appendix D.
Therefore, $PDE$ can be decomposed into the following components: $$\begin{aligned}
PDE = CDE(m_1^\ast,m_2^\ast) + INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) + INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast).\end{aligned}$$
$TDE$ can be decomposed into the following components: $$\begin{aligned}
TDE & = & PDE + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}\\
\\
& = & CDE(m_1^\ast,m_2^\ast) + INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) + INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)\\
& & + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}.\end{aligned}$$
We next focus on $SIE_{M_1}$ and decompose it into $PIE_{M_1}$ and $NatINT_{M_1M_2}$ by subtracting $PIE_{M_1}$ from $SIE_{M_1}$: $$\begin{aligned}
SIE_{M_1}-PIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & NatINT_{M_1M_2},\end{aligned}$$ where $NatINT_{M_1M_2}$ is listed in Definition 3.
Therefore, $SIE_{M_1}$ can be decomposed into the following components: $$\begin{aligned}
SIE_{M_1} = PIE_{M_1} + NatINT_{M_1M_2}.\end{aligned}$$
Combining all the derivations above, we have the decomposition of total effect as follows: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)\\
& & + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}+ NatINT_{M_1M_2}\\
& & + PIE_{M_1} + PIE_{M_2}.\end{aligned}$$
We next present the interpretation for each component assuming binary $A$, $M_1$ and $M_2$ with the conditions $a=1$, $a^\ast=0$, $m_1^\ast=0$ and $m_2^\ast=0$ for illustration purpose. While other interpretations were proposed in the literature [@v4; @b], our work represent a different and more flexible interpretation from the perspective of population averages which accounts for the distribution of the mediators in the causal structure.
controlled direct effect {#controlled-direct-effect-1 .unnumbered}
------------------------
With the specified conditions, the controlled direct effect can be written as: $$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\\
\\
\Rightarrow \quad\quad CDE(0,0) & = & Y(1,0,0)-Y(0,0,0).\end{aligned}$$
$CDE(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to neither mediation nor interaction.
reference interaction effects {#reference-interaction-effects-1 .unnumbered}
-----------------------------
With the specified conditions, the reference interaction effect between $A$ and $M_1$ can be written as: $$\begin{aligned}
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & \sum_{m_1}[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\
\\
\Rightarrow INT_{ref\mbox{-}AM_1}(0,0) & = & \sum_{m_1}[Y(1,m_1,0)-Y(0,m_1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=m_1)\\
\\
& = & [Y(1,0,0)-Y(0,0,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=0)\\
& & + [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=1)\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times I(M_1(0)=1)\\
\\
& = & [Y(1,1,0)-Y(0,1,0)-Y(1,0,0)+Y(0,0,0)]\times M_1(0).\end{aligned}$$
$INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$ and $M_1$ only.\
The sum of reference interaction effect between $A$ and $M_2$ and reference interaction effect between $A$, $M_1$ and $M_2$ can be written as: $$\begin{aligned}
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
\Rightarrow INT_{ref\mbox{-}AM_2+AM_1M_2}(0) & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)-Y(1,m_1,0)-Y(0,m_1,m_2)+Y(0,m_1,0)]\\
& & \times I(M_1(0)=m_1)\times I(M_2(0,m_1)=m_2)\\
\\
& = & \sum_{M_2}[Y(1,0,m_2)-Y(1,0,0)-Y(0,0,m_2)+Y(0,0,0)]\\& & \times I(M_1(0)=0)\times I(M_2(0,0)=m_2)\\
& & + \sum_{M_2}[Y(1,1,m_2)-Y(1,1,0)-Y(0,1,m_2)+Y(0,1,0)]\\& & \times I(M_1(0)=1)\times I(M_2(0,1)=m_2)\\
\\
& = & [Y(1,0,0)-Y(1,0,0)-Y(0,0,0)+Y(0,0,0)]\\
& & \times I(M_1(0)=0)\times I(M_2(0,0)=0)\\
& & + [Y(1,0,1)-Y(1,0,0)-Y(0,0,1)+Y(0,0,0)]\\
& & \times I(M_1(0)=0)\times I(M_2(0,0)=1)\\
& & + [Y(1,1,0)-Y(1,1,0)-Y(0,1,0)+Y(0,1,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0,1)=0)\\
& & + [Y(1,1,1)-Y(1,1,0)-Y(0,1,1)+Y(0,1,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0,1)=1)\\
\\
& = & [Y(1,0,1)-Y(1,0,0)-Y(0,0,1)+Y(0,0,0)]\\
& & \times I(M_1(0)=0)\times I(M_2(0,0)=1)\\
& & + [Y(1,1,1)-Y(1,1,0)-Y(0,1,1)+Y(0,1,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0,1)=1)\\
\\
& = & [Y(1,0,1)-Y(1,0,0)-Y(0,0,1)+Y(0,0,0)]\\
& & \times [1-I(M_1(0)=1)]\times I(M_2(0,0)=1)\\
& & + [Y(1,1,1)-Y(1,1,0)-Y(0,1,1)+Y(0,1,0)]\\
& & \times I(M_1(0)=1)\times I(M_2(0,1)=1)\\
\\
& = & [Y(1,0,1)-Y(1,0,0)-Y(0,0,1)+Y(0,0,0)]\\
& & \times [1-M_1(0)]\times M_2(0,0)\\
& & + [Y(1,1,1)-Y(1,1,0)-Y(0,1,1)+Y(0,1,0)]\\
& & \times M_1(0)\times M_2(0,1).\\\end{aligned}$$
$INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ can be interpreted as the effect due to the interaction between $A$ and $M_2$ only, conditioning on the potential value of $M_1$ with the fixed reference level $a^\ast$.\
natural counterfactual interaction effects {#natural-counterfactual-interaction-effects-1 .unnumbered}
------------------------------------------
The natural counterfactual interaction effect between $A$ and $M_1$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_1} & = & Y(a,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & - Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]\times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_2(a^\ast,m_1)=m_2)-Y(a^\ast,m_1,m_2)I(M_2(a^\ast,m_1)=m_2)]\\
& & \times [I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_2(a^\ast,m_1)=m_2)-Y(a^\ast,m_1,m_2)I(M_2(a^\ast,m_1)=m_2)\\
& & - Y(a,m_1^\ast,m_2)I(M_2(a^\ast,m_1^\ast)=m_2)-Y(a^\ast,m_1^\ast,m_2)I(M_2(a^\ast,m_1^\ast)=m_2)]\\
& & \times [I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)],\end{aligned}$$ where the last equation follows by adding two extra terms which do not change the value of $NatINT_{AM_1}$.\
With the specified conditions, $NatINT_{AM_1}$ can be written as: $$\begin{aligned}
NatINT_{AM_1} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)I(M_2(0,m_1)=m_2)-Y(0,m_1,m_2)I(M_2(0,m_1)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0,0)=m_2)+Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
\\
& = & \sum_{m_2}[Y(1,0,m_2)I(M_2(0,0)=m_2)-Y(0,0,m_2)I(M_2(0,0)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0,0)=m_2)+Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & + \sum_{m_2}[Y(1,1,m_2)I(M_2(0,1)=m_2)-Y(0,1,m_2)I(M_2(0,1)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0,0)=m_2)+Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
\\
& = & \sum_{m_2}[Y(1,1,m_2)I(M_2(0,1)=m_2)-Y(0,1,m_2)I(M_2(0,1)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0,0)=m_2)+Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
\\
& = & \sum_{m_2}[Y(1,1,m_2)I(M_2(0,1)=m_2)-Y(0,1,m_2)I(M_2(0,1)=m_2)\\
& & - Y(1,0,m_2)I(M_2(0,0)=m_2)+Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times[M_1(1)-M_1(0)],\end{aligned}$$ where the indicator functions $I(M_2(0,1)=m_2)$ and $I(M_2(0,0)=m_2)$ indicate that $M_2$ is at its potential values $M_2(0,1)$ and $M_2(0,0)$ which may vary with respect to different individuals.\
$NatINT_{AM_1}$ can be interpreted as the effect due to the mediation through $M_1$ and the interaction between $A$ and $M_1$ conditioning on the potential values of $M_2$ with the fixed reference level $a^\ast$.\
The natural counterfactual interaction effect between $A$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_2} & = & Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)\\
& & \times[I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)-Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)-Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)\\
& & - Y(a,m_1,m_2^\ast)I(M_1(a^\ast)=m_1) + Y(a^\ast,m_1,m_2^\ast)I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)],\end{aligned}$$ where the last equation follows by adding two extra terms which do not change the value of $NatINT_{AM_2}$.\
With the specified conditions, $NatINT_{AM_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_2} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)I(M_1(0)=m_1)-Y(0,m_1,m_2)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,m_1)=m_2)-I(M_2(0,m_1)=m_2)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,0)I(M_1(0)=m_1)-Y(0,m_1,0)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,m_1)=0)-I(M_2(0,m_1)=0)]\\
& & + \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
\\
& = & \sum_{m_1}[Y(1,m_1,1)I(M_1(0)=m_1)-Y(0,m_1,1)I(M_1(0)=m_1)\\
& & - Y(1,m_1,0)I(M_1(0)=m_1)+Y(0,m_1,0)I(M_1(0)=m_1)]\\
& & \times[M_2(1,m_1)-M_2(0,m_1)],\end{aligned}$$ where the indicator function $I(M_1(0)=m_1)$ indicates that $M_1$ is at its potential value $M_1(0)$ which may vary with respect to different individuals.\
$NatINT_{AM_2}$ can be interpreted as the effect due to the mediation through $M_2$ and the interaction between $A$ and $M_2$, conditioning on the potential value of $M_1$ with the fixed reference level $a^\ast$.\
The natural counterfactual interaction effect between $A$, $M_1$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{AM_1M_2} & = & Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a),M_2(a,M_1(a)))\\
& & - Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))+Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a,M_1(a),M_2(a^\ast,M_1(a)))+Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))\\
& & +Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)]I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)][I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)- Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)\\
& & + Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
& & + \sum_{m_2}\sum_{m_1}[-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)]\\
& & \times[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times[I(M_2(a,m_1^\ast)=m_2)-I(M_2(a^\ast,m_1^\ast)=m_2)],\end{aligned}$$ where the last equation follows by adding six extra terms which do not change the value of $NatINT_{AM_1M_2}$.\
With the specified conditions, $NatINT_{AM_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{AM_1M_2} & = & \sum_{m_2}\sum_{m_1}[Y(1,m_1,m_2)-Y(0,m_1,m_2)- Y(1,m_1,0)+Y(0,m_1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,m_1)=m_2)-I(M_2(0,m_1)=m_2)]\\
& & + \sum_{m_2}\sum_{m_1}[-Y(1,0,m_2)+Y(0,0,m_2)]\\
& & \times[I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times[I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& = & \sum_{m_2} [Y(1,0,m_2)-Y(0,0,m_2)- Y(1,0,0)+Y(0,0,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times[I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
& & + \sum_{m_2} [Y(1,1,m_2)-Y(0,1,m_2)- Y(1,1,0)+Y(0,1,0)+ Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=m_2)-I(M_2(0,1)=m_2)]\\
& & + \sum_{m_2}[-Y(1,0,m_2)+Y(0,0,m_2)]\\
& & \times[I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times[I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
& & + \sum_{m_2}[-Y(1,0,m_2)+Y(0,0,m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& = & \sum_{m_2} [Y(1,1,m_2)-Y(0,1,m_2)- Y(1,1,0)+Y(0,1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=m_2)-I(M_2(0,1)=m_2)]\\
& & + \sum_{m_2}[-Y(1,0,m_2)+Y(0,0,m_2)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& = & [Y(1,1,0)-Y(0,1,0)- Y(1,1,0)+Y(0,1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=0)-I(M_2(0,1)=0)]\\
& & + [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [-Y(1,0,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=0)-I(M_2(0,0)=0)]\\
& & + [-Y(1,0,1)+Y(0,0,1)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=0)-I(M_2(0,1)=0)]\\
& & + [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [-Y(1,0,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=0)-I(M_2(0,0)=0)]\\
& & + [-Y(1,0,1)+Y(0,0,1)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & -[Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0) + Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
& & + [-Y(1,0,1)+Y(0,0,1)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [Y(1,0,0)-Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
& & + [-Y(1,0,1)+Y(0,0,1)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & - [Y(1,0,1)-Y(0,0,1)-Y(1,0,0)+Y(0,0,0)]\\
& & \times[I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times[I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(1,1,1)-Y(0,1,1)- Y(1,1,0)+Y(0,1,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1,1)-M_2(0,1)]\\
& & + [-Y(1,0,1)+Y(0,0,1)+Y(1,0,0)-Y(0,0,0)]\\
& & \times[M_1(1)-M_1(0)]\\
& & \times[M_2(1,0)-M_2(0,0)],\end{aligned}$$ where the six equality follows by the facts that $I(M_2(1,1)=0)=1-I(M_2(1,1)=1)$ and $I(M_2(0,1)=0)=1-I(M_2(0,1)=1)$.\
$NatINT_{AM_1M_2}$ can be interpreted as the effect due to mediation through both $M_1$ and $M_2$, and the interaction between $A$, $M_1$ and $M_2$.\
The natural counterfactual interaction effect between $M_1$ and $M_2$ can be rewritten as: $$\begin{aligned}
NatINT_{M_1M_2} & = & Y(a^\ast,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))\\
& & -Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))+Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & + \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1} Y(a^\ast,m_1,m_2)[I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times [I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
\\
& = & \sum_{m_2}\sum_{m_1} [Y(a^\ast,m_1,m_2)-Y(a^\ast,m_1,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times [I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times [I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)]\\
& & + \sum_{m_2}\sum_{m_1} [-Y(a^\ast,m_1^\ast,m_2)]\\
& & \times [I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)]\\
& & \times [I(M_2(a,m_1^\ast)=m_2)-I(M_2(a^\ast,m_1^\ast)=m_2)],\end{aligned}$$ where the last equality follows by adding three extra terms which do not change the value of $NatINT_{M_1M_2}$.\
With the specified conditions, $NatINT_{M_1M_2}$ can be written as: $$\begin{aligned}
NatINT_{M_1M_2} & = & \sum_{m_2}\sum_{m_1} [Y(0,m_1,m_2)-Y(0,m_1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times [I(M_2(1,m_1)=m_2)-I(M_2(0,m_1)=m_2)]\\
& & + \sum_{m_2}\sum_{m_1} [-Y(0,0,m_2)]\\
& & \times [I(M_1(1)=m_1)-I(M_1(0)=m_1)]\\
& & \times [I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& = & \sum_{m_2} [Y(0,0,m_2)-Y(0,0,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times [I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
& & + \sum_{m_2}[Y(0,1,m_2)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=m_2)-I(M_2(0,1)=m_2)]\\
& & + \sum_{m_2} [-Y(0,0,m_2)]\\
& & \times [I(M_1(1)=0)-I(M_1(0)=0)]\\
& & \times [I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
& & + \sum_{m_2} [-Y(0,0,m_2)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& & = \sum_{m_2}[Y(0,1,m_2)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=m_2)-I(M_2(0,1)=m_2)]\\
& & + \sum_{m_2} [-Y(0,0,m_2)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=m_2)-I(M_2(0,0)=m_2)]\\
\\
& = & [Y(0,1,0)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=0)-I(M_2(0,1)=0)]\\
& & + [Y(0,1,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [-Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=0)-I(M_2(0,0)=0)]\\
& & + [-Y(0,0,1)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [-Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [Y(0,1,1)-Y(0,1,0)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
& & + [-Y(0,0,1)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(0,1,1)-Y(0,1,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,1)=1)-I(M_2(0,1)=1)]\\
& & + [-Y(0,0,1)+Y(0,0,0)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
& & \times [I(M_2(1,0)=1)-I(M_2(0,0)=1)]\\
\\
& = & [Y(0,1,1)-Y(0,1,0)]\times [M_1(1)-M_1(0)]\times [M_2(1,1)-M_2(0,1)]\\
& & + [-Y(0,0,1)+Y(0,0,0)] \times [M_1(1)-M_1(0)]\times [M_2(1,0)-M_2(0,0)],\end{aligned}$$ where the fifth equality follows by the facts that $I(M_2(1,1)=0)=1-I(M_2(1,1)=1)$ and $I(M_2(0,1)=0)=1-I(M_2(0,1)=1)$.\
$NatINT_{M_1M_2}$ can be interpreted as the effect due to mediation through both $M_1$ and $M_2$, and the interaction between $M_1$ and $M_2$. Since the interaction is not involved with the change in exposure $A$, the interpretation can be simply put as the effect due to the mediation through both $M_1$ and $M_2$ only.\
pure indirect effects {#pure-indirect-effects-1 .unnumbered}
---------------------
The pure indirect effect through $M_1$ can be rewritten as: $$\begin{aligned}
PIE_{M_1} & = & Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)\times [I(M_1(a)=m_1)-I(M_1(a^\ast)=m_1)] \times I(M_2(a^\ast,m_1)=m_2).\end{aligned}$$
With the specified conditions, $PIE_{M_1}$ can be written as: $$\begin{aligned}
PIE_{M_1} & = & \sum_{m_2}\sum_{m_1}Y(0,m_1,m_2)\times [I(M_1(1)=m_1)-I(M_1(0)=m_1)]\times I(M_2(0,m_1)=m_2)\\
\\
& = & \sum_{m_2} Y(0,0,m_2)\times [I(M_1(1)=0)-I(M_1(0)=0)] \times I(M_2(0,0)=m_2)\\
& & + \sum_{m_2} Y(0,1,m_2)\times [I(M_1(1)=1)-I(M_1(0)=1)]\times I(M_2(0,1)=m_2)\\
\\
& = & -\sum_{m_2} Y(0,0,m_2)\times [I(M_1(1)=1)-I(M_1(0)=1)]\times I(M_2(0,0)=m_2)\\
& & + \sum_{m_2} Y(0,1,m_2)\times [I(M_1(1)=1)-I(M_1(0)=1)]\times I(M_2(0,1)=m_2)\\
\\
& = & \sum_{m_2} [Y(0,1,m_2)I(M_2(0,1)=m_2)-Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times [I(M_1(1)=1)-I(M_1(0)=1)]\\
\\
& = & \sum_{m_2} [Y(0,1,m_2)I(M_2(0,1)=m_2)-Y(0,0,m_2)I(M_2(0,0)=m_2)]\\
& & \times [M_1(1)-M_1(0)],\end{aligned}$$ where the third equation follows by the facts that $I(M_1(1)=0)=1-I(M_1(1)=1)$ and $I(M_1(0)=0)=1-I(M_1(0)=1)$ and the indicator functions, $I(M_2(0,1)=m_2)$ and $I(M_2(0,0)=m_2)$, indicate that $M_2$ is at its potential values which may vary with respect to different individuals.\
$PIE_{M_1}$ can be interpreted as the effect due to the mediation through $M_1$ only, conditioning on the potential values of $M_2$ with the fixed reference level $a^\ast$.\
The pure indirect effect through $M_2$ can be rewritten as: $$\begin{aligned}
PIE_{M_2} & = & Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a,m_1)=m_2)\\
& & - \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)I(M_1(a^\ast)=m_1)I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}Y(a^\ast,m_1,m_2)\times I(M_1(a^\ast)=m_1) \times [I(M_2(a,m_1)=m_2)-I(M_2(a^\ast,m_1)=m_2)].\end{aligned}$$
With the specified conditions, $PIE_{M_2}$ can be written as: $$\begin{aligned}
PIE_{M_1} & = & \sum_{m_2}\sum_{m_1}Y(0,m_1,m_2)\times I(M_1(0)=m_1)\times [I(M_2(1,m_1)=m_2)-I(M_2(0,m_1)=m_2)]\\
\\
& = & \sum_{m_1}Y(0,m_1,0)\times I(M_1(0)=m_1)\times [I(M_2(1,m_1)=0)-I(M_2(0,m_1)=0)]\\
& & + \sum_{m_1}Y(0,m_1,1)\times I(M_1(0)=m_1)\times [I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
\\
& = & -\sum_{m_1}Y(0,m_1,0)\times I(M_1(0)=m_1)\times [I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
& & + \sum_{m_1}Y(0,m_1,1)\times I(M_1(0)=m_1)\times [I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
\\
& = & \sum_{m_1}[Y(0,m_1,1)\times I(M_1(0)=m_1)-Y(0,m_1,0)\times I(M_1(0)=m_1)]\\
& & \times [I(M_2(1,m_1)=1)-I(M_2(0,m_1)=1)]\\
\\
& = & \sum_{m_1}[Y(0,m_1,1)\times I(M_1(0)=m_1)-Y(0,m_1,0)\times I(M_1(0)=m_1)]\\
& & \times [M_2(1,m_1)-M_2(0,m_1)],\end{aligned}$$ where the third equation follows by the facts that $I(M_2(1,m_1)=0)=1-I(M_2(1,m_1)=1)$ and $I(M_2(0,m_1)=0)=1-I(M_2(0,m_1)=1)$ and the indicator functions, $I(M_1(0)=m_1)$, indicates that $M_1$ is at its potential values which may vary with respect to different individuals.\
$PIE_{M_2}$ can be interpreted as the effect due to the mediation through $M_2$ only, conditioning on the potential values of $M_1$ with the fixed reference level $a^\ast$.
Appendix D. Non-identifiability issues of $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$\
and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$ in a sequential two-mediator scenario {#appendix-d.-non-identifiability-issues-of-int_refmbox-am_2m_1astm_2ast-and-int_refmbox-am_1m_2m_1astm_2ast-in-a-sequential-two-mediator-scenario .unnumbered}
=======================================================================================
We show that the reference interaction effects, $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$, are non-identifiable in a sequential two-mediator scenario as shown in Figure \[fig5\].\
*Proof*:
We first decompose $INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ into $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$: $$\begin{aligned}
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)\\
& & + Y(a^\ast,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)\\
& & +Y(a^\ast,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)+Y(a,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
& = & \sum_{m_2}\sum_{m_1}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
& & +\sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2).\end{aligned}$$
Therefore, we have the following formulas: $$\begin{aligned}
INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}\sum_{m_1}[Y(a,m_1^\ast,m_2)-Y(a^\ast,m_1^\ast,m_2)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\\
\\
INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast) & = & \sum_{m_2}\sum_{m_1}[Y(a,m_1,m_2)-Y(a^\ast,m_1,m_2)-Y(a,m_1^\ast,m_2)+Y(a^\ast,m_1^\ast,m_2)\\
& & -Y(a,m_1,m_2^\ast)+Y(a^\ast,m_1,m_2^\ast)+Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2).\end{aligned}$$
It can be seen that both formulas include the following term: $$\begin{aligned}
\sum_{m_2}\sum_{m_1}Y(a,m_1^\ast,m_2)\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2),\end{aligned}$$ which can be rewritten as the counterfactual formula $Y(a,m_1^\ast,M_2(a^\ast,M_1(a^\ast)))$.
Note that $m_1^\ast$ is an arbitrary reference level of $M_2$. Let us consider an instance that there exists $a^{\ast\ast}\neq a^\ast$ such that $M_1(a^{\ast\ast})=m_1^\ast$. In this case, the counterfactual formula can be rewritten as $Y(a,M_1(a^{\ast\ast}),M_2(a^\ast,M_1(a^\ast)))$, where $M_1$ is being activated by two different values of exposure $A$ in the kite graph formed up by the path $A\rightarrow M_1\rightarrow Y$ and the path $A\rightarrow M_1\rightarrow M_2 \rightarrow Y$ in Figure \[fig5\]. Avin et al. [@a] showed that such counterfactual formulas are non-identifiable and referred to as problematic counterfactual formulas. Because the instance cannot be ruled out in any certain population, $Y(a,m_1^\ast,M_2(a^\ast,M_1(a^\ast)))$ is non-identifiable. Therefore, $INT_{ref\mbox{-}AM_2}(m_1^\ast,m_2^\ast)$ and $INT_{ref\mbox{-}AM_1M_2}(m_1^\ast,m_2^\ast)$ are non-identifiable.
Appendix E. Linear regression models with continuous outcome and continuous mediators in a sequential two-mediator scenario {#appendix-e.-linear-regression-models-with-continuous-outcome-and-continuous-mediators-in-a-sequential-two-mediator-scenario .unnumbered}
===========================================================================================================================
Suppose we have a directed acyclic graph as shown in Figure \[fig5\]. Assume the following linear models for $Y$, $M_2$ and $M_1$ are correctly specified: $$\begin{aligned}
E[Y|A,M_1,M_2,C] & = & \theta_0 + \theta_1A + \theta_2M_1 + \theta_3M_2 + \theta_4AM_1 + \theta_5AM_2 + \theta_6M_1M_2\\
& & + \theta_7AM_1M_2 + \theta_8^\prime C\\
\\
E[M_2|A,M_1,C] & = & \beta_0 + \beta_1A + \beta_2M_1 + \beta_3AM_1 + \beta_4^\prime C\\
\\
E[M_1|A,C] & = & \gamma_0 + \gamma_1A + \gamma_2^\prime C, \end{aligned}$$ where $C$ is a sufficient confounding set that satisfies the identification assumptions $(A1)$-$(A6)$; $\epsilon_Y$, $\epsilon_{M_2}$ and $\epsilon_{M_1}$ denote independent random error terms for $Y$, $M_2$ and $M_1$ and follow $N(0,\sigma_{Y}^2)$, $N(0,\sigma_{M_2}^2)$ and $N(0,\sigma_{M_1}^2)$, respectively. According to Appendix C, the total effect can be decomposed into the following components: $$\begin{aligned}
TE & = & CDE(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)+INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)\\
& & + NatINT_{AM_1} + NatINT_{AM_2}+ NatINT_{AM_1M_2}+ NatINT_{M_1M_2}\\
& & + PIE_{M_1} + PIE_{M_2}.\end{aligned}$$
The expected value of each component conditional on the sufficient confounding set are presented in the following.
Controlled direct effect {#controlled-direct-effect-2 .unnumbered}
------------------------
$$\begin{aligned}
CDE(m_1^\ast,m_2^\ast) & = & Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)\end{aligned}$$
$$\begin{aligned}
&\Rightarrow & E[CDE(m_1^\ast,m_2^\ast)|c]\\
\\
& = & E[Y(a,m_1^\ast,m_2^\ast)-Y(a^\ast,m_1^\ast,m_2^\ast)|c]\\
\\
& = & E[Y(a,m_1^\ast,m_2^\ast)|c]-E[Y(a^\ast,m_1^\ast,m_2^\ast)|c]\\
\\
& = & E[Y(a,m_1^\ast,m_2^\ast)|a,c]-E[Y(a^\ast,m_1^\ast,m_2^\ast)|a^\ast,c]\quad by\; A1\\
\\
& = & E[Y(a,m_1^\ast,m_2^\ast)|a,m_1^\ast,m_2^\ast,c]-E[Y(a^\ast,m_1^\ast,m_2^\ast)|a^\ast,m_1^\ast,m_2^\ast,c]\quad by\; A2\\
\\
& = & E[Y|a,m_1^\ast,m_2^\ast,c]-E[Y|a^\ast,m_1^\ast,m_2^\ast,c]\quad by\; consistency\\
\\
&= & (\theta_0+\theta_1a+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4am_1^\ast+\theta_5am_2^\ast+\theta_6m_1^\ast m_2^\ast+\theta_7am_1^\ast m_2^\ast+\theta_8^\prime c)\\
& & -(\theta_0+\theta_1a^\ast+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4a^\ast m_1^\ast+\theta_5a^\ast m_2^\ast+\theta_6m_1^\ast m_2^\ast+\theta_7a^\ast m_1^\ast m_2^\ast+\theta_8^\prime c)\\
\\
& = & (\theta_1a+\theta_4am_1^\ast+\theta_5am_2^\ast+\theta_7am_1^\ast m_2^\ast)-(\theta_1a^\ast+\theta_4a^\ast m_1^\ast+\theta_5a^\ast m_2^\ast+\theta_7a^\ast m_1^\ast m_2^\ast)\\
\\
& = & \theta_1\left(a-a^\ast\right)+\theta_4m_1^\ast\left(a-a^\ast\right)+\theta_5m_2^\ast\left(a-a^\ast\right)+\theta_7m_1^\ast m_2^\ast\left(a-a^\ast\right)\\
\\
& = & \left(\theta_1+\theta_4m_1^\ast+\theta_5m_2^\ast+\theta_7m_1^\ast m_2^\ast\right)\left(a-a^\ast\right).\end{aligned}$$
\
Reference interaction effect between $A$ and $M_1$ {#reference-interaction-effect-between-a-and-m_1 .unnumbered}
--------------------------------------------------
We first consider $M_1$ as a categorical random variable. $$\begin{aligned}
INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast) & = & \sum_{m_1} [Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\\\end{aligned}$$ $$\begin{aligned}
&\Rightarrow & E[INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)|c]\\
\\
& = & E\left[\sum_{m_1} [Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\times I(M_1(a^\ast)=m_1)\bigg| c\right]\\
\\
& = & \sum_{m_1}E\left[[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)]\times I(M_1(a^\ast)=m_1)|c\right]\\
\\
& = & \sum_{m_1}E\left[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)|c\right]\\
& &\times E\left[I(M_1(a^\ast)=m_1)|c\right]\quad by\; A4\\
\\
& = & \sum_{m_1}E\left[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)|c\right]\\
& &\times \Pr(M_1(a^\ast)=m_1|c)\\
\\
& = & \sum_{m_1}E\left[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)|c\right]\\
& &\times \Pr(M_1(a^\ast)=m_1|a^\ast,c)\quad by\; A3\\
\\
& = & \sum_{m_1}E\left[Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2^\ast)-Y(a,m_1^\ast,m_2^\ast)+Y(a^\ast,m_1^\ast,m_2^\ast)|c\right]\\
& &\times \Pr(M_1=m_1|a^\ast,c)\quad by\; consistency\\
\\
& = & \sum_{m_1}{E\left[Y\left(a,m_1,m_2^\ast\right)\middle| c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}-\sum_{m_1}{E\left[Y\left(a^\ast,m_1,m_2^\ast\right)\middle| c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & -\sum_{m_1}{E\left[Y\left(a,m_1^\ast,m_2^\ast\right)\middle| c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}+\sum_{m_1}{E\left[Y\left(a^\ast,m_1^\ast,m_2^\ast\right)\middle| c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
\\
& = & \sum_{m_1}{E\left[Y\left(a,m_1,m_2^\ast\right)\middle|a,m_1,m_2^\ast,c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & -\sum_{m_1}{E\left[Y\left(a^\ast,m_1,m_2^\ast\right)\middle|a^\ast,m_1,m_2^\ast, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & -\sum_{m_1}{E\left[Y\left(a,m_1^\ast,m_2^\ast\right)\middle|a,m_1^\ast,m_2^\ast, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & +\sum_{m_1}{E\left[Y\left(a^\ast,m_1^\ast,m_2^\ast\right)\middle|a^\ast,m_1^\ast,m_2^\ast, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\quad by\; A1\;A2\\
\\
& = & \sum_{m_1}{E\left[Y|a,m_1,m_2^\ast,c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}-\sum_{m_1}{E\left[Y|a^\ast,m_1,m_2^\ast, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & -\sum_{m_1}{E\left[Y|a,m_1^\ast,m_2^\ast\, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}+\sum_{m_1}{E\left[Y|a^\ast,m_1^\ast,m_2^\ast, c\right]}\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\; by\; consistency\end{aligned}$$
We next extend the formula to consider a continuous $M_1$. $$\begin{aligned}
& & E[INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)|c] \\
\\
& = & \int_{m_1}{E\left[Y\middle| a,m_1,m_2^\ast,c\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}}\\
& & -\int_{m_1}{E\left[Y\middle| a^\ast,m_1,m_2^\ast,c\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}}\\
& & -\int_{m_1}{E\left[Y\middle| a,m_1^\ast,m_2^\ast,c\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}}\\
& & +\int_{m_1}{E\left[Y\middle| a^\ast,m_1^\ast,m_2^\ast,c\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}}\\
\\
& = & \int_{m_1}(\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2^\ast+\theta_4am_1+\theta_5am_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7am_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)})\\
& & -\int_{m_1}(\theta_0+\theta_1a^\ast+\theta_2m_1+\theta_3m_2^\ast+\theta_4a^\ast m_1+\theta_5a^\ast m_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7a^\ast m_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & -\int_{m_1}(\theta_0+\theta_1a+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4am_1^\ast+\theta_5am_2^\ast\\
& & +\theta_6m_1^\ast m_2^\ast+\theta_7am_1^\ast m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & + \int_{m_1}(\theta_0+\theta_1a^\ast+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4a^\ast m_1^\ast+\theta_5a^\ast m_2^\ast\\
& & +\theta_6m_1^\ast m_2^\ast+\theta_7a^\ast m_1^\ast m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
\\
& = & \left(\theta_0+\theta_1a+\theta_3m_2^\ast+\theta_5am_2^\ast+\theta_8^\prime c\right)+\left(\theta_2+\theta_4a+\theta_6m_2^\ast+\theta_7am_2^\ast\right)\\
& & \times\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_0+\theta_1a^\ast+\theta_3m_2^\ast+\theta_5a^\ast m_2^\ast+\theta_8^\prime c\right)-\left(\theta_2+\theta_4a^\ast+\theta_6m_2^\ast+\theta_7a^\ast m_2^\ast\right)\\
& & \times\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_0+\theta_1a+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4am_1^\ast+\theta_5am_2^\ast+\theta_6m_1^\ast m_2^\ast+\theta_7am_1^\ast m_2^\ast+\theta_8^\prime c\right)\\
& & +\left(\theta_0+\theta_1a^\ast+\theta_2m_1^\ast+\theta_3m_2^\ast+\theta_4a^\ast m_1^\ast+\theta_5a^\ast m_2^\ast+\theta_6m_1^\ast m_2^\ast+\theta_7a^\ast m_1^\ast m_2^\ast+\theta_8^\prime c\right)\\
\\
& = & \left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c-m_1^\ast\right)\times\left(\theta_4+\theta_7m_2^\ast\right)\times\left(a-a^\ast\right).\end{aligned}$$\
The sum of two reference interaction effects: $INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)$ {#the-sum-of-two-reference-interaction-effects-int_refmbox-am_2am_1m_2m_2ast .unnumbered}
---------------------------------------------------------------------------------------
$$\begin{aligned}
INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast) & = & \sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\\
& & \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\end{aligned}$$
$$\begin{aligned}
&\Rightarrow & E[INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)|c]\\
\\
& = & E\left[\sum_{m_2}\sum_{m_1} [Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\right. \\
& & \left.\times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)\bigg|c\right]\\
\\
& = & \sum_{m_2}\sum_{m_1}E\left[[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)]\right.\\
& &\left. \times I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)|c\right]\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)|c]\\
& & \times E[I(M_1(a^\ast)=m_1)\times I(M_2(a^\ast,m_1)=m_2)|c]\; by\; A4\;A6\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)|c]\\
& & \times \Pr(M_1(a^\ast)=m_1|c)\times \Pr(M_2(a^\ast,m_1)=m_2)|c)\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)|c]\\
& & \times \Pr(M_1(a^\ast)=m_1|a^\ast,c)\times \Pr(M_2(a^\ast,m_1)=m_2)|a^\ast,m_1,c)\; by\; A3\;A5\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)-Y(a,m_1,m_2^\ast)-Y(a^\ast,m_1,m_2)+Y(a^\ast,m_1,m_2^\ast)|c]\\
& & \times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\; by\;consistency\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)|c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2^\ast)|c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y(a^\ast,m_1,m_2)|c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & + \sum_{m_2}\sum_{m_1}E[Y(a^\ast,m_1,m_2^\ast)|c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2)|a,m_1,m_2,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y(a,m_1,m_2^\ast)|a,m_1,m_2^\ast,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y(a^\ast,m_1,m_2)|a^\ast,m_1,m_2,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & + \sum_{m_2}\sum_{m_1}E[Y(a^\ast,m_1,m_2^\ast)|a^\ast,m_1,m_2^\ast,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\; by\;A1\;A2\\
\\
& = & \sum_{m_2}\sum_{m_1}E[Y|a,m_1,m_2,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y|a,m_1,m_2^\ast,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & - \sum_{m_2}\sum_{m_1}E[Y|a^\ast,m_1,m_2,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\\
& & + \sum_{m_2}\sum_{m_1}E[Y|a^\ast,m_1,m_2^\ast,c]\times \Pr(M_1=m_1|a^\ast,c)\times \Pr(M_2=m_2|a^\ast,m_1,c)\; by\;consistency\\
\\
& = & \int_{m_2}\int_{m_1}{E\left[Y\middle| a,m_1,m_2,c\right]}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}\\
& & - \int_{m_2}\int_{m_1}{E\left[Y\middle| a,m_1,m_2^\ast,c\right]}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}\\
& & - \int_{m_2}\int_{m_1}{E\left[Y\middle| a^\ast,m_1,m_2,c\right]}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}\\
& & + \int_{m_2}\int_{m_1}{E\left[Y\middle| a^\ast,m_1,m_2^\ast,c\right]}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}\\
\\
& = & \int_{m_1}\int_{m_2}{E\left[Y\middle| a,m_1,m_2,c\right]}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1}\int_{m_2}{E\left[Y\middle| a,m_1,m_2^\ast,c\right]}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1}\int_{m_2}{E\left[Y\middle| a^\ast,m_1,m_2,c\right]}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & + \int_{m_1}\int_{m_2}{E\left[Y\middle| a^\ast,m_1,m_2^\ast,c\right]}d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
\\
& = & \int_{m_1}\int_{m_2}(\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2+\theta_4am_1+\theta_5am_2\\
& & +\theta_6m_1m_2+\theta_7am_1m_2+\theta_8^\prime c) d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1}\int_{m_2}(\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2^\ast+\theta_4am_1+\theta_5am_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7am_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1}\int_{m_2}(\theta_0+\theta_1a^\ast+\theta_2m_1+\theta_3m_2+\theta_4a^\ast m_1+\theta_5a^\ast m_2\\
& & +\theta_6m_1m_2+\theta_7a^\ast m_1m_2+\theta_8^\prime c)d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & + \int_{m_1}\int_{m_2} (\theta_0+\theta_1a^\ast+\theta_2m_1+\theta_3m_2^\ast+\theta_4a^\ast m_1+\theta_5a^\ast m_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7a^\ast m_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_2=m_2\middle| a^\ast,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
\\
& = & \int_{m_1}\left[(\theta_0+\theta_1a+\theta_2m_1+\theta_4am_1+\theta_8^\prime c)\right.\\
& & +\left.\left(\theta_3+\theta_5a+\theta_6m_1+\theta_7am_1\right)\times\left(\beta_0+\beta_1a^\ast+\beta_2m_1+\beta_3a^\ast m_1+\beta_4^\prime c\right)\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1} (\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2^\ast+\theta_4am_1+\theta_5am_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7am_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & - \int_{m_1} \left[(\theta_0+\theta_1a^\ast+\theta_2m_1+\theta_4a^\ast m_1+\theta_8^\prime c)\right.\\
& & +\left.\left(\theta_3+\theta_5a^\ast+\theta_6m_1+\theta_7a^\ast m_1\right)\times\left(\beta_0+\beta_1a^\ast+\beta_2m_1+\beta_3a^\ast m_1+\beta_4^\prime c\right)\right]d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
& & + \int_{m_1} (\theta_0+\theta_1a^\ast+\theta_2m_1+\theta_3m_2^\ast+\theta_4a^\ast m_1+\theta_5a^\ast m_2^\ast\\
& & +\theta_6m_1m_2^\ast+\theta_7a^\ast m_1m_2^\ast+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a^\ast,c\right)}\\
\\
& = & \left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\\
& & -\left(\theta_0+\theta_1a+\theta_3m_2^\ast+\theta_5am_2^\ast+\theta_8^\prime c\right)\\
& & -\left(\theta_2+\theta_4a+\theta_6m_2^\ast+\theta_7am_2^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_0+\theta_1a^\ast+\theta_8^\prime c\right)-\left(\theta_3+\theta_5a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & -\left(\theta_2+\theta_4a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & -\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\\
& & +\left(\theta_0+\theta_1a^\ast+\theta_3m_2^\ast+\theta_5a^\ast m_2^\ast+\theta_8^\prime c\right)\\
& & +\left(\theta_2+\theta_4a^\ast+\theta_6m_2^\ast+\theta_7a^\ast m_2^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
\\
& = & \theta_1\left(a-a^\ast\right)+\theta_5\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(a-a^\ast\right)\\
& & +\theta_4\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\left(a-a^\ast\right)\\
& & +\theta_7\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\left(a-a^\ast\right)\\
& & +\theta_5\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\left(a-a^\ast\right)\\
& & +\theta_7\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\left(a-a^\ast\right)\\
& & -\left(\theta_1+\theta_5m_2^\ast\right)\left(a-a^\ast\right)-(\theta_4+\theta_7m_2^\ast)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)(a-a^\ast)\\
\\
& = & \left\{\theta_1+\theta_5\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\right.\\
& & \left.+\theta_7\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right. \\
& & \left.+\theta_5\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & \left.+\theta_7\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\right.\\
& & \left. -\left(\theta_1+\theta_5m_2^\ast\right)-\theta_7m_2^\ast\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right) \right\}(a-a^\ast).\end{aligned}$$
\
Natural counterfactual interaction effects {#natural-counterfactual-interaction-effects-2 .unnumbered}
------------------------------------------
We derive the the expected value of each interaction effect.
$$\begin{aligned}
Y(a,M_1(a),M_2(a,M_1(a))) = \sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)\times I(M_1(a)=m_1)\times I(M_2(a,m_1)=m_2)\end{aligned}$$
$$\begin{aligned}
&\Rightarrow& E[Y(a,M_1(a),M_2(a,M_1(a)))|c]\\
\\
& = & E\left[\sum_{m_2}\sum_{m_1}Y(a,m_1,m_2)\times I(M_1(a)=m_1)\times I(M_2(a,m_1)=m_2) \bigg|c\right]\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\times I\left(M_1\left(a\right)=m_1\right)\times I\left(M_2\left(a,m_1\right)=m_2\right)\middle| c\right]}\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\middle| c\right]E\left[I\left(M_1\left(a\right)=m_1\right)\middle| c\right]E\left[I\left(M_2\left(a,m_1\right)=m_2\right)\middle| c\right]}\; by\; A4\;A6\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\middle| c\right]\Pr{\left(M_1\left(a\right)=m_1\middle| c\right)}\Pr{\left(M_2\left(a,m_1\right)=m_2\middle| c\right)}}\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\middle| c\right]\Pr{\left(M_1\left(a\right)=m_1\middle| a,c\right)}\Pr{\left(M_2\left(a,m_1\right)=m_2\middle| a,m_1,c\right)}}\; by\; A3\;A5\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\middle| c\right]\Pr{\left(M_1=m_1\middle| a,c\right)}\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}}\; by\; consistency\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\left(a,m_1,m_2\right)\middle| a,m_1,m_2,c\right]\Pr{\left(M_1=m_1\middle| a,c\right)}\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}}\; by\; A1\;A2\\
\\
& = & \sum_{m_2}\sum_{m_1}{E\left[Y\middle| a,m_1,m_2,c\right]\Pr{\left(M_1=m_1\middle| a,c\right)}\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}}\; by\; consistency\\
\\
& = & \int_{m_2}\int_{m_1}{E\left[Y\middle| a,m_1,m_2,c\right]d}\Pr{\left(M_1=m_1\middle| a,c\right)}d\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}\\
\\
& = & \int_{m_2}\int_{m_1}(\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2+\theta_4am_1+\theta_5am_2\\
& & +\theta_6m_1m_2+\theta_7am_1m_2+\theta_8^\prime c)d\Pr{\left(M_1=m_1\middle| a,c\right)}d\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}\\
\\
& = & \int_{m_1}\int_{m_2}(\theta_0+\theta_1a+\theta_2m_1+\theta_3m_2+\theta_4am_1+\theta_5am_2\\
& & +\theta_6m_1m_2+\theta_7am_1m_2+\theta_8^\prime c)d\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a,c\right)}\\
\\
& = & \int_{m_1}\int_{m_2} [(\theta_0+\theta_1a+\theta_2m_1+\theta_4am_1+\theta_8^\prime c)\\
& & +\left(\theta_3+\theta_5a+\theta_6m_1+\theta_7am_1\right)m_2]d\Pr{\left(M_2=m_2\middle| a,m_1,c\right)}d\Pr{\left(M_1=m_1\middle| a,c\right)}\\
\\
& = & \int_{m_1} \left[\left(\theta_0+\theta_1a+\theta_2m_1+\theta_4am_1+\theta_8^\prime c\right)\right.\\
& & +\left.\left(\theta_3+\theta_5a+\theta_6m_1+\theta_7am_1\right)\left(\beta_0+\beta_1a+\beta_2m_1+\beta_3am_1+\beta_4^\prime c\right)\right]d\Pr\left(M_1=m_1\middle| a,c\right)\\
\\
& = & \int_{m_1} \left[\left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_2+\theta_4a\right)m_1\right.\\
& & +\left.\left(\theta_3+\theta_5a+\left(\theta_6+\theta_7a\right)m_1\right)\left(\beta_0+\beta_1a+\beta_4^\prime c+\left(\beta_2+\beta_3a\right)m_1\right)\right]d\Pr\left(M_1=m_1\middle| a,c\right)\\
\\
& = & \left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)+\left(\theta_6+\theta_7a\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a\right)\left(\beta_2+\beta_3a\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_2+\beta_3a\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)^2\right].\qquad\qquad(W1)\end{aligned}$$
Similarly, we can obtain the following expected values for the rest of the counterfactual formulas. $$\begin{aligned}
& & E[Y(a,M_1(a),M_2(a^\ast,M_1(a)))|c]\\
\\
& = & \left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)+\left(\theta_6+\theta_7a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W2)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a,M_1(a^\ast),M_2(a,M_1(a^\ast)))|c]\\
\\
& = & \left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)+\left(\theta_6+\theta_7a\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a\right)\left(\beta_2+\beta_3a\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_2+\beta_3a\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W3)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a^\ast,M_1(a),M_2(a,M_1(a)))|c]\\
\\
& = & \left(\theta_0+\theta_1a^\ast+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a^\ast\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a^\ast\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)+\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W4)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))|c]\\
\\
& = & \left(\theta_0+\theta_1a^\ast+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a^\ast\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W5)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))|c]\\
\\
& = & \left(\theta_0+\theta_1a^\ast+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a^\ast\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W6)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & \left(\theta_0+\theta_1a+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\qquad\qquad(W7)\end{aligned}$$
$$\begin{aligned}
& & E[Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & \left(\theta_0+\theta_1a^\ast+\theta_8^\prime c\right)+\left(\theta_3+\theta_5a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\\
& & +\left(\theta_2+\theta_4a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & +\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right].\qquad\qquad(W8)\end{aligned}$$
The formulas of natural counterfactual interaction effects can be obtained as follows: $$\begin{aligned}
& & E[NatINT_{AM_1}|c]\\
\\
& = & (W2)-(W6)-(W7)+(W8)\\
\\
& = & \left[\theta_4\gamma_1+\theta_7\gamma_1\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)+\theta_5\gamma_1\left(\beta_2+\beta_3a^\ast\right)\right.\\
& & +2\theta_7\gamma_1\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\left.\theta_7\gamma_1^2\left(\beta_2+\beta_3a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast)^2\end{aligned}$$
$$\begin{aligned}
& & E[NatINT_{AM_2}|c]\\
\\
& = & (W3)-(W5)-(W7)+(W8)\\
\\
& = & \left[\theta_5\beta_1+\theta_7\beta_1\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)+\theta_5\beta_3\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & \left.+\theta_7\beta_3\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right] \right](a-a^\ast)^2\end{aligned}$$
$$\begin{aligned}
& & E[NatINT_{AM_1M_2}|c]\\
\\
& = & (W1)-(W4)-(W3)+(W5)-(W2)+(W6)+(W7)-(W8)\\
\\
& = & \left[\theta_7\beta_1\gamma_1+\theta_5\beta_3\gamma_1+2\theta_7\beta_3\gamma_1\left(\gamma_0+\gamma_2^\prime c\right)+\theta_7\beta_3\gamma_1^2\left(a+a^\ast\right) \right](a-a^\ast)^3\end{aligned}$$
$$\begin{aligned}
& & E[NatINT_{M_1M_2}|c]\\
\\
& = & (W4)-(W5)-(W6)+(W8)\\
\\
& = & \left[\beta_1\gamma_1\left(\theta_6+\theta_7a^\ast\right)+\beta_3\gamma_1\left(\theta_3+\theta_5a^\ast\right)\right.\\
& & +2\beta_3\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & \left. +\beta_3\gamma_1^2\left(\theta_6+\theta_7a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast)^2.\end{aligned}$$
\
Pure indirect effects {#pure-indirect-effects-2 .unnumbered}
---------------------
The pure indirect effect through $M_1$ can be obtained by the following derivation: $$\begin{aligned}
PIE_{M_1} = Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\end{aligned}$$ $$\begin{aligned}
&\Rightarrow& E[PIE_{M_1}|c]\\
\\
& = & E[Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & E[Y(a^\ast,M_1(a),M_2(a^\ast,M_1(a)))|c]-E[Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & (W6)-(W8)
\\
& = & \left[\gamma_1\left(\theta_2+\theta_4a^\ast\right)+\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\right.\\
& & +\gamma_1\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\\
& & +2\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & \left.+\gamma_1^2\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast).\end{aligned}$$
Similarly, the pure indirect effect through $M_2$ can be obtained by the following derivation: $$\begin{aligned}
PIE_{M_2} = Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\end{aligned}$$ $$\begin{aligned}
&\Rightarrow& E[PIE_{M_2}|c]\\
\\
& = & E[Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & E[Y(a^\ast,M_1(a^\ast),M_2(a,M_1(a^\ast)))|c]-E[Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & (W5)-(W8)
\\
& = & \left[\beta_1\left(\theta_3+\theta_5a^\ast\right)+\beta_1\left(\theta_6+\theta_7a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & +\beta_3\left(\theta_3+\theta_5a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & \left.+\beta_3\left(\theta_6+\theta_7a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\right](a-a^\ast).\end{aligned}$$\
Total effect {#total-effect .unnumbered}
------------
$$\begin{aligned}
TE = Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))\end{aligned}$$
$$\begin{aligned}
&\Rightarrow& E[TE|c]\\
\\
& = & E[Y(a,M_1(a),M_2(a,M_1(a)))-Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & E[Y(a,M_1(a),M_2(a,M_1(a)))|c]-E[Y(a^\ast,M_1(a^\ast),M_2(a^\ast,M_1(a^\ast)))|c]\\
\\
& = & (W1)-(W8)
\\
& = & \left[\theta_1+\theta_5\left(\beta_0+\beta_4^\prime c\right)+\beta_1\theta_3+\theta_4\left(\gamma_0+\gamma_2^\prime c\right)+\gamma_1\theta_2\right.\\
& & +\theta_7\left(\beta_0+\beta_4^\prime c\right)\left(\gamma_0+\gamma_2^\prime c\right)+\beta_1\theta_6\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\gamma_1\theta_6\left(\beta_0+\beta_4^\prime c\right)+\theta_5\beta_2\left(\gamma_0+\gamma_2^\prime\right)+\theta_3\beta_3\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\theta_3\beta_2\gamma_1+\theta_7\beta_2\sigma_{M_1}^2+\theta_6\beta_3\sigma_{M_1}^2+\theta_7\beta_2\left(\gamma_0+\gamma_2^\prime c\right)^2\\
& & \left. +\theta_6\beta_3\left(\gamma_0+\gamma_2^\prime c\right)^2+2\gamma_1\theta_6\beta_2\left(\gamma_0+\gamma_2^\prime c\right)\right](a-a^\ast)\\
\\
& & +\left[\beta_1\theta_5+\gamma_1\theta_4+\beta_1\theta_7\left(\gamma_0+\gamma_2^\prime c\right)\right.\\
& & +\gamma_1\theta_7\left(\beta_0+\beta_4^\prime c\right)+\gamma_1\beta_1\theta_6+\theta_5\beta_3\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\theta_5\beta_2\gamma_1+\theta_3\beta_3\gamma_1+\theta_7\beta_3\sigma_{M_1}^2+\theta_7\beta_3\left(\gamma_0+\gamma_2^\prime c\right)^2\\
& & \left.+2\gamma_1\theta_7\beta_2\left(\gamma_0+\gamma_2^\prime c\right)+2\gamma_1\theta_6\beta_3\left(\gamma_0+\gamma_2^\prime c\right)+\theta_6\beta_2\gamma_1^2\right]\left(a^2-{a^\ast}^2\right)\\
\\
& & + \left[\gamma_1\beta_1\theta_7+\theta_5\beta_3\gamma_1+2\gamma_1\theta_7\beta_3\left(\gamma_0+\gamma_2^\prime c\right)+\theta_7\beta_2\gamma_1^2+\theta_6\beta_3\gamma_1^2\right]\left(a^3-{a^\ast}^3\right)\\
\\
& & + \theta_7\beta_3\gamma_1^2\left(a^4-{a^\ast}^4\right).\end{aligned}$$
\
Summary of Results {#summary-of-results .unnumbered}
------------------
In this subsection, the results of all components are listed below for a quick reference for the readers. $$\begin{aligned}
E[CDE(m_1^\ast,m_2^\ast)|c]
& = & \left(\theta_1+\theta_4m_1^\ast+\theta_5m_2^\ast+\theta_7m_1^\ast m_2^\ast\right)\left(a-a^\ast\right)\\
\\
E[INT_{ref\mbox{-}AM_1}(m_1^\ast,m_2^\ast)|c]
& = & \left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c-m_1^\ast\right)\times\left(\theta_4+\theta_7m_2^\ast\right)\times\left(a-a^\ast\right)\\
\\
E[INT_{ref\mbox{-}AM_2+AM_1M_2}(m_2^\ast)|c]
& = & \left\{\theta_1+\theta_5\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\right.\\
& & \left.+\theta_7\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right. \\
& & \left.+\theta_5\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & \left.+\theta_7\left(\beta_2+\beta_3a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\right.\\
& & \left. -\left(\theta_1+\theta_5m_2^\ast\right)-\theta_7m_2^\ast\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right) \right\}(a-a^\ast)\\
\\
E[NatINT_{AM_1}|c]
& = & \left[\theta_4\gamma_1+\theta_7\gamma_1\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)+\theta_5\gamma_1\left(\beta_2+\beta_3a^\ast\right)\right.\\
& & +2\theta_7\gamma_1\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\left.\theta_7\gamma_1^2\left(\beta_2+\beta_3a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast)^2\\
\\
E[NatINT_{AM_2}|c]
& = & \left[\theta_5\beta_1+\theta_7\beta_1\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)+\theta_5\beta_3\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & \left.+\theta_7\beta_3\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right] \right](a-a^\ast)^2\\
\\
E[NatINT_{AM_1M_2}|c]
& = & \left[\theta_7\beta_1\gamma_1+\theta_5\beta_3\gamma_1+2\theta_7\beta_3\gamma_1\left(\gamma_0+\gamma_2^\prime c\right)+\theta_7\beta_3\gamma_1^2\left(a+a^\ast\right) \right](a-a^\ast)^3\\
\\
E[NatINT_{M_1M_2}|c]
& = & \left[\beta_1\gamma_1\left(\theta_6+\theta_7a^\ast\right)+\beta_3\gamma_1\left(\theta_3+\theta_5a^\ast\right)\right.\\
& & +2\beta_3\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & \left. +\beta_3\gamma_1^2\left(\theta_6+\theta_7a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast)^2\\
\\
E[PIE_{M_1}|c]
& = & \left[\gamma_1\left(\theta_2+\theta_4a^\ast\right)+\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\beta_0+\beta_1a^\ast+\beta_4^\prime c\right)\right.\\
& & +\gamma_1\left(\theta_3+\theta_5a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\\
& & +2\gamma_1\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(\gamma_0+\gamma_2^\prime c\right)\\
& & \left.+\gamma_1^2\left(\theta_6+\theta_7a^\ast\right)\left(\beta_2+\beta_3a^\ast\right)\left(a+a^\ast\right)\right](a-a^\ast)\\
\\
E[PIE_{M_2}|c]
& = & \left[\beta_1\left(\theta_3+\theta_5a^\ast\right)+\beta_1\left(\theta_6+\theta_7a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\right.\\
& & +\beta_3\left(\theta_3+\theta_5a^\ast\right)\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)\\
& & \left.+\beta_3\left(\theta_6+\theta_7a^\ast\right)\left[\sigma_{M_1}^2+\left(\gamma_0+\gamma_1a^\ast+\gamma_2^\prime c\right)^2\right]\right](a-a^\ast)\\
\\
E[TE|c]
& = & \left[\theta_1+\theta_5\left(\beta_0+\beta_4^\prime c\right)+\beta_1\theta_3+\theta_4\left(\gamma_0+\gamma_2^\prime c\right)+\gamma_1\theta_2\right.\\
& & +\theta_7\left(\beta_0+\beta_4^\prime c\right)\left(\gamma_0+\gamma_2^\prime c\right)+\beta_1\theta_6\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\gamma_1\theta_6\left(\beta_0+\beta_4^\prime c\right)+\theta_5\beta_2\left(\gamma_0+\gamma_2^\prime\right)+\theta_3\beta_3\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\theta_3\beta_2\gamma_1+\theta_7\beta_2\sigma_{M_1}^2+\theta_6\beta_3\sigma_{M_1}^2+\theta_7\beta_2\left(\gamma_0+\gamma_2^\prime c\right)^2\\
& & \left. +\theta_6\beta_3\left(\gamma_0+\gamma_2^\prime c\right)^2+2\gamma_1\theta_6\beta_2\left(\gamma_0+\gamma_2^\prime c\right)\right](a-a^\ast)\\
\\
& & +\left[\beta_1\theta_5+\gamma_1\theta_4+\beta_1\theta_7\left(\gamma_0+\gamma_2^\prime c\right)\right.\\
& & +\gamma_1\theta_7\left(\beta_0+\beta_4^\prime c\right)+\gamma_1\beta_1\theta_6+\theta_5\beta_3\left(\gamma_0+\gamma_2^\prime c\right)\\
& & +\theta_5\beta_2\gamma_1+\theta_3\beta_3\gamma_1+\theta_7\beta_3\sigma_{M_1}^2+\theta_7\beta_3\left(\gamma_0+\gamma_2^\prime c\right)^2\\
& & \left.+2\gamma_1\theta_7\beta_2\left(\gamma_0+\gamma_2^\prime c\right)+2\gamma_1\theta_6\beta_3\left(\gamma_0+\gamma_2^\prime c\right)+\theta_6\beta_2\gamma_1^2\right]\left(a^2-{a^\ast}^2\right)\\
\\
& & + \left[\gamma_1\beta_1\theta_7+\theta_5\beta_3\gamma_1+2\gamma_1\theta_7\beta_3\left(\gamma_0+\gamma_2^\prime c\right)+\theta_7\beta_2\gamma_1^2+\theta_6\beta_3\gamma_1^2\right]\left(a^3-{a^\ast}^3\right)\\
\\
& & + \theta_7\beta_3\gamma_1^2\left(a^4-{a^\ast}^4\right).\end{aligned}$$
[^1]: \[a1\]Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87131, USA
[^2]: \[a2\]Comprehensive Cancer Center, University of New Mexico, Albuquerque, NM, 87131, USA
[^3]: \[a3\]Department of Internal Medicine, University of New Mexico, Albuquerque, NM, 87131, USA
|
---
abstract: 'We discuss some aspects of de Araujo, Coelho and Costa [@arau16; @arau17] concerning the role of a time dependent magnetic ellipticity on the pulsars’ braking indices and on the putative gravitational waves (GWs) these objects can emit. Since only nine of more than 2000 known pulsars have accurately measured braking indices, it is of interest to extend this study to all known pulsars, in particular as regards GW generation. In summary, our results show a pessimistic prospect for the detection of GWs generated by these pulsars, even for advanced detectors such as aLIGO and AdVirgo, and the planned Einstein Telescope, would not be able to detect these pulsar, if the ellipticity has magnetic origin.'
address:
- |
Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais,\
S. J. Campos, SP 1227-010, Brazil\
$^*$E-mail: jcarlos.dearaujo@inpe.br
- |
Departamento de Física, Universidade Tecnológica Federal do Paraná\
Medianeira, PR 85884-000, Brazil\
$^{**}$E-mail: jazielcoelho@utfpr.edu.br
- |
Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais,\
S. J. Campos, SP 1227-010, Brazil\
$^\dagger$E-mail: samantha.ladislau@inpe.br\
$^\ddagger$E-mail: cesar.costa@inpe.br
author:
- 'José C. N. de Araujo$^*$'
- 'Jaziel G. Coelho$^{**}$'
- 'Samantha M. Ladislau$^\dagger$ and César A. Costa$^\ddagger$'
title: Gravitational Waves From Pulsars Due To Their Magnetic Ellipticity
---
Ellipticity of Magnetic Origin and Gravitational Waves from Pulsars
===================================================================
If the magnetic field and (or) the angle between the axes of rotation and the magnetic dipole of the pulsars are independent of time, the combination of magnetic dipole and gravitational wave (GW) brakes could only explain braking index (n) in the interval $3 < n < 5$. The observations, however, show that only PSR J1640-4631 has braking index in this interval, as can be seen in Table \[ta1\]. In particular, we consider this issue in the context of magnetic ellipticity [@arau16]. It is worth stressing that the magnetic field and the angle between the axes of rotation and the magnetic dipole of the pulsars are dependent on time.
Recall that the equatorial ellipticity is given by $$\epsilon=\frac{I_{xx}-I_{yy}}{I_{zz}},$$ where $I_{xx}$, $I_{yy}$, $I_{zz}$ are the moment of inertia with respect to the rotation axis, $z$, and along directions perpendicular to it.
The pulsar is deformed by its own dipole magnetic field. Such deformation associated with the fact that the axes of rotation and of the magnetic dipole are misaligned generates an ellipticity given by (see, e.g., Bonazzola and Gourgoulhon [@bona96]; Konno et al [@konn00]; de Freitas Pacheco and Regimbau [@regi06]): $$\epsilon_B = \kappa\frac{B_0^2 R^4}{G M^2}\sin^2\phi, \label{eq:epsilonB}$$
where $B_0$ is the dipole magnetic field, $R$ and $M$ are the radius and the mass of the star respectively, $\phi$ is the angle between the rotation and magnetic dipole axes, whereas $\kappa$ is the distortion parameter, which depends on both the star equation of state (EoS) and the magnetic field configuration [@regi06]. We consider that $\kappa = 10 - 1000$, as suggested by numerical simulations [@bona96; @regi06].
$^\diamond n \equiv f_{\rm rot}\,{\ddot f}_{\rm rot}/{\dot{f}^2_{\rm rot}}$, where $f_{\rm rot} = 1/P$ is the rotating frequency, $\dot{f}_{\rm rot}$ and $\ddot{f}_{\rm rot}$ are their time derivatives.\
\[ta1\]
Recall that the power emitted by a rotating magnetic dipole is given by [@padm01] $$\dot{E}_{\rm d}= -\frac{16\pi^4}{3}\frac{B_0^2 R^6\sin^2\phi}{c^3}f_{\rm rot}^4, \label{Ed}$$ and the power loss via GW emission reads [@shap83] $$\dot{E}_{\rm GW} = -\frac{2048\pi^6}{5}\frac{G}{c^5}I^2\epsilon^2 f_{\rm rot}^6. \label{EGW}$$
Also, the total energy of the pulsar is provided by its rotational energy, $E_{\rm rot} = 2 \pi^2If_{\rm rot}^2$, and any change on it is given by $\dot{E}_{\rm d}$ and $\dot{E}_{\rm GW}$, namely $$\dot{E}_{\rm rot}\equiv \dot{E}_{\rm GW} +\dot{E}_{\rm d} \label{Erotdef}.$$
Now, from the definition of the braking index (see, e.g., the note in Table \[ta1\]), one can easily obtain that[^1] $$n=3+2\eta-2\frac{P}{\dot P}\left(1+\eta\right)\left[\frac{\dot B_0}{B_0}+\dot{\phi}\cot{\phi} \right], \label{neta}$$ where $\eta$ is defined in such a way that $\dot{E}_{\rm GW} = \eta \dot{E}_{\rm rot}$, which is interpreted as the efficiency of GW generation. In de Araujo, Coelho & Costa [@arau16] it is also shown that with Eq. \[neta\] one can explain, in principle, the braking indices of the pulsars of Table \[ta1\].
Recall that the GW amplitude generated by a pulsar reads $$h^2 = \frac{5}{2}\frac{G}{c^3}\frac{I}{r^2}\frac{\mid\dot{f}_{\rm rot}\mid}{f_{\rm rot}}.$$ This equation considers that the spindown is due to gravitational waves only, i.e., n = 5 (spindown limit - SD).
From the definition of $\eta$ one obtains that $\dot{\bar{f}}_{\rm rot} = \eta \dot{f}_{\rm rot}$, i.e., the part of the spindown related to the GW emission brake. Thus, one can obtain an equation for the GW amplitude that holds for n $ < $ 5, namely $$\bar{h}^2 = \frac{5}{2}\frac{G}{c^3}\frac{I}{r^2}\frac{\mid\dot{\bar{f}}_{rot}\mid}{f_{rot}} = \frac{5}{2}\frac{G}{c^3}\frac{I}{r^2}\frac{\mid\dot{f}_{\rm rot}\mid}{f_{\rm rot}} \, \eta . \label{heta}$$
Recall that the GW amplitude also reads $$h = \frac{16\pi^2G}{c^4} \frac{I\epsilon f_{\rm rot}^2}{r},$$ (see, e.g, Shapiro and Teukolsky[@shap83]). Combining both equations for the GW amplitude one obtains $$\epsilon = \sqrt{\frac{5}{512\pi^4} \frac{c^5}{G}\frac{\dot{P}P^3}{I}\eta}. \label{epet}$$
Now, for a purely magnetic brake we have $$\bar{B}_0\sin^2\phi = \frac{3 I c^3}{4 \pi^2 R^6} P \dot{P},$$ where $\bar{B}_0$ would be the magnetic field whether the break were purely magnetic. If there is also a GW brake contribution we have that $B_0 < \bar{B}_0$. Combining the definition of $\eta$ and Eq. \[epet\] one obtains after some algebraic manipulation the following equation for the efficiency $\eta$ $$\eta = 1 - \left(\frac{B_0}{\bar{B}_0} \right)^2,$$ which is obviously lower than one, as it should be. Substituting this last equation into Eq.\[eq:epsilonB\] we obtain $$\epsilon = \frac{3Ic^3}{4\pi^2GM^2R^2}P\dot{P} \left(1 - \eta\right) \kappa. \label{eek}$$ Finally, substituting this last equation into equation \[epet\], we obtain $$\eta = \frac{288}{5}\frac{I^3c}{GM^4R^4}\frac{\dot{P}}{P}\left( 1-\eta \right)^2 \kappa^2. \label{ek}$$
Notice that with Eqs.\[eek\] and \[ek\] we obtain $\epsilon$ and $\eta$ in terms of $M$, $R$, $I$, $P$ and $\dot{P}$ for a given value of $\kappa$. Since in practice $\eta \ll 1$, the following useful equations are obtained $$\epsilon \simeq \frac{3Ic^3}{4\pi^2GM^2R^2}P\dot{P}\kappa \label{eeka}$$ and $$\eta \simeq \frac{288}{5}\frac{I^3c}{GM^4R^4}\frac{\dot{P}}{P} \kappa^2. \label{eka}$$
We now calculate $\epsilon_{B}$ and $\eta$ for the pulsars of Table \[ta1\]. We then adopt fiducial values for M, R and I. We adopt $\kappa = 10$ and 1000, which have the same orders of magnitude of the values considered by, e.g., Regimbau and de Freitas Pacheco[@regi06] .
In Table \[ta2\] we present the result of these calculations. Even for the extremely optimistic case, the value of the ellipticity is at best $\epsilon_{B} \sim 10^{-5}$ (for ) and the corresponding efficiency $\eta \sim 10^{-8}$. Therefore, the amplitude of the GW in this case would be four orders of magnitude lower than the spindown limit ($\eta=1$). Thus, even advanced detectors such as aLIGO and AdVirgo, and the planned Einstein Telescope, would not be able to detect these pulsars.
\[ta2\]
Notice that Eqs. \[eeka\] and \[eka\] do not depend on the braking index n. Consequently, we can calculate such quantities for the pulsars of the ATNF Pulsar Catalog. We refer the reader to the paper by de Araujo, Coelho and Costa[@arau17] for details . In Fig. \[fig1\] we show an interesting histogram with the data of the ATNF Catalog, namely, the number of pulsars for $\log \epsilon_{B}$ bin. Note the high number of pulsars concentrated around $\sim 10^{-10}\, (10^{-8})$ for $ k = 10 \,(1000)$. The values of $\eta$ are also extremely small, a histogram can be found in de Araujo, Coelho & Costa[@arau17], where can be seen a peak at $10^{-16} - 10^{-15}$.
![Ellipticity histogram for the pulsars of ATNF Catalog for $\kappa = 10$.[]{data-label="fig1"}](fig1){width="8cm"}
These extremely small values of $\epsilon_{B}$ and $\eta$ imply that the GW amplitudes are at best seven orders of magnitude smaller than those obtained by assuming the spindown limit (SD), being therefore hardly detected (see Fig. \[fig2\]).
![Histogram of $\eta^{1/2} = h/h^{SD}$ (spin-down ratio) for the pulsars of ATNF Catalog for $\kappa = 10$.[]{data-label="fig2"}](fig2){width="8cm"}
Final Remarks
=============
We present an expression for the braking index considering that the ellipticity is of magnetic dipole origin and time dependent. In this context, we model the braking indices of the 9 pulsars that have such measured quantities accurately. Then we calculate the amplitudes of the GWs generated by these 9 pulsars. Summing up, we conclude that these amplitudes are too small to be detected. For example, the pulsar PSR J1846-0258 would need to be observed for over 1000 years to be detected by the Einstein Telescope.
Since the equations for $\eta$, $\epsilon_{B}$ and $h$ are independent of n, we extend our study for most of the pulsars of the “ATNF Pulsar Catalog”. Regarding detectability, the prospects remain pessimistic, since the ellipticity generated by the magnetic dipole is extremely small, the corresponding amplitude of GWs is much smaller than the amplitude obtained via the spindown limit.
Acknowledgments {#acknowledgments .unnumbered}
===============
J.C.N.A thanks FAPESP (2013/26258-4) and CNPq (307217/2016-7) for partial support. J.G.C. is likewise grateful to the support of CNPq (421265/2018-3 and 305369/2018-0). S.M.L. and C.A.C. acknowledge CAPES for financial support.
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[^1]: The detailed derivation of Eq. \[neta\] can be found in de Araujo, Coelho & Costa [@arau16].
|
---
abstract: 'We show how to realize a single-photon Dicke state in a large one-dimensional array of two-level systems, and discuss how to test its quantum properties. Realization of single-photon Dicke states relies on the cooperative nature of the interaction between a field reservoir and an array of two-level-emitters. The resulting dynamics of the delocalized state can display Rabi-like oscillations when the number of two-level emitters exceeds several hundred. In this case the large array of emitters is essentially behaving like a “mirror-less cavity”. We outline how this might be realized using a multiple-quantum-well structure and discuss how the quantum nature of these oscillations could be tested with the Leggett-Garg inequality and its extensions.'
author:
- 'Guang-Yin Chen$^{1,2}$, Neill Lambert$^2$, Che-Ming Li$^{1}$, Yueh-Nan Chen$^{1\star}$, and Franco Nori$^{2,4}$'
title: 'Delocalized single-photon Dicke states and the Leggett-Garg inequality in solid state systems'
---
Department of Physics and National Center for Theoretical Sciences, National Cheng-Kung University, Tainan 701, Taiwan
Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan
Department of Engineering Science, National Cheng-Kung University, Tainan City 701, Taiwan
Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA\
$^\star$e-mail:yuehnan@mail.ncku.edu.tw
When an ensemble of atoms interacts with a common radiation field each atom can no longer be regarded as an individual radiation source but the whole ensemble of atoms can be regarded as a macroscopic dipole moment[@Lvo; @Inouye]. This collective behavior leads to cooperative radiation, i.e. the so-called superradiance introduced by Dicke in 1954. Superradiance, and its extended effects, has also been observed in solid state systems such as quantum dots[@SRQD], quantum wells[@SRQW], and coupled cavities[@zhou]. This effect is generally characterized by an enhanced emission intensity that scales as the square of the number of atoms.
Recently, a particularly interesting consequence of this cooperative interaction was discussed by Svidzinsky *et al* [@scullyPRA; @newPRA; @scullyPRL08]. In their work they showed that there could be cooperative delocalized effects even when just a single photon is injected into a large cloud of atoms. The state that is created via this mechanism is a highly-entangled Dicke state[@wstatepaper]. An interesting open question is if such a state can be realized and manipulated in a solid-state environment.
To answer this question we analyze what happens when a single-photon is injected into a large *one-dimensional* array of two-level-emitters (TLE). We find that because of the cooperative interaction between light and matter the structure acts like an effective optical cavity without mirrors[@scullyPRL08], and realizes a one-dimensional variation of the Dicke-state discussed by Svidzinsky *et al* [@scullyPRA; @newPRA; @scullyPRL08]. We show that the delocalized state formed in this emitter-array can exhibit quantum behaviour through the coherent oscillatory dynamics of the state. We discuss how such a phenomenon might be realized in a multiple-quantum-well (MQW) array and discuss physically-realistic parameters. To show how the quantum features of such an experiment might be verified, we apply the Leggett-Garg (LG) inequality[@LG], and a Markovian extension[@NLPRL], to examine the quantum coherence of the delocalized state over the MQW structure. Finally, we discuss two other potential candidates for experimental realization.
Results {#results .unnumbered}
=======
We consider an array containing $N$ two-level emitters coupled to a photonic reservoir. A photon with wavevector $k_0$ incident on the array, as shown in Fig. 1(a). If the $N$-TLE array uniformly absorbs this incident photon (in practice, one can detune the incident photon from resonance, such that TLEs are equally likely to be excited[@scullyscience]), the $N$-TLE can be in a collective excited state with one excitation delocalized over the whole system. Post-selecting this state (since in the vast majority of cases the photon will not be absorbed) results in the superposition state $$|+\rangle_{\textbf{k}_0}=\frac{1}{\sqrt{N}}\sum_je^{ik_0z_j}|j\rangle$$ of the exciton in this $N$-TLE structure, where $z_j$ is the position of the $j$th TLE. The state $$|j\rangle=|\textrm{g}_1,\textrm{g}_2,...,\textrm{g}_{j-1},\textrm{e}_j,\textrm{g}_{j+1},...,\textrm{g}_N\rangle$$ describes the state with the $j$th TLE being in its excited state. Including the coupling between the TLE array and the 1D radiation fields, the state vector of the total system at time $t$ can be written as: $$|\Psi(t)\rangle=b_+(t)|+\rangle_{\textbf{k}_0}|0\rangle+b_{\bot}(t)|\bot\rangle_{\textbf{k}_0}|0\rangle+\sum_{k_{z}}b_{k_{z}}(t)|\textrm{g}\rangle|1_{k_{z}}\rangle,
\label{ket}$$ where $|0\rangle$ denotes the zero-photon state, $|1_{k_{z}}\rangle$ denotes one photon in the $k_{z}$-mode, and $|\textrm{g}\rangle$ is the TLE ground state. Note that the superposition state $|+\rangle_{\textbf{k}_0}$ is a Dicke state[@scullyPRL08; @scullyPRL06; @scullylaser], and $|\bot\rangle_{\textbf{k}_0}$ is a summation over all other Dicke states orthonormal to $|+\rangle_{\textbf{k}_0}$. The interaction between the TLE array and radiation fields can then be described by[@Lee; @YN] $$H_{\textrm{int}}=\sum_{k_{z}}\sum_{j=1}^{N}\hbar
g_{k_{z}}\{{\sigma^-_ja^{\dag}_{k_{z}}e^{[i(\omega_{k_{z}}-\omega_0)t-ik_{z}z_j]}}+\textrm{h.c.}\},
\label{H}$$ where $\omega_{k_{z}}$ is the frequency of the $k_z$-mode photon, $\omega_0$ is the excitation energy of the TLE, $\sigma^-_j$ is the lowering operator for the $j$th TLE, $a^{\dag}_{k_{z}}$ is the creation operator for one photon in the $k_z$-mode, and $g_{k_{z}}$ is the coupling strength between TLE and the $k_z$-mode photon.
In the limit of $k_0L\gg1$ ($L$ is the total length of the array), from the time-dependent Schrödinger equation $$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=H_\textrm{int}|\Psi(t)\rangle,
\label{sch}$$ the dynamical evolution of the Dicke state $|+\rangle_{\textbf{k}_0}$ can be written as[@scullyPRL08]: $$\dot{b}_+(t)=-\frac{1}{N}\int_0^t\!\!\!\!dt'\sum_{k_z}\sum_{i,j=1}^Ng_{k_z}^2[e^{i(\omega_{k_z}-\omega_0)(t'-t)}e^{i(k_z-k_0)(z_i-z_j)}]b_+(t').
\label{b}$$ With the approximation $g_{k_z}^2\approx g_{k_0}^2$ and $\sum_{k_z}\rightarrow \textrm{L}_{\textrm{ph}}/(2\pi)\int dq$, Eq. (\[b\]) can be expressed as: $$\dot{b}_+(t)=-\frac{1}{N}\frac{\textrm{L}_{\textrm{ph}}}{2\pi}g_{k_0}^2\int_0^t\!\!\!\!dt'\;b_+(t')\int_{-\infty}^{\infty}\!\!\!\!dq\;\{e^{ivq(t'-t)}\sum_{\xi=0}^N[(N-\xi)(e^{i\xi qh}+e^{-i\xi qh})]\},
\label{b2}$$ where $\textrm{L}_{\textrm{ph}}$ is the quantization length of the radiation field, $v$ is the speed of light, and $\xi$ is a counting index, since the value of $(z_i-z_j)$ can range between $-Nh$ and $Nh$. The dynamical evolution of the Dicke state $|+\rangle_{k_0}$ can thus be obtained by solving Eq. (\[b2\]).
For the array containing $N$ TLEs, the dynamical evolution of the state $|+\rangle_{\textbf{k}_0}$ can be enhanced by the superradiant effect, $\Gamma_{\textrm{array}}=N~\Gamma_{\textrm{TLE}}$, as shown in the red dashed and blue dotted curves shown in Fig. 1(b). For an extremely large array ($L\gg\lambda$, where $\lambda$ is the wavelength of the emitted photon), the probability to be absorbed across the whole sample is made uniform by sufficiently detuning the incident photon energy from that of the TLEs[@scullyscience]. As mentioned earlier this means that the majority of photons pass through unabsorbed. Later we will discuss how the absorbtion event can be signalled by a two-photon correlation when this scheme is realized by an array of quantum wells.
The solid curve in Fig. 1(c) represents Rabi-like oscillations together with an exponential decay. The enhanced decay rate proportional to $N$ is a quantum effect, but may also be described in a semi-classical way by regarding the $N$ TLEs as $N$ classical harmonic oscillators[@scullyPRA]. For $N\gg1$, the summation $\sum_{i,j=1}^N$ in Eq. (\[b\]) can be replaced by the integration $(N/L)^2\int\!dz\int\!dz'$, showing that the effective coupling strength $g$ between the state $|+\rangle_{\textbf{k}_0}$ and the field is $g=\sqrt{N}g_{\textbf{k}_0}$. The period of oscillations is therefore enhanced by a factor $\sqrt{N}$ compared to the bare exciton-photon coupling.
Effective two-level system
--------------------------
To illustrate that the Rabi-like oscillation is mathematically equivalent to an effective quantum coherent oscillations between two states (e.g., a spin or a single excitation cavity-QED system), we transform the Eq. (\[b2\]) into the energy representation via $\tilde{b}_+(E)=\int_0^\infty b_+(t)e^{iEt}dt$, and obtain[@Gurvitz]: $$\left\{E+\frac{1}{N}\frac{\textrm{L}_{\textrm{ph}}}{2\pi}\;g_{k_0}^2\int_{-\infty}^{\infty}~\!\!\!dq~\frac{\sum_{\xi=0}^N[(N-\xi)2\cos(\xi qh)]}{E-vq}\right\}b_+(E)=-i.
\label{dos}$$ Equation (\[dos\]) thus indicates that the density of states (DOS) $D(q)$ of the radiation field in the TLE array, $$D(q)\propto\sum_{\xi=0}^N[(N-\xi)\cos(\xi
qh)],$$ where $q\equiv k_z-k_0$, $\xi$ is a counting index, and $h$ denotes the separation between each period. The insets in Fig. 1(b) and 1(c) show the DOS for TLE array containing different number of emitters. As can be seen, when increasing the number of periods $N$, the line-shape of $D(q)$ (black solid curve in the inset of Fig. 1(c)) becomes Lorentzian-like. Therefore, the TLE array coupled to radiation fields can be interpreted as a Dicke state $|+\rangle_{\textbf{k}_0}$ coupled to a Lorentzian-like continuum, as shown in Fig. 2(a). Following the study by Elattari and Gurvitz[@Gurvitz], for large $N$, our system can be mapped to the Dicke state $|+\rangle_{\textbf{k}_0}$ coherently coupled to a resonant state $|k_0\rangle$ with a Markovian dissipation as depicted in Fig. 2(b). The remaining part of the DOS which does not fit the Lorenzian distribution can be treated as an effective excitonic polarization decay.
Realization with multiple-quantum-wells
---------------------------------------
To show that this effect can be realized in a solid-state environment we consider in detail how to use a multiple-quantum-well (MQW) structure as the two-level-emitter array. In such a MQW structure, each single quantum well can be regarded as a two-level emitter. The quantum-well exciton will be confined in the growth direction (chosen to be *z*-axis) and free to move in the *x*-*y*-plane. Due to the relaxation of momentum conservation in the *z*-axis, the coupling between the photon fields and the quantum wells is one-dimensional. Therefore, if we assume a incident photon with wavevector $k_0$ on the MQW along the *z*-axis, the interacting Hamiltonian can be written exactly the same as the form in Eq. (\[H\]). Furthermore, quantum wells have the remarkable advantage that the phase factor $ik_0z_j$ in $|+\rangle_{k_0}$ can be fixed during the quantum-well growth process, and since the photon fields travel in MQW only along the *z*-axis, a one-dimensional waveguide is not required.
To elaborate on the physical parameters necessary to realize the single-photon Dicke state we assume a MQW structure with a period of 400 nm, where each quantum well consists of one GaAs layer of thickness 5 nm (sandwiched between two AlGaAs slabs). The exciton energy $\hbar\omega_0$ of a single quantum well can take the value[@pin] we utilized in Fig. 1 (i.e., 1.514 eV), such that the resonant photon wavelength $\lambda=2\pi c/\omega_0\approx$ 820 nm. To realize the Dicke state at all we already demanded that the photon be off-resonance with the array. In principle the on-resonance regime can be reached by tuning the quantum well array energies after the Dicke state has been realized. To identify when the state has been created a pair of identical photons with wavevector $k_0$ are produced by the two-photon down-conversion crystal, as shown in Fig. 3. One of the photons is directed to the detector-1 (D1) and the other is along the growth direction of the MQW. The distance between the crystal and D1 is arranged to be the same as that between the crystal and the MQW. Once there is a click in D1, there should be one photon simultaneously sent into the MQW. The photon incident on the MQW generally passes through the MQW and registers a count in detector-2 (D2), but it could also excite one of the multiple quantum wells and form a delocalized exciton. The presence of a count in D1 and the absence of a count in D2 therefore tells us that the MQW has been prepared in the superposition state $|+\rangle_{k_0}$. Since the interaction between the photon fields and the MQW structure is identical with Eq. (\[H\]), the exciton dynamics of the $|+\rangle_{k_0}$ and the density of states of the photon fields in MQW can show the same behaviors as those in Fig. 1(b) and (c) (here one unit of time is $10$ picosecond) when the MQW contains corresponding number $N$ of the quantum wells.
For a MQW structure containing a large number of quantum wells (i.e., $N\geq200$), the dynamical evolution of the superposition state $|+\rangle_{k_0}$ shows Rabi-like oscillations. However, one should note that the Rabi-like oscillations here are different from the Rabi oscillations reported in secondary emission spectra[@koch; @kavokin] of excitons in the MQW structures. The secondary emission occurs when the MQW is illuminated by coherent light, and emission occurs in a direction different from the excitation direction. However, in our system, the incident excitation is a [*single photon*]{}, and the detector-2 (see Fig. 3) receiving the emitted photon is positioned along the excitation direction. Furthermore, the MQW system we consider is Bragg-arranged (i.e., the inter-well spacing equals half the wavelength of light at the exciton frequency), for which the Rabi oscillations in secondary emission cannot appear[@kavokin]. Therefore, the Rabi-like oscillations in Fig. 1(c) are different from those in secondary emission but are a result of the coherent oscillations between the delocalized exciton state $|+\rangle_{\textbf{k}_0}$ and the resonant photon state $|k_0\rangle$.
The Leggett-Garg Inequality
---------------------------
While we have argued that the oscillations one would observe in this large mirror-less cavity are quantum-mechanical in nature (akin to vacuum Rabi splitting), there is still some ambiguity. In the earlier work of Svidzinsky *et al* [@scullyPRA] they employ a semi-classical explanation of a similar phenomena. Thus the question remains open as to whether ${| k_0 \rangle}$ can truly be considered a single resonant state with neglible phase decoherence, and whether the Dicke state retains its long-range spatial coherent nature on a sufficient time-scale. There may be alternative classical and semi-classical explanations of the oscillations one may see in experiment. Similar problems were overcome in the field of cavity and circuit-QED by observation of other quantum features (e.g., the scaling of the energy spectrum[@CircuitQED]). However, in quantum wells we are restricted to certain types of measurements. Recently, great advances have been made in measuring the excitonic states in quantum wells via four-wave mixing techniques[@Patton]. We can also, in principle, make measurements on the emitted photons (e.g., as in Fig. 3). As a first test one could measure the emitted photons at D2 to verify the single-particle nature of the dynamics with a simple violation of the Cauchy-Schwarz inequality[@QuantumNoise], or observation of anti-bunching. We will not go into detail on this here, but essentially it corresponds to the detection of only single-photons. In other words, after we detect the single photon at D2, no more measurements will occur until another photon is injected into the sample. This is a trivial application of the Cauchy-Schwartz inequality, but indicates that we are operating in the single-excitation limit.
In order to verify the quantum coherence of the delocalized state in the MQW rigorously one could apply a test like the Leggett-Garg (LG) inequality[@LG]. The LG inequality depends on the fact that at a macroscopic level several assumptions about our observations of classical reality can be made: realism, locality, and the possibility of non-invasive measurement. In 1985, Leggett and Garg derived their inequality[@LG] to test the first and last assumptions, which when combined they called “macroscopic realism”. The experimental violations of this inequality in a “macroscopic” superconducting circuit[@Laloy], polarized photon state[@PNAS; @PRL; @SR], electron-nuclear spin pairs[@natcomm; @PRL_spin], have recently been seen.
Given a dichotomic observable $Q(t)$, which is bound by $|Q(t)|\leq1$, the Leggett-Garg inequality is: $$|L_Q(t)|\equiv|\langle Q(t_1)Q\rangle+\langle Q(t_1+t_2)Q(t_1)\rangle-\langle Q(t_1+t_2)Q\rangle|\leq1,
\label{LG}$$ where $Q\equiv Q(t=0)$, and $t_1<t_2$. A violation of this inequality suggests either the assumption of realism or of non-invasive measurements is being broken.
To apply this to the system we have been discussing we must formalize further how, for large $N$, the MQW system can be mapped to an effective two-level system \[as shown in Fig. 2(c)\]. The dynamics of this effective model can be described by a Markovian master equation: $$\dot{\rho}=\mathcal{L}[\rho]=\frac{1}{i\hbar}[\widetilde{H}_{\textrm{eff}},\rho]+\Sigma[\rho],
\label{Liou}$$ where $$\begin{array}{ll}
\widetilde{H}_{\textrm{eff}}=&\hbar g(\sigma^{-}+\sigma^{+})\\
\Sigma[\rho]=&\kappa(s\rho s^{\dag}-\frac{1}{2}s^{\dag}s\rho-\frac{1}{2}\rho s^{\dag}s)+\gamma(r\rho r^{\dagger}-\frac{1}{2}r^{\dagger}r\rho-\frac{1}{2}\rho r^{\dagger}r).
\end{array}
\label{master}$$
Here, $\mathcal{L}$ is the Liouvillian of the system, $\widetilde{H}_{\textrm{eff}}$ is the coherent interaction in this effective cavity-QED system, $\sigma^{-}=|k_0\rangle_{\textbf{k}_0}\langle+|$ ($\sigma^{+}=|+\rangle_{\textbf{k}_0}\langle k_0|$) denotes the lowering (raising) operator for the Dicke state $|+\rangle_{\textbf{k}_0}$, and $g=\sqrt{N}g_{\textbf{k}_0}$. The state $|\textrm{vac}\rangle$ is the vacuum state which in the full basis is $|\textrm{g}\rangle\otimes |0\rangle$, i.e. no excitation in the Dicke state or in the resonant state $k_0$. In the self-energy $\Sigma[\rho]$, the $s=| \mathrm{vac}\rangle \langle
k_0|$ operators describe the loss of the photon from the MQW system with rate $\kappa$, and the $r =
|\mathrm{vac}\rangle_{\textbf{k}_0}\langle+|$ operators describe the loss of excitonic polarization with rate ${\gamma}$. With this master equation, in Fig. 4(a) we plot $|L_Q(t)|$ using the observable[@PRB_NL], $$Q=|k_0\rangle\langle k_0|-|+\rangle_{\textbf{k}_0~\textbf{k}_0}\langle+|-|\mathrm{vac}\rangle\langle\mathrm{vac}|.$$ Considerable violations ($>1$) of the LG inequality \[Eq. (\[LG\])\] appear in the region above the blue dashed line in Fig. 4(a). The violations resulting from the quantum oscillations between the states $|+\rangle_{\textbf{k}_0}$ and $|k_0\rangle$ indicate the quantum coherence of the delocalized state in the MQW structure.
A direct application of this inequality to the example of a quantum well array seems extremely challenging because the measurement of a photon leaving the system, and the four-wave mixing measurements of the excitonic states [@Patton; @Schulzgen], are fundamentally invasive. To test the inequality unambiguously would require a fast projective (quantum non-demolition) measurement of the single photon state ${| k_0 \rangle}$, or the Dicke state ${| + \rangle}_{\textbf{k}_0}$. Such measurements are now in principle possible in optical[@Gleyzes; @Khalili] and microwave [@Johnson; @NLPRA] cavities, but not in the effective cavity we describe here.
Some progress can be made by making further assumptions. It was shown by Huelga et al[@huelga2; @huelga3; @huelga1] and others[@NLPRL; @NLPRA] that the assumption of Markovian dynamics eliminates the need to assume non-invasive measurement if we can reliably prepare the system in a desired state (then the invasive nature of the second measurement, e.g., because of the destruction of the photon, does not affect the inequality). Under this Markovian assumption the inequality can be written in terms of population measurements of the state we wish to measure (which in general we describe as a single-state projective operator $Q={| q \rangle}{\langle q |}$, for some measurable state of the system ${| q \rangle}$), $$|L_{P_{Q}}(t)|\equiv|2\langle P_{Q}(t)P_{Q}\rangle-\langle
P_{Q}(2t)P_{Q}\rangle|\leq \langle P_{Q}\rangle, \label{L1}$$ where $\langle P_{Q}\rangle$ is the expectation value of the zero-time population $P_{Q}\equiv P_{Q}(t=0)$, and $\langle
P_{Q}(t)P_{Q}\rangle$ is the two-time correlation function. Note that if the zero-time state is the steady state then this is equivalent to the original[@LG] LG inequality, but again demands non-invasive measurements. If the zero-time state is not the steady state, but some prepared state e.g. $\rho(0)=Q$, $P_Q(0)=1$, then a violation of this variant of the Leggett-Garg inequality indicates behaviour only beyond a classical Markovian regime, i.e. a strong indication of the quantumness of this delocalized state, though not irrefutable proof. We now consider the above inequality in two different regimes.
Initial Dicke state: Markovian test
-----------------------------------
If we can deterministically prepare the state ${| + \rangle}$ (dropping the $k_0$ subscript for brevity) as described in Fig. 3, we can construct the inequality ([Eq. (\[L1\])]{}) with ${| q \rangle}={| + \rangle}$ by preparing that state so $P_{+}(0)=1$, and then (invasively) measuring the state of the quantum wells at time $t$ later (see below). This is then equivalent to the test to eliminate purely Markovian dynamics[@huelga2; @huelga3; @huelga1]. In general such a measurement will be invasive (and can generally be described by some positive operator valued measurement (POVM)), but since we are not concerned with events after the second measurement, we can just assume that it is proportional to the probability of obtaining the Dicke state ${| + \rangle}$ at that time. In other words, we can assume the second measurement is just a normal projective measurement, $P_{+}\equiv|+\rangle\langle +|$.
The correlation function $\langle P_{+}(t)P_{+}\rangle$, where $P_{+}(0)=1$, can be calculated from $$\langle
P_{+}(t)P_{+}\rangle=\textrm{Tr}[P_{+}\exp(\mathcal{L}t){| + \rangle}{\langle + |}]$$ In Fig. 4(b), we plot $|L_{P_{+}}(t)|$ as a function of time (solid black curve). The behavior is oscillatory but damped due to the couplings to the Markovian photon dissipation and the excitonic polarization decay. A considerable violation $(>1)$ of the inequality of Eq. (\[L1\]) appears in the region above the blue dashed line in Fig. 4(b). The violation there comes from the coherent oscillations between the states $|+\rangle$ and $|k_0\rangle$, and is beyond the classical Markovian description.
The Dicke state ${| + \rangle}$ describes a particular coherent superposition of a single excitation across all $N$ quantum wells. It has been shown that four-wave mixing and pump probe techniques [@Patton; @Schulzgen] can be used to measure the state of multiple excitations across multiple wells. Thus it seems feasible that such an experiment can be used to determine the excitation density. If we assume that only the ${| + \rangle}$ plays a role here, and the other Dicke states ($|\bot\rangle$) are unoccupied, then this is sufficient for our purposes, as it will tell us if the array contains a single excitation or not. However, whether the ${| + \rangle}$ can be in general distinguished from the other Dicke states with such a measurement is an interesting open problem, and requires further study.
Initial photonic state: Markovian test
--------------------------------------
Similarly, if we could deterministically prepare the state ${| k_0 \rangle}$, we could construct the inequality ([Eq. (\[L1\])]{}, with ${| q \rangle}={| k_0 \rangle}$) by preparing that state (so $P_{k_0}(0)=1)$, and then measuring when a single photon is detected at detector D2. The second measurement needed to construct the correlation functions in [Eq. (\[L1\])]{} is then simply given by the superoperator $$\mathcal{J}(\rho)=\kappa|\mathrm{vac}\rangle_{k_0}\langle
k_0|\rho|k_0\rangle_{k_0}\langle \mathrm{vac}|,$$ where ${| \mathrm{vac} \rangle}$ is the vacuum state. Again, we can assume the second measurement is just a normal projective measurement (after rescaling by $\kappa$), $P_{k_0}\equiv|k_0\rangle\langle k_0|$. Thus, while the photon measurement is much simpler than the quantum well one described earlier, in our scheme it is not clear if we can determinstically know when ${| k_0 \rangle}$ is created in the same way that ${| + \rangle}$ is, as ${| k_0 \rangle}$ is an effective state of the field modes. In Fig. 4(b), we plot $|L_{P_{k_0}}(t)|$ as a function of time (dashed red curve). Again a considerable violation $(>1)$ of the inequality of Eq. (\[L1\]) appears, and indicates behavior beyond the classical Markovian description.
Of course, ultimately we cannot distinguish classical non-markovian dynamics from quantum dynamics with this method, though certain complex Markovian systems can produce non-monotonic and complex behavior[@NLPRL] which it is important to eliminate. To really show that the large array of quantum wells is behaving like a cavity without a mirror and exhibiting quantum Rabi oscillations more work needs to be done on full state tomography techniques and precise measurements of excitonic states, so that either the full Leggett-Garg inequality, or some other test, can be investigated.
Discussion {#discussion .unnumbered}
==========
In summary, we investigated the dynamical evolution of the delocalized state of a two-level-emitter array state. When the array contains a large number of emitters, the dynamical evolution shows Rabi-like oscillatory behavior. By showing that the DOS of the radiation field in the TLE array is Lorentzian-like, the whole system can be mapped to an effective two-level system (e.g., like a single excitation cavity-QED system). For the physical implementation we suggested a multiple-quantum-well structure and discussed relevant parameters. We also applied the original Leggett-Garg inequality, and a Markovian variation of it, to examine the quantum coherence of the MQW structure.
In addition to the MQW structure, there are other experimentally-accessible systems that can mediate one-dimensional coupling between two-level emitters and the photon fields. Below we provide two potential candidates:
(I.) *Metal nanowire*: $N$ two-level quantum dots positioned near a metal nanowire[@PRB_GY] as shown in Fig. 5(a). Due to the quantum confinement, the surface plasmons propagate along the axis direction on the surface of the nanowire. The coupling between quantum dots and the surface plasmons enable the incident surface plasmons to excite one of the $N$ quantum dots and the delocalized exciton over the $N$ dots can then be formed.
(II.) *Superconducting transmission line*: A superconducting transmission line resonator coupled to $N$ dc-SQUID-based charge qubits[@zhou] as depicted in Fig. 5(b). With proper gate voltage, the Cooper-pair box formed by the dc SQUID with two Josephson junctions can behave like a two-level system (charge qubit). The incident photon propagating in the one-dimensional transmission line would excite one of the charge qubits and form the delocalized state over the $N$ charge qubits. Recent progress in generating and measuring single microwave photons[@Mpho1; @Mpho2; @Mpho3; @Mpho4; @Mpho5] may make the generation and detection of the single-photon Dicke state feasible in the near future.
Methods {#methods .unnumbered}
=======
**Dicke states.** The state $|\bot\rangle_{k_0}|0\rangle$ in Eq. (\[ket\]) denotes a collection of single-excitation Dicke states besides $|+\rangle_{k_0}$. The set of Dicke states are listed in the Table I:
[lll]{}$|+\rangle_{k_0}=\frac{1}{\sqrt{N}}\sum_je^{ik_0z_j}|j\rangle$\
$|1\rangle_{k_0}=\frac{1}{\sqrt{2}}(e^{ik_0z_1}|1\rangle-e^{ik_0z_2}|2\rangle)$\
$|2\rangle_{k_0}=\frac{1}{\sqrt{6}}(e^{ik_0z_1}|1\rangle+e^{ik_0z_2}|2\rangle-2e^{ik_0z_3}|3\rangle)$\
\
$|N-1\rangle_{k_0}=\frac{1}{\sqrt{N(N-1)}}[e^{ik_0z_1}|1\rangle+e^{ik_0z_2}|2\rangle+\ldots+e^{ik_0z_{N-1}}|N-1\rangle-(N-1)e^{ik_0z_{N}}|N\rangle]$\
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This work is supported partially by the National Science Council, Taiwan, under the grant number NSC 98-2112-M-006-002-MY3 and NSC 100-2112-M-006-017. N.L. is supported by RIKEN’s FPR scheme. F.N. acknowledges partial support from the Laboratory of Physical Sciences, National Security Agency, Army Research Office, Defense Advanced Research Projects Agency, Air Force Office of Scientific Research, National Science Foundation Grant No. 0726909, JSPS-RFBR Contract No. 09-02-92114, Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and Funding Program for Innovative R&D on S&T (FIRST).
GYC carried out all calculations under the guidance of NL and YNC. CML and FN attended the discussions. All authors contributed to the interpretation of the work and the writing of the manuscript.
The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to Y.N.C.
**Table I** The set of all Dicke states[@scullyPRL08]. Here, $|j\rangle=|\textrm{g}_1,\textrm{g}_2,...,\textrm{g}_{j-1},\textrm{e}_j,\textrm{g}_{j+1},...,\textrm{g}_N\rangle$ describes the state with the $j$th two-level emitter in its excited state.
**Figure 1: Dynamical evolution of the Dicke state and the density of states of the radiation field in the two-level-emitter array.** (a) The schematic diagram of the two-level-emitter array. The array contains $N$ two-level emitters coupled to the one-dimensional photon reservoir. With proper excitation energy, the incident photon can excite one of the $N$ two-level emitters, and the Dicke state can be formed. The dynamical evolutions of the Dicke state $|+\rangle_{k_0}$ for the TLE array containing (b) 20 (red dashed), 60 (blue dotted), and (c) 200 (black-solid) two-level emitters. These evolutions are obtained by solving the time-dependent Schrödinger equation \[Eq. (\[sch\])$\sim$(\[b2\]) \] in the limit of $k_0L\gg1$. The period of the oscillations for the black solid curve is 0.54 time units. Here, the unit of time is normalized by the spontaneous decay rate $\Gamma_{\textrm{TLE}}$ of a single two-level emitter. The insets show that when increasing the number of periods $N$, the normalized density of states of the radiation field in the TLE array containing 20 (red dashed), 60 (blue dotted) \[the inset in (b)\], and 300 (black solid) \[the inset in (c)\] two-level emitters. The green dashed-dotted curve of the inset in (c) is a Lorentzian fit for $N$=200.
**Figure 2: The correspondence of two-level-emitter array to other systems.** (a) The two-level-emitter array coupled to the radiation field can be interpreted as the Dicke state $|+\rangle_{k_0}$ coupled to a Lorentzian-like continuum spectrum if $N$ is large enough. (b) The system can be further mapped to a Dicke state coherently coupled to a resonant state $|k_0\rangle$ with a Markovian dissipation. The coupling strength $g$ between $|+\rangle_{k_0}$ and $|k_0\rangle$ is $g=\sqrt{N}g_{k_0}$.
**Figure 3: Multiple-quantum-well structure.** A schematic diagram of the GaAs/AlGaAs MQW structure. We assume that the MQW structure is grown along the *z*-axis, with a period of 400 nm, and each quantum well consists of one GaAs layer of thickness 5 nm (sandwiched between two AlGaAs slabs). The exciton energy $\hbar\omega_0$ of a single quantum well is set to be[@pin] 1.514 eV, such that the resonant photon wavelength $\lambda=2\pi
c/\omega_0\approx$ 820 nm. A pair of identical photons with wavevector $k_0$ could be produced by a two-photon down-conversion crystal. One of the photons is directed to the detector-1 (D1) and the other is along the growth direction of the MQW.
**Figure 4: Violation of the LG inequality and its extensions.** (a) The original LG inequality for $|L_Q(t)|$ \[Eq. (\[LG\])\] as a function of time. The region above the blue dashed line indicates the violation regime. (b) The inequality $|L_{P_{Q}}(t)|$ \[Eq. (\[L1\])\] as a function of time for the state ${| q \rangle}=|k_0\rangle$ (red dashed curve) and ${| q \rangle}=|+\rangle_{\textbf{k}_0}$ (black solid curve) in a MQW system containing 200 periods. The region above the blue dashed line indicates the violation regime. In plotting both figures, the coupling constant $g=8.3$ meV, between $|+\rangle_{\textbf{k}_0}$ and $|k_0\rangle$, is determined from the period of the Rabi-like oscillations in Fig. 1(c). The photon loss $\kappa=3.3$ meV is obtained from the width of the Lorentzian fitting (the green dashed-dotted curve in the inset of Fig. 1(c). Here we have set the excitonic polarization decay rate $\gamma$ as the spontaneous emission rate of the general GaAs/AlGaAs quantum well $\gamma=\Gamma_{QW}=100$ (1/ns).
**Figure 5: Experimentally-accessible systems.** Schematics of two alternative experimentally-accessible systems which could realize single-photon Dicke states: (a) $N$ two-level quantum dots coupled to metal nanowire surface plasmons; and (b) $N$ dc-SQUID-based charge qubits coupled to a one-dimensional transmission line. A Cooper-pair box formed by a DC SQUID with two Josephson junctions can act like a two-level system by properly tuning the gate voltage. The incident surface plasmon (photon in transmission line) can excite one of the $N$ quantum dots (charge qubit), the excited quantum dot (charge qubit) can re-emit a surface plasmon (photon) which would be absorbed by another quantum dot (charge qubit), and so on. A delocalized state over the quantum-dot (charge-qubit) array can therefore be formed.
|
---
abstract: 'We study the Anderson localization in systems, in which transport channels with rather different properties are coupled together. This problem arises naturally in systems of hybrid particles, such as exciton-polaritons, where it is not obvious which transport channel dominates the coupled system. Here we address the question of whether the coupling between a strongly and a weakly disordered channel will result in localized (insulating) or delocalized (metallic) behavior. Complementing an earlier study in 1D \[H. Y. Xie, V. E. Kravtsov, and M. Müller, Phys. Rev. B **86** 014205 (2012)\], the problem is solved here on a bilayer Bethe lattice with parametrically different parameters. The comparison with the analytical solution in 1D shows that dimensionality plays a crucial role. In $D=1$ localization is in general dominated by the dirtier channel, which sets the backscattering rate. In contrast, on the Bethe lattice a delocalized channel remains almost always delocalized, even when coupled to strongly localized channels. We conjecture that this phenomenology holds true for finite dimensions $D>2$ as well. Possible implications for interacting many-body systems are discussed.'
author:
- 'Hong-Yi Xie'
- 'M. Müller'
title: Localization in coupled heterogeneously disordered transport channels on the Bethe lattice
---
Introduction {#intro}
============
In a variety of physical contexts, the situation arises that two or more propagating channels with different transport properties are coupled together, competing with each other or modifying each other’s properties. Under these circumstances it is interesting to study the resulting localization properties on the coupled system. Such a question arises in particular in the context of exciton-polaritons, which are hybrid particles: half photons, half excitons, the two channels being coupled linearly via dipolar interaction. [@savona07] Another realization of this physical situation can be found in bilayer graphene, with different degrees of disorder affecting the two layers. A recent work proposed such bi- or trilayers as field effect transistors, whereby a gate potential controls the degree of disorder sensed by the electrons in the bilayer. [@Dx]
Similar questions arise in the problem of energy or matter localization in few- or even many-body problems, where a multitude of propagation channels may exist to transport particles or energy from one place in the system to another. For example, energy may be transported in small, nearly independent units in the form of quasiparticles, or it may have a propagation channel in which a larger amount of energy is propagating in the form of blobs of several quasiparticle-like excitations that form sorts of bound states. Such “bound states" were argued to be favorable transport channels in the context of few-particle problems. The problem was especially studied in low dimensions, [@Shepelyansky; @Imry; @Xie2012] where under certain circumstances such compounds are found to have an enhanced localization length as compared to single-particle excitations. The question arises, then, as to which channel of propagation is the most favorable in transport problems containing a larger number of particles, or in the situation of particles at finite density.
In this type of problem, the various propagation channels are not independent, but couple to each other by scattering events. Understanding transport in such interacting systems is a challenging and largely unresolved problem. Here we do not aim at resolving all aspects of the many-body problem, but address one sub-question which arises in its context. Indeed, the interacting systems have a common feature with noninteracting hybrid particles: Two or more propagating channels with parametrically different localization properties are coupled together and influence each other’s transport characteristics. Under these circumstances it is interesting to study what are the resulting localization properties in the coupled system. In particular in the specific case where a less localized system is coupled to a more localized one, the question arises as to which of the two components eventually dominates the transport: Does one obtain an insulating or conducting system? A central result of our work is to show that the answer to this question depends crucially on the dimensionality of the system.
In our recent work [@XKM2012] the question of the competition between alternative propagation channels was raised in disordered one-dimensional systems. This case can be studied in great detail in the form of a single-particle problem with two parallel, coupled channels. Among others, this naturally describes the Anderson localization of exciton-polaritons in quasi-one-dimensional semiconductor heterostructures. By exactly solving the Anderson model on a two-leg ladder ($D=1$), we found two regimes whose localization properties are qualitatively different: (i) a resonant regime, where the “slow" chain (the more disordered one) dominates the localization length of the ladder; this can be understood as a manifestation of the fact that in one dimension the backscattering rate determines the localization properties of a coupled system, since in general the localization length is of the order of the mean free path; (ii) an off-resonant regime, where the “faster" chain helps to delocalize the slow chain, although with low efficiency.
In that 1D study the disorder was taken to set the smallest energy scale, which allows for a fully analytic solution of the problem. In higher dimensions ($D > 2$), however, weak disorder has no significant effect on localization. Hence, we are restricted to considering intermediate or strong disorder in order to address meaningfully the question of the role of interchannel coupling. Meanwhile, since the disorder is comparable to or stronger than the hopping strengths, resonance conditions, as in regime (i) of the weakly disordered 1D chains, are impossible. Furthermore, in contrast to the physics in one dimension, proliferation of backscattering plays a subdominant role for the Anderson transition of the coupled system, and therefore, the resulting phenomenology of coupled-channel problems turns out to be rather different.
In this paper we study two coupled Bethe lattices with different transport characteristics. This can be viewed as the limit of infinite dimensions ($D \to \infty$) of the problem of coupled channels, which we will contrast with the case of two coupled chains ($D=1$). The behavior on the Bethe lattice is suggestive of the physics to be expected in high-dimensional systems. Indeed, we believe that the qualitative behavior of coupled lattices in $D>2$ is very similar to the phenomenology found on the Bethe lattice. However, the latter has the significant advantage of being exactly solvable, which we exploit below.
![Schematic phase diagram for coupled Bethe lattices with identical hopping strength, but different disorder $W_1\neq W_2$, as inferred from the results in Figs. \[equiv-latts\] and \[gam12-equiv-hop\]. The critical disorder for uncoupled lattices, $W_{c}\approx 17.3$, is indicated by the red lines. (a) Nearest-neighbor intralattice coupling only, $\gamma=0$. The mobility edge for the middle of the band ($E=0$) of the coupled system is indicated by the black curve. In region $A$ (yellow), in the absence of the coupling $t_\perp$ the two lattices would both be localized. However, the finite $t_\perp$ pushes the system into the delocalized phase. If in the absence of coupling one lattice is delocalized and the other one localized, there are two possibilities. In region $B$ (gray), the coupled system becomes delocalized; that is, the less disordered channel dominates. Only when the delocalized lattice is very close to criticality and is coupled to a very strongly disordered lattice \[region $C$ (green)\] localization prevails. However, this regime occurs in a very narrow window of parameters. (b) Next-nearest-neighbor hopping included, $\gamma=1$. The mobility edge is indicated by the blue curve. In region $A$ (yellow) the coupling between two localized lattices induces a delocalized phase. In contrast to (a), the region $C$ is eliminated by the next-nearest-neighbor hopping: The coupled system is *always* more delocalized than either of the uncoupled lattices. []{data-label="bethe-ph-diag"}](phase-diagram.pdf){width="85.00000%"}
Statistical models on the Bethe lattice [@HansBethe; @Baxter] have attracted a lot of studies, because they admit exact solutions and reflect features of the corresponding systems in sufficiently high spatial dimensions. The Anderson model on the Bethe lattice was first introduced and solved by Abou-Chacra, Anderson, and Thouless in Refs. and , where the existence of the localization transition was proven and the location of the mobility edge was found. That work showed in particular that localization is possible in the *absence* of loops in the lattice. The model was solved by studying the self-consistency equation for the on-site self-energy, which leads to a nonlinear integral equation for the probability distribution function of that quantity. The transition from the localized phase to the delocalized phase is characterized by the instability of the fixed point distribution of real self-energies with respect to a perturbation with infinitesimal imaginary parts of the self-energies. Physically, the latter describes an infinitesimally weak coupling to a dissipative bath which allows for decay processes. The above instability signals that local excitations start coupling to a bath on sites infinitely far away, which signals their spatial delocalization.
The stationary distribution function of the self-energy can be found numerically with the help of a population dynamics, or “pool," method. [@AAT1; @MonGarel] The original work by Abou-Chacra [@AAT1; @AT2] has inspired a number of studies in both the physics [@MirlinFyov9197; @MilDer94; @MonGarel; @BiroliTar; @BiroliTar2] and the mathematics [@KunzSouillard; @AcoKlein; @Klein; @AizSimWar; @AizWar] communities. The recent work Ref. points out that the Anderson model on the Bethe lattice may have a further transition within the delocalized phase and corresponds to a transition in the level statistics. Here, we focus however on the standard delocalization transition, as discussed by Abou-Chacra
In the present work we generalize the approach by Abou-Chacra to the case of two coupled Bethe lattices. Following Refs. and we derive a recursion relation for the local Green’s functions (encoded in a $2\times2$ matrix in layer space) and study the effect of interlayer coupling on the location of the transition. We restrict ourselves to the band center ($E=0$). Furthermore, we focus on the case of lattices with identical hopping, but different disorder strengths. This choice is motivated by the one-dimensional case, where equal hoppings lead to resonance effects, which enhance the localization tendency in the coupled system. In contrast, we find that despite the choice of equal hoppings such a localizing effect almost never occurs on coupled Bethe lattices. This is illustrated by the schematic phase diagrams of Fig. \[bethe-ph-diag\], which anticipate and summarize the main results of our analysis: Under most circumstances the coupling between two layers [*enhances delocalization*]{}. Only when one couples a barely metallic layer to a strongly disordered second layer and excludes next-nearest-neighbor interlayer couplings \[$\gamma=0$ in the Hamiltonian (\[full-ham-bethe\]) below\], the coupling can induce localization. However, in the largest part of the phase diagram the coupling has a delocalizing effect. In the case of next-nearest-neighbor interlayer couplings (i.e., nearest-neighbor coupling across layers, $\gamma=1$), the coupled layers are always delocalized if one of the uncoupled layers is delocalized. Moreover, in some range of parameters a coupling between two localized lattices can induce delocalization.
Our central result may be summarized by the statement that on Bethe lattices the delocalized lattice essentially dominates the physics. In other words, if a delocalized channel exists, delocalization, diffusion, and the ability of entropy production will persist even upon coupling to more localized channels. As mentioned before this is quite opposite to the phenomenology in 1D where most often the more disordered chain dominates the localization properties.
The remainder of the paper is organized as follows. In Sec. \[bethe-recur\] we define the Anderson model on two coupled Bethe lattices and derive the recursion relation for the local Green’s functions. In Sec. \[pop-dym\] we present the population dynamics algorithm, which is used to study the statistics of the local self-energy. In Sec. \[ph-digrm\] we obtain the location of mobility edges, which gives rise to the phase diagrams shown in Fig. \[bethe-ph-diag\]. Their qualitative features will be explained by a perturbative analysis. Finally, we discuss the role of dimensionality and the possible implications of our results on the interacting particles in the Conclusion.
Bilayer Anderson model {#bethe-recur}
======================
Model
-----
![Anderson model on a bilayer Bethe lattice, described by the Hamiltonian (\[full-ham-bethe\]), shown for connectivity $K+1=3$. We consider two types of interlayer coupling: (i) Only nearest-neighbor coupling (horizontal blue lines), setting $\gamma=0$. (ii) Additional next-nearest-neighbor coupling (diagonal green lines), with $\gamma=1$.[]{data-label="bethe-latt"}](bethe-lattice.pdf){width="45.00000%" height="6cm"}
We consider two Bethe lattices labeled by $\nu=\{1,2\}$. A Bethe lattice is defined as the interior of an infinite regular Cayley tree, each vertex having the same coordination number $K+1$. The essential feature of such a lattice is the absence of loops. The Bethe lattice can be realized as the thermodynamic limit of a random regular graph of constant connectivity $(K+1)$, that is, a graph where each site connects to other $K+1$ sites, which are randomly and uniformly selected. It is known that any finite portion of such a graph is a tree, with probability tending to one as the size tends to infinity. The advantage of the random-graph construction is the explicit absence of boundary effects. A random graph can thus be viewed as a regular tree wrapped onto itself.
Analogously to the two-chain model studied in Ref. , we define the Anderson model on coupled Bethe lattices as (cf. Fig. \[bethe-latt\])
& H = \_[=1,2]{}[ ( [\_[i]{}[\_[i ]{} c\_[i ]{}\^c\_[i ]{}]{}- t\_ \_[i,j ]{}[(c\_[i ]{}\^c\_[j]{}+h.c.)]{}]{} ) ]{}\
& -t\_ ( \_[i]{}[ c\_[i 1]{}\^c\_[i 2]{}]{} + \_[i,j ]{}[(c\_[i 1]{}\^c\_[j 2]{}+ c\_[i 2]{}\^c\_[j 1]{} )]{} +h.c. ).
Here $i$ labels the coordinates of two corresponding sites in the two layers, and $\langle i,j \rangle$ denotes two nearest neighbors $i$ and $j$ on the Bethe lattice. We take the onsite energies $\ep_{i\nu}$ to be independently distributed random variables with zero mean. $t_\parallel$ is the nearest-neighbor hopping strength within each layer. As motivated above, we take the intralayer hoppings to be equal, so as to come closest to the resonant case in one dimension, which shows the strongest localizing effects. $t_\perp$ is the interlayer hopping strength. In addition to direct (nearest-neighbor) interlayer coupling, we also allow for next-nearest-neighbor hoppings of strength $\gamma t_\perp$. We will consider the two cases $\gamma=0$ and $\gamma=1$. For $K=1$, the former reduces to the 1D model studied in Ref. .
The reason to include finite next-nearest-neighbor hoppings is as follows. Consider the effect of coupling a first lattice to another one with very strong disorder or vanishing hopping. If we exclude next-nearest-neighbor hopping by setting $\gamma=0$, the only significant effect of the coupling is to increase the effective disorder on the first lattice, while the renormalization of its hopping is strongly subdominant or even absent altogether. However, the latter is not the case if we allow for next-nearest-neighbor interlayer hopping. Indeed, this introduces a weak but nonnegligible additional propagation channel, which proceeds via the disordered lattice. As the study below will show (and as anticipated in Fig. \[bethe-ph-diag\]), in the case $\gamma=1$, the renormalization of the hopping $t_\parallel$ always dominates over the enhancement of effective disorder, and hence, the coupling always has a delocalizing tendency. In the context of the more general problems of coupled parallel propagation channels in many-body systems, the case of $\gamma\neq 0$ appears to be a rather generic and natural choice. Even a rather small $\gamma$ is sufficient to avoid the phenomenology found for $\gamma=0$, which leads to atypical behavior in some small regions of the phase diagram.
In the Hamiltonian (\[full-ham-bethe\]) the two layers are subject to different random potentials, characterized by two probability distribution functions $p_\nu(\ep)$. For convenience we assume them to be box distributed: \[rand-pot-dit\] p\_() =
1/W\_, & ,\
0, & .
Our goal is to study the effect of weak interlayer coupling $t_\perp$ on the Anderson transition of the system. As mentioned above, this parallels the case of two resonant chains described in Fig. 7 of Ref. . However, as we will discuss in detail in Sec. \[ph-digrm\], the parameter range of interest on the Bethe lattice is $W_{1,2} \gtrsim t_\perp, t_\parallel$, in contrast to the weak disorder limit considered in Ref. . However, the notion of resonant interlayer coupling is meaningful only if the disorder is so weak that a well-defined dispersion relation exists, which is not the case for the regime of interest on the Bethe lattice. Therefore, the equality of the two intralayer hoppings $t_\parallel$ does not have important consequences in the present study.
Recursion relation for the local Green’s functions
--------------------------------------------------
The retarded Green’s function at energy $E$ is defined by G\_[i,j]{} (E) = i ,| |j,, where $\nu,\mu\in \{1,2\}$ are layer labels, the kets stand for |i,c\_[i ]{}\^|, and $\eta$ is an infinitesimal positive real number, representing an infinitesimally weak coupling to a dissipative bath into which particles can decay. We introduce $2 \times 2$ matrices in the layer space, $\hat{H}_i$, $\hat{G}_i$, and $\hat{T}$, whose elements are
(\_[i]{})\_= i ,| |i,, (\_i)\_= G\_[i,i]{}, and \_= -\_ t\_ - (1-\_) t\_.
$\hat{T}$ describes the hopping from one pair of sites to a neighbor pair of sites.
One can easily show that $\hat{G}_i=\hat{G}_i(E)$ satisfies the following equation: \[gre-rec-1\] \_i = , where $\partial{i}$ denotes the set of neighbors of $i$. The $\hat{G}_j^{(i)}$ are the Green’s functions at the coordinate $j$ in the absence of all bonds between the pairs of sites at $i$ and $j$. $\hat{G}_j^{(i)}=\hat{G}_j^{(i)}(E)$ satisfies the recursion relation, \[gre-rec-2\] \_j\^[(i)]{} = , where $\partial{j}\setminus i$ denotes the set of neighbors of $j$ excluding $i$. $\hat{G}_i$ and $\hat{G}_j^{(i)}$ are complex symmetric matrices. In order to obtain $\hat{G}_i$, we first solve the recursion relation (\[gre-rec-2\]), and then substitute the solution into Eq. (\[gre-rec-1\]).
The self-energies are defined via the layer-diagonal matrix elements $(\hat{G}_{j}^{(i)})_{\nu\nu}$ as \[def-self-energy\] S\_[j]{}(E)=E + i- \_[j]{} - 1/(\_[j]{}\^[(i)]{})\_. Their imaginary parts, \[im-self-energy\] \_[j]{}(E) S\_[j]{}(E), characterize the decay processes of local excitations overlapping with $|j,\nu \rangle$ and having energy $E$.[@Anderson1958]
Under the recursion (\[gre-rec-2\]) the $\Gamma_{j\nu}$ assume a stationary distribution, whose characteristics determine whether the system is in the localized phase or in the delocalized phase. [@Anderson1958; @AAT1; @AT2; @BiroliTar2] In the thermodynamic limit, one has \[crt-mit\] \_[0]{}\_[ ]{} P(\_1 > 0 \_2 > 0) =
0, ,\
>0, ,
$\mathcal{N}$ being the number of lattice sites. The thermodynamic limit, $\mathcal{N} \to \infty$, and the limit of vanishing dissipation, $\eta \to 0$, do not commute, since in a finite system, whose spectrum is discrete, $\eta \to 0$ always leads to vanishing $\Gamma_{j\nu}$’s. Note that the values $\Gamma_{\nu=1,2}$ on the two sublattices are statistically dependent in the presence of coupling; in particular, they are of the same order of magnitude.
We emphasize that the average value of $\Gamma_{\nu}$, namely $\langle \Gamma_{\nu} \rangle$, *cannot* be used to identify the Anderson transition, because in the localized phase an infinitesimal dissipation $\eta$ leads to long tails in the distribution function of $\Gamma_{\nu}$, which leads to a finite value $\langle \Gamma_{\nu} \rangle$. Instead, one needs to consider the typical value of $\Gamma_{\nu}$, as defined by the geometric average \[typ-values\] \_[,]{} = e\^[ ]{}, which depends on the lattice label $\nu$ if the two lattices are statistically not identical. However, as they are of the same order of magnitude, the localization transition can be identified by either of the two typical values, by locating the boundary between the two regimes: \[crt-mit-2\] \_[0]{}\_[ ]{} \_[,]{} =
0, ,\
>0, .
The equivalence of $\Gamma_{\text{typ},\nu=1,2}$ for the purpose of identifying the phase transition will be shown explicitly in Sec. \[pop-dym\], based on the population dynamics.
Anderson transition and population dynamics {#pop-dym}
===========================================
A convenient way to determine the mobility edge was proposed in Refs. and . It is based on analyzing the stability of the real solution of Eq. (\[gre-rec-2\]) at the energy $E$ with respect to the insertion of the infinitesimal imaginary energy shift $i\eta$. In the localized phase the real solution is stable. In contrast, in the delocalized phase, the solution develops a finite imaginary part, which implies that $\Gamma_{\text{typ},\nu}(E+i\eta) \neq 0$ as $\eta \to 0$. The physical interpretation of this criterion is as follows. For a finite but large tree, if the boundary sites are coupled to a bath with a dissipation rate $\eta$, we test whether the dissipation at the root of the tree, measured by $\Gamma_{\text{typ},\nu}{(E)}$, is vanishing or not as the tree size tends to infinity. If $E$ belongs to the localized part of the spectrum (point spectrum), particle transport is absent at large scale and there is no dissipation at the root. In contrast, in the delocalized regime, we observe finite dissipation even deep inside the tree. This procedure in fact implements the criterion (\[crt-mit\]) for the Anderson transition, as the instability of real self-energies reflects the Anderson transition as a phenomenon of spontaneous breakdown of unitarity of the scattering matrix associated with the system Hamiltonian.[@Fyorov2003]
The stability analysis can be realized by a population dynamics, which is a numerical recipe to solve the stochastic iteration Eq. (\[gre-rec-2\]). A detailed description of such an algorithm for the single-lattice case can be found in Refs. and . The basic idea is to simulate the distribution of a random variable $X$ by the empirical distribution of a large population of representatives $\{X_\alpha\}_{\alpha=1}^{\mathcal{M}}$. Here the random variable $X$ is the symmetric $2\times2$ matrix $\hat{G}_j^{(i)}$. For simplicity, we denote $\hat{G}_j^{(i)}$ by $\hat{G}$, and the population by $\{\hat{G}_\alpha\}_{\alpha=1}^{\mathcal{M}}$. The $\mathcal{M} \gg 1$ representatives can be understood as values of Green’s functions on sites at a given distance from the root on a large tree. The population dynamics consists of a number of sweeps of the population, which simulate the propagation of dissipation step by step towards the root of the tree, whereby the number of representatives is kept constant.[@MonGarel] At the $n_s^{\rm th}$ stage, we denote the population as $\{\hat{G}_{\alpha,n_s}\}_{\alpha=1}^{\mathcal{M}}$, which are obtained with the following procedure:
\(i) As an initial condition for the population we chose the Green’s functions of $\mathcal{M}$ uncoupled sites subject to a random potential and a small dissipation, that is, $\{\hat{G}_{\alpha,0}\}_{\alpha=1}^{\mathcal{M}}$ with matrix elements (\_[,0]{})\_ &=& (E - \_ +i)\^[-1]{}, {1,2},\
(\_[,0]{})\_[12]{}&=&(\_[,0]{})\_[21]{}=0,where $\ep_{\alpha\nu}$ are independently drawn from the probability distribution (\[rand-pot-dit\]). $\eta$ is taken as small positive number, representing the dissipation on the boundary sites of the tree.
\(ii) Generate the $n_s^{\rm th}$ population from the $(n_s-1)^{\rm th}$ population. For each member $\beta = 1,2, \cdots, \mathcal{M}$ of the new population, we choose $K$ matrices randomly and uniformly from the population $\{\hat{G}_{\alpha,n_s-1}\}_{\alpha=1}^{\mathcal{M}}$, called $\{\hat{G}_{\alpha_1,n_s-1}, \cdots, \hat{G}_{\alpha_K,n_s-1} \}$, and generate $2K$ random numbers according to the probability distribution function in Eq. (\[rand-pot-dit\]), called $\{ \ep_{1\nu}, \cdots, \ep_{K\nu}\}$ with $\nu =1,2$. Substitute these quantities on the right-hand side of Eq. (\[gre-rec-2\]) with $\eta =0$ since the dissipative bath only couples to the boundary sites, and obtain $\hat{G}_{\beta,n_s}$ on the left-hand side.
We calculate the typical value of $\Gamma_\nu$ in the population $\{\hat{G}_{\alpha,n_s}\}_{\alpha=1}^{\mathcal{M}}$, = \_[=1]{}\^, using Eqs. (\[def-self-energy\]) and (\[im-self-energy\]). The localization transition can be determined by studying the evolution of $\Gamma_{\text{typ},\nu}^{(n_s)}$ under sweeps. As shown in Ref. , if the $\Gamma_{j\nu}$’s are small enough, the recursion relation (\[gre-rec-2\]) leads to a *linear homogeneous* equation for $\Gamma_{j\nu}$, and the growth of the typical value of $\Gamma_{j\nu}$ under sweeps is dominated by the largest eigenvalue of the linearized recursion relation. Therefore, as long as $\Gamma_{\text{typ},\nu}^{(n_s)}$ is sufficiently small, statistically $\Gamma_{\text{typ},\nu}^{(n_s)}$ grows linearly with the growth rate \_[n\_s]{} = - . Notice that in this linear regime as long as the two lattices are coupled, the statistics of $\lambda_{n_s}$ is *independent* of the lattice index $\nu$. In other words, $\Gamma_{\text{typ},\nu=1,2}^{(n_s)}$ deviate from zero simultaneously as the system crosses into the delocalized phase, and therefore the criterion for delocalization transition (\[crt-mit-2\]) does not depend on $\nu$. The statistical analysis of $\lambda_{n_s}$ below is restricted to the linear regime where the $\Gamma_\nu$ remain small.
The average growth rate of $\Gamma_{\text{typ},\nu}^{(n_s)}$ over $n_s \gg 1$ successive sweeps is given by \[grow-r-num\] = \_[n\_s\^=1]{}\^[n\_s]{}[\_[n\_s\^]{}]{}, and the standard deviation is \[variance-lam\] = . Physically, $|\overline{\lambda}|^{-1}$ may be interpreted as a localization length in the insulating phase, or as a correlation length in the delocalized phase. The Anderson transition occurs when[@MonGarel] \[crt-mit-3\] = 0.
We obtained numerical results using a population size $\mathcal{M}=10^7$, dissipation $\eta =10^{-15}$, and $n_s=200$ sweeps. The statistics of $\lambda_{n_s}$ was collected only after about $10$ sweeps to avoid the initial transient. We checked that the $\eta$ dependence of $\overline{\lambda}$ and $\delta\lambda$ was very weak, as long as $\eta$ was taken to be small enough.
Phase diagram {#ph-digrm}
=============
Let us now analyze the effect of the interlayer coupling $t_\perp$ on the Anderson transition. For convenience we focus on the band center, $E=0$. We concentrate on relatively weak interlayer coupling $t_\perp \lesssim t_{\parallel}$, which guarantees that in the absence of disorder the energy bands are not substantially changed by the coupling. In this case a mobility edge first appears at the band center $E=0$ upon increasing the hopping strength. [@wcoup] Below we present the numerical results of the population dynamics, which lead to the phase diagrams shown in Fig. \[bethe-ph-diag\]. In Sec. \[anay-two-bethe\] a perturbative analysis is given to explain the qualitative features of the phase diagram.
Numerical results {#num-two-bethe}
-----------------
In the numerical calculations we took a connectivity $K+1=3$ and hopping strengths $t_{\perp} =t_\parallel = 1$. We analyze the two cases in turn.
### Statistically identical lattices ($W_1=W_2$) {#stat-id}
![Numerical results for the growth rates $\overline{\lambda}$ \[Eq. (\[grow-r-num\])\] at the band center for *statistically identical* Bethe lattices as functions of disorder strength $W_1=W_2$. Energies are in units of $t_\parallel =1$. The error bars correspond to $\delta{\lambda}$ \[Eq. (\[variance-lam\])\]. For the uncoupled lattices (red triangles) the critical disorder strength is $W_c(t_\perp=0) \approx 17.3$. Upon coupling the lattices we find the critical disorder strengths: (a) $W_c(t_\perp=1, \gamma=0) \approx 20.7$ (black squares); (b) $W_c(t_\perp=1, \gamma=1) \approx 37.5$ (blue circles).[]{data-label="equiv-latts"}](diag-growth-rate.pdf){width="47.00000%"}
We first analyze two statistically identical lattices with disorder strength $W=W_1=W_2$ (following the diagonal line in Fig. \[bethe-ph-diag\]). In Fig. \[equiv-latts\] we show $\overline{\lambda}\pm \delta{\lambda}$ at $E=0$ as functions of the disorder strength for uncoupled and coupled lattices. The transition point is determined by Eq. (\[crt-mit-3\]). For the uncoupled lattices we find the critical disorder $W_c(t_\perp=0) \approx 17.3$, which agrees with the results in Refs. and . For coupled lattices, the critical disorder strength increases to $W_c(t_\perp=1,\gamma=0) \approx 20.7$ with nearest-neighbor coupling only, and to $W_c(t_\perp=1,\gamma=1) \approx 37.5$ when next-nearest-neighbor coupling is included. Thus the critical disorder is enhanced by the coupling, $W_c(t_\perp \neq 0) > W_c(t_\perp=0)$. This implies that if two decoupled lattices are in the localized phase but close enough to criticality, the coupling will delocalize the system.
### Parametrically different lattices ($W_1 \neq W_2$)
Let us now study two Bethe lattices with identical hopping but different disorder strengths. We take $W_1 = W_c(t_\perp=0) \approx 17.3$ to be critical (following the red line $W_1=W_c$ in Fig. \[bethe-ph-diag\]) and analyze whether the coupling to a more disordered lattice pushes the system to a localized or delocalized phase. If the interlayer coupling is weak $t_\perp \lesssim t_\parallel$, for both $\gamma = 0$ and $\gamma = 1$, we expect that the system is delocalized when $W_2$ is not much larger than $W_c$. This is expected from the results of the preceding section. However, for $W_2 \gg W_c$ the situation may depend on the type of interlayer coupling.
In Fig. \[gam12-equiv-hop\], we show $\overline{\lambda}\pm \delta{\lambda}$ as functions of $W_2$. We observe the following features: For $\gamma=0$ a mobility edge occurs at the fairly large disorder $W_2=W_{2,c}(t_\perp=1,\gamma=0) \approx 47$. In other words, as long as $W_2 < W_{2,c}$ the band center becomes delocalized, while it is localized beyond $W_{2,c}$. However, when next-nearest-neighbor hopping is included, with relative strength $\gamma=1$, the band center becomes always delocalized upon coupling, for any value of $W_2$. As $W_2 \to \infty$, the two lattices decouple effectively, and the band center tends back to criticality, from the localized and the delocalized side, for $\gamma=0$ and $\gamma=1$, respectively. Empirically we find that $\overline{\lambda} \sim c(\gamma)/W_2$ for large $W_2$, where $c(\gamma =0)<0$ and $c(\gamma =1) >0$. As will become clear from the perturbative analysis in Sec. \[anay-two-bethe\], this is due to the suppression or enhancement of the probability of resonances between two neighboring sites on the first lattice. That effect is of the order of $1/W_2$.
![ *Statistically nonidentical* Bethe lattices: Numerical results for the growth rate $\overline{\lambda}$ \[Eq. (\[grow-r-num\])\] at the band center as functions of the disorder strength $W_2$. The disorder strength on the $1$ lattice is fixed at $W_1=W_c(t_\perp=0) \approx 17.3$ and the interlayer coupling is $t_\perp=1$. The other parameters are the same as in Fig. \[equiv-latts\]. (a) For the nearest-neighbor coupling there is a mobility edge, $W_{2,c}(t_\perp=1,\gamma=0) \approx 47$. As $W_2 \to \infty$ the system approaches criticality from the localized phase, and $\overline{\lambda} \sim -1/W_2$, as expected analytically. (b) With next-nearest-neighbor coupling the system is always in the delocalized phase and approaches the transition point like $\overline{\lambda} \sim 1/W_2$ as $W_2 \to \infty$ (best fit shown as dashed line in the log-log plot).[]{data-label="gam12-equiv-hop"}](fixw1-c.pdf){width="47.00000%"}
The results obtained in Figs. \[equiv-latts\] and \[gam12-equiv-hop\] give rise to the schematic phase diagram shown in Fig. \[bethe-ph-diag\]. One can distinguish three regions according to the effect of the interlayer coupling:
\(i) Region $A$ (yellow area). In the absence of coupling the two lattices are both localized but close enough to criticality. The coupling pushes the two nearly critical lattices into the delocalized phase.
\(ii) Region $B$ (gray area). The better conducting lattice is (sufficiently far) in the delocalized phase, while the more disordered lattice is strongly localized. The coupled system is nevertheless delocalized due to the dominance of the better channel.
\(iii) Region $C$ (green area). In the absence of coupling the less disordered lattice is delocalized but very close to criticality. The more disordered lattice is strongly localized. If there is nearest-neighbor coupling only ($\gamma=0$), it pushes the system to the localized phase. However, this atypical region is entirely absent if a strong enough next-nearest-neighbor coupling is included ($\gamma=1$).
Perturbative analysis {#anay-two-bethe}
---------------------
The salient features of the phase diagrams shown in Fig. \[bethe-ph-diag\] can be understood qualitatively by applying a perturbative analysis in the limit $W_2 \gg t_\parallel,t_\perp $. The coupling to the strongly disordered second lattice has two competing effects on the first lattice: On the one hand, the hopping strength $t_\parallel$ is effectively enhanced. On the other hand, the variance of the on-site energies on the first lattice is effectively enhanced, too. If the relative enhancement of the hopping dominates, the coupling tends to delocalize the system.
To leading order in $1/W_2$, the correction for the hopping strength between nearest-neighbor sites $|i,1\rangle$ and $|j,1\rangle$ is \[hop-correction\] \_[1,ij]{}() ( Y\_[1,ij]{} + Y\_[2,ij]{} ), where \[y1y2\]
Y\_[1,ij]{} & = W\_2 ( + ),\
Y\_[2,ij]{} & = .
Likewise, the correction of the local potential on site $|i,1\rangle$ due to self-energy effects is \[dis-correction\] \_[i1]{}() , where \[y3y4\] Y\_[3,ij]{} = , Y\_[4,ij]{} = \_[j ]{}. $Y_{1,2,3,4}$ are dimensionless random variables whose probability distributions have long tails.[@Anderson1958; @AltGef] Hence both $\delta{t}_{1,ij}$ and $\delta{\ep}_{i1}$ are dominated by rare, large values. This implies that the dominant events are those where either the hopping strength or the disorder strength is strongly enhanced, that is, a link is either strongly favored or blocked. The ratio of the probabilities of such enhancements determines which effect is dominant. Notice that a typical value of $\delta{t}_{1,ij}$ is of order $O(1/W_2^2)$ for $\gamma=0$, but $O(1/W_2)$ for $\gamma>0$, while $\delta{\ep}_{i1}$ scales as $O(1/W_2)$ regardless of the value of $\gamma$. Therefore, the enhancement of disorder is dominant when $\gamma=0$, that is, when the nearest-neighbor interchain hopping is suppressed.
Let us now discuss the Anderson transition at $E=0$. We base this discussion on two observations: First, the delocalization of wavefunctions on the Bethe lattice has recently been shown to occur along single paths. [@BiroliTar2] We should therefore study the decay rate of excitations along the best possible path for propagation. To obtain a qualitative understanding of the effects of coupling, we approximate the propagation amplitude between two remote sites of the first lattice as the product \[prod\] A\_L = \_[i=1]{}\^[L]{} = R\_L \_[i=1]{}\^[L]{}, L 1, where $L$ is the distance between the two sites. $\prod_{i=1}^{L}{t_{\parallel}/\ep_{i1}}$ is the amplitude in the absence of coupling, and \[eff-ratio\] R\_L = \_[i=1]{}\^[L]{}, represents the enhancement due to the coupling to the second lattice. The amplitude $A_L$ is the lowest order term in an expansion in the hopping. It corresponds to Anderson’s “upper limit” approximation, which neglects self-energy effects from sites lateral to the considered paths, as well as the regularization of resonances due to higher order corrections from hoppings along the path. Based on this approximation one obtains a simple approximate criterion for localization: Consider the probability $P_{|A_L| \gtrsim 1}$ that the propagation amplitude $|A_L|$ along the most favorable path exceeds some fixed finite value $O(1)$. Localization obtains so long as this probability vanishes in the thermodynamic limit, $P_{|A_L| \gtrsim 1}\to 0$ as $L\to \infty$. [@Anderson1958; @AltGef]
In order to understand the phase diagram in Fig. \[bethe-ph-diag\], let us consider critical disorder on the first lattice, $W_1=W_c$, and study how the probability $P_{|A_L| \gtrsim 1}(W_2)$ depends on $W_2$ via the correction factor $R_L$. For $W_2\to \infty$, $R_L\to 1$ (in probability), and $P_{|A_L| \gtrsim 1}(W_2 \to \infty)$ behaves critically; that is, it does not decay exponentially with $L$. For finite but large $W_2$, we need to estimate the correction factor $R_L$. As mentioned above, it is dominated by the rare events in which either the hopping strength or the local disorder are strongly enhanced, such that one of the factors in Eq. (\[eff-ratio\]) is significantly different from unity.
The probability of a strong enhancement of hopping, $P_{|\delta{t}_{1,ij}| \gtrsim t_\parallel}$, scales with $W_2$ like [@AltGef] \[prob-hop\] P\_[|\_[1,ij]{}| t\_]{} \~
c\_0 t\_\^2 , =0\
c\_1 , >0,
where $c_{0,1} = O(1)$ do not depend crucially on the parameters of the system. The probability of the strong enhancement of local disorder behaves like \[prob-dis\] P\_[|\_[i1]{}||\_[i1]{}|]{} \~c\_2() , where $c_2(\gamma) = O(1)$ depends on $\gamma$, and has a finite limit $c_2(\gamma\to 0) > 0$.
Among the $L$ factors of $R_L$ a fraction of order $P_{|\delta{t}_{1,ij}| \gtrsim t_\parallel}$ is significantly larger than unity, and a fraction of order $P_{|\delta{\ep}_{i1}|\gtrsim |\ep_{i1}|}$ terms significantly smaller than unity. Therefore, it is reasonable to assume that a typical value of $R_L$ takes the form \[type-enhance\] R\_[L,]{} \~e\^[s(,W\_2) L]{}, where the Lyapunov exponent $s(\gamma,W_2)$ is \[lyapu\] s(,W\_2) = P\_[|\_[1,ij]{}| t\_]{} - P\_[|\_[i1]{}||\_[i1]{}|]{}, with $\alpha, \beta$ of order $O(1)$. Substituting Eqs. (\[prob-hop\]) and (\[prob-dis\]) in Eq. (\[lyapu\]), we predict the scaling s(,W\_2) \~, W\_2 W\_1,t\_, with a coefficient \[c-gamma\] c() \~ ( f() - ), where $f(\gamma) \propto \gamma$ for $\gamma \ll 1$. Obviously, the condition $c(\gamma) =0$ marks the transition between enhanced and suppressed propagation. Close to that criticality, the inverse localization or correlation length follows from the growth rate $|\overline{\lambda}|$ \[cf. Eq. (\[grow-r-num\])\] which is proportional to $s(\gamma,W_2)$, \[loc-length\] s(,W\_2)\~. The scaling with $1/W_2$ is clearly observed in the numerical data of Fig. \[gam12-equiv-hop\], confirming the dominance of rare events.
Let us now discuss the $\gamma$ dependence of $c(\gamma)$. Without next-nearest-neighbor interlayer hopping, we have $c(\gamma=0) \approx -7.5< 0$, as we numerically obtain in Fig. \[gam12-equiv-hop\](a). This is due to the fact that $P_{|\delta{t}_{1,ij}| \gtrsim t_\parallel}$ is parametrically smaller than $P_{|\delta{\ep}_{i1}|\gtrsim |\ep_{i1}|}$ for $W_2 \to \infty$ \[cf. Eqs. (\[prob-hop\]) and (\[prob-dis\])\]. Therefore, for large enough $W_2 (> W_{2,c})$ the more disordered lattice drives a less disordered, critical lattice to the localized phase, as seen in regime $C$ of Fig. \[bethe-ph-diag\] (a).
However, when $\gamma > 0$, the probabilities for significant corrections $\delta{t}_{1,ij}$ and $\delta{\ep}_{i1}$ both scale as $1/W_2$. For large enough $\gamma$, $c(\gamma)$ becomes positive, as one may anticipate from Eq. (\[c-gamma\]), considering that ${t_\parallel}/{W_1}$ is numerically small at criticality. Indeed, the case $\gamma=1$ shown in Fig. \[bethe-ph-diag\](b) is already deep in this regime, with $c(\gamma =1)\approx 12.8>0$ \[cf. Fig. \[gam12-equiv-hop\](b)\].
The equation $c(\gamma=\gamma_c)=0$ has a solution for some $0< \gamma_c<1$. $\gamma_c$ determines the minimal next-nearest-neighbor interlayer hopping which assures delocalization upon coupling to a disordered lattice even in the limit $W_2\gg W_1$. A naive linear interpolation between $c(\gamma=0)$ and $c(\gamma=1)$ allows us to obtain a rough estimate \_c0.37 for the Bethe lattices of connectivity $K+1=3$ considered here.
Discussion and Conclusion
=========================
We have studied the Anderson localization problem on two coupled Bethe lattices, which represents a two-channel problem in the limit of infinite dimensions. Our main result is the finding that a conducting transport channel is hardly ever localized by the coupling to more disordered channels. Rather, transport is usually enhanced by such a coupling. This holds true except in the case where three conditions are met simultaneously: (i) the conducting channel is very close to criticality; (ii) it is coupled to a strongly localized channel; (iii) next-nearest-neighbor interlayer couplings are strongly suppressed or absent. Only in these exceptional cases the coupling to localized channels may induce a localized phase in an otherwise conducting channel. The coupling between moderately localized channels may instead induce delocalization. We believe that these trends persist also in high but finite dimensions ($D>2$) where the metal-insulator transition takes places at strong disorder. This conjecture is based on the observation that in higher dimensions, as well as on the Bethe lattice, delocalization is mostly driven by a sufficiently strong forward scattering, whereas weak localization effects and enhanced backscattering play a much less important role than in $D\leq 2$. We believe that this difference is at the root of the very different phenomenology between coupled Bethe lattices and 1D chains.
In two dimensions, the localization length becomes parametrically larger than the mean-free path at weak disorder. However, since the proliferation of weak localization and backscattering leads to complete localization (in the absence of special symmetries), we expect that a well propagating channel becomes more strongly localized upon resonant coupling to a more disordered channel, similarly as in one dimension. It might be interesting to investigate this numerically. Apart from its theoretical interest, the physics of coupled, unequally disordered 2D lattices might also have practical applications. For example, it was recently proposed [@Dx] that a sheet of bilayer graphene with different disorder strength on the two layers could be operated as a field effect transistor, whereby a perpendicular gate bias tunes the effective disorder of carriers.
The study of localization properties of few- or many-particle systems is more subtle than the toy problem which it motivated here in part. The reason is that multiple (much more than two) coupled channels with complicated substructures may exist to transport particles or energy. In particular, for few-particle problems, it has been shown that some of the channels, in which a large number of quasiparticle-like excitations are propagating in the form of “bound states,” can be more efficient for transport than others, in which the excitations propagate essentially as independent units. [@Shepelyansky; @Imry; @Xie2012] This suggests that one might have to think of the many-body problem as having a hierarchy of channels with parametrically different transport properties. It is reasonable to assume that the effective dimensionality of such channels should be the same as that of the system. Our analysis of a two-channel model in 1D has demonstrated that a “bad” channel dominates only when it is resonantly coupled to a better (“faster”) channel. If the two channels are far from resonance, or if they live in higher effective dimensionality, the fast channel almost always dominates the localization properties, as the present study suggests. Moreover, even in 1D resonance conditions are not met very easily. It either requires two channels with equal hopping strength, or an energy close to the band center or the band edges. Summarizing these considerations, we come to the qualitative conclusion that, apart from some exceptional cases, better conducting channels generically dominate the delocalization: A diffusing channel is difficult to shut down by coupling to dirtier channels.
In the context of interacting many-particle systems the above leads to the following conjecture: In order to establish that a many-particle system conducts and is not fully localized, a sufficient condition will be found by identifying the best transport channel and showing that it is delocalized. Indeed, our study suggests that the inclusion of coupling to other channels usually only enhances transport. This observation should be a central ingredient when generalizing the ideas of Refs. and to the analysis of quantum dynamics and transport of systems with several particles. However, at this stage, the application to many-particle systems remains a conjecture which needs to be tested further. For example, one should establish whether coupling to a much larger number of slow channels does not alter our qualitative findings of “the survival of the fastest".
We are grateful to D. Basko, P. Brouwer, and V. E. Kravtsov for stimulating discussions.
[99]{} V. Savona, J. Phys.: Condens. Matter **19**, 295208 (2007). D. Xue, H. Liu, V. Sacksteder IV, J. Song, H. Jiang, Q. F. Sun and X. C. Xie, J. Phys.: Condens. Matter **25**, 105303 (2013). H. Y. Xie, V. E. Kravtsov, and M. Müller, Phys. Rev. B **86**, 014205 (2012). D. L. Shepelyansky, Phys. Rev. Lett. **73**, 2607 (1994). Y. Imry, Europhys. Lett. **30** (7), 405 (1995). H. Y. Xie, *Anderson localization in disordered systems with competing channels* (LAP Lambert Academic Publishing, Saarbrücken, 2012), Chap. 2, where aspects of the few-particle problem are discussed. It is suggested there that the most efficient transport channel follows a hierarchical structure in the spatial arrangement of the particles. H. A. Bethe, Proc. R. Soc. A **150**, 552 (1935). R. J. Baxter, *Exactly solved models in statistical mechanics* (Academic Press, London, 1982). P. W. Anderson, Phys. Rev. **109**, 1492 (1958). R. Abou-Chacra, P. W. Anderson, and D. J. Thouless, J. Phys. C **6**, 1734 (1973). R. Abou-Chacra and D. J. Thouless, J. Phys. C **7**, 65 (1973). A. D. Mirlin and Y. V. Fyodorov, Nucl. Phys. B **366**, 507 (1991); Phys. Rev. B **56**, 13393 (1997). J. D. Miller and B. Derrida, J. Stat. Phys. **75**, 357 (1994). C. Monthus and T. Garel, J. Phys. A **42**, 075002 (2009). G. Biroli, G. Semerjian, and M. Tarzia, Prog. Theor. Phys. Suppl. **184**, 187 (2010). G. Biroli, A. C. Ribeiro-Teixeira, and M. Tarzia, arXiv: 1211.7334 \[cond-mat.dis-nn\]. H. Kunz and B. Souillard, J. Physique Lett. **44**, L411 (1983). V. Acosta and A. Klein, J. Stat. Phys. **69**, 277 (1992). A. Klein, Comm. Math. Phys. **177**, 755 (1996); Adv. Math. **133**, 163 (1998). M. Aizenman, R. Sims, and S. Warzel, Prob. Theor. Rel. Fields **136**, 363 (2006); Comm. Math. Phys. **264**, 371 (2006). M. Aizenman and S. Warzel, Phys. Rev. Lett. **106**, 136804 (2011). Y. V. Fyodorov, Pis’ma Zh. Eksp. Teor. Fiz. **78**, 286 (2003) \[JETP Lett. , 250 (2003)\]. As in the two-chain problem in Ref. , if $t_{\perp}$ becomes very strong as compared to $t_\parallel$ we reach the one-channel regime, where there is a gap between the two clean subbands, and thus only one propagating channel at a given energy. In this case the energy with the largest localization length is no longer at the band center (cf. Figs. 6 and 11 of Ref. ). A similar situation is expected for coupled Bethe lattices: If the coupling is too strong, the mobility edge first appears at some energy $E\neq 0$. D. J. Thouless, J. Phys. C: Solid St. Phys. **3**, 1559 (1970); B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. **78**, 2803 (1997).
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abstract: 'We investigate identical pion HBT intensity interferometry in central [Au+Au collisions at 1.23]{}. High-statistics $\pi^-\pi^-$ and $\pi^+\pi^+$ data are measured with HADES at SIS18/GSI. The radius parameters, derived from the correlation function depending on relative momenta in the longitudinally comoving system and parametrized as three-dimensional Gaussian distribution, are studied as function of transverse momentum. A substantial charge-sign difference of the source radii is found, particularly pronounced at low transverse momentum. The extracted source parameters agree well with a smooth extrapolation of the center-of-mass energy dependence established at higher energies, extending the corresponding excitation functions down towards a very low energy.'
author:
- |
J. Adamczewski-Musch$^{4}$, O. Arnold$^{10,9}$, C. Behnke$^{8}$, A. Belounnas$^{16}$, A. Belyaev$^{7}$, J.C. Berger-Chen$^{10,9}$, J. Biernat$^{3}$, A. Blanco$^{2}$, C. Blume$^{8}$, M. Böhmer$^{10}$, P. Bordalo$^{2}$, S. Chernenko$^{7,\dag}$, L. Chlad$^{17}$, C. Deveaux$^{11}$, J. Dreyer$^{6}$, A. Dybczak$^{3}$, E. Epple$^{10,9}$, L. Fabbietti$^{10,9}$, O. Fateev$^{7}$, P. Filip$^{1}$, P. Fonte$^{2,a}$, C. Franco$^{2}$, J. Friese$^{10}$, I. Fröhlich$^{8}$, T. Galatyuk$^{5,4}$, J. A. Garzón$^{18}$, R. Gernhäuser$^{10}$, M. Golubeva$^{12}$, R. Greifenhagen$^{6,c}$, F. Guber$^{12}$, M. Gumberidze$^{4,b}$, S. Harabasz$^{5,3}$, T. Heinz$^{4}$, T. Hennino$^{16}$, S. Hlavac$^{1}$, C. Höhne$^{11,4}$, R. Holzmann$^{4}$, A. Ierusalimov$^{7}$, A. Ivashkin$^{12}$, B. Kämpfer$^{6,c}$, T. Karavicheva$^{12}$, B. Kardan$^{8}$, I. Koenig$^{4}$, W. Koenig$^{4}$, B. W. Kolb$^{4}$, G. Korcyl$^{3}$, G. Kornakov$^{5}$, R. Kotte$^{6}$, A. Kugler$^{17}$, T. Kunz$^{10}$, A. Kurepin$^{12}$, A. Kurilkin$^{7}$, P. Kurilkin$^{7}$, V. Ladygin$^{7}$, R. Lalik$^{3}$, K. Lapidus$^{10,9}$, A. Lebedev$^{13}$, L. Lopes$^{2}$, M. Lorenz$^{8}$, T. Mahmoud$^{11}$, L. Maier$^{10}$, A. Mangiarotti$^{2}$, J. Markert$^{4}$, T. Matulewicz$^{19}$, S. Maurus$^{10}$, V. Metag$^{11}$, J. Michel$^{8}$, D.M. Mihaylov$^{10,9}$, S. Morozov$^{12,14}$, C. Müntz$^{8}$, R. Münzer$^{10,9}$, L. Naumann$^{6}$, K. Nowakowski$^{3}$, M. Palka$^{3}$, Y. Parpottas$^{15,d}$, V. Pechenov$^{4}$, O. Pechenova$^{4}$, O. Petukhov$^{12}$, K. Piasecki$^{19}$, J. Pietraszko$^{4}$, W. Przygoda$^{3}$, S. Ramos$^{2}$, B. Ramstein$^{16}$, A. Reshetin$^{12}$, P. Rodriguez-Ramos$^{17}$, P. Rosier$^{16}$, A. Rost$^{5}$, A. Sadovsky$^{12}$, P. Salabura$^{3}$, T. Scheib$^{8}$, H. Schuldes$^{8}$, E. Schwab$^{4}$, F. Scozzi$^{5,16}$, F. Seck$^{5}$, P. Sellheim$^{8}$, I. Selyuzhenkov$^{4,14}$, J. Siebenson$^{10}$, L. Silva$^{2}$, Yu.G. Sobolev$^{17}$, S. Spataro$^{e}$, S. Spies$^{8}$, H. Ströbele$^{8}$, J. Stroth$^{8,4}$, P. Strzempek$^{3}$, C. Sturm$^{4}$, O. Svoboda$^{17}$, M. Szala$^{8}$, P. Tlusty$^{17}$, M. Traxler$^{4}$, H. Tsertos$^{15}$, E. Usenko$^{12}$, V. Wagner$^{17}$, C. Wendisch$^{4}$, M.G. Wiebusch$^{8}$, J. Wirth$^{10,9}$, D. Wójcik$^{19}$, Y. Zanevsky$^{7,\dag}$, P. Zumbruch$^{4}$\
(HADES collaboration)
date: Received
title: 'Identical pion intensity interferometry in central Au+Au collisions at 1.23 GeV'
---
Two-particle intensity interferometry of hadrons is widely used to study the spatio-temporal size, shape and evolution of their sources created in heavy-ion collisions or other reactions involving hadrons (for a review see Ref.[@Lisa05]). The technique, pioneered by Hanbury Brown and Twiss [@HBT56] to measure angular radii of stars, later on named HBT interferometry, is based on the quantum-statistical interference of identical particles. Goldhaber et al. [@GGLP60] first applied intensity interferometry to hadrons. In heavy-ion collisions, the intensity interferometry does not allow to measure directly the reaction volume, as the emission source, changing in shape and size in the course of the collision, is affected by density and temperature gradients and dynamically generated space-momentum correlations ([*e.g.*]{} radial expansion after the compression phase or resonance decays). Thus, intensity interferometry generally does not yield the proper source size, but rather an effective “length of homogeneity” [@Lisa05]. It measures source regions in which particle pairs are close in momentum, so that they are correlated as a consequence of their quantum statistics or due to their two-body interaction. In general, the sign and strength of the correlation is affected by (i) the strong interaction, (ii) the Coulomb interaction if charged particles are involved, and (iii) the quantum statistics in the case of identical particles (Fermi-Dirac suppression for fermions, Bose-Einstein enhancement for bosons). In the case of $\pi\pi$ correlations, the mutual strong interaction was found to be minor [@Bowler88] compared to the effects (ii) and (iii).
Pion freeze-out dynamics may be relevant to ongoing searches for the QCD critical point in the $T-\mu_B$ plane, where $T$ and $\mu_B$ are the temperature and the baryon-chemical potential. Systems with $\mu_B$ above the critical point are expected to undergo a first-order phase transition which might be visible in a non-monotonic behavior of various source parameters. However, it is also conceivable that the initial temperatures of the system, which can be reached in heavy-ion collisions at high $\mu_B$, are not high enough to create a deconfined partonic state. In this scenario a first order phase boundary cannot be reached experimentally. A recently published excitation function of HBT source radii [@STAR2015] from the domain of the Relativistic Heavy Ion Collider (RHIC) down to lower collision energies indicates such a non-monotonic energy dependence around center-of-mass energies of $\sqrt{s_\mathrm{NN}} < 10$ GeV. Even though a part of this behavior can be related to the strong impact of different pair transverse momentum intervals involved in the source parameter compilation of Ref.[@STAR2015], to a certain extent the deviation of the data points from a monotonic trend remains at low energies. Here, new precision data, especially at low collision energies of $\sqrt{s_\mathrm{NN}} < 5$ GeV, can contribute to the clarification of this exciting observation before definite conclusions on a change in physics can be drawn.
It is worth emphasizing that only preliminary data [@hbt_fopi_1995] of identical-pion HBT data exist for a large symmetric collision system (like Au+Au or Pb+Pb) at a beam kinetic energy of about $1{\mbox{$A$~GeV}}$ (fixed target, $\sqrt{s_\mathrm{NN}} = 2.3$ GeV)[^1]. For the somewhat smaller system La+La, studied at $1.2A$ GeV with the HISS spectrometer at the Lawrence Berkeley Laboratory (LBL) Bevalac, pion correlation data were reported by Christie et al. [@Christie93; @Christie92]. An oblate shape of the pion source and a correlation of the source size with the system size were found. Also, pion intensity interferometry for small systems (Ar+KCl, Ne+NaF) was studied at 1.8$A$ GeV at the LBL Bevalac using the Janus spectrometer by Zajc et al. [@Zajc84]. Both groups made first attempts to correct the influence of the pion-nuclear Coulomb interaction on the pion momenta. The effect on the source radii, however, were found negligible for their experiments.
In this letter we report on the first investigation of $\pi^-\pi^-$ and $\pi^+\pi^+$ correlations at low relative momenta in [Au+Au collisions at 1.23]{}, continuing our previous femtoscopic studies of smaller collisions systems [@hades_pLambda_ArKCl; @hades_hbt_ArKCl; @hades_pLambda_pNb]. The experiment was performed with the [**H**]{}igh [**A**]{}cceptance [**D**]{}i-[**E**]{}lectron [**S**]{}pectrometer (HADES) at the Schwerionensynchrotron SIS18 at GSI, Darmstadt. HADES [@Agakishiev:2009am], although primarily optimized to measure di-electrons [@HADES-PRL07], offers also excellent hadron identification capabilities [@hades_kpm_phi_arkcl; @hades_xi_arkcl; @hades_K0_ArKCl; @hades_Lambda_ArKCl]. HADES is a charged particle detector consisting of a six-coil toroidal magnet centered around the beam axis and six identical detection sections located between the coils and covering polar angles between $18^{\circ}$ and $85^{\circ}$. Each sector is equipped with a Ring-Imaging Cherenkov (RICH) detector followed by four layers of Mini-Drift Chambers (MDCs), two in front of and two behind the magnetic field, as well as a scintillator Time-Of-Flight detector (TOF) ($45^{\circ}$ – $85^{\circ}$) and Resistive Plate Chambers (RPC) ($18^{\circ}$ – $45^{\circ}$). Both timing detectors, TOF and RPC, allow for good particle identification, i.e. proton-pion separation. (Due to their low yield, kaons hardly affect the pion selection at SIS energies.) TOF, RPC, and Pre-Shower detectors (behind RPC, for e$^\pm$ identification) were combined into a Multiplicity and Electron Trigger Array (META). Several triggers are implemented. The minimum bias trigger is defined by a signal in a diamond START detector in front of the 15-fold segmented gold target. In addition, online Physics Triggers (PT) are used, which are based on hardware thresholds on the TOF signals, proportional to the event multiplicity, corresponding to at least 20 (PT3) hits in the TOF. About 2.1 billion PT3 triggered Au+Au collisions corresponding to the 40% most central events are taken into account for the correlation analysis. The centrality determination is based on the summed number of hits detected by the TOF and the RPC detectors. The measured events are divided in centrality classes corresponding to successive $10\,\%$ regions of the total cross section [@hades_centrality:2018am]. Here, we report only on results of the $0-10\,\%$ class; the entire centrality dependence of pion source parameters analysed as function of azimuthal angle w.r.t. the reaction plane will be part of an extended forthcoming paper, while yields and phase-space distributions of charged pions are to be presented in a separate report.
Generally, the two-particle correlation function is defined as the ratio of the probability $P_2({\bm p}_1,{\bm p}_2)$ to measure simultaneously two particles with momenta ${\bm p}_1$ and ${\bm p}_2$ and the product of the corresponding single-particle probabilities $P_1({\bm p}_1)$ and $P_1({\bm p}_2)$ [@Lisa05], $$C({\bm p}_1, {\bm p}_2) = \frac{P_2({\bm p}_1,{\bm p}_2)}{P_1({\bm p}_1) P_1({\bm p}_2)}.
\label{def_theo_corr_fct}$$ Experimentally this correlation is formed as a function of the momentum difference between the two particles of a given pair and quantified by taking the ratio of the yields of ’true’ pairs ($Y_\mathrm{true}$) and uncorrelated pairs ($Y_\mathrm{mix}$). $Y_\mathrm{true}$ is constructed from all particle pairs in the selected phase space interval from the same event. $Y_\mathrm{mix}$ is generated by event mixing, where particle 1 and particle 2 are taken from different events. Care was taken to mix particles from similar event classes in terms of multiplicity, vertex position and reaction plane angle. The events are allowed to differ by not more than 10 units in the number of the RPC+TOF hit multiplicity of $\ge182$ (i.e. corresponding to the uncertainty of the centrality class $0-10\,\%$ [@hades_centrality:2018am]), 1.2mm in the $z$-vertex coordinate (amounting to less than one third of the spacing between target segments), and 30 degrees in azimuthal angle (to be compared to the event plane resolution of $\langle \cos{\Phi}\rangle=0.612$), respectively.
The momentum difference is decomposed into three orthogonal components as suggested by Podgoretsky [@Podgoretsky83], Pratt [@Pratt86] and Bertsch [@Bertsch89]. The three-dimensional correlation functions are projections of equation (\[def\_theo\_corr\_fct\]) into the (out, side, long)-coordinate system, where ‘out’ means along the pair transverse momentum, ${\bm k}_\mathrm{t}=({\bm p}_\mathrm{t,\,1}+ {\bm p}_\mathrm{t,\,2})/2$, ‘long’ is parallel to the beam direction z, and ‘side’ is oriented perpendicular to the other directions. The particles forming a pair are boosted into the longitudinally comoving system (LCMS), where the z-components of the momenta cancel each other, $p_\mathrm{z_1}+p_\mathrm{z_2}=0$. Note that in other publications also the pair comoving system (${\bm p}_1+{\bm p}_2=0$) is frequently used. The LCMS choice allows for an adequate comparison with correlation data taken at very different, usually much higher, collision energies, where the distribution of the rapidity, $y=\tanh^{-1}{(\beta_\mathrm{z})}$, of produced particles is found to be not as narrow as in the present case but largely elongated. (Here, $\beta_\mathrm{z}=p_\mathrm{z}/E$, $E=\sqrt{p^2+m_0^2}$ and $m_0$ are the longitudinal velocity, the total energy and the rest mass of the particle, respectively. We use units with $\hbar=c^2=1$.) Hence, the experimental correlation function is given by $$C({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}}) = {\cal N} \,\frac{Y_{\mathrm{true}}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}})}{Y_{\mathrm{mix}}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}})},
\label{def_exp_corr_fct}$$ where $q_i=(p_{1,\,i}-p_{2,\,i})/2$ ($i$=’out’,’side’,’long’) are the relative momentum components, and ${\cal N}$ is a normalization factor which is fixed by the requirement $C \rightarrow 1$ at large relative momenta, where the correlation function is expected to flatten out at unity. Note that, as in our previous intensity interferometry analyses [@hades_pLambda_ArKCl; @hades_hbt_ArKCl; @hades_pLambda_pNb], we use the above low-energy convention of $q$ which is common also in studies of proton-proton correlations, in contrast to the high-energy convention of $\pi\pi$ correlations, $Q=2q$. The statistical errors of equation (\[def\_exp\_corr\_fct\]) are dominated by those of the true yield, since the mixed yield is generated with much higher statistics.
Two-track reconstruction defects (e.g. track splitting and merging effects) that are particularly important to HBT analyses were corrected by appropriate selection conditions on the META-hit and MDC-layer levels, i.e. by discarding pairs which hit the same META cell, and by excluding for particle2 three successive wires symmetrically around the MDC wire fired by particle1. This method was tested with simulations carrying neither quantum-statistical nor Coulomb effects, based on UrQMD [@UrQMD], Geant[@GEANT] and a detailed description of the detector response, to firmly exclude any close-track effect. Also broader exclusion windows have been tested, but no significant improvement was found. These simulations also showed that there are no significant long-range correlations, usually attributed either to energy-momentum conservation in correlation analyses of small systems or to minijet-like phenomena at high energies.
The data are divided into seven $k_\mathrm{t}$ bins from 50 to 400 ${\mbox{MeV$/c$}}$. The three-dimensional experimental correlation function is then fitted with the function $$\begin{aligned}
& C_\mathrm{fit}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}}) = \nonumber \\
& N \big[(1-\lambda) + \lambda\, K_\mathrm{C}(\hat q,{\ensuremath{R_{{\mbox{\textrm{\scriptsize inv}}}}}}) \,C_\mathrm{qs}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}})\big],
\label{pipi_fit_fct_3dim}\end{aligned}$$ where $$\begin{aligned}
& C_\mathrm{qs}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}})= 1\,+ \nonumber \\
& \exp{(-(2{\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}}{\ensuremath{R_{{\mbox{\textrm{\scriptsize out}}}}}})^2-(2{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}}{\ensuremath{R_{{\mbox{\textrm{\scriptsize side}}}}}})^2-(2{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}}{\ensuremath{R_{{\mbox{\textrm{\scriptsize long}}}}}})^2)} \hfill
\label{pipi_fit_be_3dim}\end{aligned}$$ represents the quantum-statistical part of the correlation function. The parameters $N$ and $\lambda$ in Eq.(\[pipi\_fit\_fct\_3dim\]) are a normalization constant and the fraction of correlated pairs, respectively, and $\hat q= {\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}({\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize side}}}}}},\,{\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}},\,k_\mathrm{t})$ is the average value of the invariant momentum difference, ${\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}=\frac{1}{2}\sqrt{({\bm p}_1 - {\bm p}_2)^2-(E_1-E_2)^2}$, for given intervals of the relative momentum components and $k_\mathrm{t}$. The range of the one- and three-dimensional fits extends in ${\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}$ from $6\,{\mbox{MeV$/c$}}$ to $80\,{\mbox{MeV$/c$}}$. Log-likelihood minimization [@Ahle02] was used in all fits to the correlation functions. The influence of the mutual Coulomb interaction in Eq.(\[pipi\_fit\_fct\_3dim\]) is separated from the Bose-Einstein part by including in the fits the commonly used Coulomb correction by Sinyukov et al.[@Sinyukov98]. The Coulomb factor $K_\mathrm{C}$ results from the integration of the two-pion Coulomb wave function squared over a spherical Gaussian source of fixed radius. This radius is iteratively approximated by the result of the corresponding fit to the correlation function. In Eq.(\[pipi\_fit\_fct\_3dim\]), the non-diagonal elements comprising the combinations ’out’-’side’ and ’side’-’long’ vanish for symmetry reasons [@UHeinz02] when azimuthally and rapidity integrated correlations functions are studied [@e895_2000; @HBT_in_UrQMD], as it is done in the present investigation. The ’out’-’long’ component, however, can have a finite value depending on the degree of symmetry of the detector-accepted rapidity distribution w.r.t. midrapidity ($y_\mathrm{cm}=0.74$). We studied this effect by including in Eq.(\[pipi\_fit\_be\_3dim\]) an additional term $-2 {\ensuremath{q_{{\mbox{\textrm{\scriptsize out}}}}}}(2{\ensuremath{R_{{\mbox{\textrm{\scriptsize out\,long}}}}}})^2 {\ensuremath{q_{{\mbox{\textrm{\scriptsize long}}}}}}$, where the prefactor accounts for both non-diagonal terms, ’out’-’long’ and ’long’-’out’. We found only marginal differences in the fits which delivered, for all transverse-momentum classes, rather small values of $R^2_\mathrm{out\,long}< 1$fm$^2$. For all results presented here, we restricted the pair rapidity to an interval $\vert y - y_\mathrm{cm} \vert<0.35$, within which $dN/dy$ does not vary by more than 10%, and limited ourselves to the fit function with the Bose-Einstein part (Eq.(\[pipi\_fit\_be\_3dim\])) consisting of diagonal elements only and added the small deviations to the systematic errors. The effect of finite momentum resolutions of the HADES tracking system is studied with dedicated simulations. Typical Gaussian resolution values of $\sigma_q({\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}=20\,{\mbox{MeV$/c$}}) \simeq 2\,{\mbox{MeV$/c$}}$ are estimated. Incorporating a corresponding correction into the fit function by convolution of Eq.(\[pipi\_fit\_fct\_3dim\]) with a Gaussian resolution function leads to radius shifts of about $\delta R/R \simeq +2$%.
Figure \[onedim\_projections\] shows one-dimensional projections of the Coulomb-corrected ${\ensuremath{\pi^{-}}}{\ensuremath{\pi^{-}}}$ correlation function together with corresponding fits with Eq.(\[pipi\_fit\_fct\_3dim\]) for various $k_\mathrm{t}$ intervals. (Due to the permutability of particles 1 and 2, one of the $q$ projections can be restriced to positive values.) The peak due to the Bose-Einstein enhancement becomes evident at low $\vert q \vert$. Its width increases with increasing $k_\mathrm{t}$. The correlation functions for ${\ensuremath{\pi^{+}}}{\ensuremath{\pi^{+}}}$ pairs look similar.
![Projections of the Coulomb-corrected three-dimensional ${\ensuremath{\pi^{-}}}{\ensuremath{\pi^{-}}}$ correlation function (dots) and of the respective fits (dashed curves) for the $k_\mathrm{t}$ intervals of $100-150~{\mbox{MeV$/c$}}$ (top), $200-250~{\mbox{MeV$/c$}}$ (middle), and $300-350~{\mbox{MeV$/c$}}$ (bottom). The left, center, and right panels give the ’out’, ’side’, and ’long’ directions, respectively. The unplotted $q$ components are integrated over $\pm 12~{\mbox{MeV$/c$}}$.[]{data-label="onedim_projections"}](quality_plot_ptclass246-crop.pdf){width="0.9\linewidth"}
The main systematic uncertainties of the results presented below arise from the slight fluctuations of the fit results when varying the fit ranges ($\sim 0.1 - 0.3$fm), from the forward-backward differences of the fit results w.r.t. midrapidity within similar transverse momentum intervals ($\sim 0.03 - 0.1$(0.2)fm for $R_{\mathrm{inv}}$, $R_{\mathrm{side}}$, $R_{\mathrm{long}}$ ($R_{\mathrm{out}}$)), and from the differences when switching on/off the ’out’-’long’ component in the fit function ($\sim 0.05 - 0.2$fm). Finally, all systematic error contributions are added quadratically. In Fig.\[Radien\_pi0pi0extract\_spline\_centr0to10\_pt0to900\] they are shown as hatched bands.
To separate a potential source radius bias introduced by the Coulomb force the charged pions experience in the field of the charged fireball, we follow the ansatz used in Ref.[@Baym_1996], $$E({\bm p}_\mathrm{f}) = E({\bm p}_\mathrm{i}) \pm V_\mathrm{eff}({\bm r}_\mathrm{i}),
\label{eq:baym_1}$$ where $E$ is the total energy, ${\bm p}_\mathrm{i}$ (${\bm p}_\mathrm{f}$) is the initial (final) momentum and ${\bm r}_\mathrm{i}$ is the inital position of the pion in the Coulomb potential $V_{\mathrm{eff}}$ with positive (negative) sign for $\pi^+$ ($\pi^-$). With $$\begin{aligned}
\frac{R_{\pi^{\pm}\pi^{\pm}}}{R_{\tilde\pi^0\tilde\pi^0}} \approx \frac{q_\mathrm{i}}{q_\mathrm{f}} = \frac{|{\bm p}_\mathrm{i}|}{|{\bm p}_\mathrm{f}|} = \sqrt{ 1 \mp 2 \frac{V_{\mathrm{eff}}}{|{\bm p}_\mathrm{f}|} \sqrt{1 + \frac{m_{\pi}^2}{{\bm p}_\mathrm{f}^2}} + \frac{V^2_{\mathrm{eff}}}{{\bm p}_\mathrm{f}^2}},
\label{eq:baym_2}\end{aligned}$$ where $q_\mathrm{i}$ ($q_\mathrm{f})$ is the initial (final) relative momentum, and with $V_{\mathrm{eff}} / k_\mathrm{t} \ll 1$, it turns out that the constructed squared source radius for pairs of neutral pions (denoted by $\tilde{\pi}^0\tilde{\pi}^0$ in the following in contrast to the case where $\pi^{-}\pi^{-}$ and $\pi^{+}\pi^{+}$ data are combined) is simply the arithmetic mean of the corresponding quantities of the charged pions, $$R_{\tilde\pi^0\tilde\pi^0}^2 = \frac{1}{2} \big(R_{\pi^+\pi^+}^2 + R_{\pi^-\pi^-}^2\big),
\label{eq:baym_3}$$ which is valid for all radius components (even though in the ’out’ direction, Eq.(\[eq:baym\_2\]) looks slightly different). Finally, the constructed $\pi^{0}\pi^{0}$ correlation radii are derived from cubic spline interpolations of the $k_\mathrm{t}$ dependence of both the corresponding experimental $\pi^{-}\pi^{-}$ and $\pi^{+}\pi^{+}$ data. This interpolation is necessary because - as result of different detector acceptances - the charged pion pairs exhibit slightly different average transverse momenta, even though they are measured in identical $k_\mathrm{t}$ intervals.
Figure\[Radien\_pi0pi0extract\_spline\_centr0to10\_pt0to900\] shows the dependence on average $k_\mathrm{t}$ (determined for ${\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}<50~{\mbox{MeV$/c$}}$) of the one-dimensional (invariant) and three-dimensional source radii for $\pi^-\pi^-$ (black squares) and $\pi^+\pi^+$ (red circles) pairs. While for low transverse momentum the Coulomb interaction with the fireball leads to an increase (a decrease) of the source size derived for negative (positive) pion pairs, at large transverse momentum apparently the Coulomb effect fades away. The effect is smallest for ${\ensuremath{R_{{\mbox{\textrm{\scriptsize out}}}}}}$. Note that the charge splitting of the source radii was early predicted by Barz [@Barz96; @Barz99] who investigated the combined effects of nuclear Coulomb field, radial flow, and opaqueness on two-pion correlations for a large collision system such as Au+Au in the $1 {\mbox{$A$~GeV}}$ energy regime. Earlier experimental works at the Bevalac employing a three-body Coulomb correction found the effect negligible for their studies of smaller systems [@Zajc84; @Christie92; @Christie93]. The parameter $\lambda$ derived from the fits with Eq.(\[pipi\_fit\_fct\_3dim\]) appears rather independent of charge sign and decreases only slightly with increasing transverse momentum, cf. lower right panel of Fig.\[Radien\_pi0pi0extract\_spline\_centr0to10\_pt0to900\]. It fits well into a preliminary evolution with $\sqrt{s_\mathrm{NN}}$ established previously [@STAR2015], except the lowest E895 data point. In contrast, $\lambda$ resulting from the fits to the one-dimensional (${\ensuremath{q_{{\mbox{\textrm{\scriptsize inv}}}}}}$-dependent) correlation function, exhibits a significant decrease with $k_\mathrm{t}$ (cf. lower left panel), probably pointing to the fact that the one-dimensional fit function is not adequate. Note that deviations from Gaussian source shapes will be studied in a forthcoming paper by applying the method of source imaging [@BrownDanielewicz97; @pipi_imaging_E895], or by using Lévy source parameterizations [@Levy_PHENIX2018].
![Source parameters as function of pair transverse momentum, $k_\mathrm{t}$, for central ($0-10\,\%$) [Au+Au collisions at 1.23]{}. The upper left, upper right, center left, and center right panels display the invariant, out, side, and long radii, respectively. The lower left and lower right panels show the corresponding $\lambda$ parameters resulting from the fits to the one- and three-dimensional correlation functions, respectively. Black squares (red circles) are for pairs of negative (positive) pions. Blue dashed lines represent constructed radii of neutral pion pairs (see text). Error bars and hatched bands represent the statistical and systematic errors, respectively. []{data-label="Radien_pi0pi0extract_spline_centr0to10_pt0to900"}](Radien_pi0pi0extract_spline_centr0to10_pt100to800_syserrorband_withlambda-crop.pdf){width="0.9\linewidth"}
The excitation functions of $R_{\mathrm{out}}$, $R_{\mathrm{side}}$, and $R_{\mathrm{long}}$ for pion pairs produced in central collisions are displayed in Fig.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\]. All shown radius parameters have been obtained by interpolating the existing measured data points to the same transverse mass of $m_\mathrm{t}=\sqrt{k_\mathrm{t}^2+m_{\pi}^2}=260\,{\mbox{MeV}}$ at which data points by STAR at RHIC [@STAR2015] are available. The statistical errors are properly propagated and quadratically added with systematic differences of linear and cubic-spline interpolations. Extrapolations were not necessary at this $m_\mathrm{t}$ value. Corresponding excitation functions at other transverse masses show similar dependencies. Surprisingly, $R_{\mathrm{out}}$ and $R_{\mathrm{side}}$ vary hardly more than 40% over three orders of magnitude in center-of-mass energy. Only $R_{\mathrm{long}}$ exhibits a systematical increase by about a factor of two to three when going in energy from SIS18 via AGS, SPS, RHIC to LHC. Note that in the excitation functions of Ref.[@STAR2015] not all, particularly AGS, data points were properly corrected for their $k_\mathrm{t}$ dependence. While the HADES $R_{\mathrm{out}}$ and $R_{\mathrm{side}}$ data for negative pions completely agree with the lowest E895 data at $2 {\mbox{$A$~GeV}}$, $R_{\mathrm{long}}$ deviates from the corresponding E895 data point. Both data are, however, in accordance with the overall smooth trend within $2~\sigma$. (The low-energy CERES data of $R_{\mathrm{out}}$ and the E866 data point of $R_{\mathrm{long}}$ for $\pi^-\pi^-$ pairs appear to be outliers.)
The combination of $R_{\mathrm{out}}^2$ and $R_{\mathrm{side}}^2$ can be related to the emission time duration [@Lisa2016], $(\Delta\tau)^2\approx(R_{\mathrm{out}}^2-R_{\mathrm{side}}^2)/\langle\beta_\mathrm{t}^2\rangle$, where $\beta_\mathrm{t}$ is the transverse pair velocity. The excitation function of $R_{\mathrm{out}}^2-R_{\mathrm{side}}^2$ is shown in Fig.\[Rout2\_Rside2\_mt260\_central\_pimpim\_pippip\]. Up to now almost all measurements below 10GeV are characterized by large errors and scatter sizeably. (Here, the outlying low-energy CERES data are solely caused by the deviation in $R_{\mathrm{out}}$, cf. top panel of Fig.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\].) The new HADES data show that the difference of source parameters in the transverse plane almost vanishes at low collision energies. With increasing energy, it reaches a maximum at $\sqrt{s_{\mathrm{NN}}}\sim 20-30$GeV and afterwards decreases towards zero at LHC energies. One would conclude that in the $1 {\mbox{$A$~GeV}}$ energy region pions are emitted into free space during a short time span of less than one to two fm/$c$. However, also the opaqueness of the source affects $R_{\mathrm{out}}^2-R_{\mathrm{side}}^2$ which could cause it to become negative, thus compensating the positive contribution from the emission time [@Barz99].
The excitation function of the freeze-out volume, $V_\mathrm{fo}=(2\pi)^{3/2}R^2_{\mathrm{side}}R_{\mathrm{long}}$, is given in Fig.\[Vfreezeout\_mt160\_central\_pimpim\_pippip\]. Note that this definition of a three-dimensional Gaussian volume does not incorporate ${\ensuremath{R_{{\mbox{\textrm{\scriptsize out}}}}}}$ since generally this length is potentially extended due to a finite value of the aforementioned emission duration. From the above HADES data, we estimate a volume of about 1,300fm$^3$ for pairs of constructed neutral pions. The volume of homogeneity steadily increases with energy, but is merely a factor four larger at LHC. Extrapolating $V_\mathrm{fo}$ to $k_\mathrm{t}=0$ yields a value of about 3,900fm$^3$.
![Excitation function of the source radii $R_{\mathrm{out}}$ (upper panel), $R_{\mathrm{side}}$ (central panel), and $R_{\mathrm{long}}$ (lower panel) for pairs of identical pions with transverse mass of $m_\mathrm{t}=260\,{\mbox{MeV}}$ in central collisions of Au+Au or Pb+Pb. Squares represent data by ALICE at LHC ($\pi^-\pi^-$+$\pi^+\pi^+$) [@ALICE2016], full triangles STAR at RHIC ($\pi^-\pi^-$+$\pi^+\pi^+$) [@STAR2015], diamonds are for CERES at SPS ($\pi^-\pi^-$+$\pi^+\pi^+$) [@CERES_2003], open triangles are for NA49 at SPS ($\pi^-\pi^-$) [@na49_2008], open circles are $\pi^-\pi^-$ data by E895 at AGS [@e895_2000; @Lisa05], and open (full) crosses involve $\pi^-\pi^-$ ($\pi^+\pi^+$) data of E866 at AGS [@e866_1999], respectively. The present data of HADES at SIS18 for pairs of $\pi^-\pi^-$ ($\pi^+\pi^+$) are given as open (full) stars. Statistical errors are displayed as error bars; if not visible, they are smaller than the corresponding symbols. []{data-label="Rosl_exctfct_mt260_central_pimpim_pippip"}](Rosl_exctfct_mt260_central_pimpim_pippip_colored_fixedscale-crop.pdf){width="0.9\linewidth"}
![Excitation function of $R_{\mathrm{out}}^2-R_{\mathrm{side}}^2$, as calculated from the data points shown in Fig.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\]. []{data-label="Rout2_Rside2_mt260_central_pimpim_pippip"}](Rout2_Rside2_mt260_central_pimpim_pippip_colored-crop.pdf){width="0.9\linewidth"}
![Excitation function of the freeze-out volume, $V_{\mathrm{fo}}=(2\pi)^{3/2}R^2_{\mathrm{side}}R_{\mathrm{long}}$, as calculated from the data points shown in Fig.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\]. []{data-label="Vfreezeout_mt160_central_pimpim_pippip"}](Vfreezeout_mt260_central_pimpim_pippip_colored-crop.pdf){width="0.9\linewidth"}
The large scatter of data points in Fig.\[Vfreezeout\_mt160\_central\_pimpim\_pippip\] below $\sqrt{s_{\mathrm{NN}}}= 10$GeV is intriguing and might indicate a non-trivial energy dependence of the radius parameters in this region. However, the simplest interpretation would be to assume instead that the energy dependence is smooth. (Note that the difference of the HADES $\pi^-\pi^-$ data and the lowest E895 data point at $2{\mbox{$A$~GeV}}$ is primarily caused by the deviation in $R_{\mathrm{long}}$.) If, however, the variation of the data at low energies, most prominently seen in the non-monotonicity of $R_{\mathrm{side}}$ (cf. Fig.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\]), is to be taken seriously, new experimental and theoretical efforts are needed to clarify the situation, as could be done with the future experiments CBM at SIS100/FAIR in Darmstadt [@CBM] and MPD at NICA in Dubna [@NICA] or with the STAR fixed-target program [@KMeehan_STAR_2017]. Finally, we want to recall that in Figs.\[Rosl\_exctfct\_mt260\_central\_pimpim\_pippip\], \[Rout2\_Rside2\_mt260\_central\_pimpim\_pippip\], and \[Vfreezeout\_mt160\_central\_pimpim\_pippip\] we display statistical uncertainties only; the systematic ones were not available for all experiments.
In summary, we presented high-statistics $\pi^-\pi^-$ and $\pi^+\pi^+$ HBT data for central [Au+Au collisions at 1.23]{}. The three-dimensional Gaussian emission source is studied in dependence on transverse momentum and found to follow the trends observed at higher collision energies, extending the corresponding excitation functions down to the very low part of the energy scale. Substantial differences of the source radii for pairs of negative and positive pions are found, especially at low transverse momenta, an effect which is not observed at higher collision energies. A clear hierarchy of the three Gaussian radii is seen in our data, i.e. $R_{\mathrm{long}}<R_{\mathrm{side}}\approx R_{\mathrm{out}}$, independent of transverse momentum. Furthermore, a surprisingly small variation of the space-time extent of the pion emission source over three orders of magnitude in center-of-mass energy, $\sqrt{s_{\mathrm{NN}}}$, is observed. Our data indicate that the very smooth trends observed at ultra-relativistic energies continue towards very low energies. While both $R_{\mathrm{out}}$ and $R_{\mathrm{long}}$ steadily decrease with decreasing $\sqrt{s_\mathrm{NN}}$, a weak non-monotonic energy dependence of $R_{\mathrm{side}}$ can not be excluded.
The HADES Collaboration gratefully acknowledges the support by the grants SIP JUC Cracow, Cracow (Poland), National Science Center, 2016/23/P/ST2/040 POLONEZ, 2017/25/N/ST2/00580, 2017/26/M/ST2/00600; TU Darmstadt, Darmstadt (Germany) and Goethe-University, Frankfurt (Germany), ExtreMe Matter Institute EMMI at GSI Darmstadt; TU München, Garching (Germany), MLL München, DFG EClust 153, GSI TMLRG1316F, BMBF 05P15WOFCA, SFB 1258, DFG FAB898/2-2; NRNU MEPhI Moscow, Moscow (Russia), in framework of Russian Academic Excellence Project 02.a03.21.0005, Ministry of Science and Education of the Russian Federation 3.3380.2017/4.6; JLU Giessen, Giessen (Germany), BMBF:05P12RGGHM; IPN Orsay, Orsay Cedex (France), CNRS/IN2P3; NPI CAS, Rez, Rez (Czech Republic), MSMT LM2015049, OP VVV CZ.02.1.01/0.0/0.0/16 013/0001677, LTT17003.
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[^1]: Throughout this publication ${\mbox{$A$~GeV}}$ refers to the mean kinetic beam energy.
|
---
abstract: 'We study the algebraic structure of the Poisson algebra $P({{\mathcal O}})$ of polynomials on a coadjoint orbit ${{\mathcal O}}$ of a semisimple Lie algebra. We prove that ${P({{\mathcal O}})}$ splits into a direct sum of its center and its derived ideal. We also show that ${P({{\mathcal O}})}$ is simple as a Poisson algebra iff ${{\mathcal O}}$ is semisimple.'
author:
- |
[**Mark J. Gotay**]{}[^1]\
Department of Mathematics\
University of Hawai‘i\
2565 The Mall\
Honolulu, HI 96822 USA\
[^2]\
Institute of Mathematics\
University of Warsaw\
ul. Banacha 2\
02-097 Warsaw, Poland\
[^3]\
Department of Mathematics\
University of Hawai‘i\
2565 The Mall\
Honolulu, HI 96822 USA
date: 'August 19, 2000'
title: POLYNOMIAL ALGEBRAS ON COADJOINT ORBITS OF SEMISIMPLE LIE GROUPS
---
[Structure Theorems]{}
Let ${{\mathfrak g}}$ be a real (finite-dimensional) semisimple Lie algebra with corresponding 1-connected Lie group $G$. It is well known that the dual space ${{{\mathfrak g}}^{\ast}}$ carries the structure of a linear Poisson manifold under the Lie-Poisson bracket. The symplectic leaves of this Poisson structure are the orbits of the coadjoint representation of $G$ on ${{{\mathfrak g}}^{\ast}}$.
As the elements of ${{\mathfrak g}}$ may be regarded as linear functions on ${{{\mathfrak g}}^{\ast}}$, the symmetric algebra ${S({{\mathfrak g}})}$ may be identified with the algebra of polynomial functions on ${{{\mathfrak g}}^{\ast}}$. Consequently, ${S({{\mathfrak g}})}$ can be realized as a Poisson subalgebra of $C^\infty({{{\mathfrak g}}^{\ast}})$. (Equivalently, the Poisson bracket ${\{\; ,\,\}}$ on ${S({{\mathfrak g}})}$ can be obtained by setting $\{\xi,\eta\} =
[\xi,\eta]$ for $\xi, \eta \in {{\mathfrak g}}$ and extending to all of ${S({{\mathfrak g}})}$ via the Leibniz rule.)
Let ${S({{\mathfrak g}})}'=\{{S({{\mathfrak g}})}, {S({{\mathfrak g}})}\}$ be the derived ideal, and let ${C({{\mathfrak g}})}$ denote the Lie center of the Poisson algebra ${S({{\mathfrak g}})}$.
${S({{\mathfrak g}})}= {C({{\mathfrak g}})}\oplus {S({{\mathfrak g}})}'.$ \[s\]
[[*Proof.* ]{}]{}We have the decomposition $${S({{\mathfrak g}})}= \bigoplus^{\infty}_{n=0}{S_n({{\mathfrak g}})}$$ of the symmetric algebra into the finite-dimensional subspaces ${S_n({{\mathfrak g}})}$ of homogeneous elements of degree $n$. Each ${S_n({{\mathfrak g}})}$ is invariant with respect to the adjoint action of ${{\mathfrak g}}$ on ${S({{\mathfrak g}})}$. Since every finite-dimensional representation of a semisimple Lie algebra is completely reducible, it follows that the adjoint action of ${{\mathfrak g}}$ on ${S({{\mathfrak g}})}$ is itself completely reducible. According to [@Di §1.2.10] we can then split $${S({{\mathfrak g}})}= {C({{\mathfrak g}})}\oplus \{{{\mathfrak g}},{S({{\mathfrak g}})}\}.$$ So we need only show that $\{{{\mathfrak g}},{S({{\mathfrak g}})}\}
= {S({{\mathfrak g}})}'$.
Now, applying the identity $$\{fg,h\} = \{f,gh\} + \{g,fh\}$$ to $f,g \in {{\mathfrak g}}$ and $h \in {S_n({{\mathfrak g}})}$, we see that $\{S_2({{\mathfrak g}}),S_n({{\mathfrak g}})\} \subset \{{{\mathfrak g}},S_{n+1}({{\mathfrak g}})\}$. Arguing recursively, we obtain $$\{S_m({{\mathfrak g}}),S_n({{\mathfrak g}})\} \subset \{{{\mathfrak g}},S_{n+m-1}({{\mathfrak g}})\},$$ from which the desired result follows.
Let ${{\mathcal O}}$ be an orbit in ${{{\mathfrak g}}^{\ast}}$. We can restrict polynomials on ${{{\mathfrak g}}^{\ast}}$ to functions on ${{\mathcal O}}$ thereby obtaining the (orbit) polynomial algebra ${P({{\mathcal O}})}$ (which, however, may not be freely generated as an associative algebra). We may identify ${P({{\mathcal O}})}$ with the quotient algebra ${S({{\mathfrak g}})}/ {I({{\mathcal O}})}$, where ${I({{\mathcal O}})}$ is the ideal of elements vanishing on ${{\mathcal O}}$, with the canonical projection $$\rho_{{{\mathcal O}}} \colon {S({{\mathfrak g}})}\to {S({{\mathfrak g}})}/ {I({{\mathcal O}})}\cong {P({{\mathcal O}})}.$$ Since ${{\mathcal O}}$ is a symplectic leaf of the Poisson structure on ${{{\mathfrak g}}^{\ast}}$, ${I({{\mathcal O}})}$ is a Lie ideal as well. Thus ${I({{\mathcal O}})}$ is a Poisson ideal (i.e., an associative ideal which is also a Lie ideal) and hence ${P({{\mathcal O}})}$ is a Poisson algebra of polynomial functions on the symplectic leaf ${{\mathcal O}}$. Note that since ${{\mathcal O}}$ is symplectic, the Lie center $Z(P({{\mathcal O}})) = {{\mathbb R}}.$
We now show that the decomposition in Proposition \[s\] projects to a similar decomposition of ${P({{\mathcal O}})}$.
${P({{\mathcal O}})}= {{\mathbb R}}\oplus {P({{\mathcal O}})}'.$ \[p\]
[[*Proof.* ]{}]{}It is clear that ${C({{\mathfrak g}})}$ projects onto constants on ${{\mathcal O}}$ and $\rho_{{{\mathcal O}}}({S({{\mathfrak g}})}') = {P({{\mathcal O}})}'$, so that ${{\mathbb R}}+
{P({{\mathcal O}})}' = {P({{\mathcal O}})}$ by Proposition \[s\]. It remains to show that ${P({{\mathcal O}})}' \cap {{\mathbb R}}= \{0\}$.
Now the restriction of the adjoint action of ${{\mathfrak g}}$ on ${S({{\mathfrak g}})}$ to the invariant subspace ${I({{\mathcal O}})}$ is also completely reducible, so we can again use [@Di §1.2.10] to split ${I({{\mathcal O}})}= {I({{\mathcal O}})}_{1} \oplus {I({{\mathcal O}})}_{2}$, where ${I({{\mathcal O}})}_{1} =
{I({{\mathcal O}})}\cap {C({{\mathfrak g}})}$ and ${I({{\mathcal O}})}_{2} = \{{{\mathfrak g}},{I({{\mathcal O}})}\} \subset {S({{\mathfrak g}})}'$.
If ${P({{\mathcal O}})}' \cap {{\mathbb R}}\neq \{0\}$, then there is an $f \in {I({{\mathcal O}})}$ such that $1+f \in {S({{\mathfrak g}})}'$. Decomposing $f =
f_{1} + f_{2}$ with $f_{1} \in {I({{\mathcal O}})}_{1}$ and $f_{2} \in
{I({{\mathcal O}})}_{2}$, we get $(1+f_{1}) + f_{2} \in {S({{\mathfrak g}})}'$, so $1+f_{1}
\in {C({{\mathfrak g}})}\cap
{S({{\mathfrak g}})}' = \{0\}$. Hence $f_{1} = -1$, and this contradicts the fact that $f_{1} \in {I({{\mathcal O}})}_1 \subset {I({{\mathcal O}})}.$
Theorem \[p\] was already known in the case when ${{\mathcal O}}$ is compact [@GGG]. In the $C^\infty$ context, one knows that if $M$ is a compact symplectic manifold, then its [Poisson algebra]{}$$C^\infty(M) = {{\mathbb R}}\oplus C^\infty(M)'$$ [@Av], while if $M$ is noncompact $$C^\infty(M) = C^\infty(M)'$$ [@Li]. Since in the smooth case $f\in C^\infty(M)'$ if and only if $f\eta$ is an exact form, where $\eta$ is the Liouville volume form, Theorem \[p\] thus suggests that the polynomial Poisson (resp. de Rham) cohomology of a noncompact coadjoint orbit ${{\mathcal O}}$ may differ from its smooth Poisson (resp. de Rham) cohomology. For example, take ${{\mathcal O}}\subset sl(2,{{\mathbb R}})^*$ to be the one-sheeted hyperboloid $x^2+y^2-z^2=1$. The Poisson tensor $$\Lambda=x{\partial}_y{\wedge}{\partial}_z+y{\partial}_z{\wedge}{\partial}_x - z{\partial}_x{\wedge}{\partial}_y$$ on $sl(2,{{\mathbb R}})^*$ is polynomial and the induced symplectic form $$\omega=x\,{\textrm{d}}y{\wedge}{\textrm{d}}z+y\,{\textrm{d}}z{\wedge}{\textrm{d}}x+z\,{\textrm{d}}x{\wedge}{\textrm{d}}y$$ on ${{\mathcal O}}$ is also polynomial. As $\omega$ is a volume form on the non-compact manifold ${{\mathcal O}}$, it is exact in the smooth category. It is, however, not exact in the polynomial category. Indeed, if $\omega={\textrm{d}}\alpha$ for some polynomial 1-form $\alpha$ on ${{\mathcal O}}$, then, according to the well-known isomorphism between Poisson and de Rham cohomology on a symplectic manifold, we would have $[\Lambda, i_\alpha\Lambda]=\Lambda$, where $[\ ,\ ]$ is the Schouten bracket and $[\Lambda,\cdot]$ is the Poisson cohomology differential [@Va]. Writing $\alpha=f\,{\textrm{d}}x+g\,{\textrm{d}}y+h\,{\textrm{d}}z$, where $f,g,h$ are polynomials, this gives $$\Lambda=H_x{\wedge}H_f+H_y{\wedge}H_g+H_z{\wedge}H_h,$$ where $H_a$ is the Hamiltonian vector field of $a$. Contracting $\Lambda$ with $\omega$ then yields $1=\{x,f\}+\{y,g\}+\{z,h\}$—a contradiction with Theorem \[p\].
We remark that Theorem \[p\] need not hold if ${{\mathfrak g}}$ is not semisimple. For instance, ${{\mathbb R}}^{2n}$ with its standard symplectic structure is a coadjoint orbit of the Heisenberg group H($2n$), but in this case $P({{\mathbb R}}^{2n}) =
P({{\mathbb R}}^{2n})'$.
[A Characterization of ${P({{\mathcal O}})}$]{}
We call a Lie algebra $L$ *essentially simple* if every Lie ideal of $L$ is either contained in the center $Z(L)$ of $L$ or contains the derived ideal $L'=[L,L]$. We say that a [Poisson algebra]{} $P$ is *simple* if the only Poisson ideals of $P$ are $P$ and $\{0\}$.
Let $P$ be a unital [Poisson algebra]{} which has no nilpotent elements with respect to the associative structure. If $P$ is simple, then it is essentially simple. \[es\]
[[*Proof.* ]{}]{}In view of [@Gr Thm.1.10], if $L$ is a Lie ideal of a unital [Poisson algebra]{} $P$ then $$\{P , {ad^{-1}(L)}\} \subset {r(J(L))},
\label{r}$$ where $${ad^{-1}(L)}= \{f \,|\, \{f, P \} \subset L \},$$ ${J(L)}$ is the largest associative ideal of $P$ contained in ${ad^{-1}(L)}$, and ${r(J(L))}$ is its radical, $${r(J(L))}= \{f \,|\, f^n \in {J(L)}\textrm{
for some } n=1,2,\ldots \}.$$ We recall from [@Gr Thm. 1.6] that ${J(L)}$ is in fact a Poisson ideal of $P.$
Suppose that $P' \not \subset L$. Then ${ad^{-1}(L)}\neq P$, so ${J(L)}\neq P$, and thus ${J(L)}= \{0\}$ as $P$ is simple. Then ${r(J(L))}=
\{0\}$ since by assumption $P$ has no associative nilpotents. Then (\[r\]) gives $$\{P, L\} \subset \{P, {ad^{-1}(L)}\} = \{0\},$$ i.e., $L \subset Z (P)$.
In particular, the hypotheses of Proposition \[es\] are satisfied by the polynomial algebra ${P({{\mathcal O}})}$. We now use this Proposition to prove our main result.
The Lie algebra ${P({{\mathcal O}})}$ is essentially simple iff the orbit ${{\mathcal O}}$ is semisimple.
[[*Proof.* ]{}]{}$(\Leftarrow)$ Assume ${{\mathcal O}}$ is semisimple and let ${{{\mathcal O}}_{{{\mathbb C}}}}$ be the complexification of ${{\mathcal O}}$, i.e., the orbit in ${{{\mathfrak g}}_{{{\mathbb C}}}}^{\:\:\ast}$ with respect to the complexified Lie group ${G_{{{\mathbb C}}}}$ which contains ${{\mathcal O}}$ in its real part. It is well known that ${{{\mathcal O}}_{{{\mathbb C}}}}$ is semisimple and that ${{{\mathcal O}}_{{{\mathbb C}}}}$ is an algebraic set in ${{{\mathfrak g}}_{{{\mathbb C}}}}^{\:\:\ast}$ [@Ko §3.8]. If ${P({{\mathcal O}})}$ were not essentially simple, then by Proposition \[es\] we would have a proper Poisson ideal $I$ in ${P({{\mathcal O}})}$, and so, after complexification, a proper Poisson ideal ${I_{{{\mathbb C}}}}$ in ${{P({{\mathcal O}})}_{{{\mathbb C}}}}:= P_{{{\mathbb C}}}({{{\mathcal O}}_{{{\mathbb C}}}})$.
Let ${V({I_{{{\mathbb C}}}})}$ be the set of zeros of ${I_{{{\mathbb C}}}}$ in ${{{\mathcal O}}_{{{\mathbb C}}}}$. Since ${{{\mathcal O}}_{{{\mathbb C}}}}$ is algebraic, ${V({I_{{{\mathbb C}}}})}\neq \emptyset$, and since ${I_{{{\mathbb C}}}}$ is a Lie ideal, ${V({I_{{{\mathbb C}}}})}$ is ${G_{{{\mathbb C}}}}$-invariant and hence consists of orbits. This forces ${V({I_{{{\mathbb C}}}})}= {{{\mathcal O}}_{{{\mathbb C}}}}$ and so ${I_{{{\mathbb C}}}}= \{0\}$. Hence we have a contradiction, since ${I_{{{\mathbb C}}}}$ is proper.
$(\Rightarrow)$ Assume that ${{\mathcal O}}$ is not semisimple. Complexifying as before, we get the complexified orbit ${{{\mathcal O}}_{{{\mathbb C}}}}$ which is not semisimple. Now there exists a semisimple orbit $\mathcal S$ in the Zariski closure of ${{{\mathcal O}}_{{{\mathbb C}}}}$ [@Ko §3.8]. Consider the Poisson ideal $K$ of elements of ${P({{\mathcal O}})}$ which vanish on $\mathcal S$. We claim that this ideal is proper. Indeed, $K = \{0\}$ implies that $I(\mathcal S) =
I({{{\mathcal O}}_{{{\mathbb C}}}})$ ,whence $\mathcal S = {\rm cl}({{{\mathcal O}}_{{{\mathbb C}}}})$ as $\mathcal S$ is an algebraic set. But this is impossible as $\mathcal S$ and ${{{\mathcal O}}_{{{\mathbb C}}}}$ are distinct orbits. As well, $K = {{P({{\mathcal O}})}_{{{\mathbb C}}}}$ is impossible as $\mathcal S \neq \emptyset.$
Now we will show that the existence of the proper Poisson ideal $K$ in the complex Poisson algebra ${{P({{\mathcal O}})}_{{{\mathbb C}}}}$ implies the existence of a proper Poisson ideal $I$ in ${P({{\mathcal O}})}$. First, put $${K_{{{\mathbb R}}}}= \{f \in {P({{\mathcal O}})}\,|\, f+ig \in K \textrm{ for some
}g \in {P({{\mathcal O}})}\}.$$ Since for $h \in {P({{\mathcal O}})}$, $f+ig \in K$ implies $(hf) +
i(hg) \in K$ and $\{h,f \} + i\{h,g \} \in K$, ${K_{{{\mathbb R}}}}$ is a Poisson ideal of ${P({{\mathcal O}})}$. Clearly $K \subset {K_{{{\mathbb R}}}}+ i{K_{{{\mathbb R}}}}$ so that if ${K_{{{\mathbb R}}}}= \{0\}$ then $K = \{0\}$. We can thus take $I = {K_{{{\mathbb R}}}}$ as long as ${K_{{{\mathbb R}}}}\neq {P({{\mathcal O}})}$.
If ${K_{{{\mathbb R}}}}= {P({{\mathcal O}})}$, then there is $g
\in {P({{\mathcal O}})}$ such that $1 + ig \in K$. Let $$K_0 = \{f \in {P({{\mathcal O}})}\,|\, if \in K \}.$$ Similarly as for ${K_{{{\mathbb R}}}}$, we can prove that $K_0$ is a Poisson ideal. Now $K_0
\neq {P({{\mathcal O}})}$, for otherwise $K = {{P({{\mathcal O}})}_{{{\mathbb C}}}}$. We can then take $I = K_0$ provided $K_0
\neq \{0\}$. But in fact $K_0 \neq \{0\}$: Since $$\{{P({{\mathcal O}})}, 1+ig\} = i\{{P({{\mathcal O}})}, g\}
\subset K,$$ $\{{P({{\mathcal O}})}, g\} \subset K_0$, and so $K_0 = \{0\}$ implies that $g \in Z({P({{\mathcal O}})}) = {{\mathbb R}}$. So $1+ig \in K$ is a nonzero constant, whence again $K = {{P({{\mathcal O}})}_{{{\mathbb C}}}}$.
In any eventuality, we now have a proper Poisson ideal $I$ of ${P({{\mathcal O}})}$. Of course, $I \not \subset Z({P({{\mathcal O}})}) = {{\mathbb R}}$. However, it may happen that $I \supset {P({{\mathcal O}})}'$, in which case Theorem \[p\] forces $I = {P({{\mathcal O}})}'$. In this circumstance we pass to the associative ideal $I^2$. Since $$\{{P({{\mathcal O}})}, I^2\} \subset \{{P({{\mathcal O}})}, I\} I \subset I^2,$$ $I^2$ is also a Lie ideal. If $I^2 \neq I$, then $I^2$ is a proper Lie ideal which neither is contained in the center nor contains the derived ideal.
To see that $I^2 \neq I$ for $I$ proper, we can use the following.
If $P$ is a commutative unital ring with no zero divisors and $I$ is a proper ideal which is finitely generated, then $I^2
\neq I$.
[[*Proof.* ]{}]{}Assume that $x_1,\ldots,x_n$ are generators of $I$ and $I^2 = I$. Then $x_i = \sum_{j=1}^n a_{ij}x_j$ for some $a_{ij}
\in I$, so that $\sum_{j=1}^n b_{ij}x_j = 0$ where $b_{ij} =
\delta_{ij} -a_{ij}$. Setting $B = \left(b_{ij}\right)$, Cramer’s Rule gives $x_i \det B = 0$ whence $\det B = 0$. But $\det B \in \{1\} + I$ so $\det B \neq 0$. $\blacktriangledown$
Thus ${P({{\mathcal O}})}$ is not essentially simple.
The last part of this proof provides a converse to Proposition \[es\] when $P = {P({{\mathcal O}})}$. In particular, we conclude that ${P({{\mathcal O}})}$ is simple if and only if ${{\mathcal O}}$ is semisimple.
One can also see explicitly that ${P({{\mathcal O}})}$ is not essentially simple when ${{\mathcal O}}$ is nilpotent as follows. Since a nilpotent orbit is conical [@Br], $I({{\mathcal O}})$ is a homogeneous ideal. As a consequence, the notion of homogeneous polynomial makes sense in $P({{\mathcal O}})$. Let $P_k({{\mathcal O}})$ denote the subspace consisting of all homogeneous polynomials of degree $k$. By virtue of the commutation relations of ${{\mathfrak g}}$, $$\{P_k({{\mathcal O}}),P_l({{\mathcal O}})\} \subset
P_{k+l-1}({{\mathcal O}}),$$ whence each $P_{(k)}({{\mathcal O}}) = \oplus_{\ell \geq k}P_\ell({{\mathcal O}})$ for $k \geq 1$ is a proper Poisson ideal of ${P({{\mathcal O}})}$.
[99]{}
A. Avez, *Remarques sur les automorphismes infinitésimaux des variétés symplectiques compactes,* Rend. Sem. Mat. Univers. Politecn. Torino [**33**]{} (1974–1975), 5–12.
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J. Dixmier, [*Enveloping Algebras,*]{} North-Holland, Amsterdam (1977).
M. J. Gotay, J. Grabowski, and H. B. Grundling, *An obstruction to quantizing compact symplectic manifolds,* Proc. Amer. Math. Soc. [**28**]{} (2000), 237–243.
J. Grabowski, *The Lie structure of $\,C^*$ and Poisson algebras*, Studia Math. [**81**]{} (1985), 259–270.
B. Kostant, *Lie group representations on polynomial rings,* Amer. J. Math. [**85**]{} (1963), 327–404.
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[^1]: Supported in part by NSF grants 96-23083 and 00-72434. E-mail: gotay@math.hawaii.edu
[^2]: Supported by KBN, grant No. 2 P03A 031 17. E-mail: jagrab@mimuw.edu.pl
[^3]: E-mail: bryon@math.hawaii.edu
|
---
abstract: 'The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced *broadcast domination* in graphs, a covering problem using balls of various radii where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by ${\ensuremath{\gamma_b}}(G)$. The dual (in the sense of linear programming) of broadcast domination is *multipacking*: a multipacking is a set $P \subseteq V(G)$ such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking in a graph $G$ is denoted by ${\ensuremath{\mathrm{mp}}}(G)$. Naturally, ${\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G)$. Hartnell and Mynhardt proved that (whenever ${\ensuremath{\mathrm{mp}}}(G)\geq 2$). In this paper, we show that . Moreover, we conjecture that this can be improved to ${\ensuremath{\gamma_b}}(G) \leq 2{\ensuremath{\mathrm{mp}}}(G)$ (which would be sharp).'
author:
- 'L. Beaudou[^1]'
- 'R. C. Brewster[^2]'
- 'F. Foucaud'
title: |
Broadcast domination and multipacking:\
bounds and the integrality gap
---
Introduction {#introduction .unnumbered}
============
The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence respectively. Typically we are interested in minimum (cost) coverings and maximum packings. Natural questions to ask are for what graph do these dual problems have equal (integer) values, and in the case they are not equal, can we bound the difference between the two values? The second question is the focus of this paper.
The particular covering problem we study is broadcast domination. Let $G=(V,E)$ be a graph. Define the *ball of radius $r$ around $v$* by $N_r(v) = \{ u : d(u,v) \leq r \}$. A *dominating broadcast* of $G$ is a collection of balls $N_{r_1}(v_1), N_{r_2}(v_2), \dots, N_{r_t}(v_t)$ (each $r_i > 0$) such that $\bigcup_{i=1}^t N_{r_i}(v_i) = V$. Alternatively, a dominating broadcast is a function $f: V \to \mathbb{N}$ such that for any vertex $u \in V$, there is a vertex $v \in V$ with $f(v)$ positive and $\mathrm{dist}(u,v) \leq f(v)$. (The ball around $v$ with radius $f(v)$ belongs to the covering.) The *cost* of a dominating broadcast $f$ is $\sum_{v \in V} f(v)$ and the minimum cost of a dominating broadcast in $G$, its *broadcast number*, is denoted by ${\ensuremath{\gamma_b}}(G)$.[^3]
When broadcast domination is formulated as an integer linear program, its dual problem is *multipacking* [@Brewster2013; @Teshima2012]. A multipacking in a graph $G$ is a subset $P$ of its vertices such that for any positive integer $r$ and any vertex $v$ in $V$, the ball of radius $r$ centered at $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking of $G$, its *multipacking number*, is denoted by ${\ensuremath{\mathrm{mp}}}(G)$. Broadcast domination was introduced by Erwin [@Erwin2001; @Erwin2004] in his doctoral thesis in 2001. Multipacking was then defined in Teshima’s Master’s Thesis [@Teshima2012] in 2012, see also [@Brewster2013] (and [@Brewster2017; @hartnell_2014; @Yang2015] for subsequent studies). As we have already mentioned, this work fits into the general study of coverings and packings, which has a rich history in Graph Theory: Cornuéjols wrote a monograph on the topic [@Cornuejols2001].
In early work, Meir and Moon [@MeirMoon1975] studied various coverings and packings in trees, providing several inequalities relating the size of a minimum covering and a maximum packing. Giving such inequalities connecting the parameters ${\ensuremath{\gamma_b}}$ and ${\ensuremath{\mathrm{mp}}}$ is the focus of our work. Since broadcast domination and multipacking are dual problems, we know that for any graph $G$, $${\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G).$$
This bound is tight, in particular for strongly chordal graphs, see [@Farber84; @Lubiw87; @Teshima2012]. (In a recent companion work we prove equality for grids [@Beaudou2018].) A natural question comes to mind. How far apart can these two parameters be? Hartnell and Mynhardt [@hartnell_2014] gave a family of graphs $(G_k)_{k \in
\mathbb{N}}$ for which the difference between both parameters is $k$. In other words, the difference can be arbitrarily large. Nonetheless, they proved that for any graph $G$ with ${\ensuremath{\mathrm{mp}}}(G)\geq 2$, $${\ensuremath{\gamma_b}}(G) \leq 3 {\ensuremath{\mathrm{mp}}}(G) - 2$$ and asked [@hartnell_2014 Section 5] whether the factor $3$ can be improved. Answering their question in the affirmative, our main result is the following.
\[thm:bounding\] Let $G$ be a graph. Then, $${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G) + 3.$$
Moreover, we conjecture that the additive constant in the bound of Theorem \[thm:bounding\] can be removed.
\[conj:fac2\] For any graph $G$, ${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G)$.
In Section \[sec:bound\], we prove Theorem \[thm:bounding\]. In Section \[sec:discussion\], we show that Conjecture \[conj:fac2\] holds for all graphs with multipacking number at most $4$. We conclude the paper with some discussions in Section \[sec:remarks\].
Proof of Theorem \[thm:bounding\] {#sec:bound}
=================================
We want to bound the broadcast number of a graph by a function of its multipacking number. We first state a key counting result which is used throughout the remainder of this paper.
For any two relative integers $a$ and $b$ such that $a \leq b$, $\llbracket a, b\rrbracket$ denotes the set $\mathbb{Z} \cap [a,b]$.
\[lem:path\] Let $G$ be a graph, $k$ be a positive integer and $(u_0,\ldots,u_{3k})$ be an isometric path in $G$. Let be the set of every third vertex on this path. Then, for any positive integer $r$ and any ball $B$ of radius $r$ in $G$, $$|B \cap P| \leq \left\lceil \frac{2r+1}{3} \right\rceil.$$
Let $B$ be a ball of radius $r$ in $G$, then any two vertices in $B$ are at distance at most $2r$. Since the path $(u_0,\ldots,u_{3k})$ is isometric the intersection of the path and $B$ is included in a subpath of length $2r$. This subpath contains at most $2r+1$ vertices and only one third of those vertices can be in $P$.
Any positive integer $r$ is greater than or equal to $\left\lceil
\frac{2r+1}{3} \right\rceil$. Thus, Lemma \[lem:path\] ensures that $P$ is a valid multipacking of size $k+1$. We have the following (see also [@dun_al_2006]):
For any graph $G$, $${\ensuremath{\mathrm{mp}}}(G) \geq \left\lceil\frac{{\text{diam}}(G)+1}{3}\right\rceil.$$
Building on this idea, we have the following result.
\[thm:main\] Given a graph $G$ and two positive integers $k$ and $k'$ such that , if there are four vertices $x,y,u$ and $v$ in $G$ such that $$d_G(x,u) = d_G(x,v) = 3k \text{, } d_G(u,v) = 6k \text{ and }d_G(x,y) = 3k
+ 3k',$$ then $${\ensuremath{\mathrm{mp}}}(G) \geq 2k + k'.$$
Let $(u_{-3k},\ldots,u_0,\ldots,u_{3k})$ be the vertices of an isometric path from $u$ to $v$ going through $x$. Note that $u =
u_{-3k}$, $x = u_0$ and $v = u_{3k}$. We shall select every third vertex of this isometric path and let $P_1$ be the set $\{u_{3i}
| i \in \llbracket -k, k \rrbracket\}$.
We thus have already selected $2k+1$ vertices. In order to complete our goal, we need $k'-1$ additional vertices. Let $(x_0,\ldots,x_{3k + 3k'})$ be the vertices of an isometric path from $x$ to $y$. Note that $x =
x_0$ and $y = x_{3k+3k'}$. We shall select every third vertex on this isometric path starting at $x_{3k+6}$. Formally, we let $P_2$ be the set $\{ x_{3k+3(i+2)} | i \in \llbracket 0,k'-2
\rrbracket\}$. Finally, we let $P$ be the union of $P_1$ and $P_2$. An illustration of this is displayed in Figure \[fig:firstscheme\].
\(u) at (-3,0) ; (v) at (3,0) ; (x) at (0,0) ; (y) at (0,-5) ; (x3k6) at (0,-3.4) ; (x3k) at (0,-2.6) ; (x3k3) at (0,-3) ; (um3) at (-.4,0) ; (u3) at (.4,0) ; (x3) at (0,-.4) ; (u) – (v); (x) – (x3) – (x3k) – (x3k3) – (y); at ($(x) + (0,.2)$) [$x = x_0 = u_0$]{}; at ($(u) + (0,.2)$) [$u = u_{-3k}$]{}; at ($(v) + (0,.2)$) [$v = u_{3k}$]{}; at ($(y) + (0,-.2)$) [$y = x_{3k+3k'}$]{}; at ($(x3) + (0,0)$) [$x_3$]{}; at (x3k) [$x_{3k}$]{}; at ($(x3k6)+(.1,0)$) [$x_{3k+6}$]{}; at (x3k3) [$x_{3k+3}$]{}; at (-.1,-4.2) [$P_2$]{}; at (-1.5,-0.2) [$P_1$]{}; (-3.1,.1) rectangle (3.1,-.1); (-.1,-3.3) rectangle (.1,-5.1);
Since every vertex of $P_2$ is at distance at least $3k + 6$ from $x$, while every vertex of $P_1$ is at distance at most $3k$ from $x$, we infer that $P_1$ and $P_2$ are disjoint. Thus $|P| =
2k+k'$. We shall now prove that $P$ is a valid multipacking.
Let $r$ be an integer between 1 and $|P| - 1$, and let $B$ be a ball of radius $r$ in $G$ (we do not care about the center of the ball). If this ball $B$ intersects only $P_1$ or only $P_2$, then we know by Lemma \[lem:path\] that it cannot contain more than $r$ vertices of $P$. We may then consider that the ball $B$ intersects both $P_1$ and $P_2$. Let $l$ denote the greatest integer $i$ such that $x_{3k+3(i+2)}$ is in $B$ and in $P_2$. Let us name this vertex $z$. From this, we may say that $$\label{eq:P2}
|B \cap P_2| \leq l + 1$$
Before ending this preamble, we state an easy inequality. For every integer $n$, $$\label{eq:mod3}
\left\lceil\frac{n}{3}\right\rceil \leq \frac{n}{3} + \frac{2}{3}$$
We now split the remainder of the proof into two cases.
#### Case 1: $3(l+2) \leq r$.
In this case, we just use Lemma \[lem:path\] for $P_1$. We have $$|B \cap P_1 | \leq \left\lceil \frac{2r + 1}{3} \right\rceil,$$ and by Inequality , this quantity is bounded above by $\frac{2r+1}{3} + \frac{2}{3}$. We obtain with Inequality ,
&&|B P| & l+1 + + &&\
&& & l+2 + &&\
&& & + &&\
&& & r.&&
Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
#### Case 2: $3(l+2) > r$.
Here we need some more insight. Recall that $l + 2 $ cannot exceed $k'$ and that $k' \leq k$. Thus $$\begin{aligned}
r & < 3(l+2) \\
& < 2k' + l +2\\
& < 2k + l + 2,
\end{aligned}$$ and since $r$ is an integer, we get $$\label{eq:nice}
r \leq 2k + l + 1.$$
We also note that any vertex $u_i$ for $|i| \leq 3k + 3(l+2) -
(2r+1)$ is at distance at least $2r+1$ from $z$. By the triangle inequality $d(z,u_i) \geq d(z,x)-d(u_i,x)$, where $d(z,x)=3k + 3(l+2)$, and $d(u_i,x) = |i|$. Since the ball $B$ has radius $r$, no such vertex can be in $B$. Since we assumed that $B$ intersects $P_1$, not all the vertices of the $uv$-path are excluded from $B$. This means that $$\label{eq:nonzero}
3k > 3k + 3(l+2) - (2r+1).$$ We partition the vertices of $P_1$ into three sets: $U_L, U_M, U_R$. The vertex $u_i$ belongs to: $U_L$ if $i < -3k - 3(l+2) + 2(r+1)$; $U_M$ if $|i| \leq 3k + 3(l+2) - (2r+1)$; and $U_R$ if $i > 3k + 3(l+2) - (2r+1)$. See Figure \[fig:case21\]. The distance from $u = u_{-3k}$ to the first vertex (smallest positive index) in $U_R$ is then $6k + 3(l+2) - (2r+1) + 1$. We compare this distance with $2r+1$.
##### Case 2.1: $6k + 3(l+2) - (2r+1) + 1 \geq 2r+1$.
We match $U_L$ with $U_R$ so that each pair is at distance at least $2r+1$ (match $u_{-3k}$ with the first vertex in $U_R$ and so on, as pictured in Figure \[fig:case21\]). Therefore the ball $B$ contains at most one vertex of each matched pair. In other words, $B$ contains at most $\lceil |U_R|/3 \rceil$ vertices from $U_L \cup U_R$, and so $$|B \cap P_1| \leq \left\lceil \frac{3k - (3k + 3(l+2) -
2r) + 1}{3} \right\rceil.$$ By using Inequality again, $$\begin{aligned}
|B \cap P| & \leq l+1 + \left\lceil \frac{2r+1}{3} \right\rceil - (l+2)\\
& \leq r.
\end{aligned}$$ Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
##### Case 2.2: $6k + 3(l+2) - (2r+1) + 1 < 2r+1$.
We partition each of $U_L$ and $U_R$ as shown in Figure \[fig:case22\]. The vertices that are distance at least $2r+1$ from a vertex in $U_L \cup U_R$ are the sets $U'_L$ and $U'_R$, and those that are close to all other vertices are $U''_L$ and $U''_R$. We can match pairs of vertices $U'_L \cup U'_R$. This allows us to say that the extremities of $P_1$ will contribute at most $\left\lceil \frac{6k - (2r+1) + 1}{3}
\right\rceil$ which equals $2k + \lceil\frac{-2r}{3}\rceil$. Using again Inequality , this is bounded above by $2k -
\frac{2r}{3} + \frac{2}{3}$.
For any integer $i$ between $6k + 3(l+2) - (2r+1) + 1$ and $2r$, vertices $u_{-i}$ and $u_{i}$ belong to $U''_L$ and $U''_R$ respectively. Such vertices may be in $B$. Since $P_1$ contains every third vertex on these two subpaths, this amounts to at most $$2 \left\lceil\frac{2r - 6k - 3(l+2) + (2r+1)}{3}\right\rceil$$ such vertices. This quantity is equal to $$2\left\lceil \frac{4r+1}{3} \right\rceil -4k - 2(l+2),$$ which in turn, using Inequality is bounded above by $$\frac{8r}{3} + 2 -4k -2(l+2).$$
By putting everything together, we derive that
&& |B P| & (l+1) + (2k - + ) + ( +2 -4k - 2(l+2)) &&\
&&& 2r - 2k - l - .&&
But since $|B \cap P|$ is an integer, we may rewrite this last inequality as
&& |B P| & r + (r - 2k - l - 1) &&\
&&& r. &&
Thus, $|B \cap P|$ cannot exceed $r$ and the ball $B$ contains at most $r$ vertices of $P$, as required. This concludes the proof of Theorem \[thm:main\].
Theorem \[thm:main\] allows us to give a lower bound on the size of a maximum multipacking in a graph in terms of its diameter and radius.
\[coro:diam-rad\] For any graph $G$ of diameter $d$ and radius r, $${\ensuremath{\mathrm{mp}}}(G) \geq \frac{d}{6} + \frac{r}{3} - \frac{3}{2}.$$
We just pick the integer $k$ such that $d$ can be expressed as $6k + \alpha$ where $\alpha$ is in $\llbracket 0,5 \rrbracket$ and the integer $k'$ such that $r$ can be expressed as $3k +
3k'+\beta$ where $\beta$ is in $\llbracket 0,2\rrbracket$.
We must have two vertices at distance $6k$ in $G$. On a shortest path of length $6k$, the middle vertex has some vertex at distance $3k+3k'$. We can then apply Theorem \[thm:main\]. $$\begin{aligned}
{\ensuremath{\mathrm{mp}}}(G) &\geq 2k + k'\\
& \geq \frac{1}{3}(d - \alpha) + \frac{1}{3} \left(r - \beta - \frac{1}{2}(d - \alpha)\right)\\
& \geq \frac{d}{6} + \frac{r}{3} - \frac{9}{6}.\qedhere
\end{aligned}$$
We can now finalize the proof of our main theorem.
Since the diameter of a graph is always greater than or equal to its radius, we conclude from Corollary \[coro:diam-rad\] that $$\frac{{\text{rad}}(G)-3}{2} \leq {\ensuremath{\mathrm{mp}}}(G) \leq {\ensuremath{\gamma_b}}(G) \leq {\text{rad}}(G).$$ Hence, for any graph $G$, $${\ensuremath{\gamma_b}}(G) \leq 2 {\ensuremath{\mathrm{mp}}}(G) + 3,$$ proving Theorem \[thm:bounding\].
Note that in our proof, we chose the length of the long path to be a multiple of $6$ for the reading to be smooth. We think that the same ideas implemented with more care would work for multiples of $3$. This might slightly improve the additive constant in our bound, but we believe that it would not be enough to prove Conjecture \[conj:fac2\] (while adding too much complexity to the proof).
Proving Conjecture \[conj:fac2\] when ${\ensuremath{\mathrm{mp}}}(G)\leq 4$ {#sec:discussion}
===========================================================================
The following collection of results shows that Conjecture \[conj:fac2\] holds for graphs whose multipacking number is at most $4$.
\[lemma-distances\] Let $G$ be a graph and $P$ a subset of vertices of $G$. If, for every subset $U$ of at least two vertices of $P$, there exist two vertices of $U$ that are at distance at least $2|U|-1$, then $P$ is a multipacking of $G$.
We prove the contrapositive. Let $G$ be a graph and $P$ a subset of its vertices which is not a multipacking. Then there is a ball $B$ of radius $r$ which contains $r+1$ vertices of $P$.
Let $U$ be the set $B \cap P$, then $U$ has size at least $r+1$. Moreover, any two vertices in $U$ are at distance at most $2r$ which is stricly smaller than $2|U|-1$.
\[prop:mp=3\] Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)=3$, then ${\ensuremath{\gamma_b}}(G)\leq 6$.
We prove the contrapositive again. Let $G$ be a graph with broadcast number at least 7. Then, the eccentricity of any vertex is at least 7 (otherwise we could cover the whole graph by broadcasting with power 6 from a single vertex).
Let $x$ be any vertex of $G$. There must be a vertex $y$ at distance 7 from $x$. Let $u$ be any vertex at distance 3 from $x$ and on a shortest path from $x$ to $y$. Then $u$ is at distance 4 from $y$. But $u$ has also eccentricity at least 7. So there is a vertex $v$ at distance 7 from $u$. By the triangle inequality, $v$ is at distance at least 4 from $x$ and at least 3 from $y$. Therefore the set $\{u,v,x,y\}$ satisfies the condition of Lemma \[lemma-distances\] and the multipacking number of $G$ is at least 4 (and so it is not equal to 3).
The following proposition improves Theorem \[thm:bounding\] for graphs $G$ with ${\ensuremath{\mathrm{mp}}}(G) \leq 6$ and shows that Conjecture \[conj:fac2\] holds when ${\ensuremath{\mathrm{mp}}}(G) = 4$.
\[prop:mp=4\] Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)\geq 4$, then ${\ensuremath{\gamma_b}}(G)\leq 3{\ensuremath{\mathrm{mp}}}(G)-4$.
For a contradiction, let $G$ be a counterexample, that is a graph with multipacking number $p$ at least 4 while ${\ensuremath{\gamma_b}}(G)\geq
3p-3$. Then, the eccentricity of any vertex of $G$ is at least $3p-3$ (otherwise we could broadcast at distance $3p-4$ from a single vertex). Let $x$ be a vertex of $G$ and let $V_i$ denote the set of vertices at distance exactly $i$ of $x$. By our previous remark, $V_{3p-3}$ is non-empty. Let $y$ be a vertex in $V_{3p-3}$ and consider a shortest path $P_{xy}$ from $x$ to $y$ in $G$. Let $v_0=x$, and for $1\leq i\leq p-1$, let $v_i$ be the vertex on $P_{xy}$ belonging to $V_{3i}$ (thus $v_{p-1}=y$).
Now, since ${\ensuremath{\gamma_b}}(G)\geq 3p-3$, there must be a vertex $u$ at distance at least $3p-3$ of $v_{p-2}$ (otherwise we could broadcast from that single vertex). Note that the triangle inequality ensures that the distance between $u$ and $v_i$ is at least $3+3i$ for $i$ between $0$ and $p-2$. The distance from $u$ to $v_{p-1}$ is at least $3p-6$ which is at least 6 since $p$ is at least 4. Consider the set $P=\{u,v_0,\ldots, v_{p-1}\}$. We claim that $P$ is a multipacking of $G$ of size $p+1$, which is a contradiction.
Let $B$ be a ball of radius $r$. Since $P_{xy}$ is an isometric path, Lemma \[lem:path\] ensures us that $B$ contains at most $$\left\lceil \frac{2r+1}{3} \right\rceil$$ vertices from $P \cap P_{xy}$ which is smaller than $r$. When $B$ does not include $u$, the ball is satisfied. For balls that contain vertex $u$, the maximum size of $P \cap B$ is $$\left\lceil \frac{2r+1}{3} \right\rceil + 1.$$ Whenever $r$ is 4 or more, this quantity does not exceed $r$. So every ball with radius $4$ or more is satisfied. We still need to check balls of radius 1,2, and 3 which contain $u$.
- Balls of radius 1 are easy to check since every vertex of $P_{xy}$ is at distance at least 3 from $u$.
- For balls of radius 2, it is enough to check that there is only one vertex at distance 4 or less from $u$ in $P \cap P_{xy}$.
- For balls of radius 3, there is only one way to select $u$ and three vertices in $P \cap P_{xy}$ within distance 6 from $u$. We should take $v_0, v_1$ and $v_{p-1}$. But since $v_0$ and $v_{p-1}$ are at distance $3p-3$ from each other, they cannot appear simultaneously in a ball of radius 3 (since $p$ is at least 4, $3p-3$ is at least 9).
Therefore $P$ is a multipacking of size $p+1$, which is a contradiction.
Let $G$ be a graph. If ${\ensuremath{\mathrm{mp}}}(G)\leq 4$, then ${\ensuremath{\gamma_b}}(G)\leq 2{\ensuremath{\mathrm{mp}}}(G)$.
When ${\ensuremath{\mathrm{mp}}}(G)\leq 2$, this is shown in [@hartnell_2014]. The case ${\ensuremath{\mathrm{mp}}}(G)=3$ is implied by Proposition \[prop:mp=3\], and the case ${\ensuremath{\mathrm{mp}}}(G)=4$ follows from Proposition \[prop:mp=4\].
Concluding remarks {#sec:remarks}
==================
We conclude the paper with some remarks.
The optimality of Conjecture \[conj:fac2\]
------------------------------------------
We know a few examples of connected graphs $G$ which achieve the conjectured bound, that is, ${\ensuremath{\gamma_b}}(G)=2{\ensuremath{\mathrm{mp}}}(G)$. For example, one can easily check that $C_4$ and $C_5$ have multipacking number $1$ and broadcast number $2$. In Figure \[fig:twoFour\], we depict three examples having multipacking number $2$ and broadcast number $4$. By making disjoint unions of these graphs, we can build further extremal graphs with arbitrary multipacking number. However, if we only consider connected graphs, we do not even know an example with multipacking number $3$ and broadcast number $6$. Hartnell and Mynhardt [@hartnell_2014] constructed an infinite family of connected graphs $G$ with ${\ensuremath{\gamma_b}}(G)=\tfrac{4}{3}{\ensuremath{\mathrm{mp}}}(G)$, but we do not know any construction with a higher ratio. Are there arbitrarily large connected graphs that reach the bound of Conjecture \[conj:fac2\]?
An approximation algorithm
--------------------------
The computational complexity of broadcast domination has been extensively studied, see for example [@Dabney2009; @HeggernesLokshtanov2006] and references of [@Brewster2013; @Teshima2012; @Yang2015]. It is particularly interesting to note that, unlike most other natural covering problems, broadcast domination is solvable in polynomial (sextic) time [@HeggernesLokshtanov2006]. It is not known whether this is also the case for multipacking, but a cubic-time algorithm exists for strongly chordal graphs [@Brewster2017; @Yang2015], as well as a linear-time algorithm for trees [@Brewster2013; @Brewster2017; @Yang2015]. We note that our proof of Theorem \[thm:bounding\], being constructive, implies the existence of a $(2+o(1))$-factor approximation algorithm for the multipacking problem.
There is a polynomial-time algorithm that, given a graph $G$, constructs a multipacking of $G$ of size at least $\frac{{\ensuremath{\mathrm{mp}}}(G)-3}{2}$.
To construct the multipacking, one first needs to compute the radius $r$ and diameter $d$ of the graph $G$. Then, as described in the proof of Corollary \[coro:diam-rad\], we compute $\alpha$ and $k$, and find the four vertices $x$, $y$, $u$, $v$ and the two isometric paths $P_1$ and $P_2$ described in Theorem \[thm:main\]. Finally, we proceed as in the proof of Theorem \[thm:main\], that is, we essentially select every third vertex of these two paths to obtain the multipacking $P$. All distances and paths can be computed in polynomial time using classic methods. By Corollary \[coro:diam-rad\], $P$ has size at least $\frac{{\text{rad}}(G)-3}{2}$. Since ${\ensuremath{\mathrm{mp}}}(G)\leq {\text{rad}}(G)$, the approximation factor follows.
[00]{} L. Beaudou and R. C. Brewster, On the multipacking number of grid graphs, manuscript. arXiv e-prints:1803.09639.
R. C. Brewster, C. M. Mynhardt and L. Teshima, New bounds for the broadcast domination number of a graph, *Central European Journal of Mathematics*, [**11**]{} (2013), 1334–1343.
R. C. Brewster, G. MacGillivray and F. Yang, Broadcast domination and multipacking in strongly chordal graphs, submitted.
G. Cornuéjols. [*Combinatorial Optimization: packing and covering.*]{} CBMS-NSF regional conference series in applied mathematics, vol. 74. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001.
J. Dabney, B. C. Dean, S. T. Hedetniemi, A linear-time algorithm for broadcast domination in a tree, *Networks*, **53** (2009), 160–169.
J. E. Dunbar, D. J. Erwin, T. W. Haynes, S. M. Hedetniemi and S. T. Hedetniemi, Broadcasts in graphs, *Discrete Applied Mathematics*, **154** (2006), 59–75.
D. J. Erwin, *Cost domination in graphs*, PhD Thesis, Department of Mathematics, Western Michigan University, 2001.
D. J. Erwin, Dominating broadcasts in graphs, *Bulletin of the ICA*, [**42**]{} (2004), 89–105.
M. Farber, Domination, Independent Domination, and Duality in Strongly Chordal Graphs, *Discrete Applied Mathematics*, [**7**]{} (1984), 115–130.
B. L. Hartnell and C. M. Mynhardt, On the difference between broadcast and multipacking numbers of graphs, *Utilitas Mathematica*, **94** (2014), 19–29.
P. Heggernes and D. Lokshtanov, Optimal broadcast domination in polynomial time, *Discrete Mathematics*, [**306**]{} (2006), 3267–3280.
A. Lubiw, Doubly Lexical Orderings of Matrices, *SIAM Journal on Computing*, [**16**]{} (1987), 854–879.
A. Meir and John W. Moon, Relations between packing and covering numbers of a tree, *Pacific Journal of Mathematics*, [**61**]{} (1975), 225–233.
L. Teshima, [*Broadcasts and multipackings in graphs*]{}, Master’s Thesis, Department of Mathematics and Statistics, University of Victoria, 2012.
F. Yang, [*New results on broadcast domination and multipacking*]{}, Master’s Thesis, Department of Mathematics and Statistics, University of Victoria, 2015.
[^1]: LIMOS, Université Clermont Auvergne, Aubière (France). E-mails: laurent.beaudou@uca.fr, florent.foucaud@gmail.com
[^2]: Department of Mathematics and Statistics, Thompson Rivers University, Kamloops (Canada). E-mail: rbrewster@tru.ca
[^3]: One may consider the cost to be any function of the powers (for example the sum of the squares), see e.g. [@HeggernesLokshtanov2006]. We shall stick to the classical convention of linear cost.
|
[**System of roots of very special sandwich algebras**]{}[^1]\
Richard Cushman
This paper continues the study of very special sandwich algebras begun in Cushman [@cushman17a]. We show that a class of very special sandwich algebras has an analogue of a root system of a semisimple Lie algebra [@humphreys p.42]. This leads to an analogue of a Weyl group, which we study in another paper [@cushman17b].
System of roots {#sec1}
===============
[**Summary §1**]{} \[summary5\]
In §1.1 we give the axioms of a system of roots, which unlike the axioms for a root system does *not* use an inner product. In §1.2 we prove some consequences of these axioms, one of which verifies that every root system is a system of roots.
Axioms for a system of roots {#sec1sub1}
----------------------------
Let $V$ be a finite dimensional real vector space with $\Phi $ a finite subset of nonzero vectors, which satisfy the following axioms.
1\. $V = {{\mathop{\rm span}\nolimits}}_{{\mathbb{R}}}\Phi $, using addition $+$ of vectors in $V$.
2\. $\Phi = - \Phi $, where $-$ is the additive inverse of $+$.
3.
4\. Fix $\alpha \in \Phi $ and suppose that ${\beta }_1$, ${\beta }_2$, and ${\beta }_1+ {\beta }_2 \in \Phi \cup \{ 0 \}$. Then $$\langle {\beta }_1+{\beta }_2, \alpha \rangle = \langle {\beta }_1, \alpha \rangle + \langle {\beta }_2, \alpha \rangle .
\label{eq-sec1subsec1one}$$ 5. For every $\alpha \in \Phi $ we have $\langle \alpha , \alpha \rangle = 2$.
We call $\Phi \cup \{ 0 \} $ a *system of roots* in $V$.
The main point in the axioms for a system of roots is that there is *no* Euclidean inner product on $V$. In fact, this distinguishes a system of roots from a root system of a semisimple complex Lie algebra.
Some consequences {#sec1subsec2}
-----------------
In this section we draw some consequences from the axioms of a system of roots.
**Lemma 1.2.1** Axiom $3$ is follows from axiom $1$.
**Proof.** Axiom 3 holds because for every $\beta $, $\alpha \in \Phi $ the affine line ${\mathbb{R}}\rightarrow V: t \mapsto \beta +t \alpha $ through $\beta $ in the direction of $\alpha $ intersects $\Phi \cup \{ 0 \} $ in at most a finite number of distinct values of the parameter $t \in {\mathbb{Z}}$. Since $\beta \in \Phi $ it follows that $0$ is such a parameter value. Thus there is a maximal $q$ and $p
\in {{\mathbb{Z}}}_{\ge 0}$ such that for every $j \in {\mathbb{Z}}$ with $- q \le j \le p$ we have $\beta + j \alpha \in
\Phi \cup \{ 0 \} $. So an extremal root chain through $\beta $ in the direction $\alpha $ exists. By definition $\langle \beta , \alpha \rangle = q-p$. Thus axiom $3$ is a consequence of axiom $1$.
**Claim 1.2.2** Let $\alpha $, $\beta \in \Phi $. If $\langle \beta , \alpha \rangle < 0$, then $\beta + \alpha \in \Phi \cup \{ 0 \}$; while if $\langle \beta , \alpha \rangle > 0 $, then $\beta - \alpha
\in \Phi \cup \{ 0 \} $.
**Proof.** From axiom $3$ we deduce that the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ through $\beta $ in the direction $\alpha $ with integer pair $(q,p)$ exists. If $q-p = \langle \beta , \alpha \rangle
< 0$, then $p > q \ge 0$. So ${\mathcal{S}}^{\beta }_{\alpha }$ contains $\beta + \alpha $. Therefore $\beta + \alpha \in \Phi \cup \{ 0 \} $. If $q -p = \langle \beta , \alpha \rangle > 0$, then $q > p \ge 0$. So the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ through $\beta $ in the direction $\alpha $ contains $\beta - \alpha $. Therefore $\beta - \alpha \in \Phi \cup \{ 0 \} $. $\square $
**Lemma 1.2.3** For every $\beta $, $\alpha \in \Phi $ we have $\langle \beta , - \alpha \rangle
= - \langle \beta , \alpha \rangle $ and $\langle - \beta , \alpha \rangle = - \langle \beta , \alpha \rangle $.
**Proof.** From axiom $3$ the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }:
\beta -q \alpha , \ldots , \beta + p \alpha $ through $\beta $ in the direction $\alpha $ with integer pair $(q,p)$ exists. From axiom $2$ it follows that $-\beta $ and $-\alpha \in \Phi $. The following argument shows that the extremal root chain ${\mathcal{S}}^{\beta }_{-\alpha }$ has integer pair $(p,q)$. From the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ we see that $$\beta -p(-\alpha ), \, \beta -(p-1)(-\alpha ) , \ldots , \beta +(q-1)(-\alpha ) , \, \beta + q(-\alpha )$$ is an extremal root chain through $\beta $ in the direction $-\alpha $ with integer pair $(p,q)$. In other words, $\langle \beta , -\alpha \rangle = p -q = - \langle \beta , \alpha \rangle $.
Next we show that the extremal root chain ${\mathcal{S}}^{-\beta }_{\alpha }$ through $-\beta $ in the direction $\alpha $ has integer pair $(p,q)$. Multiplying the elements of the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ by $-1$ and using axiom $2$, we see that $$-\beta - p \alpha , \, -\beta -(p-1)\alpha , \ldots , -\beta +(q-1)\alpha , \, -\beta + q\alpha$$ is an extremal root chain ${\mathcal{S}}^{-\beta }_{\alpha }$ through $-\beta $ in the direction $\alpha $ with integer pair $(p,q)$. In other words, $\langle - \beta , \alpha \rangle = p-q = - \langle \beta , \alpha \rangle $. $\square $.
We now consider root systems, see [@humphreys p.42]. Let $\big( U, (\, \, , \, \, ) \big) $ be a finite dimensional real vector space with a Euclidean inner product $(\, \, , \, \, ) $. Let $\Phi $ be a finite subset of nonzero vectors in $U$. Suppose that the following axioms hold.
1\. $U = {{\mathop{\rm span}\nolimits}}_{{\mathbb{R}}}\Phi $.
2\. If $\alpha \in \Phi $ and $\lambda \in {\mathbb{R}}$, then $\lambda \alpha \in \Phi $ if and only if $|\lambda | =1$.
3\. If $\alpha \in \Phi $, then the reflection ${\sigma }_{\alpha }: U \rightarrow U: v \mapsto v -
{\mbox{$\frac{{\scriptstyle 2(v,\alpha )}}{{\scriptstyle (\alpha , \alpha )}}$}} \alpha $ is an orthogonal real linear mapping of $\big( U, (\, \, , \, \, ) \big)$ into itself, which preserves $\Phi $.
4\. If $\beta $, $\alpha \in \Phi $, then $\langle \beta , \alpha \rangle = \frac{2(v,\alpha )}{(\alpha , \alpha )} \in {\mathbb{Z}}$.
Then $\Phi $ is a *root system*.
**Claim 1.2.4** Every root system is a system of roots.
**Proof.** Suppose that $\Phi $ is a root system. Then axiom $1$ of root system is the same as axiom $1$ of a system of roots. If $\alpha \in \Phi $, then from axiom $2$ of a root system it follows that $-\alpha \in \Phi $ and thus $\alpha = -(-\alpha )$. So $\Phi = - \Phi $, which is axiom $2$ of a system of roots. From lemma 5.2.1 it follows that for every $\beta $, $\alpha \in \Phi $ there is an extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ through $\beta $ in the direction $\alpha $ with integer pair $(q,p)$. Thus the first statement in axiom $3$ for a system of roots holds. This does not complete its verification, because we still have to show that $\langle \beta , \alpha \rangle = q-p$, where $\langle \beta , \alpha \rangle = \frac{2(\beta , \alpha )}{(\alpha , \alpha )}$. Axiom $5$ of a system of roots follows because for every $\alpha \in \Phi $, which is nonzero by hypothesis, we have $\langle \alpha , \alpha \rangle = \frac{2(\alpha , \alpha )}{(\alpha , \alpha )} = 2$. From the definition $\langle \beta , \alpha \rangle = \frac{2(\beta , \alpha )}{(\alpha , \alpha )}$ it follows that for each fixed $\alpha \in \Phi $ the function $K_{\alpha }: \Phi \rightarrow {\mathbb{Z}}: \beta \mapsto \langle \beta , \alpha \rangle $ is linear and is ${\mathbb{Z}}$-valued by axiom $4$ of a root system. This proves axiom 4 of a system of roots.
We now finish the proof of axiom $3$ of a system of roots. Using axiom $3$ of a root system we show that for every $\alpha \in \Phi $ the orthogonal reflection ${\sigma }_{\alpha }:U \rightarrow U:
\beta \mapsto \langle \beta , \alpha \rangle $ maps the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ into itself. For every $j \in {\mathbb{Z}}$ with $-q \le j \le p$ let $\beta +j \alpha \in {\mathcal{S}}^{\beta }_{\alpha }$. Then $$\begin{aligned}
{\sigma }_{\alpha }(\beta +j \alpha ) & = \beta +j \alpha -K_{\alpha }(\beta +j \alpha ) \alpha \notag \\
& = \beta +j \alpha -K_{\alpha }(\beta )\alpha - jK_{\alpha }(\alpha ) \alpha \notag \\
& = \beta - (\langle \beta , \alpha \rangle +j) \alpha . \tag*{$\square $} \end{aligned}$$ Since $\beta +j \alpha \in \Phi \cup \{ 0 \} $ and ${\sigma }_{\alpha }$ maps $\Phi $ into itself and $0$ into $0$, it follows that $\beta - (\langle \beta , \alpha \rangle +j) \alpha \in \Phi \cup \{ 0 \} $. Because $\langle \beta , \alpha \rangle \in {\mathbb{Z}}$, for every $j \in {\mathbb{Z}}$ with $-q \le j \le p$ we have $-(\langle \beta , \alpha \rangle +j ) \in F = \big\{ j \in {\mathbb{Z}}, \, -q \le j \le p {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\beta + j \alpha \in \Phi \cup \{ 0 \} \big\} $. So the orthogonal reflection ${\sigma }_{\alpha }$ maps the extremal root chain ${\mathcal{S}}^{\beta }_{\alpha }$ into itself. Moreover $-(\langle \beta , \alpha \rangle +p) $ is the smallest element of $F$. But ${\mathcal{S}}^{\beta }_{\alpha }$ is an extremal root chain with integer pair $(q,p)$. So $-q = -\langle \beta , \alpha \rangle -p$, that is, $\langle \beta , \alpha \rangle = q-p$. This proves axiom $3$ of a system of roots and thereby the claim. $\square $
Very special sandwich algebras of class $\mathcal{C}$ {#sec2}
=====================================================
Recall that a *complex sandwich algebra* $\widetilde{\mathfrak{g}}= \mathfrak{g}\oplus \widetilde{\mathfrak{n}}$ is a complex Lie algebra where $\mathfrak{g}$ is a simple Lie algebra, $\widetilde{\mathfrak{n}}$ is the nilpotent radical of $\widetilde{\mathfrak{g}}$, which is a sandwich, that is, $[\widetilde{\mathfrak{n}}, [\widetilde{\mathfrak{n}}, \widetilde{\mathfrak{n}}]] =0$, and for $\mathfrak{h}$, a Cartan subalgebra of $\mathfrak{g}$, the set ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$ is a maximal toral subalgebra of ${\mathop{\mathrm{gl}}\nolimits}(\widetilde{\mathfrak{n}}, {\mathbb{C}})$. The sandwich algebra $\widetilde{\mathfrak{g}}$ is *very special* if $\widetilde{\mathfrak{g}}$ is a subalgebra of a complex simple Lie algebra $\underline{\mathfrak{g}}$, which has rank $1$ greater than the rank of the simple Lie algebra $\mathfrak{g}$. Let $\underline{\mathfrak{h}}$ be a Cartan subalgebra of $\underline{\mathfrak{g}}$. We assume that the Cartan subalgebras $\mathfrak{h}$ and $\underline{\mathfrak{h}}$ are *aligned*, that is, there is a vector ${\underline{H}}^{\ast } \in \underline{\mathfrak{h}}$ such that $\mathcal{R} = {\mathrm{R}}^0 =
\{ \underline{\alpha } \in \underline{\mathcal{R}} {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\underline{\alpha }({\underline{H}}^{\ast }) =0 \} $. Here $\mathcal{R}$ and $\underline{\mathcal{R}}$ are the root systems associated to the Cartan subalgebras $\mathfrak{h}$ and $\underline{\mathfrak{h}}$ of the complex simple Lie algebras $\mathfrak{g}$ and $\underline{\mathfrak{g}}$, respectively. For later purposes let ${\mathrm{R}}^{-} = \{ \underline{\alpha } \in
\underline{\mathcal{R}} {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\underline{\alpha } ({\underline{H}}^{\ast }) < 0 \} $. A very special complex sandwich algebra is of *class $\mathcal{C}$* if the center $Z$ of the nilradical $\widetilde{\mathfrak{n}}$ has complex dimension $1$. see Cushman [@cushman17a].
[**Summary of §2**]{} \[sec2subsec1\] In §2.1 we recall the definition of the collection of roots $\widehat{\mathcal{R}}$ for the nilpotent radical $\widetilde{\mathfrak{n}}$. We define an operation of addition on $\widehat{\mathcal{R}}$ and construct a set of positive simple roots $\widehat{\Pi }$, which is, an basis for the real vector space $V = {{\mathop{\rm span}\nolimits}}_{{\mathbb{R}}}\{ \widehat{\alpha } {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\widehat{\alpha } \in \widehat{\Pi } \} $. This allows us to show that axioms 1 and 2 for a system of roots holds for the set $\Phi $ of nonzero roots in $\widehat{\mathcal{R}}$. Associated to a positive root $\widehat{\alpha} \in \widehat{\Pi }$ is a root algebra ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$, which is isomorphic to a $3$-dimensional Heisenberg algebra ${\mathrm{h}}_3$. In §2.3 we associate to every extremal root chain ${\mathcal{S}}^{\widehat{\beta }}_{\widehat{\alpha }}$ the space ${\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }}$, formed by taking the direct sum of root spaces corresponding to elements of ${\mathcal{S}}^{\widehat{\beta }}_{\widehat{\alpha }}$. In claim 2.3.1 we show that the adjoint representation of ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ on $\widetilde{\mathfrak{n}}$ is completely reducible and classify its irreducible summands. The goal of §2.4 is to verify that $\Phi $ satisfies axioms 4 and 5 of a system of roots. Lemm 2.4.1 shows that $\Phi $ satisfies axiom $5$ of a system of roots. The main result §2 is claim 2.4.2, which shows that for every fixed $\widehat{\alpha } \in \Phi $ the function $K_{\widehat{\alpha }}: \widehat{\mathcal{R}} \rightarrow {\mathbb{Z}}: \widehat{\beta }\mapsto \langle \widehat{\beta} ,
\widehat{\alpha } \rangle $ is linear. In other words, $\Phi $ satisfies axiom $4$ of a system of roots. Our main tool in proving claim 2.4.2 is claim 2.4.4, which states that if for $i=1,2$ the extremal root chain ${\mathcal{S}}^{{\widehat{\beta }}_i}_{\widehat{\alpha }}$ has an integer pair $(q_i,p_i)$ and if ${\mathcal{S}}^{{\widehat{\beta }}_1 +
{\widehat{\beta }}_2}_{\widehat{\alpha }}$ is also an extremal root chain with integer pair $(r,s)$, then the Killing integer $r-s$ of ${\mathcal{S}}^{{\widehat{\beta }}_1+ {\widehat{\beta }}_2}_{\widehat{\alpha }}$ is equal to $(q_1+q_2)-(p_1+p_2)$.
A list of very special sandwich algebras of class $\mathcal{C}$ is given in the appendix.
Roots and root spaces {#sec2subsec1}
---------------------
Let $\widetilde{\mathfrak{g}} = \mathfrak{g} \oplus \widetilde{\mathfrak{n}}$ be a very special sandwich algebra with nilpotent radical $\widetilde{\mathfrak{n}}$ and a complex simple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$. By definition of sandwich algebra ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$ is a maximal torus of ${\mathop{\mathrm{gl}}\nolimits}(\widetilde{\mathfrak{n}}, {\mathbb{C}})$. Thus $\widetilde{\mathfrak{n}} =
\sum_{\widehat{\alpha } \in \widehat{\mathcal{R}}} \oplus {\widehat{\mathfrak{g}}}_{\widehat{\alpha }}$, where ${\widehat{\mathfrak{g}}}_{\widehat{\alpha }}$ is the ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$-invariant subspace $ \{ X \in \widetilde{\mathfrak{n}} {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}[H,X] = \widehat{\alpha }(H) X, \, \mbox{for all $H \in \mathfrak{h}$} \} $, which is called the *root space* corresponding to the *root* $\widehat{\alpha } \in \widehat{\mathcal{R}}$. The set $\widehat{\mathcal{R}}$ of *roots* associated to $\mathfrak{h}$ is finite. Since ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$ is a maximal torus, ${\dim }_{{\mathbb{C}}}\, {\widehat{\mathfrak{g}}}_{\widehat{\alpha }} = 1$ for every $\widehat{\alpha } \in \widehat{\mathcal{R}}$. So ${\widehat{\mathfrak{g}}}_{\widehat{\alpha }}$ is spanned by the nonzero *root vector* $X_{\widehat{\alpha }}$. Since $\widetilde{\mathfrak{g}}$ is a very special sandwich algebra of class $\mathcal{C}$, the center $Z$ of its nilpotent radical $\widetilde{\mathfrak{n}}$ is ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$ -invariant subspace of $\widetilde{\mathfrak{n}}$, which is $1$-dimensional. Hence $Z$ is spanned by the nonzero root vector $X_{\widehat{\zeta }}$ for some $\widehat{\zeta } \in \widehat{\mathcal{R}}$. Let $\mathcal{Z} = \{ \widehat{\zeta } \} $ and set $\mathcal{Y} = \widehat{\mathcal{R}} \setminus \mathcal{Z}$. Let $Y = \sum_{\widehat{\alpha } \in \mathcal{Y}} \oplus {\widehat{\mathfrak{g}}}_{\widehat{\alpha }}$. Then $Y$ is ${{\mathop{\mathrm{ad}}\nolimits}}_{\mathfrak{h}}$-invariant and $\widetilde{\mathfrak{n}} = Y \oplus Z$. Because $\widetilde{\mathfrak{n}}$ is a sandwich, it follows that $[Y,Y] = Z$.
Define an addition operation ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$ on $\widehat{\mathcal{R}}$ by saying that if $\widehat{\alpha }, \widehat{\beta } \in \widehat{\mathcal{R}}$, then $\widehat{\alpha } {\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}\widehat{\beta } \in \widehat{\mathcal{R}}$ if there are $\underline{\alpha }$, $\underline{\beta } \in {\mathrm{R}}^{-} \subseteq \underline{\mathcal{R}}$ such that $\underline{\alpha } + \underline{\beta } \in {\mathrm{R}}^{-}$ and $\widehat{\alpha } =
\underline{\alpha } | \mathfrak{h}$ and $\widehat{\beta } = \underline{\beta } | \mathfrak{h}$. In particular, we have $(\widehat{\alpha } {\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}\widehat{\beta })|\mathfrak{h} =
(\underline{\alpha } + \underline{\beta })|\mathfrak{h}$. Clearly the operation of addition ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$ is commutative and associative. An inspection of the table for the very special sandwich algebras of class $\mathcal{C}$ in the appendix shows that
**Observation 2.1.1** For every very special sandwich algebra the linear function $\widehat{\zeta } \in \widehat{\mathcal{R}}$ is identically zero on $\mathfrak{h}$.
Thus $\widehat{\zeta }$ is the additive identity element for the addition operation ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$. In fact, this observation shows that the operation ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$ on $\widehat{\mathcal{R}}$ the ordinary addition operation of addition of linear functions on $\mathfrak{h}$. Thus $\widehat{\mathcal{R}}$ is an abelian group under ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$. We will use the notation $+$ for ${\mbox{$\, {+}^{\raisebox{-7pt}{$\hspace{-7pt}\widehat{\, \, \, }$}} \, $}}$ and $\widehat{0}$ interchangeably with $\widehat{\zeta }$. We now show that
**Lemma 2.1.2** The set ${\widetilde{\mathcal{R}}}_{\widehat{\zeta}} =
\{ (\widehat{\alpha }, \widehat{\beta }) \in \mathcal{Y} \times \mathcal{Y} {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\widehat{\alpha } + \widehat{\beta } = \widehat{\zeta } \} $ is nonempty.
**Proof.** Suppose that $x = \sum_{\widehat{\alpha } \in \mathcal{Y}} a_{\widehat{\alpha }} X_{\widehat{\alpha }}$ and $y = \sum_{\widehat{\beta } \in \mathcal{Y}} b_{\widehat{\alpha }} X_{\widehat{\beta }}$ are nonzero vectors in $Y$. Then $$\begin{aligned}
[x,y] & = \sum_{\widehat{\alpha } + \widehat{\beta } \in \widehat{R}} c_{\widehat{\alpha }, \widehat{\beta} } \,
a_{\widehat{\alpha }} b_{\widehat{\beta }} X_{\widehat{\alpha } + \widehat{\beta }}, \notag \\
& \hspace{.25in} \mbox{where $[X_{\widehat{\alpha }}, X_{\widehat{\beta }}] = c_{\widehat{\alpha }, \widehat{\beta} }X_{\widehat{\alpha }+\widehat{\beta }}$ for some $c_{\widehat{\alpha }, \widehat{\beta} } \in {\mathbb{C}}\setminus \{ 0 \} $}, \notag \\
& = \sum_{\widehat{\alpha } + \widehat{\beta } \ne \widehat{\zeta }} c_{\widehat{\alpha }, \widehat{\beta} } \,
a_{\widehat{\alpha }} b_{\widehat{\beta }} X_{\widehat{\alpha } + \widehat{\beta }} +
\big( \hspace{-3pt} \sum_{\widehat{\alpha } + \widehat{\beta } \widehat{\zeta }}c_{\widehat{\alpha }, \widehat{\beta} } \,
a_{\widehat{\alpha }} b_{\widehat{\beta }} \big) X_{\widehat{\zeta }} \notag \end{aligned}$$ If the second term of the last equality is vacuuous, then $[x,y] \in Z = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \{ \widehat{\zeta } \} $ is a linear combination of elements of other root spaces of $\widetilde{\mathfrak{n}}$. This contradicts the fact that $\widetilde{\mathfrak{n}}$ is a direct sum of its root spaces. Thus the set ${\widetilde{\mathcal{R}}}_{\widehat{\zeta}}$ is nonempty. $\square $
Construct the finite set ${\mathcal{R}}_{\widehat{\zeta }}$ as follows. Let ${\widehat{\alpha }}_1 \in {\mathcal{R}}_{\widehat{\zeta }}$ if there is $({\widehat{\alpha }}_1, {\widehat{\beta }}_1)
\in {\widetilde{\mathcal{R}}}_{\widehat{\zeta}}$. Note that $({\widehat{\alpha }}_1, {\widehat{\beta }}_1)
\in {\widetilde{\mathcal{R}}}_{\widehat{\zeta}}$ implies that $({\widehat{\beta }}_1, {\widehat{\alpha }}_1)
\in {\widetilde{\mathcal{R}}}_{\widehat{\zeta}}$, because addition is commutative. Recursively let ${\widehat{\alpha }}_{i+1} \in
{\mathcal{R}}_{\widehat{\zeta }}$ if there is $$({\widehat{\alpha }}_{i+1}, {\widehat{\beta }}_{i+1})
\in {\widetilde{\mathcal{R}}}_{\widehat{\zeta}} \setminus \big\{ ({\widehat{\alpha }}_j, {\widehat{\beta }}_j) \, \, \& \, \,
({\widehat{\beta }}_j, {\widehat{\alpha }}_j) \in \mathcal{Y} \times \mathcal{Y} \, \mbox{for $1 \le j \le i $} \big\} .$$ This recursion stops after a finite number of repetitions because ${\widetilde{\mathcal{R}}}_{\widehat{\zeta}}$ is a finite set. Then ${\mathcal{R}}_{\widehat{\zeta }} = \{ {\widehat{\alpha }}_i \in \mathcal{Y} {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}1 \le i \le M \} $.
**Lemma 2.1.3** $\mathcal{Y} = {\mathcal{R}}_{\widehat{\zeta }} \amalg (-{\mathcal{R}}_{\widehat{\zeta }})$.
**Proof.** Suppose that there is a root $\widehat{\alpha } \in \mathcal{Y} \setminus
\big( {\mathcal{R}}_{\widehat{\zeta }} \amalg -{\mathcal{R}}_{\widehat{\zeta }} \big) $. Then $-\widehat{\alpha }$ is an element of $\widehat{\mathcal{R}}$ by definition of addition on $\widehat{\mathcal{R}}$. But $\widehat{\alpha } +(-\widehat{\alpha }) = \widehat{0}$, that is, $\widehat{\alpha } \in
{\mathcal{R}}_{\widehat{\zeta }}$ by definition. This is a contradiction. So $Y = {\mathcal{R}}_{\widehat{\zeta }}\amalg
(-{\mathcal{R}}_{\widehat{\zeta }})$. $\square $
The elements of ${\mathcal{R}}_{\widehat{\zeta }}$ are *simple*. For if for some $j \in \{ 1, \ldots , M \} $ the vector ${\widehat{\alpha }}_j$ can be written as $\widehat{\alpha } + \widehat{\beta }$ for some $\widehat{\alpha }$, $\widehat{\beta } \in \mathcal{Y}$, then for some nonzero $c_{\widehat{\alpha }, \widehat{\beta }} \in {\mathbb{C}}$ we have $X_{\widehat{\alpha } + \widehat{\beta }} = c_{\widehat{\alpha}, \widehat{\beta }} \,
[X_{\widehat{\alpha }}, X_{\widehat{\beta }} ] \in Z$, since $[Y,Y] = Z$. So ${\widehat{\alpha }}_j \in \mathcal{Z}$, which contradicts the fact that ${\widehat{\alpha }}_j \in {\mathcal{R}}_{\widehat{\zeta }} \subseteq \mathcal{Y}$ by construction and $\mathcal{Y} \cap \mathcal{Z} = \varnothing $ by definition. We call ${\mathcal{R}}_{\widehat{\zeta }}$ the set $\widehat{\Pi }$ of *simple positive roots* of $\widehat{\mathcal{R}}$.
Using lemma 2.1.3 we get
**Corollary 2.1.4** $Y = \sum_{\widehat{\alpha } \in \widehat{\Pi }} \oplus \big( {\widehat{\mathfrak{g}}}_{\widehat{\alpha }} \oplus {\widehat{\mathfrak{g}}}_{-\widehat{\alpha }} \big) = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} { \big\{ X_{{\widehat{\alpha }}_j}, X_{-{\widehat{\alpha }}_j} \big\} }^M_{j=1} $.
Since $\widetilde{\mathfrak{n}} = Y \oplus {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \{ X_{\widehat{\zeta }} \} $, it follows that the real vector space $V = {{\mathop{\rm span}\nolimits}}_{{\mathbb{R}}} \{ {\widehat{\alpha }}_j \in \mathcal{Y}, \, 1 \le j \le M \} $ with nonzero roots $\Phi = \mathcal{Y}$ satisfies the axiom 1 and axiom 2 (and thus axiom 3) of a system of roots.
Root subalgebras {#sec2subsec2}
----------------
For each root $\widehat{\alpha } \in \widehat{\Pi }\subseteq \widehat{\mathcal{R}}$, the *root subalgebra* ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha})} = {\mathop{\rm span}\nolimits}\{ X_{\widehat{\alpha }}, \,
X_{-\widehat{\alpha}}, \, H_{\widehat{\zeta}} = X_{\widehat{\zeta }} \} $ is the Heisenberg algebra ${\mathrm{h}}_3$ with bracket relations $$[H_{\widehat{\zeta }}, X_{\widehat{\alpha }}] =
[H_{\widehat{\zeta}}, X_{-\widehat{\alpha} }] \, = \, 0 \, \, \& \, \,
[X_{\widehat{\alpha }}, X_{-\widehat{\alpha }}] = H_{\widehat{\zeta }}.$$
**Claim 2.2.1** For every $\alpha \in \widehat{\Pi }$ the adjoint representation of ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ on $\widetilde{\mathfrak{n}}$ decomposes into a finite sum of *irreducible* ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ representations. This decomposition is unique up to a reordering of the summands.
**Proof**. We argue as follows. Since $\{ X,Y, H \} $ span the nilpotent Lie algebra ${\mathrm{h}}_3$ with bracket relations $$[H, X] = [H,Y] = 0, \, \, \, \mathrm{and} \, \, \, [X,Y] = H,$$ their image $\rho (X)$, $\rho (Y)$, and $\rho (H)$ under the finite dimensional Lie algebra representation $\rho : {\mathrm{h}}_3 \rightarrow {\mathop{\mathrm{gl}}\nolimits}(\widetilde{\mathfrak{n}}, {\mathbb{C}})$ are nilpotent linear maps of the finite dimensional complex vector space $\widetilde{\mathfrak{n}}$ into itself. Let $v \in \ker \rho (Y)$ and let $\mathfrak{f} = \{ w, \rho (Y)w, \ldots , ({\rho }(Y))^n w =v \}$ be the longest Jordan chain in $\widetilde{\mathfrak{n}}$, which ends at $v$. Suppose that $n \ge 2$. Then the matrix of $\rho (Y)$ with respect to the basis $\mathfrak{f}$ of the vector space $U = {\mathop{\rm span}\nolimits}\{ w, \rho (Y)w, \ldots , ({\rho }(Y))^n w =
v \} $ is the lower $(n+1) \times (n+1)$ Jordan block. With respect to the basis $\mathfrak{f}$ the matrices of $\rho (Y)$, $\rho (X)$, and $\rho (H)$ are $$\mbox{{\tiny $\begin{pmatrix}
0 & & & & & & \\ 1 & 0 & & && & \\ & & & & & & \\
& \ddots & & & & & & \\ & & & & \ddots & & \\
& & & & \ddots & 0& \\ & & & & & 1 & 0 \end{pmatrix} $}}, \,
\mbox{{\tiny $\begin{pmatrix}
0 & & & & & \\
\vdots & \ddots & & & & \\
\vdots & & \ddots & & & \\
0 & & & \ddots & & \\
-1 & 0 & \cdots & \cdots & 0 & \\
0 & 1 & 0 &\cdots & 0 & 0 \end{pmatrix} $}} , \,
\mbox{{\tiny $\left( \begin{array}{ccclr} 0 & & & & \\
0 & \ddots & & & \\
\vdots & \ddots & \ddots & & \\
0 & \cdots & \ddots & 0 & \\
2 & 0 & \cdots & 0 & 0
\end{array} \right) $,}}$$ respectively. When $n=0$ let $\rho (Y)$, $\rho (X)$, and $\rho (H)$ be the zero matrix; while when $n=1$ let $\rho (Y) =$[ $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$]{}, and let $\rho (X)$ and $\rho (H)$ be the $2\times 2$ zero matrix. For every $n \ge 0$ it is straightforward to check that the following bracket relations hold $$[\rho (H), \rho (X) ] = [\rho (H), \rho (Y) ] = 0 \, \, \,
\mathrm{and} \, \, \, [\rho (X), \rho (Y)] = \rho (H).$$ This verifies that on $U$ the mapping $\rho $ is an $n+1$-dimensional representation of ${\mathrm{h}}_3$. It is irreducible. Suppose not. Then the only subspace of $U$, which is invariant under $\rho ({\mathrm{h}}_3)$, is ${\mathop{\rm span}\nolimits}\{ v \} $. By hypothesis there is a $\rho ({\mathrm{h}}_3)$-invariant subspace $V$ such that $U = V \oplus {\mathop{\rm span}\nolimits}\{ v \}$. But $(\rho (Y))^n V \subseteq {\mathop{\rm span}\nolimits}\{ v \} $, which contradicts the $\rho ({\mathrm{h}}_3)$-invariance of $V$. Therefore on $U$ the representation $\rho $ is irreducible.
Repeat the above construction for each vector in a basis $\{ v, v_2, \ldots ,$ $v_m \} $ of $\ker \rho (Y)$ in $\widetilde{\mathfrak{n}}$. This determines the Jordan normal form of $\rho (Y)$ and decomposes the representation $\rho $ into a sum of finite dimensional irreducible representations of ${\mathrm{h}}_3$. The summands are unique up to a reordering, because the Jordan blocks of $\rho (Y)$ are unique up to a reordering. $\square $
Extremal root chains {#sec2subsec3}
--------------------
In this section we identify the representation space of the irreducible summands of the adjoint representation of the root subalgebra ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$, $\widehat{\alpha} \in \widehat{\Pi }$, on $\widetilde{\mathfrak{n}}$ with an extremal root chain ${\mathcal{S}}^{\widehat{\beta }}_{\widehat{\alpha }}$ through $\widehat{\beta } \in \Phi $ in the direction $\widehat{\alpha}$.
We begin by recalling the concept of an extremal root chain. Let $\widehat{\alpha }, \widehat{\beta } \in \widehat{\mathcal{R}}$ and let $V
= {{\mathop{\rm span}\nolimits}}_{{\mathbb{R}}} \{ \widehat{\alpha } {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\, \widehat{\alpha } \in \Phi \}$. A collection of the form $\{ \widehat{\beta } + j\, \widehat{\alpha } \in V {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\, j \in F \} $, where $F$ is a finite subset of ${\mathbb{Z}}$, is called a *chain*. If a chain is of the form $$\widehat{\beta } - q\, \widehat{\alpha} , \, \, \widehat{\beta } - (q-1) \widehat{\alpha} , \ldots ,
\widehat{\beta } - \widehat{\alpha } , \, \, \widehat{\beta }, \, \, \widehat{\beta} + \widehat{\alpha} , \ldots ,
\widehat{\beta} + p\, \widehat{\alpha} ,
\label{eq-sec2subsec3one}$$ where $q,p \in {{\mathbb{Z}}}_{\ge 0}$ and $\ell \in F$ if and only if $\ell \in {\mathbb{Z}}$ and $-q \le \ell \le p$, then (\[eq-sec2subsec3one\]) is an *unbroken chain through* $\widehat{\beta }$ *in the direction* $\widehat{\alpha }$ with *with integer pair* $(q,p)$. Its *length* is $q+p+1$. Given a chain $\{ \widehat{\beta} + j\, \widehat{\alpha } \in V{\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\, j \in F \} $, a chain of the form $\{ \widehat{\beta} + j\, \widehat{\alpha} \in V {\, \mathop{\rule[-4pt]{.5pt}{13pt}\, }\nolimits}\, j \in F' \subseteq F \} $ is a *subchain* of the given chain. This subchain is *proper* if $F'$ is a proper subset of $F$. If each element of a chain lies in $\Phi \cup \{ 0 \} $, then the chain is called a *root chain* through $\widehat{\beta }$ in the direction $\widehat{\alpha }$. If (\[eq-sec2subsec3one\]) is an unbroken root chain with integer pair $(q,p)$ chosen as large as possible so that this chain is not a proper subchain of another unbroken root chain through $\widehat{\beta }$ in the direction $\widehat{\alpha }$, then the unbroken root chain ${\mathcal{S}}^{\widehat{\beta }}_{\widehat{\alpha }}$ (\[eq-sec2subsec3one\]) is called an *extremal root chain* through $\widehat{\beta }$ in the direction $\widehat{\alpha }$ with integer pair $(q,p)$. For every $\widehat{\beta }$, $\widehat{\alpha} \in
\widehat{\mathcal{R}}$ there is an extemal roots chain with a suitable integer pair $(q,p)$ an extremal root chain exists.
We now identify the irreducible summands of the adjoint representation of the root subalgebra ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha})}$, $\widehat{\alpha } \in \widehat{\Pi }$ on the nilradical $\widetilde{\mathfrak{n}}$. Corresponding to the extremal root chain (\[eq-sec2subsec3one\]) is the subspace $${\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }} =
\sum _{\stackrel{-q \le j \le p}{\rule{0pt}{6pt}{\scriptscriptstyle j\in {\mathbb{Z}}}}} \oplus
{\widehat{\mathfrak{g}}}_{\widehat{\beta} + j\, \widehat{\alpha }}.
\label{eq-irred}
\vspace{-.1in}$$ of $\widetilde{\mathfrak{n}}$. If $\widehat{0}$ appears in the root chain (\[eq-sec2subsec3one\]), then ${\widehat{\mathfrak{g}}}_{\widehat{0}} = Z = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \{ \widehat{\zeta } \} $. If $\widehat{\beta } \pm \widehat{\alpha } \notin \widehat{\mathcal{R}}$, then ${\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }} = {\widehat{\mathfrak{g}}}_{\widehat{\beta }}$. Using the fact that $[{\widehat{\mathfrak{g}}}_{\widehat{\alpha }}, {\widehat{\mathfrak{g}}}_{\widehat{\beta }}] =
{\widehat{\mathfrak{g}}}_{\widehat{\alpha } +\widehat{\beta }}$ if $\widehat{\alpha } +\widehat{\beta } \in \widehat{\mathcal{R}}$ and $0$ if $\widehat{\alpha } +\widehat{\beta } \notin \widehat{\mathcal{R}}$ and the definition of extremal root chain, it follows that ${\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }}$ is an ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\alpha )}}$-invariant subspace of $\widetilde{\mathfrak{n}}$.
**Claim 2.3.1** For every $\widehat{\beta } \in \widehat{\mathcal{R}}$ and every $\widehat{\alpha }\in \widehat{\Pi }$ the subspace ${\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }}$ (\[eq-irred\]) of $\widetilde{\mathfrak{n}}$ is a representation space for the adjoint representation of ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$, which is irreducible.
**Proof**. Let $(q,p)$ be the integer pair associated to the extremal root chain through $\widehat{\beta }$ in the direction of $\widehat{\alpha }$. Then ${\widehat{\mathfrak{g}}}_{\widehat{\beta} + p\, \widehat{\alpha }}$ is the top root space corresponding to the top root $\widehat{\beta} + p\, \widehat{\alpha }$ in this chain. The linear operator ${{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}$ steps down the extremal root chain from ${\widehat{\mathfrak{g}}}_{\widehat{\beta} + j\, \widehat{\alpha }}$ to $$\left\{ \begin{array}{rl} {\widehat{\mathfrak{g}}}_{\widehat{\beta} + (j-1)\widehat{\alpha }}, &
\mbox{when $-q < j \le p$, $j \in {\mathbb{Z}}$ } \\
{\rule{0pt}{16pt}}\{ 0 \} , & \mbox{when $j =-q$.} \end{array} \right.$$ Since ${{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}{\widehat{\mathfrak{g}}}_{\widehat{\beta} + q \, \widehat{\alpha }}v = \{ 0 \}$, the linear map ${{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}:{\widehat{\mathfrak{g} }}^{\widehat{\beta }}_{\widehat{\alpha }} \rightarrow
{\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }}$ is nilpotent of height $p+q$, because ${{\mathop{\mathrm{ad}}\nolimits}}^{p+q}_{X_{-\widehat{\alpha }}}{\widehat{\mathfrak{g}}}_{\widehat{\beta } + p\, \widehat{\alpha }} =
{\widehat{\mathfrak{g}}}_{\beta + q \, \alpha }$, while ${{\mathop{\mathrm{ad}}\nolimits}}^{p+q+1}_{X_{-\widehat{\alpha }}}
{\widehat{\mathfrak{g}}}_{\widehat{\beta} + p\, \widehat{\alpha }} = \{ 0 \} $. Taking a nonzero vector $v \in {\widehat{\mathfrak{g}}}_{\widehat{\beta } + p \, \widehat{ \alpha }}$, the above argument shows that $\mathfrak{f} = \{ v, {{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}v, \ldots ,
{{\mathop{\mathrm{ad}}\nolimits}}^{p+q}_{X_{-\widehat{\alpha }}}v \} $ is a Jordan chain in ${\mathfrak{g}}^{\widehat{\beta }}_{\widehat{\alpha }}$ of length $p+q+1$. So $\mathfrak{f}$ is a basis of $W = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \{ v, {{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}v, \ldots ,
{{\mathop{\mathrm{ad}}\nolimits}}^{p+q}_{X_{-{\widehat{\alpha }}}} v \} $ with respect to which the matrix of ${{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}|W$ is a $(p+q+1) \times
(p+q+1)$ lower Jordan block. By the classification of irreducible ${\widetilde{\mathfrak{g}}}^{(\alpha )}$-representations, $W$ is a representation space for a $p+q+1$-dimensional irreducible representation of ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$. But ${\dim }_{{\mathbb{C}}}\, {\widehat{\mathfrak{g}}}_{\widehat{\beta} +j\, \widehat{ \alpha }} = 1$ for every $-q \le j \le p$, $j\in {\mathbb{Z}}$. Therefore ${\dim }_{{\mathbb{C}}}\, {\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha } }= p+q+1 = \dim W$, which shows that $W = {\widehat{\mathfrak{g}}}^{\widehat{\beta }}_{\widehat{\alpha }}$. $\square $
Verification of axioms 4 and 5 {#sec2subsec4}
------------------------------
In this subsection we verify that axioms 4 and 5 for a system of roots holds for the set of roots $\widehat{\mathcal{R}}$ of a very special sandwich algebra.
First we verify that axiom 5 holds. In other words,
**Lemma 2.4.1** For every $\widehat{\alpha} \in \Phi = \mathcal{Y}$ the Killing integer $\langle \widehat{\alpha }, \widehat{\alpha } \rangle $ associated to the extremal root chain through $\widehat{\alpha }$ in the direction of $\widehat{\alpha } $ is $2$.
**Proof**. From lemma 1.2.3 it follows that for every $\widehat{\beta }$, $\widehat{\alpha} \in \mathcal{Y}$ we have $$\langle \widehat{\beta }, - \widehat{\alpha }\rangle = - \langle \widehat{\beta },\widehat{\alpha } \rangle \, \,
\mathrm{and}\, \, \langle - \widehat{\beta }, \widehat{\alpha }\rangle = - \langle \widehat{\beta }, \widehat{\alpha} \rangle ,
\label{eq-ktwonew}$$ respectively. Thus it suffices to assume that $\widehat{\beta }, \widehat{\alpha }\in \widehat{\Pi }$. We now determine the extremal root chain in $\widehat{\mathcal{R}}$ through $\widehat{\alpha} \in \widehat{\Pi }$ in the direction $\widehat{\alpha }$. Such an unbroken root chain is $$\widehat{\alpha } - 2\, \widehat{\alpha } = -\widehat{\alpha }, \, \, \widehat{\alpha }- 1\widehat{\alpha } = \widehat{0},
\, \, \widehat{\alpha } + 0 \, \widehat{\alpha } =\widehat{\alpha }.$$ This root chain is extremal, since $\pm 2\widehat{\alpha } = \pm \widehat{\alpha } \pm \widehat{\alpha }
\in \mathcal{Z} $ and thus does not lie in $\mathcal{Y}$. It has associated integer pair $(2,0)$. Hence by definition its Killing integer $\langle \alpha , \alpha \rangle$ is $2$. $\square $
We now verify that axiom 4 of a system of roots holds for the collection of roots $\Phi $. This amounts to proving
**Claim 2.4.2** For each fixed $\widehat{\alpha } \in
\Phi = \mathcal{Y}$ the function $$K_{\widehat{\alpha }}: \widehat{\mathcal{R}} \rightarrow {\mathbb{Z}}:
\widehat{\beta} \mapsto \langle \widehat{\beta }, \widehat{\alpha} \rangle$$ is linear. By linear we mean: if $\widehat{\gamma } , \widehat{\delta } \in \mathcal{R} $ such that $\widehat{\gamma }+ \widehat{\delta } \in \mathcal{R} $, then $K_{\widehat{\alpha }}(\widehat{\gamma } +\widehat{\delta } ) =
K_{\widehat{\alpha }}(\widehat{\gamma }) + K_{\widehat{\alpha }}(\widehat{\delta })$.
We will need a few preliminary results. Let $\widehat{\alpha } \in \widehat{\Pi }$. Suppose that ${\widehat{\beta }}_1$, ${\widehat{\beta }}_2$ and ${\widehat{\beta }}_1 + {\widehat{\beta }}_2$ lie in $\mathcal{R}$. For $i=1,2$ let $${\widehat{\beta }}_i - q_i \, \widehat{\alpha }, \ldots , \, {\widehat{\beta }}_i - 1\alpha ,
\, {\widehat{\beta }}_i, \, {\widehat{\beta }}_i + 1\widehat{\alpha }, \ldots , {\widehat{\beta }}_i + p_i \, \widehat{\alpha }
\label{eq-sec2subsec4two}$$ be an extremal root chain through ${\widehat{\beta }}_i$ in the direction $\widehat{\alpha }$ with integer pair $(q_i,p_i) \in ({{\mathbb{Z}}}_{\ge 0})^2$. Let $$\mbox{\footnotesize ${\widehat{\beta }}_1 + {\widehat{\beta }} - r \, \widehat{\alpha } , \ldots ,
{\widehat{\beta }}_1+ {\widehat{\beta }} _2 - (r-1) \widehat{\alpha }, \ldots
{\widehat{\beta }}_1 + {\widehat{\beta }}_2 +(s-1) \widehat{\alpha } ,
{\widehat{\beta }}_1 + {\widehat{\beta }}_2 + s\, \widehat{\alpha }$}
\label{eq-sec2subsec4three}$$ be an extremal root chain through ${\widehat{\beta }}_1+ {\widehat{\beta }}_2$ in the direction $\widehat{\alpha }$ with integer pair $(r,s)$. Consider the chain $$\mbox{\footnotesize ${\widehat{\beta }}_1 + {\widehat{\beta }}_2 - (q_1+q_2) \widehat{\alpha }, \ldots ,
{\widehat{\beta }}_1+ {\widehat{\beta }}_2 - 0\, \widehat{\alpha} , \ldots ,
{\widehat{\beta }_1} + {\widehat{\beta }}_2 + (p_1 +p_2)\widehat{\alpha }$}
\label{eq-sec2subsec4four}$$
**Lemma 2.4.3** The chain (\[eq-sec2subsec4three\]) is an extremal root subchain of the unbroken chain (\[eq-sec2subsec4four\]).
**Proof.** The following argument shows that the chain (\[eq-sec2subsec4four\]) is unbroken. Write $$\begin{aligned}
\mbox{{\footnotesize ${\widehat{\beta }}_2 - q_2\widehat{\alpha } + {\widehat{\beta }}_1 - q_1 \widehat{\alpha }, \,
{\widehat{\beta }}_2 - q_2\widehat{\alpha } + {\widehat{\beta }}_1 - (q_1-1) \widehat{\alpha }, \ldots ,
{\widehat{\beta }}_2 - q_2\widehat{\alpha } + {\widehat{\beta }}_1 + p_1 \widehat{\alpha }$}} & \notag \\
&\hspace{-3in} \mbox{{\footnotesize ${\widehat{\beta }}_2- (q_2-1)\widehat{\alpha } + {\widehat{\beta }}_1 + p_1 \widehat{\alpha }, \ldots , {\widehat{\beta }}_2 + p_2\widehat{\alpha } + {\widehat{\beta }}_1 + p_1 \widehat{\alpha} $.}}
\label{eq-sec2subsec4five}\end{aligned}$$ Using the fact that the chains in (\[eq-sec2subsec4two\]) are unbroken, it follows that the chain (\[eq-sec2subsec4five\]) is unbroken. Clearly the chains (\[eq-sec2subsec4four\]) and (\[eq-sec2subsec4five\]) are equal.
To prove that the chain (\[eq-sec2subsec4three\]) is a subchain of the chain (\[eq-sec2subsec4four\]) suppose that $$q_1+q_2 < r \quad \mathrm{or} \quad p_1+p_2 < s.
\label{eq-sec2subsec4six}$$ Recall that ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ is the root subalgebra of $\widetilde{\mathfrak{n}}$ associated to the positive simple root $\widehat{\alpha }\in \widehat{\Pi }$. Then ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ is equal to ${\mathop{\rm span}\nolimits}\{ X_{\widehat{\alpha } },
X_{-\widehat{\alpha }}, H_{\widehat{\zeta }} \} $, which is isomorphic to ${\mathrm{h}}_3$. Let $$W = \sum_{\stackrel{-(q_1+q_2) \le \ell \le (p_1+p_2)}{\rule{0pt}{6pt}{\scriptscriptstyle \ell \in {\mathbb{Z}}}}} \hspace{-15pt} \oplus \, {\widehat{\mathfrak{g}}}_{{\widehat{\beta }}_1+ {\widehat{\beta }}_2 + \ell \widehat{\alpha }},$$ where ${\widehat{\mathfrak{g}}}_{\widehat{\gamma }}$ is the root space of $\widetilde{\mathfrak{n}}$ corresponding to $\widehat{\gamma }$ in the chain (\[eq-sec2subsec4four\]). Since the root chains in (\[eq-sec2subsec4two\]) are unbroken and extremal, we have $${{\mathop{\mathrm{ad}}\nolimits}}^{q_i+p_i+1}_{X_{-\widehat{\alpha}}} {\widehat{\mathfrak{g}}}_{{\widehat{\beta }}_i +p_i\, \widehat{\alpha }}= \{ 0 \}$$ and therefore $${{\mathop{\mathrm{ad}}\nolimits}}^{p_1+p_2+q_1+q_2+1}_{X_{-\widehat{\alpha}}}
{\widehat{\mathfrak{g}}}_{{\widehat{\beta }}_1 + {\widehat{\beta}}_2 +(p_1+p_2)\widetilde{\alpha }} = \{ 0 \} .$$ This implies that $W$ is ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widehat{\mathfrak{g}}}^{(\widehat{\alpha })}}$-invariant. By claim 2.3.1 $${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1 +{\widehat{\beta }}_2}_{\widehat{\alpha }} =
\sum_{\stackrel{-r \le m \le s}{{\scriptscriptstyle m \in {\mathbb{Z}}}}} \oplus
\, {\widehat{\mathfrak{g}}}_{{\widehat{\beta }}_1 + {\widehat{\beta }}_2 + m \widehat{\alpha }}
\vspace{-.1in}$$ is a representation space for an irreducible representation of ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$ on ${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1 +{\widehat{\beta }}_2}_{\widehat{\alpha }}$, since (\[eq-sec2subsec4three\]) is an extremal root chain. From the hypothesis (\[eq-sec2subsec4six\]) it follows that $W$ is a proper ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$-invariant subspace of ${\widetilde{\mathfrak{g}}}^{{\widehat{\beta }}_1 + {\widehat{\beta }}_2}_{\widehat{\alpha }}$. (Recall that equation (\[eq-irred\]) implies that ${\widehat{\mathfrak{g}}}_{\widehat{\gamma }} = \{ 0 \} $, if $\widehat{\gamma }= {\widehat{\beta }}_1+{\widehat{\beta }}_2 + m \, \widehat{\alpha }$ lies in the chain (\[eq-sec2subsec4four\]) and either $m< -r$ or $m >s$.) But this contradicts the irreducibility of the ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$-representation on ${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1 + {\widehat{\beta }}_2}_{\widehat{\alpha }}$. Therefore the statement in (\[eq-sec2subsec4six\]) is false, that is, $$q_1+q_2 \ge r \quad \mathrm{and} \quad p_1+p_2 \ge s .
\label{eq-sec2subsec4seven}$$ This shows that the chain (\[eq-sec2subsec4three\]) is a subchain of the chain (\[eq-sec2subsec4four\]). Since the chain (\[eq-sec2subsec4three\]) is extremal by hypothesis, we have proved the lemma. $\square $
We now define the notion of the sum of the extremal root chains $${\widehat{\beta }}_i - q_i\, \widehat{\alpha} , \ldots , {\widehat{\beta }}_i + p_i \, \widehat{\alpha}
\label{eq-sec2subsec4eight}$$ with integer pair $(q_i,p_i)$ for $i=1,2$. Suppose that ${\widehat{\beta }}_1 + {\widehat{\beta }}_2 \in \Phi $. The *sum* of the extremal root chains (\[eq-sec2subsec4eight\]) is the root chain obtained from the unbroken chain $${\widehat{\beta }}_1 + {\widehat{\beta }}_2 - (q_1+q_2)\widehat{\alpha }, \ldots ,
{\widehat{\beta }}_1 + {\widehat{\beta }}_2 + (p_1+p_2)\widehat{\alpha }
\label{eq-sec2subsec4nine}$$ by removing all the elements ${\widehat{\beta }}_1 + {\widehat{\beta }}_2 + j \, \widehat{\alpha }$ of (\[eq-sec2subsec4nine\]) which do not lie in $\Phi \cup \{ 0 \}$. By lemma 2.4.3 the extremal root chain $${\widehat{\beta }}_1 + {\widehat{\beta }}_2 - r\, \widehat{\alpha} , \ldots ,
{\widehat{\beta }}_1 + {\widehat{\beta }}_2 + s \, \widehat{\alpha}
\label{eq-sec2subsec4ten}$$ with integer pair $(r,s)$ is an unbroken root subchain of (\[eq-sec2subsec4nine\]). From the definition of the nonnegative integer $r$ it follows that ${\widehat{\beta }}_1 + {\widehat{\beta }}_2 - k \, \widehat{\alpha }
\notin \Phi \cup \{ \widehat{0} \} $ for every $k \in {\mathbb{Z}}$ with $r+1 \le k \le (q_1+q_2)$. Similarly, from the definition of the nonnegative integer $s$ it follows that $\{ {\widehat{\beta }}_1 + {\widehat{\beta }}_2 + m\, \widehat{\alpha }
\notin \Phi \cup \{ 0 \} \} $ for every $m \in {\mathbb{Z}}$ with $s+1 \le m \le (p_1+p_2)$. Consequently, the sum of the extremal root chains (\[eq-sec2subsec4eight\]) is the extremal root chain (\[eq-sec2subsec4ten\]).
Next we prove
**Claim 2.4.4** The integer $(q_1+q_2)-(p_1+p_2)$ associated to the unbroken chain (\[eq-sec2subsec4nine\]) with integer pair $(q_1+q_2,p_1+p_2)$ is equal to the Killing integer $r-s$ of the extremal root chain (\[eq-sec2subsec4ten\]).
**Proof.** For $i=1,2$ let $\widehat{\alpha} \in \widehat{\Pi }$ and ${\widehat{\beta }}_i \in \widehat{\mathcal{R}}$. Recall that ${\widehat{\beta }}_i - q_i \, \widehat{\alpha } , \ldots , {\widehat{\beta }}_i + p_i\, \widehat{\alpha }$ is an extremal root chain with integer pair $(q_i,p_i)$ and length $q_i+p_i+1$. Since (\[eq-sec2subsec4eight\]) is an extremal root chain, ${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_i}_{\widehat{\alpha }}$ is a complex $N_i=q_i+p_i+1$-dimensional representation space for an irreducible ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$ representation. Let $n_i =$[$ \left\{ \begin{array}{rl} {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(q_i+p_i), &
\mbox{if $N_i$ is odd} \\
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(q_i+p_i+1), & \mbox{if $N_i$ is even.} \end{array} \right. $]{} Then $n_i$ is a nonnegative integer.
Recall that the root subalgebra ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ is isomorphic to ${\mathrm{h}}_3$. Let $\iota :\widetilde{\mathfrak{g}} \rightarrow \underline{\mathfrak{g}}$ be the inclusion map, which is defined since $\widetilde{\mathfrak{g}}$ is a subalgebra of the simple Lie algebra $\underline{\mathfrak{g}}$. The image of the root vector $X_{-\widehat{\alpha }}$ under $\iota $ is the root vector $X_{-\underline{\alpha }}$ in $\underline{\mathfrak{g}}$. This latter root vector embeds into the root subalgebra ${\underline{\mathfrak{g}}}^{(\underline{\alpha })} =
{\mathop{\rm span}\nolimits}\{ X_{-\underline{\alpha }}, X_{\underline{\alpha }}, H_{\underline{\alpha }} \} $ of $\underline{\mathfrak{g}}$, which is isomorphic to ${\mathop{\mathrm{sl}}\nolimits}(2, {\mathbb{C}})$. Since the root chains in (\[eq-sec2subsec4eight\]) are extremal, it follows that the complex $N_i =q_i+p_i+1$-dimensional ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$ representation on ${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_i}_{\widehat{\alpha}}$ is irreducible. Thus there is a vector $v_i \in {\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_i}_{\widehat{\alpha}}$ such that $V_i =
{\mathop{\rm span}\nolimits}\{ v_i, {{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha }}}v_i, \ldots , {{\mathop{\mathrm{ad}}\nolimits}}^{q_i+p_i}_{X_{-{\widehat{\alpha }}}} v_i \} =
{\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_i}_{\widehat{\alpha}}$. Consider the vector $w_i = \iota (v_i) \in \underline{\mathfrak{g}}$. Let $W_i = {\mathop{\rm span}\nolimits}\{ w_i, {{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\underline{\alpha }}}w_i, \ldots ,$ ${{\mathop{\mathrm{ad}}\nolimits}}^{q_i+p_i}_{X_{-\underline{\alpha }}}w_i \} $. Because $X_{-\underline{\alpha }} = \iota (X_{-\widehat{\alpha }})$ and ${{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\underline{\alpha }}} {\raisebox{0pt}{$\scriptstyle\circ \, $}}\iota =
\iota {\raisebox{0pt}{$\scriptstyle\circ \, $}}{{\mathop{\mathrm{ad}}\nolimits}}_{X_{-\widehat{\alpha}}}$, it follows that $W_i = \iota (V_i)$ and ${{\mathop{\mathrm{ad}}\nolimits}}^{q_i+p_i+1}_{X_{-\underline{\alpha }}}w_i =0$. Since $V_i$ is an $N_i$-dimensional irreducible representation of ${\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}$ on $\widetilde{\mathfrak{n}}$, we find that the root subalgebra ${\underline{\mathfrak{g}}}^{(\underline{\alpha})} $ of $\underline{\mathfrak{g}}$, which is isomorphic to ${\mathop{\mathrm{sl}}\nolimits}(2, {\mathbb{C}})$ and contains $\iota ({\widehat{\mathfrak{g}}}^{(\widehat{\alpha } )})$, acts irreducibly on $W_i$. Therefore ${{\mathop{\mathrm{ad}}\nolimits}}_{H_{\underline{\alpha }}}w_i = n_i w_i$, where $n_i=$[$\left\{ \begin{array}{rl} {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(q_i+p_i), &
\mbox{if $N_i$ is odd} \\ {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(q_i+p_i+1), & \mbox{if
$N_i$ is even} \end{array} \right. $]{} is a nonnegative integer. This shows that $$-n_i, \, \, -n_i+2, \, \, -n_i+2(2), \ldots , -n_i+2(n_i-1) =
n_i -2, \, \, n_i
\label{eq-sec2subsec4twelveb}$$ lists the elements of the extremal root chains in (\[eq-sec2subsec4eight\]). The root chain (\[eq-sec2subsec4ten\]) with integer pair $(r,s)$ and length $M =r+s+1$ is extremal. Therefore ${\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1 + {\widehat{\beta }}_2}_{\widehat{\alpha }}$ is an $M$-dimensional representation space for an ${{\mathop{\mathrm{ad}}\nolimits}}_{{\widetilde{\mathfrak{g}}}^{(\widehat{\alpha })}}$-irreducible representation. The eigenvalues of ${{\mathop{\mathrm{ad}}\nolimits}}_{H_{\underline{\alpha }}}$ on $i({\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1 +{\widehat{\beta }}_2}_{\widehat{\alpha }})$ are listed as follows. $$-m, \, \, -m+2, \, \, -m+2(2), \ldots , -m+2(m-1) = m-2, \, \, m .
\label{eq-sec2subsec4elevenstarf}$$ Since the vector spaces $i({\widehat{\mathfrak{g}}}^{{\widehat{\beta }}_1+ {\widehat{\beta }}_2}_{\widehat{\alpha }})$ and ${\underline{\mathfrak{g}}}^{{\underline{\beta }}_1 + {\underline{\beta }}_2}_{\underline{\alpha }}$ are isomorphic, equation (\[eq-sec2subsec4elevenstarf\]) lists the elements of the extremal root chain (\[eq-sec2subsec4ten\]) provided that the positive integer $m=$[$\left\{ \begin{array}{rl}
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(r+s), & \mbox{if $M$ is odd} \\ {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(r+s+1), &
\mbox{if $M$ is even} \end{array} \right. $.]{}
From claim 2.4.3 the extremal root chain (\[eq-sec2subsec4ten\]) is a subchain of the unbroken chain (\[eq-sec2subsec4nine\]). Thus the list (\[eq-sec2subsec4elevenstarf\]) is a sublist of the list $$\begin{aligned}
&-\ell =-(n_1+n_2), \, \, -\ell +2, \, \, -\ell +2(2), \ldots , \notag \\
&\hspace{1in} -\ell +2(n_1+n_2-1) = \ell -2, \, \, \ell ,
\label{eq-sec2subsec4thirteen}\end{aligned}$$ which labels the elements of the unbroken chain (\[eq-sec2subsec4nine\]). Suppose that the nonnegative integers $\ell $ and $m$ have a different parity. In particular, suppose that $\ell $ is even and $m$ is odd. Then $1$ appears in the list (\[eq-sec2subsec4elevenstarf\]) but not in the list (\[eq-sec2subsec4thirteen\]). This contradicts the fact that (\[eq-sec2subsec4elevenstarf\]) is a sublist of (\[eq-sec2subsec4thirteen\]). Suppose that $\ell $ is odd and $m$ is even. Then $0$ appears in the list (\[eq-sec2subsec4elevenstarf\]) but not in the list (\[eq-sec2subsec4thirteen\]). Again this contradicts the fact that (\[eq-sec2subsec4elevenstarf\]) is a sublist of (\[eq-sec2subsec4thirteen\]). Therefore $\ell $ and $m$ must have the same parity. Conversely, if $\ell $ and $m$ have the same parity, then the list (\[eq-sec2subsec4elevenstarf\]) is a sublist of the list (\[eq-sec2subsec4thirteen\]). Consequently, for $i=1,2$ there is a $j_i \in {\mathbb{Z}}$, $0 \le j_i \le \ell $, such that $$-\ell +2j_1 = -m \quad \mathrm{and} \quad \ell -2j_2 =m.$$ Therefore $j_1=j_2 =j$. Thus $j$ is the number of elements of the unbroken chain (\[eq-sec2subsec4nine\]) which need to be removed from its left and right ends to obtain the extremal root chain (\[eq-sec2subsec4ten\]). Hence the integer $(q_1+q_2) - (p_1+p_2)$ associated to the unbroken chain (\[eq-sec2subsec4nine\]) with integer pair $(q_1+q_2, p_1+p_2)$ is equal to the Killing integer $r-s$ of the extremal root chain (\[eq-sec2subsec4ten\]) with integer pair $(r,s)$. This proves claim 2.4.4. $\square $
Claim 2.4.4 may be reformulated as
**Claim 2.4.5** Let $\widehat{\alpha } \in \widehat{\Pi }$ with ${\widehat{\beta }}_1$, and ${\widehat{\beta }}_2 \in
\widehat{\mathcal{R}}$ such that ${\widehat{\beta }}_1 + {\widehat{\beta }} _2 \in \widehat{\mathcal{R}}$. For $i=1,2$ let $${\widehat{\beta }}_i - q_i \, \widehat{\alpha }, \ldots , {\widehat{\beta }}_i + p_i \, \widehat{\alpha }
\label{eq-sec2subsec4twelvec}$$ be extremal root chains with integer pair $(q_i,p_i)$ and Killing integer $\langle {\widehat{\beta }}_i, \widehat{\alpha } \rangle = q_i-p_i$. Suppose that the extremal root chain $${\widehat{\beta }}_1+ {\widehat{\beta}}_2 - r \, \widehat{\alpha}, \ldots , {\widehat{\beta }}_1+ {\widehat{\beta }}_2 + s \, \widehat{\alpha}$$ with integer pair $(r,s)$ and Killing integer $\langle {\widehat{\beta }}_1 + {\widehat{\beta }}_2, \widehat{\alpha } \rangle = r-s$ is the sum of the extremal root chains in (\[eq-sec2subsec4twelvec\]). Then $$\langle {\widehat{\beta }}_1+ {\widehat{\beta }}_2, \widehat{\alpha} \rangle =
\langle {\widehat{\beta }}_1 , \widehat{\alpha } \rangle +
\langle {\widehat{\beta }}_2 , \widehat{\alpha } \rangle .
\label{eq-sec2subsec4thirteend}$$
**Proof.** If $-\widehat{\beta } \in \widehat{\Pi }$ and $\widehat{\alpha } \in \widehat{\Pi }$, then by lemma 1.2.3 we have $\langle \widehat{\beta }, \widehat{\alpha } \rangle = - \langle -\widehat{\beta }, \widehat{\alpha} \rangle
=-K_{\widehat{\alpha }}(-\widehat{\beta })$. Similarly, if $\widehat{\beta } \in \widehat{\Pi }$ and $\widehat{\alpha } \in
\widehat{\Pi }$, then by lemma 1.2.3 $\langle \widehat{\beta }, \widehat{\alpha }\rangle
= - \langle \widehat{\beta }, - \widehat{\alpha} \rangle = -K_{-\widehat{\alpha }}(\widehat{\beta })$. Therefore it suffices to prove the claim when $\widehat{\alpha}$, $\widehat{\beta } \in \widehat{\Pi }$. Using claim 2.4.4 we deduce that for every root $\widehat{\alpha }$ in $\widehat{\Pi }$ the function $K_{\widehat{\alpha }}:\widehat{\mathcal{R}} \rightarrow {\mathbb{Z}}: \widehat{\beta }\mapsto
\langle \widehat{\beta }, \widehat{\alpha }\rangle $ is linear. $\square $
This completes the verification of
**Theorem 2.4.6** Let $\widetilde{\mathfrak{g}} = \mathfrak{g} \oplus \widetilde{\mathfrak{n}}$ be a very special sandwich algebra of class $\mathcal{C}$. Then the collection $\Phi $ of nonzero roots in $\widehat{\mathcal{R}}$ associated to the nilradical $\widetilde{\mathfrak{n}}$ is a system of roots.
Because simple sandwich algebras of class $\mathcal{C}$ are Lie algebras which are not semisimple and have a system of roots, the notion of a system of roots is a conservative generalization of the concept of a root system of a semisimple Lie algeba.
[**Appendix**]{} \[appendix\] From the classification of very special sandwich algebras, see [@cushman17a], a very special sandwich algebra of $\mathcal{C}$ is one of the following: ${\widetilde{\mathbf{C}}}_{\ell +1}$, ${\widetilde{\mathbf{G}}}^1_2$, ${\widetilde{\mathbf{G}}}^2_2$, ${\widetilde{\mathbf{F}}}_4$, ${\widetilde{\mathbf{E}}}_6$ or ${\widetilde{\mathbf{E}}}_7$. The following list gives $\widetilde{\mathfrak{g}}$, $\mathfrak{g}$, $\underline{\mathfrak{g}}$, the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, and the roots $\widehat{\mathcal{R}}$ of the adjoint action of $\mathfrak{h}$ on $\widetilde{\mathfrak{n}}$. By ${\underline{\mathbf{X}}}^{(\ell )}_n$ we mean the simple Lie algebra of Cartan type $\underline{\mathbf{X}}$ of rank $n-1$ whose Dynkin diagram is obtained by removing the node numbered $\ell $ from the Dykin diagram of the simple Lie algebra of Cartan type $\mathbf{X}$ of rank $n$. We use the following notation. Let ${\{ {\varepsilon }_i \} }^n_{i=1}$ be the standard basis for $({{\mathbb{R}}}^n)^{\ast }$ and let ${\{ e_i \} }^n_{i=1}$ be the standard dual basis of ${{\mathbb{R}}}^n$. ${\mathrm{h}}_{2n+1}$ denotes the the Lie algebra of the $2n+1$ dimensional Heisenberg group.
[**Very special sandwich algebras of class $\mathcal{C}$**]{}
[ll]{} $1$. & ${\widetilde{\mathbf{C}}}_{\ell +1}$, ${\mathbf{C}}^{(1)}_{\ell +1}$, ${\underline{\mathbf{C}}}_{\ell +1}$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_{2\ell +1}$;\
------------------------------------------------------------------------
& $\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \big\{ h_i = e_i-e_{i+1}, \, 2 \le i \le \ell ; \, h_{\ell +1} = e_{\ell +1} \big\} $\
------------------------------------------------------------------------
& $\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = -2{\varepsilon }_1|\mathfrak{h}, \\
{\widehat{\alpha }}_k & = -({\varepsilon }_1-{\varepsilon }_{k+1})|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, k} & = -({\varepsilon }_1+{\varepsilon }_{k+1})|\mathfrak{h}, \, 1 \le k \le \ell \big \}
\end{array} \right. $\
$2.$ & ${\widetilde{\mathbf{G}}}^1_2$, ${\mathbf{G}}^{(1)}_2$, ${\underline{\mathbf{G}}}_2$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_3 $;\
& $\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \big\{ h_2 = -2e_1-e_2-e_3 \big\} $\
------------------------------------------------------------------------
& $\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = ({\varepsilon }_2-{\varepsilon }_3)|\mathfrak{h}, \\
{\widehat{\alpha }}_1 & = ({\varepsilon }_1-{\varepsilon }_3)|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, 1} & = ({\varepsilon }_2 -{\varepsilon }_3)|\mathfrak{h}
\end{array} \right. $
[ll]{} $3. $ &${\widetilde{\mathbf{G}}}^2_2$, ${\mathbf{G}}^{(2)}_2$, ${\underline{\mathbf{G}}}_2$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_5 $;\
& $\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \big\{ h_1 = e_1-e_2 \big\} $\
------------------------------------------------------------------------
& $\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = ({\varepsilon }_1+{\varepsilon }_2-2{\varepsilon }_3)|\mathfrak{h}, \\
{\widehat{\alpha }}_1 & = (-{\varepsilon }_1+2{\varepsilon}_2-{\varepsilon }_3)|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, 1} & = (2{\varepsilon }_1- {\varepsilon }_2 -{\varepsilon }_3)|\mathfrak{h} \\
{\widehat{\alpha }}_2 & = ({\varepsilon }_2-{\varepsilon }_3)|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, 2} & = ({\varepsilon }_1-{\varepsilon }_3)|\mathfrak{h} \end{array} \right. $
------ --------------------------------------------------------------------------------------------------------------------------------------------------------------
$4.$ $\widetilde{\mathbf{F}}_4$, ${\mathbf{F}}^{(1)}_4$, ${\underline{\mathbf{F}}}_4$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_{15} $;
$\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \{ h_2 = e_3-e_4, \, h_3 = 2e_4, \,
h_4 = e_1-e_2-e_3-e_4 \} $
$\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = ({\varepsilon }_1+{\varepsilon }_2)|\mathfrak{h}, \\
{\widehat{\alpha }}_1 & = -{\varepsilon }_1|\mathfrak{h}, \, \, {\widehat{\alpha }}^{\, 1} =
-{\varepsilon }_2|\mathfrak{h} \\
{\widehat{\alpha }}_{k+1} & = (-{\varepsilon }_1+{\varepsilon}_{k+2})|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, k+1} & = (-{\varepsilon }_2- {\varepsilon }_{k+2})|\mathfrak{h}, \, \, k=1,2 \\
{\widehat{\alpha }}_{k+3} & = (-{\varepsilon }_1-{\varepsilon}_{k+2})|\mathfrak{h}, \\
{\widehat{\alpha }}^{\, k+3} & = (-{\varepsilon }_2+ {\varepsilon }_{k+2})|\mathfrak{h}, \, \, k=1,2 \\
{\widehat{\alpha }}_6 & =
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(-{\varepsilon }_1 -{\varepsilon }_2+{\varepsilon }_3+{\varepsilon}_4 )|\mathfrak{h}, \\
\rule{0pt}{12pt} {\widehat{\alpha }}^{\, 6} & =
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(-{\varepsilon }_1 -{\varepsilon }_2-{\varepsilon }_3-{\varepsilon}_4 )|\mathfrak{h}, \\
\rule{0pt}{12pt} {\widehat{\alpha }}_7& =
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(-{\varepsilon }_1 -{\varepsilon }_2-{\varepsilon }_3+{\varepsilon}_4 )|\mathfrak{h}, \\
\rule{0pt}{12pt} {\widehat{\alpha }}^{\, 7} & =
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(-{\varepsilon }_1 -{\varepsilon }_2+{\varepsilon }_3-{\varepsilon}_4 )|\mathfrak{h} \end{array} \right. $
------ --------------------------------------------------------------------------------------------------------------------------------------------------------------
[ll]{} $5.$ &${\widetilde{\mathbf{E}}}_6$, ${\mathbf{E}}^{(2)}_6$, ${\underline{\mathbf{E}}}_6$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_{21}$;\
& $\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \big\{
h_1 = {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(e_1-\sum^5_{i=2}e_i +3e_6), \, h_i = e_{i-1}-e_{i-2}, \, 3 \le i \le 6 \big\} $\
------------------------------------------------------------------------
& $\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = -{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(\sum^6_{i=1}{\varepsilon }_i)|\mathfrak{h}, \\
{\widehat{\alpha }}_{i<j} & = -({\varepsilon }_i+{\varepsilon}_j)|\mathfrak{h}, \, \, 1 \le i < j \le 5 \\
{\widehat{\alpha }}_{j\ell m} & = {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}( \sum^5_{i=1} (-1)^{k(i)} {\varepsilon}_i - {\varepsilon }_6)|\mathfrak{h} \\
&\hspace{.25in} \parbox[t]{3.5in}{$k(j) = k(\ell ) = k(m) =1$; $k(i) =0$ \\ if $i \notin \{ j , \ell , m \} $;
$j, \ell , m \in \{ 1, \ldots , 5 \} $ distinct} \end{array} \right. $
[ll]{} $6.$ &${\widetilde{\mathbf{E}}}_7$, ${\mathbf{E}}^{(1)}_7$, ${\underline{\mathbf{E}}}_7$; $\widetilde{\mathfrak{n}} = {\mathrm{h}}_{33}$;\
& $\mathfrak{h} = {{\mathop{\rm span}\nolimits}}_{{\mathbb{C}}} \big\{ h_2 = e_2-e_1, \, h_3 = e_2+e_1, \, h_i =
e_{i-1}-e_{i-2}, \, 4 \le i \le 7 \big\} $\
------------------------------------------------------------------------
& $\widehat{\mathcal{R}} = \left\{ \begin{array}{rl}
\widehat{\zeta } & = -{\varepsilon }_7|\mathfrak{h}, \\
{\widehat{\alpha }}_1 & =
{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}( \sum^6_{i=1}{\varepsilon}_i - {\varepsilon }_7)|\mathfrak{h} \\
\rule{0pt}{12pt} {\widehat{\alpha }}^{\, 1} & = -{\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}(\sum^7_{i=1}{\varepsilon }_i)|\mathfrak{h}, \\
\rule{0pt}{12pt} {\widehat{\alpha }}_{j\ell } & = {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}( \sum^6_{i=1}(-1)^{k(i)}{\varepsilon}_i - {\varepsilon }_7)|\mathfrak{h} \\
&\hspace{.25in} \parbox[t]{3.5in}{$k(j) = k(\ell ) =1$; $k(i) =0$ \\ if $i \notin \{ j , \ell \} $;
$j, \ell \in \{ 1, \ldots , 5 \} $ distinct} \\
\rule{0pt}{12pt} {\widehat{\alpha }}^{j\ell } & = {\mbox{$\frac{\scriptstyle 1}{\scriptstyle 2}\,$}}( \sum^6_{i=1}(-1)^{k(i)}{\varepsilon}_i - {\varepsilon }_7)|\mathfrak{h} \\
&\hspace{.25in} \parbox[t]{3.5in}{$k(j) = k(\ell ) =0$; $k(i) =1$ \\ if $i \notin \{ j , \ell \} $;
$j, \ell \in \{ 1, \ldots , 5 \} $ distinct} \end{array} \right. $
\[references\]
[99]{}
Cohen, A.M., Ivanyos, Gábor, and Roozemond, Dan, Simple Lie algebras having extremal elements, *Indag. Math* **19** (2008) 177–188.
Cushman, R., Very special sandwich algebras,\
`ArXiv.math.1708.02172`.
Cushman, R., The Weyl group of a very special sandwich algebra of class $\mathcal{C}$, (in preparation).
Humphreys, J.E., *Introduction to Lie Algebras and Representation Theory*, Graduate Texts in Mathematics, vol. **9**, Springer Verlag, New York, 1972.
Wikipedia, ${\mathbf{E}}_6$, ${\mathbf{E}}_7$, ${\mathbf{E}}_8$, and “semidirect product” `URL: www.wikiped-`\
`ia.org`.
[^1]: version:
|
---
author:
- 'C. Ricci, Y. Ueda, K. Ichikawa, S. Paltani, R. Boissay, P. Gandhi, M. Stalevski, and H. Awaki'
bibliography:
- 'iron\_cf.bib'
date: 'Received; accepted'
title: 'The narrow FeK$\alpha$ line and the molecular torus in active galactic nuclei - an IR/X-ray view'
---
Introduction
============
The unification model of active galactic nuclei (AGN) predicts that the supermassive black hole (SMBH) in their centre is surrounded by a molecular toroidal-like structure [@Antonucci:1993kb]. Anisotropic obscuration was originally required to explain the detection of broad lines in polarised light found in the optical spectrum of the Seyfert2 [@Antonucci:1985qo; @Miller:1983lr], and is now considered to be one of the fundamental ingredients needed to explain the structure of AGN. According to this paradigm Seyfert1s (Sy1s) are observed pole-on with respect to the molecular torus, while Seyfert2s (Sy2s) are seen edge-on. The radiation produced by the central engine and absorbed by the torus is mainly re-emitted in the mid-IR band (MIR, 5–30$\mu$m). The first direct observation of the dusty torus was carried out using MIR interferometry, for [@Jaffe:2004fr]. This work was then followed by several others (e.g., [@Prieto:2004uq], [@Prieto:2005kx], [@Meisenheimer:2007fk], [@Tristram:2007ys], [@Raban:2009vn]), and all of them detected a clear compact structure within few parsecs from the SMBH.
The fluorescent iron K$\alpha$ line is possibly the most important tracer of the material surrounding the SMBH. The FeK$\alpha$ line is made of two components, K$\alpha_1$ (E=6.404keV) and K$\alpha_2$ (E=6.391keV), with a branching ratio of K$\alpha_1$:K$\alpha_2$=2:1, and it is produced when one of the two K-shell electrons of an iron atom is ejected following photoelectric absorption of an X-ray photon. After the photoelectric event, the excited state can decay in two ways. i) An L-shell electron drops into the K-shell releasing a photon. ii) The excess energy is carried away through the ejection of an L-shell electron (Auger effect). The fluorescent yield ($Y$) determines the probability of fluorescence versus the Auger effect. The iron line is the strongest X-ray line produced from the reprocessing of the primary continuum, because of the Fe relative abundance, and because the fluorescent yield is proportional to the fourth power of atomic number ($Y\propto Z^4$).
Amongst the other lines produced by X-ray reflection from neutral material the strongest are the iron K$\beta$ line at 7.06keV ($\sim13.5\%$ of the flux of the FeK$\alpha$, [@Palmeri:2003qf]), and the nickel K$\alpha$ line at $\sim7.47$keV (e.g., [@Yaqoob:2011kx]). The first evidence of an FeK$\alpha$ line in the X-ray spectrum of an AGN was found by @Mushotzky:1978kx when studying [*OSO-8*]{} observations of CentaurusA. Ten years later @Guilbert:1988oq and @Lightman:1988qy predicted that fluorescent emission from neutral iron should be common in the X-ray spectra of Seyfert galaxies. Following this, @Nandra:1989uq found evidence of an emission line at $E\sim6$keV in the [*EXOSAT*]{} spectrum of . In the same year, using a larger sample @Pounds:1989fj found significant iron K$\alpha$ emission lines in the spectra of three more Seyfert1 galaxies: , , and . Since then, thanks to the enormous progress in the development of X-ray detectors, iron lines have been found to be almost ubiquitous in AGN (e.g., [@Fukazawa:2011fk]).
The FeK$\alpha$ line is made of two components. While the narrow core of the line, with a full width at half maximum (FWHM) of $\simeq 2,000\rm\,km\,s^{-1}$ [@Shu:2011fk], is observed in almost all AGN, in $\simeq 35-45\%$ of the cases [@de-La-Calle-Perez:2010fk], an additional broadened component due to relativistic effects (e.g., [@Fabian:2003dp]) or to distortion of the continuum caused by clumpy ionised absorbers in the line-of-sight (e.g., [@Turner:2009fk], [@Miyakawa:2012vn]) is found. The size of the narrow FeK$\alpha$ emitting region is on average $\sim 3$ times larger than that of the broad line region [@Shu:2011fk], which seems to point towards most of the narrow core originating in the molecular torus. Another argument in favour of this scenario is the weak variability of reflection-dominated Compton-thick (CT, $N_{\rm\,H}\geq 10^{24}\rm\,cm^{-2}$) AGN (e.g., [@Bianchi:2012dq] and references therein). A torus origin of the FeK$\alpha$ line would imply that its equivalent width (EW) is directly linked to the half-opening angle of the torus $\theta_{\rm\,OA}$ [@Krolik:1994fk] and to its equatorial column density $N_{\mathrm{\,H}}^{\mathrm{\,T}}$ (e.g., [@Ikeda:2009nx], [@Murphy:2009uq]).
An anti-correlation between the equivalent width of the FeK$\alpha$ line and the X-ray luminosity of AGN has been found by a large number of studies (e.g., [@Iwasawa:1993ys], [@Bianchi:2007vn], [@Shu:2010zr]). Such a trend is known as the X-ray Baldwin effect, for analogy with the Baldwin effect [@Baldwin:1977fk], i.e. the decrease of the CIV$\,\lambda 1549$ EW with the luminosity. Several explanations have been proposed for the X-ray Baldwin effect: i) a luminosity-dependent variation in the ionisation state of the iron-emitting material [@Nandra:1997fk; @Nayakshin:2000uq]; ii) the effect of the delay between the primary X-ray emission and the reflection component [@Jiang:2006vn; @Shu:2012fk]; iii) the decrease in the number of continuum photons in the iron line region with the Eddington ratio ($\lambda_{\mathrm{Edd}}$, [@Ricci:2013vn]), as an effect of the correlation between the photon index ($\Gamma$) of the continuum and $\lambda_{\mathrm{Edd}}$ (e.g., [@Shemmer:2008fk]); iv) the decrease of the covering factor of the torus with the luminosity (e.g., [@Page:2004kx], [@Zhou:2005ys]), as expected by luminosity-dependent unification models (e.g., [@Ueda:2003qf]). In a recent paper [@Ricci:2013fk], we have shown that the decrease of the covering factor of the torus with the luminosity is able to reproduce the slope of the X-ray Baldwin effect for a wide range of equatorial column densities of the torus.
The thermal MIR continuum is produced by circumnuclear dust (e.g., [@Stalevski:2012fk]), which is heated by the optical/UV/X-ray photons produced in the disk and in the warm corona. The first attempts to model the MIR spectral energy distribution (SED) of AGN using a torus with a smooth density distribution (e.g., [@Pier:1992ys; @Pier:1993zr]) were able to model part of the SED but not to produce realistic MIR spectra. This was probably the first evidence that dust in the torus does not have a smooth distribution. Already years earlier, @Krolik:1988ly hypothesised that the torus is made of optically thick dusty clouds, because a smooth distribution could not survive close to the SMBH. Several other pieces of evidence of a clumpy structure of the torus have been added in past years. From interferometric observations of Circinus galaxy @Tristram:2007ys found that the data could not support a smooth-distribution scenario, but rather pointed towards the dust having a clumpy or filamentary structure. The discovery that Seyfert1s and Seyfert2s follow the same X-ray/MIR luminosity correlation (e.g., [@Gandhi:2009uq], [@Ichikawa:2012fk]), and the detection of silicate emission in Seyfert2s [@Sturm:2006qf] also provide strong arguments for the clumpy scenario. Important information about the structure of the torus (e.g., its covering factor and number of clouds, see Section\[Sect:midIR\_prop\]) can be obtained by modelling the IR spectra of AGN using clumpy torus models such as the one developed by @Nenkova:2002fk. In the past few years, several studies (e.g., [@Mor:2009fk], [@Alonso-Herrero:2011zr]) have carried out detailed analyses of the MIR properties of AGN for a significant number of objects.
The aim of this work is to compare the properties of the narrow component of the iron K$\alpha$ line with those of the torus obtained by recent MIR studies. The paper is organised as follows. In Sect.\[Sect:midIR\_prop\], we present our MIR/X-ray AGN sample and describe the [*XMM-Newton*]{}/EPIC data analysis; in Sect.\[Sect:spectral\_analysis\] we illustrate the X-ray spectral analysis; in Sect.\[Sect:renorm\] we discuss how to remove the effects of degeneracy caused by the X-ray photon index, the observing angle, the half-opening angle of the torus, and the torus equatorial column density on the values of FeK$\alpha$ EW; and in Sect.\[Sect:EWvsf2\] we study the relation between the FeK$\alpha$ EW and the physical characteristics of the torus. In Sect.\[Sect:summary\] we discuss our findings and present our conclusions.
5 ![Schematic representation of the angles considered. $\theta_{\rm\,i}$ is the inclination angle of the observer, while $\theta_{\rm\,OA}$ ($\pi/2-\sigma_{\rm\,tor}$) is the half-opening angle of the torus. $N_{\rm\,H}^{\rm\,T}$ is the equatorial column density of the torus, i.e. the maximum value of $N_{\rm\,H}$ for any value of $\theta_{\rm\,i}$.[]{data-label="fig:geometry"}](geometry.ps "fig:"){height="9cm"}
Sample and X-ray data analysis {#Sect:midIR_prop}
==============================
To study the relation between the iron K$\alpha$ line and the properties of the torus obtained from MIR studies, we used the sample reported in the recent work of [@Elitzur:2012vn], which includes the works of [@Mor:2009fk], @Nikutta:2009kx, @Alonso-Herrero:2011zr, @Deo:2011vn, and @Ramos-Almeida:2011ly. All these works used the IR clumpy torus model of @Nenkova:2002fk [@Nenkova:2008uq; @Nenkova:2008kx], which allows fundamental characteristics of the torus to be deduced by fitting MIR spectra. In the model, the optical luminosity is used to estimate the bolometric emission of the accreting system irradiating the torus. The free parameters of this model are the torus width parameter ($\sigma_{\mathrm{tor}}=\pi/2-\theta_{\rm\,OA}$, see Fig.\[fig:geometry\]), the mean number of clouds along the equatorial line ($N_0$), the $5500\,\AA$ dust optical depth of a single cloud ($\tau_{\mathrm{V}}$), the inclination angle of the torus with respect to the line of sight ($\theta_{\mathrm{\,i}}$), the ratio between the outer and inner radius of the torus ($Y$), and the index of the radial power-law distribution of clouds (q, where the number of clouds is given by $N(r)\propto r^{\mathrm{-q}}$).
We cross-correlated the sample of [@Elitzur:2012vn] with the [*XMM-Newton*]{} [@Jansen:2001fk] public data archive (as of November 2012), selecting only type-I AGN to avoid uncertainties in the estimates of the FeK$\alpha$ and continuum flux due to absorption. Amongst the sources with public [*XMM-Newton*]{} observations, PG1700+518 was detected with a very low S/N, which did not allow constraining the parameters of the Fe K$\alpha$ line, so that its spectrum was not used for our study. The final sample contains a total of 49 observations of 24 objects. Most of the sources (19) in our final sample are from the work of @Mor:2009fk, while four sources are taken from @Alonso-Herrero:2011zr and only one from @Ramos-Almeida:2011ly. None of the sources reported in the works of @Nikutta:2009kx and @Deo:2011vn were observed by [*XMM-Newton*]{}. Although all the works use the clumpy torus model of @Nenkova:2002fk [@Nenkova:2008uq; @Nenkova:2008kx], some differences exist in the approach they followed. @Mor:2009fk fitted the [*Spitzer*]{}/IRS $\sim2-35\,\mu$m spectra using a three-component model, which includes a dusty clumpy torus, a clumpy narrow-line region (NLR), and black-body emission from hot dust. This last component accounts for the near-IR (NIR, $\lambda\lesssim 5\mu$m) excess observed when fitting the spectra using only the first two components. In a recent work @Mor:2011kx studied a large sample of $\sim 15,000$ AGN and show that most AGN need this hot dust component to explain their NIR spectra. @Alonso-Herrero:2011zr combined the IR photometric SED with MIR ground-based spectroscopic data in the $8-13\,\mu$m, while @Ramos-Almeida:2011ly only used photometric data. Both @Alonso-Herrero:2011zr and @Ramos-Almeida:2011ly fitted the data using only the clumpy torus model, because the high angular resolution data they use in their work allows contamination from NLR dust to be ignored.
To study the Fe K$\alpha$ EW we used the data obtained by the PN [@Struder:2001uq] and MOS [@Turner:2001fk] cameras on-board [*XMM-Newton*]{}. The original data files (ODFs) were downloaded from the [*XMM-Newton*]{} Science Archive (XSA)[^1] and then reduced using the [*XMM-Newton*]{} Standard Analysis Software (SAS) version 12.0.1 [@Gabriel:2004fk]. The raw PN and MOS data files were processed using the `epchain` and `emchain` tasks, respectively. For each observation we checked the background light curve in the 10–12 keV energy band of the data sets in order to detect and filter the exposures for periods of high background activity. We selected only patterns that correspond to single and double events (PATTERN $\leq 4$) for PN, and to single, double, triple, and quadruple events for MOS (PATTERN $\leq 12$), as suggested by the standard guidelines. The source spectra were extracted from the final filtered event list using circular regions centred on the object (with a typical radius of 30arcsec), while the background was estimated from regions close to the source (preferably on the same CCD), where no other source was present (with a radius of 40arcsec). For sources detected with a low S/N, we extracted the spectra using a smaller radius (10arcsec). We checked for pile-up with the `epatplot` task, and for those observations where it was significant (see Table\[tab:obslog\]), we used annular regions centred on the source, with an inner radius of 5 to 15arcsec, depending on the strength of the pile-up. We added a multiplicative factor to the models to account for cross-calibration between PN and MOS. We fixed the factor to 1 for EPIC/PN and left the MOS1 and MOS2 factors free. For all the spectra the value of the factor turned out to be close to one within a few percentage points. The ancillary response matrices (ARFs) and the detector response matrices (RMFs) were generated using the tasks `arfgen` and `rmfgen`, respectively. The spectra were grouped to have at least 20 counts per bin, in order to use $\chi^{2}$ statistics.
The list of AGN used, together with the values of their redshift ($z$), of the Galactic column density in their direction ($N_{\rm\,H}^{\rm\,G}$), and their X-ray observation log is reported in Table\[tab:obslog\].
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X-ray spectral analysis {#Sect:spectral_analysis}
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The X-ray spectral analysis was carried out using XSPEC12.7.1b [@Arnaud:1996kx]. Since we are dealing with objects that may have different characteristics in the X-rays, we started the analysis from a simple baseline model and then added absorbing or emitting components to improve the $\chi ^{2}$. More complex models were adopted based on the results of the F-test, using a probability of $p=95\%$ as a threshold. The baseline model consists of a power-law continuum absorbed by Galactic absorption plus a Gaussian line to account for the iron K$\alpha$ emission (`wa_{\mathrm{G}}*(zpo+zgauss)` in XSPEC).
The most common features observed in the X-ray spectra of type-I AGN are ionised absorbers (often called warm absorbers) and a soft excess. Ionised absorption is believed to be produced in disk outflows (e.g., [@Turner:2009fk]) and was accounted for using the `zxipcf` model [@Reeves:2008kx]. This multiplicative model uses a grid of XSTAR [@Kallman:2001fk; @Bautista:2001uq] photoionised absorption models, and its free parameters are the covering factor of the ionised absorber $f_{\rm\,W}$, its column density ($N_{\rm\,H}^{\rm\,W}$) and its ionisation parameter ($\xi$). The ionisation parameter is given by $\xi=L_{\mathrm{ion}}/nr^2$, where $L_{\mathrm{ion}}$ and $r$ are the 5eV-300keV luminosity and distance from the absorber of the ionising source, respectively, while $n$ is the density of the absorber. The origin of the soft excess is still controversial and might be related to blurred reflection (e.g., [@Crummy:2006vn]), to Comptonisation of ultraviolet disk photons in a plasma cooler than the one responsible for the primary continuum (e.g., [@Mehdipour:2011ys], [@Noda:2013fk]), or to smeared absorption (e.g., [@Gierlinski:2004zr]). Since we are not interested in a detailed analysis of the soft excess, we adopted a simple phenomenological model (`bremsstrahlung`) to account for this feature. All the sources of the sample require more complex models than the baseline. The models we applied to fit the X-ray spectra are the following (listed in the order in which they were applied):[**Model A**]{}. Baseline model and a bremsstrahlung component at low energies to represent the soft excess. In XSPEC this is written as `wa_{\mathrm{G}}*(zpo + bremss + zgauss)`. The free parameters of this model are the photon index of the power-law continuum ($\Gamma$), the temperature of the bremsstrahlung (kT), the energy of the Fe K$\alpha$ line ($E_{\rm\,K\alpha}$), and the normalisations of the three components. This model was used for 19observations.[**Model B**]{}. Baseline model absorbed by a partially covering warm absorber: `wa_{\mathrm{G}}*zxipcf(zpo +zgauss)`. The free parameters are those of the baseline model, plus the parameters of the warm absorber ($\xi$, $N_{\rm\,H}^{\rm\,W}$ and $f_{\rm\,W}$). This model was adopted for three observations.[**Model C**]{}. Baseline model and a soft excess, absorbed by a partially covering warm absorber: `wa_{\mathrm{G}}*zxipcf(zpo + bremss+zgauss)`.The free parameters are the same as in modelA, plus the parameters of the warm absorber. A total of 18observations were fitted using this model.[**Model D**]{}. Baseline model absorbed by two partially covering warm absorbers: `wa_{\mathrm{G}}*zxipcf*zxipcf(zpo +zgauss)`. The free parameters are the same as in modelB, with the addition of the parameters of the second ionised absorber ($\xi^2$, $N_{\rm\,H,2}^{\rm\,W}$, $f_{\rm\,W}^2$). Four observations were fitted using this model.[**Model E**]{}. Baseline model plus a soft excess and a warm absorber, absorbed by neutral material: `wa_{\mathrm{G}}*zwabs*zxipcf(zpo + bremss+zgauss)`. The free parameters are the same as in modelC, plus the column density of the neutral absorber ($N_{\rm\,H}^{\rm\,C}$). This model was used for one observation.[**Model F**]{}. Baseline model and a soft excess absorbed by two partially covering warm absorbers: `wa_{\mathrm{G}}*zxipcf*zxipcf(zpo + bremss+zgauss)`. The free parameters are the same as in modelD, with the addition of the temperature and normalisation of the bremsstrahlung. Three observations were fitted using this model.[**Model G**]{}. Baseline model and a soft excess, obscured by a neutral and two partially covering ionised absorbers: `wa_{\mathrm{G}}*zwabs*zxipcf*zxipcf(zpo + bremss+zgauss)`. The free parameters are the same as in modelD, with the addition of $N_{\rm\,H}^{\rm\,C}$ and of the temperature and normalisation of the bremsstrahlung. This model was used for one observation.
For the 14 observations for which it was not possible to constrain the energy of the FeK$\alpha$ line, we fixed the parameter to $E_{\rm\,K\alpha}=6.4\rm\,keV$ (in the rest frame of the AGN). We fixed the width of the Gaussian line to $\sigma=1\rm\,eV$, a value below the energy resolution of EPIC/PN and MOS, in order to only consider the narrow core of the iron K$\alpha$ line. We used the values of the Galactic hydrogen column density $N_{\rm\,H}^{\rm\,G}$ obtained by @Dickey:1990uq mapping the HI emission of the Galaxy (see Table\[tab:obslog\]).
For all the sources, we tested whether adding a broad component of the iron K$\alpha$ line would significantly improve the fit. This was done using the broad-line profile of @Laor:1991fk (in XSPEC `laor2`). Similar to what was done by @Nandra:2007ly, we fixed the internal (for $r\leq R_{\mathrm{break}}$) emissivity indices to $\beta_1=0$, and the external one (for $r>R_{\mathrm{break}}$) to $\beta_2=3$. The inclination angle was fixed to the value obtained by MIR studies ($i=\theta_{\rm\,i}$). We fixed the energy of the broad line to $E_{\rm\,K\alpha}^{\rm\,broad}=6.4$keV (in the reference frame of the AGN) and tried two scenarios: one in which the inner radius of the iron K$\alpha$-emitting region is $r_{\mathrm{in}}=6\,r_{\mathrm{g}}$ (equivalent to the non-rotating Schwarzschild black hole case), where $r_{\mathrm{g}}=GM_{\mathrm{BH}}/c^2$ is the gravitational radius, and the other in which $r_{\mathrm{in}}=1.24\,r_{\mathrm{g}}$ (equivalent to the rotating Kerr black hole scenario). The outer radius of the FeK$\alpha$ emitting region was set in both cases to $r_{\mathrm{out}}=400\,r_{\mathrm{g}}$, while $R_{\mathrm{break}}$ was left as a free parameter. We performed an F-test using the results obtained with and without relativistic FeK$\alpha$ emission, and rejected the presence of a broad line if the probability was $p<95\%$. We found that a broad component of the line is needed for 14 observations and 6 objects (25% of the total sample). In the following we use only the narrow component of the Fe K$\alpha$ line. For 21 observations additional emission lines (such as OVII, NeIX, FeXXV, or FeXXVI) were needed to obtain a good reduced $\chi^{2}$. In Appendix\[Appendix0\] we report the values of the main parameters obtained by our spectral analysis, while all the details of the fits are reported in Appendix\[Appendix1\]. As an example we illustrate in Fig.\[fig:xrayspec\] two typical fits to the X-ray spectra of the sources of our sample. Several of the PG quasars of our sample have been studied by [@Jimenez-Bailon:2005fk] (see also [@Piconcelli:2005ly]), and the values of the EW we obtained are consistent with those reported in their paper.
The flux of the power-law continuum in the 2–10keV band ($F_{\mathrm{\,2-10}}$) was obtained using the convolution model `cflux` in XSPEC. The $k$-corrected continuum luminosities ($L_{\mathrm{\,2-10}}$) were calculated using $$L_{\mathrm{\,2-10}}=4\pi d_{\rm\,L}^{2}\frac{F_{\mathrm{\,2-10}}}{(1+z)^{2-\Gamma}},$$ where $d_{\rm\,L}$ is the luminosity distance. We used standard cosmological parameters ($H_{0}=70\rm\,km\,s^{-1}\,Mpc^{-1}$, $\Omega_{\mathrm{m}}=0.3$, $\Omega_{\Lambda}=0.7$). The iron K$\alpha$ luminosities ($L_{\rm\,K\alpha}$) were calculated in a similar fashion, excluding the $1/(1+z)^{2-\Gamma}$ $k$-correction.
Renormalising the values of EW {#Sect:renorm}
==============================
There are at least six elements that could affect the value of the FeK$\alpha$ EW and introduce a significant scatter in the correlations with the torus properties obtained by MIR studies: i) *Variability*. The delayed response of the reprocessing material to flux changes of the continuum is expected to have a significant impact on the observed values of EW. ii) The *photon index* of the X-ray emission. Higher values of $\Gamma$ imply a steeper continuum and fewer photons at the energy of the iron K$\alpha$ line, which results in lower values of the EW (e.g., [@Ricci:2013vn]). iii) The *inclination angle* of the observer with respect to the torus. For the geometries considered here, lower values of $\theta_{\rm\,i}$ produce higher values of EW because the observer is able to see more of the reflected flux (e.g., [@Ikeda:2009nx], see Fig.\[fig:ew\_thetaoi\]). iv) The *equatorial column density of the torus* ($N_{\rm\,H}^{\rm\,T}$). EW increases with $N_{\rm\,H}^{\rm\,T}$ up to $\log N_{\rm\,H}^{\rm\,T}\simeq 24$, and above this value is roughly constant (e.g., [@Ghisellini:1994uq]). v) The *half-opening angle of the torus*. The EW of the line decreases for increasing values of $\theta_{\rm\,OA}$ (e.g., [@Ikeda:2009nx]). vi) *Metallicity*. Lower values of the metallicity produce lower values of EW.
While the impact on the FeK$\alpha$ EW of the first five elements can be reduced, our knowledge of the metallicity of the circumnuclear material of AGN is still poor, so that it is not possible to take this factor into account. In the following, we describe our procedure for renormalising and correcting the values of EW.
![Ratio of the iron K$\alpha$ EW at $\theta_{\mathrm{\,i}}=5^{\circ}$ and the $EW(\theta_{\rm\,i})$ for different values of the equatorial column density of the torus $N_{\rm\,H}^{\rm\,T}$ obtained using the model of @Ikeda:2009nx for $\theta_{\rm\,OA}=70^{\circ}$ and $\Gamma=1.9$. The scatter in the figure is intrinsic to the Monte Carlo simulations.[]{data-label="fig:ew_thetaoi"}](plot_ew_thetai_norm_oa70.ps){width="9cm"}
Variability
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Continuum variability is expected to affect the FeK$\alpha$ EW measured by single observations of AGN. The bulk of the material responsible for the FeK$\alpha$ line is in fact thought to be located several light years from the X-ray source, so that variations in the continuum do not correspond to simultaneous variations in the line emission. This implies that if a source enters a high-flux state, then the flux of the FeK$\alpha$ line relative to that of the continuum (i.e., its EW) is lower than the real value, while it would be higher in a low-flux state. This has been confirmed by the recent work of @Shu:2012fk, who found an anti-correlation between EW and flux for different observations of the same sources. @Shu:2010zr show that variability might also play a role in the X-ray Baldwin effect and that the anti-correlation is attenuated when the values of EW are averaged over several observations. To account for this effect we averaged, when possible, the results obtained by different observations of the same source. Since in several cases the same source was observed at an interval of a few days, we averaged all the parameters obtained by observations carried out within one month. Comparing the fluxes of the narrow Fe K$\alpha$ line for the sources for which several observations were available, we found that in almost all cases the fluxes are consistent within the uncertainties. The only exceptions are IC4329A and NGC4151. The flux of the narrow line in IC4329A varies from $1.0^{+0.3}_{-0.4}\times 10^{-12}$ to $5.1^{+0.8}_{-0.6}\times 10^{-13}\rm\,erg\,cm^{-2}\,s^{-1}$ on a time span of 2.5 years, while in NGC4151 the line was at its maximum in May 2006 ($2.6^{+0.1}_{-0.1}\times 10^{-12}\rm\,erg\,cm^{-2}\,s^{-1}$) and at its minimum in December 2000 ($1.8^{+0.1}_{-0.1}\times 10^{-12}\rm\,erg\,cm^{-2}\,s^{-1}$). The low variability of the flux of the narrow Fe K$\alpha$ line agrees with the idea that the cold material where the X-ray radiation is reprocessed is located far away from the X-ray source.
Removing the dependence on $\Gamma$, $\theta_{\rm\,i}$, and $N_{\rm\,H}^{\rm{\,T}}$ {#sect:corrections}
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Thanks to the advent of physical torus models such as those developed by @Murphy:2009uq ([-@Murphy:2009uq]; `MYTorus`[^2]), @Ikeda:2009nx, and @Brightman:2011oq, it is now possible to reproduce the reflection features produced in the dusty torus surrounding the X-ray source. These models can also be used to remove the dependence of EW on $\Gamma$, $\theta_{\rm\,i}$, and $N_{\rm\,H}^{\mathrm{\,T}}$, renormalising EW to the same set of values for each source. Removing the degeneracy on EW introduced by these parameters is important to assess the real dependence of EW on the covering factor of the torus (and thus on $\theta_{\rm\,OA}$). This can be done by using the average values of the photon index obtained by the X-ray spectral fitting ($\Gamma^{\,\mathrm{obs}}$), and the values of the inclination angle ($\theta_{\rm\,i}^{\mathrm{\,obs}}$) and half-opening angle of the torus ($\theta_{\rm\,OA}^{\mathrm{\,obs}}$) obtained by the MIR analysis. The values of the equatorial column density of the torus ($N_{\rm\,H}^{\mathrm{\,T\,,obs}}$) can be extrapolated from the results of the MIR spectral fitting. From the relation between the extinction in the $V$ band and the optical depth ($A_{\mathrm{V}}=1.086\,\tau_{\mathrm{V}}$), and from the Galactic relation between extinction and column density found by [*ROSAT*]{} ($N_{\rm\,H}=A_{\mathrm{V}}\cdot 1.79\times 10^{21}\rm\,cm^{-2}$, [@Predehl:1995ly]), for the torus one obtains: $$\label{Eq:nhtcalc}
N_{\rm\,H}^{\mathrm{\,T,\,obs}}=1.086\,N_{0}\cdot\tau_{\mathrm{V}}\cdot1.79\times 10^{21}\rm\,cm^{-2}.$$
Equation\[Eq:nhtcalc\] assumes a Galactic $A_{\mathrm{V}}/N_{\rm\,H}$ ratio. This might, however, represent a crude approximation of the real value. [@Maiolino:2001oq; @Maiolino:2001nx] have shown that the $E(B-V)/N_{\rm\,H}$ ratio in AGN (for $\log L_{2-10}\geq 42$) ranges from $\sim 1\%$ to $\sim 40\%$ of the Galactic value. As discussed by [@Maiolino:2001nx], this is likely to imply that the $A_{\mathrm{V}}/N_{\rm\,H}$ ratio is also significantly lower in AGN than in our Galaxy. To take this effect into account, we used the average value of the $E(B-V)/N_{\rm\,H}$ ratio found by [@Maiolino:2001oq] (for $\log L_{2-10}\geq 42$) to recalculate the equatorial column density of the torus ($N_{\rm\,H,\,Av.}^{\mathrm{\,T,\,obs}}$). To convert the values of $E(B-V)/N_{\rm\,H}$ into $A_{\mathrm{V}}/N_{\rm\,H}$, we adopted the Galactic ratio ($A_{\mathrm{V}}/E(B-V)=3.1$). The value of $N_{\rm\,H,\,Av.}^{\mathrm{\,T,\,obs}}$ is then calculated similarly to what was done in Eq.\[Eq:nhtcalc\]:
$$\label{Eq:nhtcalcav}
N_{\rm\,H,\,Av.}^{\mathrm{\,T,\,obs}}=1.086\,N_{0}\cdot\tau_{\mathrm{V}}\cdot1.1\times 10^{22}\rm\,cm^{-2}.$$
In the following we consider both values of $N_{\rm\,H}^{\mathrm{\,T}}$ obtained using Eqs.\[Eq:nhtcalc\] and \[Eq:nhtcalcav\]. The values of $f_{\rm\,obs}$ and $N_{\rm\,H}^{\mathrm{\,T,\,obs}}$ are reported in Table\[tab:fitresults\].
Simulations of X-ray absorption and reflection from a clumpy torus have not been carried out yet, and the models listed above consider a smooth dust distribution, different from that of the model of @Nenkova:2008uq [@Nenkova:2008kx]. However, assuming that most of the Fe K$\alpha$ line is produced in the outer skin of the torus, the differences between the two geometries are likely to be small for type-I AGN. To correct the values of FeK$\alpha$ EW, we used the model developed by @Ikeda:2009nx. This model considers a spherical-toroidal geometry for the reprocessing material and has the advantage, with respect to `MYTorus`, of having the half-opening angle of the torus $\theta_{\rm\,OA}$ as a free parameter. The model of @Brightman:2011oq assumes a similar geometry, but considers a less realistic line-of-sight column density, which is constant for all values of $\theta_{\rm\,i}$. The other free parameters of the model of @Ikeda:2009nx are $\Gamma$, $N_{\rm\,H}^{\mathrm{\,T}}$, and $\theta_{\rm\,i}$. Using the model of @Ikeda:2009nx, we simulated, for each source, a spectrum with the parameters fixed to the values obtained by the observations, and one with a set of parameters arbitrarily chosen ($\Gamma=1.9,\theta_{\rm\,i}=5^{\circ},N_{\rm\,H}^{\mathrm{\,T}}=10^{24}\rm\,cm^{-2}$) and with $\theta_{\rm\,OA}$ fixed to the value obtained by the fit to the MIR spectra ($\theta_{\rm\,OA}^{\mathrm{\,obs}}$). We obtained the value of the Fe K$\alpha$ EW of the simulated spectrum ($EW_{\rm\,mod}$) following what was done in @Ricci:2013fk. For each source we calculated the corrections $K _{\mathrm{I}}$ with
$$\label{Eq:corr_ik}
K _{\mathrm{I}}=\frac{EW_{\rm\,mod}(\Gamma=1.9,\theta_{\rm\,i}=5^{\circ},N_{\rm\,H}^{\mathrm{\,T}}=10^{24}\rm\,cm^{-2},\theta_{\rm\,OA}^{\mathrm{\,obs}})}{EW_{\rm\,mod}(\Gamma^{\,\mathrm{obs}},\theta_{\rm\,i}^{\mathrm{\,obs}},N_{\rm\,H}^{\mathrm{\,T,\,obs}},\theta_{\rm\,OA}^{\mathrm{\,obs}})}.$$
The renormalised equivalent widths ($EW_{\mathrm{corr}}^{\rm\,I}$) can be easily calculated from the corrections and the observed values of the equivalent width ($EW_{\rm\,obs}$): $$\label{Eq:corr}
EW_{\mathrm{corr}}^{\rm\,I}=K_{\mathrm{I}}\times EW_{\rm\,obs}.$$ As an example of the corrections used, in Fig.\[fig:ew\_thetaoi\] we report the trend of EW($5^{\circ}$)/EW($\theta_{\rm\,i}$) for different values of $N_{\rm\,H}^{\mathrm{\,T}}$.
Removing the dependence on $\Gamma$, $\theta_{\rm\,i}$, and $\theta_{\rm\,OA}$ {#sect:corrections2}
------------------------------------------------------------------------------
To study the intrinsic relation between the FeK$\alpha$ EW and the equatorial column density of the torus, one can use a procedure similar to the one adopted in Sect.\[sect:corrections\]. Since we are now interested in $N_{\rm\,H}^{\mathrm{\,T}}$, we fixed the half-opening angle to an arbitrary value ($\theta_{\rm\,OA}=30^{\circ}$) for the reference value of EW and set $N_{\rm\,H}^{\mathrm{\,T}}$ to its observed value. The corrections $K _{\mathrm{II}}$ become $$\label{Eq:corr_ik2}
K _{\mathrm{II}}=\frac{EW(\Gamma=1.9,\theta_{\rm\,i}=5^{\circ},N_{\rm\,H}^{\mathrm{\,T}}=N_{\rm\,H}^{\mathrm{\,T\,,obs}},\theta_{\rm\,OA}=30^{\circ})}{EW(\Gamma^{\,\mathrm{obs}},\theta_{\rm\,i}^{\mathrm{\,obs}},N_{\rm\,H}^{\mathrm{\,T,\,obs}},\theta_{\rm\,OA}^{\mathrm{\,obs}})},$$ while the renormalised equivalent width ($EW_{\mathrm{corr}}^{\,II}$) can be obtained by $$\label{Eq:corr2}
EW_{\mathrm{corr}}^{\rm\,II}=K_{\mathrm{II}}\times EW_{\rm\,obs}.$$
The relation between the FeK$\alpha$ EW and the properties of the molecular torus {#Sect:EWvsf2}
=================================================================================
Covering factor {#section:ewvscf}
---------------
As discussed in @Mor:2009fk, the real covering factor of the torus ($f_{\rm\,obs}$) should be calculated by taking the number of clouds, the half-opening angle of the torus, and the inclination angle of the AGN into account. In the clumpy torus model the probability that the radiation from the central source escapes the torus at a given angle $\beta$ without interacting with the obscuring material is $$\label{eq:prob_esc}
P_{\mathrm{esc}}(\beta)=e^{-N_{\mathrm{0}}\mathrm{exp}\left(-\frac{\beta ^2}{\sigma _{\rm\,tor} ^2}\right)},$$ where $\beta=\pi/2-\theta_{\rm\,i}$. The geometrical covering factor of the molecular torus is given by integrating $P_{\mathrm{esc}}$ over all angles $$\label{eq:frac_obsc}
f_{\rm\,obs}=1-\int_{0}^{\pi/2}P_{\mathrm{esc}}(\beta)\cos(\beta)\mathrm{d} \beta.$$
![[*Top panel*]{}: values of the FeK$\alpha$ EW versus the geometrical covering factor of the torus ($f_{\rm\,obs}$) obtained by fitting Mid-IR spectra with the clumpy torus model. The black filled diamonds are the corrected values (obtained using the model of [@Ikeda:2009nx], see Eqs.\[Eq:corr\_ik\] and \[Eq:corr\]), while the empty red ones are the uncorrected values multiplied by an arbitrary constant factor for comparison. The data were rebinned to have six values per bin, and the uncertainties on the Fe K$\alpha$ EW were calculated using the standard error of the mean. The dotted line represents the best fit to the non-binned data obtained by applying Eq.\[eq:EWcorrvsf2\], and it has a slope of $\overline{B}=0.44\pm0.21$. The red dashed line represents the expected $EW-f_{\rm\,obs}$ trend for the set of parameters chosen for the renormalisation, calculated using the model of @Ikeda:2009nx. The intercept of the expected trend was obtained by fitting the data with the slope fixed to the expected value ($B_{\rm\,exp}\simeq0.4$). [*Bottom panel*]{}: 2–10keV luminosities versus covering factor of the torus for our sample. The dashed line represents the best fit to the data (Eq.\[eq:logLcvsFobs\]). []{data-label="fig:ew_f2"}](EW_f2_LX_rebinned_ikeda.ps){width="9cm"}
If the FeK$\alpha$ line is produced in the torus and the results obtained by applying the clumpy torus model to the MIR spectra of AGN are correct, then a positive correlation between the EW of the line and the real covering factor of the torus would be expected. To study the relation between FeK$\alpha$ EW and $f_{\rm\,obs}$ for our sample, which includes several upper limits, we followed the approach of @Guainazzi:2006fk and @Bianchi:2007vn, which is an extension of the regression method for left-censoring data described by @Schmitt:1985kx and @Isobe:1986uq. We performed 10000 Monte-Carlo simulations for each value of the Fe K$\alpha$ EW, taking the two following requirements into account: i) the values of EW of the detections were substituted with a random Gaussian distribution, whose mean is given by the value obtained by the fit, and the standard deviation by its error; ii) the upper limits U were substituted with a random uniform distribution in the interval \[0,U\]. To reduce the degeneracy introduced by different values of $\Gamma$, $\theta_{\rm\,i}$ and $N_{\rm\,H}^{\rm\,T}$, we used the values of the FeK$\alpha$ EW corrected as described in Sect.\[sect:corrections\]. For each Monte-Carlo run we fitted the values with a log-linear relationship of the type $$\label{eq:EWcorrvsf2}
\log EW_{\mathrm{corr}}^{\rm\,I}=A+B\cdot f_{\rm\,obs},$$ using the ordinary least squares (OLS\[Y|X\]) method. We used the average value of the simulations ($\overline{B}$) as a slope, and as uncertainty their standard deviation. To quantify the significance of the correlation, for each simulation we calculated the Spearman’s rank coefficient ($\rho$) and the null hypothesis probability of the correlation ($P_{\mathrm{\,n}}$), and used the values averaged over all the simulations. Applying Eq.\[eq:EWcorrvsf2\], we obtained a slope of $\overline{B}=0.44\pm0.21$. With the model of @Ikeda:2009nx, it is possible to deduce the expected $EW-f_{\rm\,obs}$ trend for the set of parameters we used to renormalise the values of EW. The correct formulation of $f_{\rm\,obs}$ is given by Eq.\[eq:frac\_obsc\], but for $\log N_{\rm\,H}^{\rm\,T}=24$ at 6.4keV, the escaping probability is $P_{\rm\,esc}\sim 0.08$ for $\beta < \sigma_{\rm\,tor}$, so that we can approximate the relation to $f_{\rm\,obs}\simeq \cos \theta_{\rm\,OA}$. We found that for $\Gamma=1.9$, $N_{\rm\,H}^{\rm\,T}=10^{24}\rm\,cm^{\rm\,-2}$, and $\theta_{\rm\,i}=5^{\circ}$, the expected slope is $B_{\rm\,exp}\simeq 0.4$, consistent with the result of the fit. The scatter plot of $EW_{\mathrm{corr}}^{\rm\,I}$ and $EW_{\rm\,obs}$ versus $f_{\rm\,obs}$ is illustrated in the top panel of Fig.\[fig:ew\_f2\]. For graphical clarity the data were rebinned to have six values per bin. Performing the statistical tests described above, we found that however the correlation is statistically not significant, with a null hypothesis probability of $P_{\mathrm{\,n}}=35\%$ and a Spearman’s rank coefficient of $\rho=0.22$. Correcting the values of EW using the equatorial column density of the torus obtained by assuming the average $E(B-V)/N_{\rm\,H}$ ratio of [@Maiolino:2001nx] does not alter significantly the results ($\overline{B}=0.46\pm0.23$, $\rho=0.23$, $P_{\mathrm{\,n}}=37\%$).
We took random Gaussian errors on $f_{\rm\,obs}$ into account, as done for EW, using the errors reported in @Alonso-Herrero:2011zr and @Ramos-Almeida:2011ly and considering uncertainties of $30\%$ for the sources of @Mor:2009fk. This does not increase the significance of the correlation, giving a null hypothesis probability of $P_{\mathrm{\,n}}=38\%$. We verified whether the fact that the MIR fitting procedures of @Mor:2009fk, @Alonso-Herrero:2011zr, and @Ramos-Almeida:2011ly differ might alter the results. We fitted the data taking only the 19 sources from @Mor:2009fk into account, and found that the correlation is still not significant ($P_{\mathrm{\,n}}=40\%$). Consistent results ($P_{\rm\,n}=34\%$, $\rho=0.24$) were obtained not considering the observations affected by pile-up in the fit.
In the bottom panel of Fig.\[fig:ew\_f2\] we show the scatter plot of the 2–10keV luminosity versus the covering factor of the torus. The two parameters are not significantly correlated ($P_{\mathrm{\,n}}=33\%$), and by fitting the data we obtained $$\label{eq:logLcvsFobs}
\log L_{\,2-10}\propto (-1.44\pm0.78)f_{\rm\,obs}.$$
Equatorial column density of the torus {#section:ewvsnht}
--------------------------------------
With the values of the FeK$\alpha$ EW corrected to remove the dependence on $\Gamma$, $\theta_{\rm\,i}$, and $\theta_{\rm\,OA}$ (Sect.\[sect:corrections2\]), we searched for a correlation with the equatorial column density of the torus. Monte Carlo simulations (e.g., [@Ikeda:2009nx], [@Murphy:2009uq]) have shown that this parameter is expected to play an important role on the iron K$\alpha$ line EW. Following the same procedure as discussed in Sect.\[section:ewvscf\], we found that the correlation is statistically not significant ($\rho=0.25$, $P_{\rm\,n}=29\%$). Fitting the data with a log-linear relation of the type $$\label{eq:EWvsNHT}
\log EW_{\mathrm{corr}}^{\rm\,II}= \alpha+ \beta \cdot \log N_{\rm\,H}^{\rm\,T},$$ we obtained $\overline{\beta}=0.16\pm0.08$. However, due to self-absorption for large values of the column density of the torus, the FeK$\alpha$ EW is expected to saturate for $\log N_{\rm\,H}^{\rm\,T} \gtrsim 24$ (e.g., [@Ghisellini:1994uq]), so that a linear increment is expected only up to this value. Considering only the data for $\log N_{\rm\,H}^{\rm\,T} \leq 24$ resulted in a slope ($\overline{\beta}=0.32\pm0.23$) which is consistent to the expected value ($\beta_{\rm\,exp}=0.53$) for the same range of $N_{\rm\,H}^{\rm\,T}$. The trend is, however, statistically non-significant ($\rho=0.30$, $P_{\rm\,n}=31\%$). In Fig.\[fig:ew\_NHT\] we show the scatter plot of $EW_{\mathrm{corr}}^{\rm\,II}$ and $EW_{\mathrm{obs}}$ versus $N_{\rm\,H}^{\rm\,T}$, together with the expected trend calculated using the model of @Ikeda:2009nx for the set of parameters used for the re-normalisation. Both the values of $EW_{\mathrm{corr}}^{\rm\,II}$ and $EW_{\mathrm{obs}}$ agree with the predicted trend, as would be expected if the line was produced in the molecular torus, although their large associated uncertainties do not allow us to draw a firm conclusion.
![Values of the FeK$\alpha$ EW versus the equatorial column density of the torus ($N_{\rm\,H}^{\rm\,T}$) obtained by fitting the mid-IR spectra. Diamonds and circles represent values of $N_{\rm\,H}^{\rm\,T}$ obtained using the Galactic $E(B-V)/N_{\rm\,H}$ ratio and the average value of [@Maiolino:2001nx], respectively. The black (blue) filled diamonds (circles) are the corrected values (obtained using the model of [@Ikeda:2009nx], see Eqs.\[Eq:corr\_ik2\] and \[Eq:corr2\]), while the empty red (cyan) diamonds (circles) are the uncorrected values. The values of EW were multiplied by an arbitrary constant factor for comparison. The data were rebinned to have six values per bin, and the uncertainties on the Fe K$\alpha$ EW are calculated using the standard error of the mean. The dashed lines represent the expected $EW-N_{\rm\,H}^{\rm\,T}$ trend for the set of parameters chosen for the renormalisation, calculated using the model of @Ikeda:2009nx. The expected trends were normalised to be compatible with the corrected data.[]{data-label="fig:ew_NHT"}](EW_NHT_rebinned.ps){width="9cm"}
Summary and discussion {#Sect:summary}
======================
Reflection of the power-law continuum from circumnuclear material in AGN is mainly observed through the narrow FeK$\alpha$ line and a reflection hump peaking at $\sim 30$keV. The fraction of continuum X-ray flux reflected (hence the FeK$\alpha$ EW) is likely to depend strongly on the covering factor of the torus ($f_{\rm\,obs}$). This implies that $f_{\rm\,obs}$, and its evolution with the physical properties of the AGN, is fundamental for a correct understanding of the cosmic X-ray background (CXB, e.g., [@Gilli:2007qf]). The maximum emission of the CXB is in fact observed at $\sim30$keV (e.g., [@Marshall:1980kx]), and a large fraction of CT AGN has often been invoked to correctly reproduce its shape (e.g., [@Gilli:2007qf]). Because it is observed through a large amount of obscuring material, most of the continuum in these objects is absorbed, which enhances the apparent reflected-to-incident flux ratio, and it makes their observed spectra peak at $\sim 30$keV. However, the fraction of Compton-thick sources needed to explain the peak is strongly linked to the fraction of reflected continuum [@Gandhi:2007fk; @Treister:2009uq], and thus to $f_{\rm\,obs}$. High values of $f_{\rm\,obs}$ have been invoked to explain the characteristics of buried AGN [@Ueda:2007fk; @Eguchi:2009kx; @Eguchi:2011uq], which are type-II objects that have a strong reflection component and a low fraction of scattered continuum. A large covering factor of the torus might explain the strong reflection observed in the hard X-ray spectrum of mildly obscured ($23 \leq \log N_{\rm\,H} < 24$) AGN found by stacking [*INTEGRAL*]{} IBIS/ISGRI data [@Ricci:2011vn] and recently confirmed by @Vasudevan:2013ys using [*Swift*]{}/BAT. The covering factor of the torus is believed to decrease with luminosity. The original idea of such a relation was put forward by @Lawrence:1982bh to explain the decrease in the fraction of obscured sources with the luminosity. In the past decade, this trend has been confirmed by several studies carried out at different wavelengths (e.g., [@Ueda:2003qf], [@Beckmann:2009ys]), and it has been shown that it would also be able to straightforwardly explain the X-ray Baldwin effect [@Ricci:2013fk]. Thus a correct understanding of the relation between reflected X-ray radiation and the covering factor of the torus is of the utmost importance for a complete understanding of the X-ray spectral evolution of AGN.
In this work we have studied the relation between the FeK$\alpha$ EW and important physical properties of the molecular torus, such as its covering factor and equatorial column density ($N_{\rm\,H}^{\rm\,T}$), for a sample of 24 AGN. This was done by combining [*XMM-Newton*]{}/EPIC observations in X-rays with the results obtained by recent MIR spectral studies carried out using the clumpy torus models of @Nenkova:2008uq [@Nenkova:2008kx]. The physical torus model of @Ikeda:2009nx was used to correct the values of the FeK$\alpha$ EW, in order to remove the degeneracy introduced by different values of $\Gamma$, $\theta_{\rm\,i}$, and $N_{\rm\,H}^{\rm\,T}$. We found that, although the correlation between the FeK$\alpha$ EW and the covering factor of the torus is statistically non-significant, the slope obtained ($\overline{B}=0.44\pm0.21$) is in very good agreement with the expected value ($B_{\rm\,exp}\simeq 0.4$, see Fig.\[fig:ew\_f2\]). A similar result is obtained when studying the relation between FeK$\alpha$ EW and $N_{\rm\,H}^{\rm\,T}$ for $\log N_{\rm\,H}^{\rm\,T} \leq 24$: the slope obtained ($\overline{\beta}=0.32\pm0.23$) is consistent with the predicted value ($\overline{\beta}_{\rm\,exp}=0.53$, see Fig.\[fig:ew\_NHT\]), although the correlation is statistically non-significant.
The fact that the correlation between EW and $f_{\rm\,obs}$ is statistically not significant is probably related to the large errors of FeK$\alpha$ EW, and to the large number of PG quasars in the sample, which skews the luminosity distribution towards high values. We also cannot exclude the effect of systematic errors introduced by the technique used to fit the MIR spectra. As argued by @Mor:2009fk, there are two main uncertainties associated to their treatment: variability between the non-simultaneous optical and MIR observations and their choice of the bolometric corrections. These uncertainties could introduce scatter into the torus parameters obtained, so that larger samples, with a more uniform luminosity distribution, are probably needed to find a clear trend between FeK$\alpha$ EW, $f_{\rm\,obs}$, and $N_{\rm\,H}^{\rm\,T}$. Another possible source of uncertainty could be introduced by the fact that the model of @Nenkova:2008uq [@Nenkova:2008kx] assumes the inner radius of the torus given by $$\label{eq:inner_radius_torus}
R_{\rm\,d}=0.4 \left(\frac{L_{\rm\,bol}}{10^{\,45}\rm\,erg\,s^{-1}}\right)^{1/2}\left(\frac{1500\,K}{T_{\rm\,sub}}\right)\rm\,pc,$$ where $L_{\rm\,bol}$ is the bolometric luminosity of the AGN and $T_{\rm\,sub}$ is the dust sublimation temperature. NIR reverberation studies of the torus have shown that Eq.\[eq:inner\_radius\_torus\] overestimates the value of $R_{\rm\,d}$, which is found to be systematically smaller by a factor of three [@Kishimoto:2007zr]. @Kawaguchi:2010fk show that the discrepancy is probably related to the fact that the accretion disk emits anisotropically, so that the effective inner radius is smaller than predicted by Eq.\[eq:inner\_radius\_torus\]. Furthermore, it has been suggested that the observed NIR excess may be due to the fact that silicate and graphite grains sublimate at different temperatures and that large grains are cooling more efficiently, leading to a sublimation zone rather than a sublimation radius (e.g., [@Kishimoto:2007zr], [@Mor:2012ys]). Although the hot graphite-only zone is a plausible source of an additional NIR emission, this component still has to be consistently included in the radiative transfer modelling. [@Schartmann:2009ve] models do account for the latter effect by separating the grains of different sizes, but no NIR bump is seen. Other suggestions for the source of the NIR excess include an additional component of low-density interclump dust [@Stalevski:2012fk; @Stalevski:2013ly] or assume that NIR and MIR emission are coming from two spatially very distinct regions [@Honig:2013bh]. Moreover, modelling of the dusty tori IR emission comes with caveats of its own [@Hoenig:2013qf], and model parameters are often degenerate, sometimes resulting in similarly good fits for different combinations of parameters. This inevitably introduces additional uncertainties in any analysis that relies on the properties of the torus obtained from fitting their IR SEDs.
An additional source of scatter might be related to the fact that FeK$\alpha$ emission originating in the torus is subject to a significant contamination from other regions of the AGN, such as the BLR or the outer part of the accretion disk. This degeneracy will be broken in a few years with the advent of [*ASTRO-H*]{} [@Takahashi:2010uq]. Thanks to the unprecedented energy resolution in the FeK$\alpha$ energy band of its X-ray calorimeter (SXS, 5eV FWHM at 6keV), [*ASTRO-H*]{} will be able to disentangle the emission produced in the torus from that arising from different regions of the AGN.
We thank the anonymous referee for his/her comments that helped improve the paper. We thank Chin Shin Chang and Almudena Alonso Herrero for their comments on the manuscript. CR is a Fellow of the Japan Society for the Promotion of Science (JSPS). This work was partly supported by the Grant-in-Aid for Scientific Research 23540265 (YU) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). PG acknowledges support from STFC grant reference ST/J00369711. MS acknowledges support of the Ministry of Education, Science and Technological Development of the Republic of Serbia through the projects Astrophysical Spectroscopy of Extragalactic Objects (176001) and Gravitation and the Large Scale Structure of the Universe (176003), and by FONDECYT through grant No. 3140518. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center, and of the SIMBAD Astronomical Database, which is operated by the Centre de Données astronomiques de Strasbourg. Based on observations obtained with [*XMM-Newton*]{}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
X-ray spectral results {#Appendix0}
======================
In Tables\[tab:fitresults\] and \[tab:fitresults2\] we report the results of the spectral fitting described in Sect.\[Sect:spectral\_analysis\] for our [*XMM-Newton*]{}/EPIC sample.
Notes on the individual sources {#Appendix1}
===============================
In the following we report the details on the emission lines added to the best fits. [**IC 4329a – II.**]{} A narrow ($\sigma=1$eV) emission feature is detected in the spectrum. The narrow line is found at $E=7.00^{+0.02}_{-0.03}$keV ($EW=20^{+2}_{-3}$eV), and is likely to be FeXXVI. [**NGC 4151 – I.**]{} The spectrum shows evidence of three narrow emission features at low energies. These features were fitted using Gaussian emission lines at $E=0.561^{+0.003}_{-0.003}$keV ($EW=91^{+3}_{-4}$eV), $E=0.903^{+0.003}_{-0.004}$keV ($EW=83^{+4}_{-3}$eV) and $E=1.33^{+0.01}_{-0.01}$keV ($EW=37^{+3}_{-6}$eV). These lines are consistent with OVII, NeIX, and MgXI, respectively. [**NGC 4151 – II.**]{} Three narrow emission features at low energies are detected. The narrow lines were found to be at $E=0.557^{+0.004}_{-0.002}$keV ($EW=89^{+4}_{-2}$eV), $E=0.903^{+0.003}_{-0.004}$keV ($EW=78^{+4}_{-3}$eV), and $E=1.34^{+0.01}_{-0.01}$keV ($EW=33^{+3}_{-4}$eV), and they are consistent with OVII, NeIX, and MgXI, respectively. [**NGC 4151 – III.**]{} Three narrow emission features at low energies are detected. The three lines are at $E=0.557^{+0.004}_{-0.003}$keV ($EW=78^{+3}_{-4}$eV), $E=0.897^{+0.007}_{-0.005}$keV ($EW=75^{+4}_{-3}$eV), and $E=1.31^{+0.02}_{-0.02}$keV ($EW=37^{+4}_{-6}$eV), and they are consistent with OVII, NeIX, and MgXI, respectively. [**NGC 4151 – IV.**]{} Two narrow emission features at low energies are detected. The narrow features are found at $E=0.558^{+0.003}_{-0.003}$keV ($EW=103^{+13}_{-1}$eV), and $E=0.897^{+0.007}_{-0.005}$keV ($EW=76^{+4}_{-5}$eV), and they are consistent with OVII and NeIX, respectively. [**NGC 4151 – V.**]{} Two narrow emission features at low energies are detected. The narrow features are located at $E=0.558^{+0.003}_{-0.002}$keV ($EW=108^{+7}_{-8}$eV), and $E=0.892^{+0.009}_{-0.007}$keV ($EW=63^{+4}_{-4}$eV), and they are consistent with OVII and NeIX, respectively. [**NGC 4151 – VI.**]{} Two narrow emission features at $E=0.559^{+0.003}_{-0.002}$keV ($EW=100^{+13}_{-7}$eV) and $E=0.895^{+0.008}_{-0.005}$keV ($EW=71^{+4}_{-3}$eV) are detected. The two lines are consistent with OVII and NeIX, respectively. [**NGC 4151 – VII.**]{} Five narrow emission features at low energies are detected. The energies of the narrow lines are $E=0.563^{+0.002}_{-0.002}$keV ($EW=86^{+4}_{-3}$eV), $E=0.896^{+0.004}_{-0.003}$keV ($EW=77^{+3}_{-3}$eV), $E=1.34^{+0.01}_{-0.01}$keV ($EW=35^{+3}_{-3}$eV), $E=1.82^{+0.02}_{-0.02}$keV ($EW=69^{+4}_{-5}$eV), and $E=7.04^{+0.03}_{-0.07}$keV ($EW= 32^{+5}_{-4}$eV). These lines are consistent with being due to OVII, NeIX, MgXI, SiXIII, and to FeXXVI, respectively. [**NGC 4151 – VIII.**]{} Three narrow emission features at low energies are detected. The narrow lines have energies of $E=0.596^{+0.005}_{-0.005}$keV ($EW=77^{+3}_{-2}$eV), $E=0.897^{+0.006}_{-0.006}$keV ($EW=59^{+4}_{-3}$eV), and $E=1.78^{+0.04}_{-0.02}$keV ($EW=50^{+5}_{-4}$eV), and are consistent with OVII, NeIX, and SiXIII, respectively. [**PG 0050+124 - I.**]{} The spectrum requires an additional line at $E=6.96_{-0.10}^{+0.09}$ keV, likely due to FeXXVI, with an equivalent width $EW=106\pm24\rm\,eV$. [**PG 0050+124 - II.**]{} Besides the FeXXVI line at $E=6.97^{+0.06}_{-0.04}$ keV ($EW=56\pm13\rm\,eV$), we found evidence of another unresolved ionised iron line (likely FeXXV) at $E=6.66^{+0.05}_{-0.04}$ keV ($EW=50_{-10}^{+12}\rm\,eV$). [**PG 1116+215 – I.**]{} Two lines at $6.7\rm\,keV$ ($EW=63^{+38}_{-34}\rm\,eV$) and $6.97\rm\,keV$ ($EW=128^{+49}_{-45}\rm\,eV$) are needed. The lines are consistent with being produced by emission of ionised iron (FeXXV and FeXXVI, respectively). [**PG 1116+215 – III.**]{} Two lines at $6.7\rm\,keV$ ($EW=68^{+35}_{-34}\rm\,eV$) and $6.97\rm\,keV$ ($EW=124^{+51}_{-45}\rm\,eV$) are needed. The two lines are consistent with being produced by emission of ionised iron (FeXXV and FeXXVI, respectively). [**PG 1126$-$041 – I.**]{} Two emission lines at low energies were also needed. The lines are located at $E=0.56\pm0.01\rm\,keV$ ($EW=107^{+12}_{-27}\rm\,eV$) and $E=0.90\pm0.02\rm\,keV$ ($EW=68^{+37}_{-33}\rm\,eV$), and are consistent with being due to OVII and NeIX, respectively. [**PG 1126$-$041 – II.**]{} Two emission lines at $E=0.60\pm0.03\rm\,keV$ ($EW=70^{+6}_{-38}\rm\,eV$) and $E=0.89\pm0.03\rm\,keV$ ($EW\leq 280\rm\,eV$) are also needed. The lines are consistent with being due to OVII and NeIX, respectively. [**PG 1126$-$041 – III.**]{} The spectrum shows an additional emission line at $E=0.60\pm0.02\rm\,keV$ ($EW=115^{+53}_{-63}\rm\,eV$), consistent with the OVII line. [**PG 1126$-$041 – IV.**]{} Two lines at low energy were also found. The line at $E=0.58\pm0.01\rm\,keV$ (EW=$65^{+12}_{-9}\rm\,eV$), is likely due to OVII emission, while that at $E=0.93\pm0.02\rm\,keV$ ($EW=89^{+22}_{-31}\rm\,eV$) is consistent with being NeIX. Another emission line at $E=7.82_{-0.08}^{+0.13}\rm\,keV$ (EW=$87^{+24}_{-40}\rm\,eV$), consistent with being the He$\beta$ form of FeXXV, was also found. [**PG 1229+204.**]{} An emission line at $E=6.72\pm 0.05$ keV ($EW=98_{-25}^{+28}\rm\,eV$), consistent with being due to the He$\alpha$ state of FeXXV, is required. [**PG 1440+356 – II .**]{} An emission line at $E=6.72\pm 0.09$ keV ($EW=67_{-36}^{+19}\rm\,eV$) was detected, and is likely the He$\alpha$ form of FeXXV. [**PG 1440+356 – III.**]{} The spectrum shows evidence of emission due to the He$\alpha$ form of FeXXV at $E=6.73\pm 0.10$ keV ($EW=106\pm41\rm\,eV$). [**PG 1440+356 – IV.**]{} We found evidence of an emission feature at $E=6.79\pm 0.07$keV ($EW=197\pm48\rm\,eV$), which is probably due to the He$\alpha$ state of FeXXV.
[^1]: http://xmm.esac.esa.int/xsa/
[^2]: http://www.mytorus.com/
|
---
abstract: 'We obtain results on the growth sequences of the differential for iterations of circle diffeomorphisms without periodic points.'
address: ' Department of Mathematics, School of Commerce, Waseda University, Shinjuku, Tokyo 169-8050, Japan.'
author:
- Nobuya Watanabe
title: Growth sequences for circle diffeomorphisms
---
Introduction and statement of results
=====================================
Let $f:S^1 \rightarrow S^1$ be a $C^{1}$-diffeomorphism where $S^1={\Bbb R}/{\Bbb Z}$. We define the [*growth sequence*]{} for $f$ by $$\Gamma_n(f)=
\max \{ \lVert Df^n\rVert,
\lVert Df^{-n}\rVert \},\ \ \ n \in \Bbb{N},$$ where $f^n$ is the $n$-th iteration of $f$ and $\lVert Df^n\rVert = {\displaystyle \max_{x \in S^1}}|Df^n(x)|$.
If $f$ has periodic points, then the study of growth sequences reduces to the case of interval diffeomorphisms which was studied in [@B],[@PS],[@W].
If $f$ has no periodic points, then by the theorem of Gottschalk-Hedlund $\Gamma_n(f)$ is bounded if and only if $f$ is $C^1$-conjugate to a rotation. Notice that if $\Gamma_n(f)$ is bounded then $f$ is minimal. So it is natural to ask how rapidly could the sequence $\Gamma_n(f)$ grow if it is unbounded.
In this paper we give an answer to this question:
Let $f:S^1 \rightarrow S^1$ be a $C^2$-diffeomorphism without periodic points. Then $$\lim_{n\rightarrow \infty}\frac{\Gamma_n(f)}{n^2}=0.$$
For any increasing unbounded sequence of positive real numbers $\theta_n =o(n^2)$ as $n \rightarrow \infty$ and any $\varepsilon >0$ there exists an analytic diffeomorphism $f:S^1 \rightarrow S^1$ without periodic points such that $$1- \varepsilon \le \limsup_{n\rightarrow \infty}
\frac{\Gamma_n(f)}{\theta_n} \le 1.$$
Preliminaries
=============
Given an orientation preserving homeomorphism $f:S^1\rightarrow S^1$, its [*rotation number*]{} is defined by $$\rho (f) = \lim_{n \rightarrow \infty}\frac{\tilde{f}^n(x)-x}{n}
\mod {\Bbb Z}$$ where $\tilde{f}$ denotes a lift of $f$ to $\Bbb{R}$. The limit exists and is independent on $x \in \Bbb{R}$ and a lift $\tilde{f}$.
Put $\alpha = \rho(f)$. Let $R_{\alpha}$ be the rigid rotation by $\alpha$ $$R_{\alpha}(x)=x+\alpha \mod {\Bbb Z}.$$
For the basic properties of circle homeomorphisms and the combinatorics of orbits of the rotation of the circle, general references are [@MS] chapter I and [@KH] chapter 11, 12.
By Poincar$\acute{{\rm e}}$ the order structure of orbits of $f$ and $R_{\alpha}$ on $S^1$ are almost same. In particular if $\rho (f) = \frac{p}{q} \in \Bbb{Q}/\Bbb{Z}$ then $f$ has periodic points of period $q$ and every periodic orbits of $f$ have the same order as orbits of $R_{\frac{p}{q}}$ on $S^1$. $\rho (f) \in
(\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}$ if and only if $f$ has no periodic points, in this case, if $f$ is of class $C^2$ then by the well known theorem of Denjoy $f$ is topologically conjugate to $R_{\alpha}$.
Suppose $\alpha \in (\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}$. Let $$\alpha = [a_1,a_2,a_3,\ldots]=\dfrac{1}{a_1+
\dfrac{1}{a_2+\dfrac{1}{a_3+{}_{\ddots}}}}\ \ \ ,
a_{i} \ge 1, a_i \in \Bbb{N}$$ be the continued fraction expansion of $\alpha$, and $$\frac{p_n}{q_n}=[a_1,a_2,\ldots,a_n]$$ be its $n$-th convergent. Then $p_n$ and $q_n$ satisfy $$p_{n+1}=a_{n+1}p_n+p_{n-1},~~p_0=0,~p_1=1,$$ $$q_{n+1}=a_{n+1}q_n+q_{n-1},~~q_0=1,~q_1=a_1,$$ $$\frac{p_0}{q_0} < \frac{p_2}{q_2} < \frac{p_4}{q_4} < \cdots < \alpha < \cdots
< \frac{p_5}{q_5} < \frac{p_3}{q_3} < \frac{p_1}{q_1}.$$
The sequence of rational numbers $\{ \frac{p_n}{q_n}\}$ is the best rational approximation of $\alpha$. This can be expressed using the dynamics of $R_{\alpha}$ as follows. $R^{q_n}_{\alpha}(0) \in [0,R^{-q_{n-1}}_{\alpha}(0)]$, and if $k >q_{n-1}$, $R^{k}_{\alpha}(0) \in [R^{q_{n-1}}_{\alpha}(0),R^{-q_{n-1}}_{\alpha}(0)]$ then $k \ge q_{n}$. Note that for $0 \le k \le a_{n+1},
R_{\alpha}^{kq_n}(0) \in [0,R^{-q_{n-1}}_{\alpha}(0)]$, and $R_{\alpha}^{(a_{n+1}+1)q_n}(0) \notin [0,R_{\alpha}^{-q_{n-1}}(0)]$.
For $\alpha \in (\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}$ the continued fraction expansion is unique. On the other hand for $\beta \in \Bbb{Q}/\Bbb{Z}$ expressions by continued fractions are not unique, $\beta = [b_1,b_2,\ldots,b_n+1] = [b_1,b_2,\ldots,b_n,1]$.
For $\alpha = [a_1,a_2,\ldots]$ and $i, j \in \Bbb{N},
1 \le i \le j$ we denote $\alpha|[i,j] = [a_i,a_{i+1},\ldots,a_j]$. In case we emphasize $\alpha$ we denote $a_{i}(\alpha), p_i(\alpha),
q_i(\alpha)$.
For $x \in S^1$, $I_n(x)$ denotes the smaller interval with endpoints $x$ and $f^{q_n}(x)$ and for an interval $J \subset S^1$, $|J|$ the length of $J$.
The following is well known. See [@MS] chapter I section 2a.
[(Denjoy)]{} Let $f$ be a $C^1$-diffeomorphism of $S^1$ without periodic points and $\log Df:S^1\rightarrow {\Bbb R}$ has bounded variation. Then there exists a positive constant $C_1 = C_1(f)$ satisfying the following properties.
$(1)$ For any $0\le l \le q_{n+1}$ and for every $x_1,x_2 \in I_n(x)$ $$\frac{1}{C_1}\le \frac{Df^l(x_1)}{Df^l(x_2)}\le C_1.$$
$(2)$ [(Denjoy inequality)]{} For every $n \in {\Bbb N}$, $$\frac{1}{C_1}\le \lVert Df^{q_n} \rVert \le C_1.$$
As stated in section 1, the growth sequences play a significant role in the problem of the smooth linearization of circle diffeomorphisms, where the arithmetic property of rotation numbers and the regularity of diffeomorphisms are important. This problem has a rich history, see e.g. [@A], [@H], [@Y], [@KS], [@St], [@KO].
In this paper, particularly we need the following improvement of Denjoy inequality which is due to Katznelson and Ornstein. The statement of Lemma 2 is obtained by merging results in [@KO], for $(1)$, (1.16), lemma 3.2 (3.6) and proposition 3.3 (a), for $(2)$, theorem 3.7.
Let $f$ be a $C^2$-diffeomorphism of $S^1$ without periodic points. Set $$E_n = \max \{ \lVert \log Df^{q_n}\rVert,~
\max_{x\in S^1} \{|D\log Df^{q_n}(x)||I_{n-1}(x)|\} \}.$$ Then the following hold.
$(1)$ $\lim_{n \rightarrow \infty} E_n = 0.$
$(2)$ If $f$ is of class $C^{2+\delta}, \delta > 0$ then there exist $C>0$ and $0 < \lambda <1$ such that $\lVert \log Df^{q_n}\rVert \le C\lambda^n$ for any $n \in \Bbb{N}$.
The conclusion of Lemma 2 (2) plus some arithmetic condition of $\rho (f)$ are sufficient to provide the $C^{1}$-linearization of $f$. We need the following which is a special case of the main theorem in [@KO]. For $C^{3+\delta}$-diffeomorphisms it is originally due to Herman [@H].
[**Corollary of Lemma 2 (2).**]{} [*If*]{} $f$ [*is of class*]{} $C^{2+\delta}$ [*and the rotation number*]{} $\alpha = \rho(f)$ [*is of bounded type i.e.*]{} $a_i(\alpha)$ [*is uniformly bounded then*]{} $\lVert Df^n \rVert$ [*is uniformly bounded.*]{}
Proof of Theorem 1
==================
Let $f:S^1 \rightarrow S^1$ be a $C^2$-diffeomorphism without periodic points with the rotation number $\rho(f) = [a_{1},a_{2},\ldots]$ and its convergents $\{\frac{p_{n}}{q_{n}}\}$.
The following crucial and fundamental lemma is due to Polterovich and Sodin ([@PS] lemma 2.3).
[(Growth lemma)]{} Let $\lbrace A(k) \rbrace_{k\ge 0}$ be a sequence of real numbers such that for each $k \ge 1$ $$2A(k)-A(k-1)-A(k+1)\le C\exp (-A(k)),\quad C > 0,$$ and $A(0)=0$. Then either for each $k \ge 0$ $$A(k) \le 2\log \left ( k\sqrt{\frac{C}{2}}+1 \right ),
~ or
~~\liminf_{k\rightarrow \infty}\frac{A(k)}{k}>0.$$
For $0 \le k \le a_{n+1}+1$ we set $A_n(k)=\log \lVert Df^{kq_n}\rVert$. Then there exists a positive constant $C=C(f)$ independent with $n$ such that for $1\le k \le a_{n+1}$, $$2A_n(k)-A_n(k-1)-A_n(k+1)\le CE_n \exp(-A_n(k)).$$
Let $A_n(k) = \log Df^{kq_n}(x_0)$ and $x_i=f^{iq_n}(x_0)$. Then we have, $$2A_n(k)-A_n(k-1)-A_n(k+1)$$ $$\le
2\log Df^{kq_n}(x_0)-\log Df^{(k-1)q_n}(x_1)
-\log Df^{(k+1)q_n}(x_{-1})$$ $$\le |\log Df^{q_n}(x_0)-\log Df^{q_n}(x_{-1})|
= |D\log Df^{q_n}(y_0)||I_n(x_{k-1})|\frac{|I_n(x_{-1})|}{|I_n(x_{k-1})|} ,$$ where $y_0\in I_n(x_{-1})$.
Notice that the intervals $I_n(x_{-1}), I_n(x_{0}), I_n(x_{1}),\ldots,
I_n(x_{a_{n+1}-1})$ are adjacent in this order and ${\displaystyle \cup_{i=0}^{a_{n+1}-1}}
I_n(x_i) \subset
I_{n-1}(f^{-q_{n-1}}(x_0))$. Since $y_0 \in I_n(x_{-1})$, we have for $1\le k \le a_{n+1}-1$, $I_n(x_{k-1})\subset I_{n-1}(f^{-q_{n-1}}(y_0))$. So by Denjoy inequality (Lemma 1 (2)) we have $$|I_n(x_{k-1})|\le C_1^2|I_{n-1}(y_0)|,$$ and using lemma 1 (1) we have $$\frac{|I_n(x_{-1})|}{|I_n(x_{k-1})|} \le C_1\frac{1}{Df^{kq_n}(x_0)}.$$ Hence we have $$2A_n(k)-A_n(k-1)-A_n(k+1)$$ $$\le C_1^3|D\log Df^{q_n}(y_0)||I_{n-1}(y_0)| \frac{1}{Df^{kq_n}(x_0)}
\le C_1^3E_n\exp(-A_n(k)).$$
We extend $A_n(k)$ for $k \ge a_{n+1}+2$ by $A_n(k)=A_n(a_{n+1}+1)$. Then by Lemma 1 (2) and the definition of $E_{n}$ we have $$2A_n(a_{n+1}+1)-A_n(a_{n+1})-A_n(a_{n+1}+2)$$ $$\le \log Df^{(a_{n+1}+1)q_n}(x_0)-\log Df^{a_{n+1}q_n}(x_0)
\le \lVert \log Df^{q_n}\rVert$$ $$\le E_n \exp (-A_n(a_{n+1}+1))\lVert Df^{(a_{n+1}+1)q_n}\rVert$$ $$\le E_n \exp (-A_n(a_{n+1}+1))\lVert Df^{q_{n+1}}\rVert
\lVert Df^{q_{n}}\rVert \lVert Df^{-q_{n-1}}\rVert$$ $$\le C_1^3E_n \exp (-A_n(a_{n+1}+1)).$$
For $k \ge a_{n+1}+2$, $2A_n(k)-A_n(k-1)-A_n(k+1)=0$.
Then since $A_n(k)$ satisfy the condition of Lemma 3 with the constant $C=C_1^3$ and obviously $ \lim_{k \rightarrow \infty}\frac{A_n(k)}{k}=0$, we have $$\lVert Df^{kq_n}\rVert \le \left (\sqrt{\frac{CE_n}{2}}k+1 \right) ^2,
~~0\le k \le a_{n+1}.$$
For $q_n\le l < q_{n+1}$, we define $0\le k_{i+1}\le a_{i+1},
(i=0,1,\ldots,n)$ inductively by $$r_{n+1}=l,~~ r_{i+1}=k_{i+1}q_i+r_i,~~ 0\le r_i < q_i.$$
Then, using $\frac{q_{i+1}}{q_i} \ge a_{i+1} \ge k_{i+1}$, $$\frac{\lVert Df^l\rVert}{l^2}\le
\frac{\prod^n_{i=0}\lVert Df^{k_{i+1}q_i}\rVert}{(k_{n+1}q_n)^2}
\le
\frac{\prod^n_{i=0}\left (\sqrt{\frac{CE_i}{2}}
k_{i+1}+1\right )^2}
{\left (k_{n+1}\prod^{n-1}_{i=0}\frac{q_{i+1}}{q_i}\right )^2}$$ $$\le \left ( \sqrt{\frac{CE_n}{2}}+1\right ) ^2
\prod^{n-1}_{i=0}\left( \sqrt{\frac{CE_i}{2}}
+\frac{q_i}{q_{i+1}}\right)^2.$$
Since $\frac{q_i}{q_{i+2}}<\frac{1}{2}$, for sufficiently small $E_i$ and $E_{i+1}$ $$\left( \sqrt{\frac{CE_i}{2}}+\frac{q_i}{q_{i+1}}\right)
\left( \sqrt{\frac{CE_{i+1}}{2}}+\frac{q_{i+1}}{q_{i+2}}\right)
\le \frac{1}{2}.$$ By Lemma 2 (1), $E_n \rightarrow 0$ as $n \rightarrow \infty$. Consequently we have $$\lim_{l \rightarrow \infty}\frac{\lVert Df^l\rVert}{l^2}=0.$$
For the case $\lVert Df^{-l} \rVert, l >0$, the argument is the same.
Proof of Theorem 2
==================
Let $\{\theta_n\}_{n\ge 1}$ be any increasing unbounded sequence of positive real numbers such that $\theta_n=o(n^2)$ as $n \rightarrow \infty$.
We consider the two-parameter family of rational functions on the Riemann sphere $\hat{\Bbb{C}}= \Bbb{C}\cup \{\infty\}$, $$J_{a,t}: \hat{\Bbb{C}} \rightarrow \hat{\Bbb{C}}, \ \
J_{a,t}(z)= \exp (2\pi i t)z^2\frac{z+a}{az+1}$$ where $a \in \Bbb{R}, a > 3$ and $t \in \Bbb{R}/\Bbb{Z}$.
For each $a, t$ the map $J_{a,t}$ makes invariant the unit circle $\partial \Bbb{D} = \{z\in \Bbb{C}; \lvert z \rvert =1\},
J_{a,t}(\partial \Bbb{D})= \partial \Bbb{D}$, moreover the restriction of $J_{a,t}$ to $\partial \Bbb{D}$ is an orientation preserving diffeomorphism. The set of critical points of $J_{a,t}$ consists of four elements containing $0$ and $\infty$ which are fixed by $J_{a,t}$. Notice that if $a \rightarrow \infty$ then on a compact tubular neighbourhood of the unit circle in $\Bbb{C}\setminus \{0\}$ $J_{a.t}$ uniformly converges to the rotation $z
\mapsto \exp (2\pi i t)z$.
Put $\psi : \Bbb{R}/\Bbb{Z} \rightarrow \partial\Bbb{D},
\psi (x)= \exp (2\pi i x)$. Conjugating $J_{a,t}|\partial \Bbb{D}$ by $\psi$ we obtain the family of analytic circle diffeomorphisms $\{f_{a,t}\}$, $$f_{a,t}: \Bbb{R}/\Bbb{Z} \rightarrow \Bbb{R}/\Bbb{Z},~~
f_{a,t}(x)=\psi^{-1}\circ J_{a,t}\circ \psi (x)
=f_{a,0}(x)+t \mod{\Bbb{Z}}.$$ Temporarily we fix $a > 3$ and abbreviate as $f_{a,t}= f_{t}$.
The following properties of this family are standard. See e.g. [@MS] chapter I, section 4, where Arnold family $x \mapsto x+ a\sin (2\pi x) +t$ is mainly dealt with but the argument is valid for our family. Also see [@KH] chapter 11, section 1.
The map $F: S^1 \rightarrow S^1 , t \mapsto \rho (f_t)$ is continuous and monotone increasing. We set $$K=\{t\in S^1; \rho (f_t) ~~{\rm is~ irrational} \}.$$ We denote Cl($K$) the closure of $K$. $F|K$ is a one-to-one map. For $t \in K$ with $F(t)=\alpha$, we denote $f_{t}=\hat{f}_{\alpha}$. Notice that $f_{t}$ never conjugate to a rational rotation. Hence for $\frac{p}{q} \in \Bbb{Q}/\Bbb{Z}$, $F^{-1}(\frac{p}{q})$ is a closed interval, say, $[\frac{p}{q}_{-},\frac{p}{q}_{+}] $.
Moreover, $F^{-1}|(\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}:
(\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}
\rightarrow K$ is continuous and $$\lim_{\alpha \rightarrow \frac{p}{q}-0}F^{-1}|
(\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}(\alpha)
= \frac{p}{q}_{-},
~\lim_{\alpha \rightarrow
\frac{p}{q}+0}F^{-1}|(\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}(\alpha)
= \frac{p}{q}_{+}.$$
Note that for every $\frac{p}{q}\in \Bbb{Q}/\Bbb{Z}$ and every $x \in S^{1}$, there exists $t \in [\frac{p}{q}_{-},\frac{p}{q}_{+}]$ such that $f_{t}^{q}(x)=x$. For $\frac{p}{q} \in \Bbb{Q}/\Bbb{Z}$, put $t_{*} = \frac{p}{q}_{-}$. The case $t_{*}= \frac{p}{q}_{+}$ is similar. Then the graph of $f^q_{t_{*}}(x)$ touches from below to the graph of the identity map, in particular, there exists $x_0 \in S^1$ such that $$f^q_{t_{*}}(x_0)=x_0, ~~Df^q_{t_{*}}(x_0)=1.$$ Then the following holds.
$D^2f^q_{t_{*}}(x_0) \ne 0.$
By contradiction, we suppose $D^2f_{t_{*}}^q(x_0)=0$. Then in our case $D^3f_{t_{*}}^q(x_0)=0$, otherwise $x_0$ is a topologically transversal fixed point of $f^q_{t_{*}}$ and persists under perturbation of $f_{t_{*}}$, which contradicts $t_{*} \in \mathrm{Cl}(K)\setminus K$. Set $z_{0} =
\psi (x_{0}) \in \partial \Bbb{D}$. Since the order of tangency to the identity map is an invariant of $C^{\infty}$-conjugacy [@T], we have for $J_{t_{*}}=J_{a,t_{*}}$ $$J_{t_{*}}^{q}(z_{0})=z_{0}, \ DJ_{t_{*}}^{q}(z_{0})=1,
\ D^{2}J_{t_{*}}^{q}(z_{0})=
D^{3}J_{t_{*}}^{q}(z_{0})= 0.$$ So $z_{0}$ is a parabolic fixed point for $J_{t_{*}}^{q}$ with multiplicity at least four. See [@M] chapter 7. By the Laeu-Fatou flower theorem ([@M] th.7.2) $z_0$ has at least three basins of attraction for $J^q_{t_{*}}$. Let $B$ be one of the immediate attracting basins of $z_0$ for $J^q_{t_{*}}$. Then $B$ must contain at least one critical point of $J_{t_{*}}^{q}$ ([@M] corollary 7.10). So each basin of the cycle $\{z_0,J_{t_{*}}(z_0),\ldots,
J_{t_{*}}^{q-1}(z_{0})\}$ contains at least one critical point of $J_{t_{*}}$. But $J_{t_{*}}$ has exactly four critical points and two of them are fixed points. We obtain a contradiction.
Hence, for example, by comparing a fractional linear transformation (see also [@B] thorem 1 (A)), we can see that there exist $C >0$ and $\{x_l\}_{l\ge 1} \subset S^1 $ with $\lim_{l\rightarrow \infty}x_l=x_0$ such that $$Df^{lq}_{t_{*}}(x_l)\ge Cl^2 , ~~
\mathrm{for~any}~l\in \Bbb{N}.$$ Since $\theta_n=o(n^2)$ , we have
[**Corollary of Lemma 5.**]{} [*For sufficiently large*]{} $l$, [*we have*]{} $\lVert Df^{lq}_{t_{*}} \rVert > \theta_{lq}$.
[**Remark.**]{} For each $k \in {\Bbb N}$ we set $$U_k=\{ t\in \mathrm{Cl}(K); \mathrm{There~ exist}\ m\ge k ~
\mathrm{and}~ x\in S^1
\ \mathrm{such\ that}\ Df^m_t(x) > m\sqrt{\theta_m} \}.$$ Obviously $U_k$ is open set in Cl($K$). By the corollary and the denseness of preimages of rational numbers by $F$ in Cl($K$), $U_k$ is dense in Cl($K$). So the following set is a residual subset of Cl($K$), $$\{t \in \mathrm{Cl}(K); \limsup_{n\rightarrow \infty}
\frac{\Gamma_n(f_{t})}{\theta_n}=\infty\}.$$
We seek a desired diffeomorphism in this family $\{ f_t \}$ by specifying its rotation number $\alpha_{\infty} = \rho(f_{t_{\infty}})
\in (\Bbb{R}\setminus \Bbb{Q})/\Bbb{Z}$. We will define an increasing sequence of even numbers $0 < n_1 < n_2 < n_3< \cdots$, and a sequence of positive integers $A_1,A_2,A_3,\ldots$ inductively. The continued fraction expansion of $\alpha_{\infty}$ is the following. $$\alpha_{\infty}
=[a_1(\alpha_{\infty}),a_2(\alpha_{\infty}),a_3(\alpha_{\infty}),\ldots]$$ $$= [1,1,\ldots,1,A_{1},1,\ldots,1,A_{2},1,\ldots,1,A_{k},1,\ldots]$$ where if $i=n_k$ then $a_i(\alpha_{\infty})=A_k$ and if $i \ne n_k$ for any $k$ then $a_i(\alpha_{\infty})=1$.
For $m, A \ge 1, m, A \in \Bbb{N}$, we set $$\alpha_{m}^{A}
= [a_1(\alpha_{m}^{A}),a_2(\alpha_{m}^{A}),a_{3}(\alpha_{m}^{A}),\ldots]$$ $$= [1,1,\ldots,1,A_{1},1,\ldots,1,A_{m-1},1,\ldots,1,A,1,1,1,\ldots]$$ where $a_i(\alpha_{m}^{A})=A_k$ if $i= n_k \le n_{m-1}$ and $a_i(\alpha_{m}^{A})=A$ if $i=n_m$ and $a_i(\alpha_{m}^{A})=1$ otherwise.
Set $\alpha_m = \alpha_{m}^{A_m}$. Notice that $\alpha_{m}^{A}|[1,n_{m}-1] = \alpha_{\infty}|[1,n_{m}-1]$ and $\alpha_{m}^{A}$ is of bounded type. Unless otherwise stated we use the symbols $p_{n}, q_{n}$ as $p_{n}(\alpha_{\infty}), q_{n}(\alpha_{\infty})$.
There exist a sequence of even numbers $0 < n_1 < n_2 < n_3< \cdots$, and a sequence of positive integers $A_1,A_2,A_3,\ldots$ such that for each $m \ge 1$ the following properties hold.
$(1)$ For any $j \in \Bbb{Z}$ with $q_{n_{m}-1} \le |j| \le A_mq_{n_{m}-1}$, $\lVert D\hat{f}_{\alpha_{m}}^{j} \rVert < \theta_{|j|}$.
$(2)$ There exists $j_{m}\in \Bbb{Z}$ such that $$q_{n_{m}-1} \le |j_{m}| \le (A_m+1)q_{n_{m}-1} ,\
\lVert D\hat{f}_{\alpha_{m}^{A_{m}+1}}^{j_{m}} \rVert \ge \theta_{|j_{m}|}.$$
$(3)$ For any $t \in F^{-1}(\alpha)$ with $\alpha|[1,n_{m+1}-1] = \alpha_{m}|[1,n_{m+1}-1]$ and any $j \in \Bbb{Z}$ with $\lvert j \rvert \le q_{n_{m}}$, $$\lVert Df_{t}^{j} \rVert-1 \le \lVert D\hat{f}_{\alpha_{m}}^{j}\rVert \le
\lVert Df_{t}^{j}\rVert +1 .$$
Let $\alpha_0 = [1,1,1,\ldots] = \frac{\sqrt{5}-1}{2}$. Since $\alpha_0$ is of bounded type by Corollary of Lemma 2 (2) there exists $C_0 > 0$ such that for any $l \in \Bbb{Z}$ , $\lVert D\hat{f}^{l}_{\alpha_{0}}\rVert \le C_{0}$. Let $n_1$ be a sufficiently large even number such that if $|i| \ge q_{n_1-1}(\alpha_0)$ then $\theta_{|i|} \ge C_0$.
Let $\beta_1 = \alpha_0|[1,n_{1}-1] =
\frac{p_{n_1-1}(\alpha_0)}{q_{n_1-1}(\alpha_0)} =
[1,1,\ldots1]=[1,1,\ldots1,\infty] \in \Bbb{Q}/\Bbb{Z}$. Then by Corollary of Lemma 5 there exists $d \in \Bbb{N}$ such that $\lVert Df_{\beta_{1-}}^{dq_{n_1-1}}\rVert > \theta_{dq_{n_1-1}}$, where $F^{-1}(\beta_{1})=[\beta_{1-},\beta_{1+}]$. Since $\alpha_{1}^{A} \rightarrow \beta_1-0$ as $A \rightarrow \infty$, $F^{-1}(\alpha_{1}^{A}) \rightarrow \beta_{1-}$ as $A \rightarrow \infty$. So for sufficiently large $A$ we have $\lVert D\hat{f}_{\alpha_{1}^{A}}^{dq_{n_1-1}}\rVert > \theta_{dq_{n_1-1}}$. Hence the following is well defined. $$A_{1}=\max\{ A; \mathrm{for\ any\ } j \in \Bbb{Z}\ \mathrm{with}\
q_{n_1-1} \le |j| \le Aq_{n_1-1},\
\lVert D\hat{f}_{\alpha_{1}^{A}}^{j}\rVert < \theta_{\lvert j\rvert}\}.$$ Therefore there exists $j_{1} \in \Bbb{Z}$ such that $$q_{n_1-1} \le |j_{1}| \le (A_{1}+1)q_{n_1-1}, \
\lVert D\hat{f}_{\alpha_{1}^{A_{1}+1}}^{j_{1}}\rVert \ge \theta_{\lvert j_{1}\rvert}.$$
Suppose we have $n_{1}, n_{2},\ldots,n_{m-1}$ and $A_{1},A_{2},\ldots,A_{m-1}$ satisfying conditions of Lemma. Notice that $\alpha_{m-1}$ is of bounded type and that (3) is satisfied by only requiring that $n_{m}-n_{m-1}$ is sufficiently large. So by the exactly same procedure as above we choose a sufficiently large even number $n_{m}$ and set $$A_{m}=\max\{ A; \mathrm{for\ any\ } j \in \Bbb{Z}\ \mathrm{with}\
q_{n_m-1} \le |j| \le Aq_{n_m-1},\
\lVert D\hat{f}_{\alpha_{m}^{A}}^{j}\rVert < \theta_{\lvert j\rvert}\}.$$
Let $\beta_{0}, \beta_{1}, \beta_{2} \in \Bbb{Q}/\Bbb{Z}$ be $$\beta_{i} = [b_{1}(\beta_{i}),b_{2}(\beta_{i}),\ldots,b_{2n}(\beta_{i})]
= \frac{p_{2n}(\beta_{i})}{q_{2n}(\beta_{i})}, \ i=0,1,2$$ such that $\beta_{0}|[1,2n-1]=\beta_{1}|[1,2n-1]=\beta_{2}|[1,2n-1] $ and for some $B \ge 1, B \in \Bbb{N},\ b_{2n}(\beta_{i})=B+i$.
Then for any $s_{1}, s_{2} \in F^{-1}((\beta_{0},\beta_{2}))$ and any $x \in S^1$ we have $$\sum_{i=1}^{q_{2n}(\beta_{2})}\lvert ( f_{s_{1}}^{i}(x),
f_{s_{2}}^{i}(x) ) \rvert \le 7.$$
The argument of the proof is same as the Światek’s of lemma 3 in [@Sw]. We recall Farey interval. A Farey interval is an interval $I = (\frac{p}{q},\frac{p'}{q'}), p,p',q,q'\in \Bbb{Z}, q,q' > 0$ with $pq'-p'q=1$. Then the following holds.
$(*)$ All rational in $I$ have the form $\displaystyle \frac{kp+lp'}{kq+lq'}, \ k, l \ge 1, k,l \in \Bbb{N}$.
Since $q_{2n}(\beta_{i})=(B+i)q_{2n-1}(\beta_{0})+q_{2n-2}(\beta_{0})$ and $p_{2n}(\beta_{i})=(B+i)p_{2n-1}(\beta_{0})+p_{2n-2}(\beta_{0})$ two intervals $(\beta_{0},\beta_{1}), (\beta_{1},\beta_{2})$ are Farey intervals and $q_{2n}(\beta_{0})<q_{2n}(\beta_{1})<q_{2n}(\beta_{2})$ and by $(*)$ the cardinality of the set of rationals in $(\beta_{0},\beta_{2})$ with denominator less than $2q_{2n}(\beta_{2})$ is at most six (three if $B \ge 3$).
For given $x \in S^1$ we define $$t_{1}=\sup\{t \in [\beta_{0-},\beta_{0+}]; f_{t}^{q_{2n}(\beta_{0})}(x)=x\},$$ $$t_{2}=\inf\{t \in [\beta_{2-},\beta_{2+}]; f_{t}^{q_{2n}(\beta_{2})}(x)=x\}.$$
We define a diffeomorphism $G: S^{1}\times [t_{1},t_{2}] \rightarrow
S^{1}\times [t_{1},t_{2}]$ by $G(y, t)=(f_{t}(y),t)$. Then we have $$DG^{i}(y,t)=
\left(
\begin{array}{@{\,}cc@{\,}}
Df_{t}^{i}(y)&\frac{d}{dt}(f_{t}^{i}(y))\\
0&1
\end{array}
\right)
=
\left(
\begin{array}{@{\,}cc@{\,}}
Df_{t}^{i}(y)&1+\sum_{k=1}^{i-1}Df_{t}^{i-k}(f_{t}^{k}(y))\\
0&1
\end{array}
\right).$$ So $G$ monotonically twists $S^1$-direction to the right. More precisely, let $\tilde{G}: \Bbb{R}\times [t_{1},t_{2}] \rightarrow
\Bbb{R}\times [t_{1},t_{2}], \tilde{G}(\tilde{y},t)=
(\tilde{f_{t}}(\tilde{y}),t)$ be a lift of $G$, then for any $i \ge 1$ the slope of the image of a vertical segment $\{ \tilde{y}\} \times
[t_{1},t_{2}]$ by $\tilde{G}^{i}$ is everywhere positive finite. Let $P : S^{1}\times [t_{1},t_{2}] \rightarrow S^1$ be the projection on the first coordinate.
By contradiction we assume $\sum_{i=1}^{q_{2n}(\beta_{2})}
\lvert ( f_{s_{1}}^{i}(x),
f_{s_{2}}^{i}(x) ) \rvert > 7$. We consider the interval $\gamma = \{x\}\times [t_{1},t_{2}]$ and its images by $G^{i}$. Since $[s_{1},s_{2}] \subset (t_{1},t_{2})$, intervals $P(G^{i}(\gamma)), 1 \le i \le q_{2n}(\beta_{2})$ overlap somewhere with multiplicity at least eight. Then, by the twist condition of $G$ there exist distinct natural numbers $i_{k}$, ($0 \le k \le 7, k \in \Bbb{Z}$) with $1 \le i_{k} \le q_{2n}(\beta_{2})$ such that for each $k$ ($1 \le k \le 7$), $$(\{f_{t_{2}}^{i_{0}}(x)\} \times [t_{1},t_{2}])
\cap G^{i_{k}}(\gamma)\ne \emptyset.$$ Moreover, using the preservation of order by $\tilde{f_{t}}: \Bbb{R}\times \{t\} \rightarrow \Bbb{R}\times \{t\}$ and the twist condition of $G$, we can see that for any $j \ge 0$, $$(\{f_{t_{2}}^{i_{0}+j}(x)\} \times [t_{1},t_{2}])
\cap G^{i_{k}+j}(\gamma)\ne \emptyset.$$ In particular for $j=q_{2n}(\beta_{2})-i_{0}$ by the definition of $t_{2}$ we have $$\gamma \cap G^{i_{k}+q_{2n}(\beta_{2})-i_{0}}(\gamma)\ne \emptyset.$$ This imply that there exists a parameter value $u_{k} \in (t_{1},t_{2})$ such that $f_{u_{k}}^{q_{2n}(\beta_{2})+i_{k}-i_{0}}(x)=x$. For each $k$ ($1 \le k \le 7$) the denominator of $\rho(f_{u_{k}})$ which divides $q_{2n}(\beta_{2})+i_{k}-i_{0}$ is less than $2q_{2n}(\beta_{2})$. This is a contradiction.
[*Proof of Theorem 2.*]{}
$\star$ [**Lower bound.**]{} Let $j_{m} \in \Bbb{Z}$ be in Lemma 6 (2). Then $\lvert j_{m}\rvert \le (A_{m}+1)q_{n_{m-1}} <
q_{n_{m}}(\alpha_{m}^{A_{m}+2})$. We assume $j_{m} > 0$. Then since three rational numbers $$\alpha_{m}^{A_{m}}|[1,n_{m}], \alpha_{m}^{A_{m}+1}|[1,n_{m}],
\alpha_{m}^{A_{m}+2}|[1,n_{m}]$$ satisfy the condition of Lemma 7 and $$\alpha_{\infty} \in
(\alpha_{m}^{A_{m}}|[1,n_{m}], \alpha_{m}^{A_{m}+1}|[1,n_{m}]),$$ $$\alpha_{m}^{A_{m}+1} \in
(\alpha_{m}^{A_{m}+1}|[1,n_{m}], \alpha_{m}^{A_{m}+2}|[1,n_{m}]),$$ we have for any $x \in S^{1}$ $$\lvert \log D\hat{f}_{\alpha_{\infty}}^{j_{m}}(x) -
\log D\hat{f}_{\alpha_{m}^{A_{m}+1}}^{j_{m}}(x) \rvert$$ $$=
\left \lvert \sum_{i=1}^{j_{m}-1}
\log Df_{0}(\hat{f}_{\alpha_{\infty}}^{i}(x)) -
\sum_{i=1}^{j_{m}-1}\log Df_{0}(\hat{f}_{\alpha_{m}^{A_{m}+1}}^{i}(x)) \right \rvert$$ $$\le
\rVert D\log Df_{0} \rVert
\sum_{i=1}^{j_{m}-1}\
\lvert ( \hat{f}_{\alpha_{\infty}}^{i}(x), \hat{f}_{\alpha_{m}^{A_{m}+1}}^{i}(x) ) \rvert
\le
7\lVert D\log Df_{0} \rVert .$$ Since there exists $x_{*} \in S^1$ such that $\lvert D\hat{f}_{\alpha_{m}^{A_{m}+1}}^{j_{m}}(x_{*})\rvert \ge \theta_{j_{m}}$ we have $$\frac{\lVert D\hat{f}_{\alpha_{\infty}}^{j_{m}}\rVert}{\theta_{j_{m}}}
\ge
\frac{\lvert D\hat{f}_{\alpha_{\infty}}^{j_{m}}(x_{*})\rvert}
{\lvert D\hat{f}_{\alpha_{m}^{A_{m}+1}}^{j_{m}}(x_{*})\rvert }
\ge
\exp (-7\lVert D\log Df_{0} \rVert).$$
For the case $j_{m} < 0$, using the chain rule $D\hat{f}_{\alpha}^{j_{m}}(x) =
(D\hat{f}_{\alpha}^{-j_{m}}
(\hat{f}_{\alpha}^{j_{m}}(x)))^{-1}$ we can obtain the same estimates .
As stated above by making the parameter $a$ sufficiently large we can assume that $\lVert D\log Df_{0} \rVert = \lVert D\log Df_{a,0} \rVert$ is smaller than any given positive value.
$\star$ [**Upper bound.**]{} Let $l \in \Bbb{Z}$ with $q_{n} \le l < q_{n+1}$. The case $q_{n} \le -l < q_{n+1}$ is similar. Let $n_{m} =\max \{n_{i}; n_{i}\le n\}$. As in the proof of Theorem 1 we expand $l$ as follows, $$l = k_{n+1}q_{n} + \cdots + k_{n_{m}+1}q_{n_{m}}+cq_{n_{m}-1}+r,$$ where $0 \le k_{i} \le a_{i}(\alpha_{\infty})=1$ ($n_{m}+1 \le i \le n+1$) and we choose $c \in \{-1,0,1\}$ so that $q_{n_{m}-1} \le r \le A_{m}q_{n_{m}-1}$.
By Lemma 2 (2) and Lemma 6 (1), (3) we have $$\lVert D\hat{f}_{\alpha_{\infty}}^{l}\rVert \le
\lVert D\hat{f}_{\alpha_{\infty}}^{q_{n}}\rVert \cdots
\lVert D\hat{f}_{\alpha_{\infty}}^{cq_{n_{m}-1}}\rVert
\lVert D\hat{f}_{\alpha_{\infty}}^{r}\rVert$$ $$\le
\exp(C\sum_{i=n_{m}-1}^{n}\lambda^{i})(1+\lVert D\hat{f}_{\alpha_{m}}^{r}\rVert)
\le
\exp(C\sum_{i=n_{m}-1}^{n}\lambda^{i})(1+\theta_{r}).$$ Therefore we have $$\limsup_{l \rightarrow \infty}
\frac{\lVert D\hat{f}_{\alpha_{\infty}}^{l}\rVert}{\theta_{l}}
\le
\limsup_{l \rightarrow \infty}
\frac{\exp(C\sum_{i=n_{m}-1}^{n}\lambda^{i})(1+\theta_{r})}{\theta_{l}}
\le 1.$$
[\[AA\]]{}
V. I. Arnold, Small denominators I, on the mapping of a circle into itself, Izv. Akad. Nauk. serie Math. 25 (1) (1961), 21-86. Translation Amer. Math. Soc. 2nd series, 46,213-284.
A. Borichev, Distortion growth for iterations of diffeomorphisms of the interval, Geom.Funct.Anal. 14(2004), no.5, 941-964.
M. R. Herman, Sur la conjugation diff$\acute{{\rm e}}$rentiable des diff$\acute{{\rm e}}$omorphismes du cercle $\grave{{\rm a}}$ des rotations, Publ. Math. I.H.E.S. 49 (1979), 5-234.
A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
Y. Katznelson, D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9(1989), no.4, 643-680.
K. M. Khanin, Ya. G. Sinai, New proof of M. Herman’s theorem, Commun. Math. Phys. 112 (1987), 89-101.
J. Milnor, Dynamics in one complex variable: Introductory lectures, Math.ArXiv, math.DS/9201272. http://front.math.ucdavis.edu/math.DS/9201272 W. de Melo, S. van Strien, One-dimensional Dynamics. Springer, New York, 1993.
L. Polterovich, M. Sodin, A growth gap for diffeomorphisms of the interval, J.Anal.Math. 92(2004), 191-209.
J. Stark, Smooth conjugacy and renormalization for diffeomorphisms of the circle, Nonlinearity 1 (4) (1988), 541-575.
G. Światek, Rational rotation numbers for maps of the circle, Commun. Math. Phys. 119 (1988),109-128.
F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier 23, no.2 (1973), 163-195.
N. Watanabe, Growth sequences for flat diffeomorphisms of the interval, Nihonkai Math.J. 15(2004), no.2, 137-140.
J.-C. Yoccoz, Conjugaison diff$\acute{{\rm e}}$rentiable des diff$\acute{{\rm e}}$omorphismes du cercle dont le nomble de rotation v$\acute{{\rm e}}$rifie une condition diophantienne, Ann. Sci. École Norm. Sup. 4 17 (1984), 333-359.
|
---
author:
- 'Alex Tong Lin[^1]'
- Yat Tin Chow
- 'Stanley J. Osher'
title: '[[A Splitting Method For Overcoming the Curse of Dimensionality in Hamilton-Jacobi Equations Arising from Nonlinear Optimal Control and Differential Games with Applications to Trajectory Generation]{}]{}[^2]'
---
[^1]: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095 (, [www.math.ucla.edu/\~atlin](www.math.ucla.edu/~atlin)), (, [www.math.ucla.edu/\~ytchow](www.math.ucla.edu/~ytchow)) (, [www.math.ucla.edu/\~sjo](www.math.ucla.edu/~sjo)).
[^2]:
|
---
abstract: 'In this study, we treat the Fermi bubbles as a scaled-up version of supernova remnants (SNRs). The bubbles are created through activities of the super-massive black hole (SMBH) or starbursts at the Galactic center (GC). Cosmic-rays (CRs) are accelerated at the forward shocks of the bubbles like SNRs, which means that we cannot decide whether the bubbles were created by the SMBH or starbursts from the radiation from the CRs. We follow the evolution of CR distribution by solving a diffusion-advection equation, considering the reduction of the diffusion coefficient by CR streaming. In this model, gamma-rays are created through hadronic interaction between CR protons and the gas in the Galactic halo. In the GeV band, we can well reproduce the observed flat distribution of gamma-ray surface brightness, because some amount of gas is left behind the shock. The edge of the bubbles is fairly sharp owing to the high gas density behind the shock and the reduction of the diffusion coefficient there. The latter also contributes the hard gamma-ray spectrum of the bubbles. We find that the CR acceleration at the shock has started when the bubbles were small, and the time-scale of the energy injection at the GC was much smaller than the age of the bubbles. We predict that if CRs are accelerated to the TeV regime, the apparent bubble size should be larger in the TeV band, which could be used to discriminate our hadronic model from other leptonic models. We also present neutrino fluxes.'
author:
- Yutaka Fujita
- Yutaka Ohira and Ryo Yamazaki
title: 'The Fermi Bubbles as a Scaled-up Version of Supernova Remnants'
---
Introduction
============
The Fermi bubbles are huge gamma-ray bubbles discovered with [*Fermi Gamma-ray Space Telescope*]{} in the direction of the Galactic center (GC) in the GeV band (@su10a; see also @dob10). They are symmetric about the Galactic plane and the size is $\sim 50^\circ$ ($\sim 10$ kpc). Their surface brightness is relatively uniform, and they have sharp edges and hard spectrum [@su10a].
Several models have been proposed for the origin of the bubbles. These models assume that activities of the super-massive black hole (SMBH) or starbursts at the GC created the bubbles. Some models indicated that cosmic-rays (CR) that are accelerated via star formation activities are conveyed into the bubbles [@cro11a; @cro12a]. Others pointed out that the CRs are originated from jets launched by the central black hole [@guo12a], or accelerated inside the bubbles [@mer11a; @che11a]. Gamma-rays can be generated by interaction between CR protons and ambient gas (hadronic models), or by inverse Compton scattering by CR electrons (leptonic models).
In this study, we treat the Fermi bubbles as a scaled-up version of supernova remnants (SNRs). CRs are accelerated at the forward shock front like SNRs. We explicitly solve a diffusion-advection equation to study the evolution of CR distribution. We also focus on the reduction of the diffusion coefficient around the bubbles, which slows CR diffusion and is crucial to explain the sharp edge of the bubbles [@guo12b]. The reduction has been indicated and studied for SNRs [@tor08b; @fuj09c; @li10a; @ohi11b; @fuj11a; @yan12b; @nav13a; @mar13a]; it could be caused by a CR streaming instability or anisotropic diffusion. For the Fermi bubbles, @yan12a studied the reduction by the latter. In this study, we investigate the former. We refer to protons as CRs unless otherwise mentioned.
Models
======
For the sake of simplicity, we assume a spherically symmetric bubble, and mainly focus on the high-galactic-latitude part of the Fermi bubbles (large $|b|$ and small $|l|$ in the Galactic coordinate). Before the bubble is born, the gas in the Galactic halo is static and has a distribution of $\rho_0(r)=\rho_1(r/r_1)^{-\omega}$, where $r$ is the distance from the GC, and $\rho_1$, $r_1$, and $\omega$ are the parameters. We assume that the adiabatic index of the gas is $\gamma=5/3$. For hydrodynamic evolution of the halo gas, we adopt the Sedov-Taylor solution [e.g. @mih84; @ost88]. If an energy is injected at $t=0$ at the GC, the radius of the shock front of the bubble can be written as $$\label{eq:Rs}
R_{\rm sh}(t) = \xi\left(\frac{E_{\rm tot}}
{\rho_1 r_1^\omega}\right)^{1/(5-\omega)}
t^{2/(5-\omega)}\;,$$ where $\xi\sim 1$, and $E_{\rm tot}$ is the injected energy. On the other hand, if the energy is continuously injected at a rate of $L_w$ at the GC, it should be $$\label{eq:Rsw}
R_{\rm sh}(t) = \xi\left(\frac{L_w}
{\rho_1 r_1^\omega}\right)^{1/(5-\omega)}
t^{3/(5-\omega)}\;.$$ We ignore the effect of CR pressure on the gas. We do not care about the energy source: it may be the SMBH or starburst activities at the GC. If the gas has a finite temperature, the Mach number of the shock gradually decreases as the velocity, $V_{\rm sh}=dR_{\rm sh}/dt$, decreases. The gas density $\rho$ and velocity $u$ for $r<R_{\rm sh}$ follow the Sedov-Taylor solution.
Our CR model is based on the one in @fuj10a for the evolution of SNRs. However, while we adopted a Monte Carlo approach in @fuj10a to calculate the CR distribution, in this study we explicitly solve a diffusion-advection equation to follow the evolution of a CR distribution function $f(r,p,t)$, where $p$ is the momentum of CRs. The equation is $$\label{eq:diff}
\frac{\partial f}{\partial t}
= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \kappa\frac{\partial
f}{\partial r}\right)
- (u+u_w)\frac{\partial f}{\partial r}
+ \frac{1}{3 r^2}\left[\frac{\partial}{\partial r}(r^2 (u+u_w))\right]
p\frac{\partial
f}{\partial p} + Q\:,$$ where the source $Q$ describes particle injection, and $u_w$ is the velocity of the waves that scatter CRs. We assume that $u_w=v_A$ for $r>R_{\rm sh}$, where $v_A$ is the Alfvén velocity, and that $u_w=0$ for $r<R_{\rm sh}$ because the waves would isotropically propagate for $r<R_{\rm sh}$.
CRs are accelerated at the shock front of the bubble ($r=R_{\rm
sh}$). We do not consider the details of particle acceleration. CRs are accelerated in the shock neighborhood, where some nonlinear effects generate strong magnetic waves or cause strong amplification of magnetic fields [@luc00a; @bel04a]. In this region, particle diffusion would follow the so-called Bohm diffusion and CR acceleration is effective. The spatial scale of the region is much smaller than $R_{\rm
sh}$ and we ignore the width. Thus, we assume that $$\label{eq:Q}
Q(r,p,t)=\left\{\begin{array}{ll}
K_Q^{-1} p^{-q} \rho(R_{\rm sh,+})V_{\rm sh}^3
\delta(r-R_{\rm sh}) & \mbox{if $p_{\rm min}<p<p_{\rm max}$} \\
0 & \mbox{otherwise,}
\end{array}
\right.$$ where $q$ is the parameter, and $R_{\rm sh,+}$ is the radius just outside the shock. At a strong shock, the standard diffusive shock acceleration model predicts that $q\sim 4$ [@dru83a]. For instant energy injection (equation (\[eq:Rs\])), the coefficient is written as $$K_Q = 16\pi^2 c\: \xi^{5-\omega}\left(\frac{2}{5-\omega}\right)^3
\frac{E_{\rm tot}}{E_{\rm cr,tot}}\ln\left(\frac{t_f}{t_0}\right)
\int_{p_{\rm min}}^{p_{\rm max}}p'^{\: 2-q}
\sqrt{p'^{\: 2}+m_p^2 c^2}\: dp' \:,$$ while for constant energy injection (equation (\[eq:Rsw\])), it is written as $$K_Q = 16\pi^2 c\: \xi^{5-\omega}\left(\frac{3}{5-\omega}\right)^3
\frac{L_w}{E_{\rm cr,tot}}(t_f-t_0)
\int_{p_{\rm min}}^{p_{\rm max}}p'^{\: 2-q}
\sqrt{p'^{\: 2}+m_p^2 c^2}\: dp' \:,$$ where $c$ is the light velocity, and $m_p$ is the proton mass. CRs are accelerated and injected into the Galactic halo space between $t=t_0$ and $t_f$, and $E_{\rm cr,tot}$ is the total energy of the CRs accelerated during that period. The maximum momentum $p_{\rm max}(t)$ is determined by the condition of $t_{\rm acc}=t_{\rm age}$, where $t_{\rm acc}$ is the acceleration time-scale and $t_{\rm age}$ is the age of the bubble. In our case, $t_{\rm age}=t$ and $$\label{eq:pmax}
p_{\rm max} \approx\frac{3}{20}\frac{e B_0}{\eta_g c^2}V_{\rm sh}^2 t\:,$$ where $e$ is the proton charge, $B_0$ is the background magnetic field, and $\eta_g$ is the gyro-factor [@aha99a; @ohi10a]. We assume that the shock is strong. Unless otherwise mentioned, we assume $\eta_g=1$ (Bohm diffusion). Instead of equation (\[eq:pmax\]), $p_{\rm max}$ is often determined by an escape condition for SNRs [@ptu05a; @rev09]. This is because the characteristic spatial length of particles penetrating into the shock upstream region can be comparable to the size of SNRs [@ohi10a]. However, this cannot be applied to the Fermi bubbles, because the size of the bubbles is much larger than the characteristic length. We fix the minimum momentum at $p_{\rm min}=m_p c$.
The CRs escaped from the shock neighborhood may amplify magnetic fluctuations (Alfvén waves) in the Galactic halo through a streaming instability [@wen74a; @ski75c]. Since CRs are scattered by the fluctuations, the diffusion coefficient $\kappa$ in equation (\[eq:diff\]) is reduced. At the rest frame and outside the bubble ($r\geq R_{\rm sh}$), the wave growth is given by $$\label{eq:psi}
\frac{\partial\psi}{\partial t}
=\frac{4\pi}{3}\frac{v_A p^4 v}{U_M}|\nabla f|\:,$$ where $\psi(r,p,t)$ is the energy density of Alfvén waves per unit logarithmic bandwidth (which are resonant with particles of momentum $p$) relative to the ambient magnetic energy density $U_M$ [@bel78a], and $v$ is the particle velocity. The diffusion coefficient is simply given as $$\label{eq:kappa}
\kappa(r,p,t) = \frac{4}{3\pi}\frac{p v c}{e B_0 \psi}\:.$$ Within the bubble, the evolution of the waves could be complicated, because it could be controlled by something like turbulence. Thus for $r<R_{\rm sh}$, we simply assume that $$\label{eq:kappa_b}
\kappa(r,p,t)=\kappa_B\rho(R_{\rm sh})/\rho(r)\:,$$ where $\kappa_B=\eta_g p c^2/(3 e B)$ is a Bohm-type diffusion coefficient [@ber94a], and $B\approx 4B_0$ for a strong shock. The results do not much depend on the diffusion coefficient inside the bubble if it is small enough. Although there is no strong observational constraint on magnetic fields in the Galactic halo, we assume that they are given by $B_0(r)=B_1(r/r_1)^{-\omega/2}$. For the value of $B_1=10\:\rm \mu G$, the Alfvén velocity has a constant value of $v_A=B_0/\sqrt{4\pi\rho}=100\rm\: km\: s^{-1}$. The field strength we assumed is comparable to or smaller than the value adopted by @su10a or $B=30\: e^{-r/{\rm 2\: kpc}}\rm\: \mu G$ for $r\lesssim 10$ kpc. Gamma-ray and neutrino fluxes created through hadronic interactions between CR protons and gas protons are calculated using the code provided by @kar08b.
We consider four models. Model FD is our fiducial model; an energy of $E_{\rm tot}=2.5\times 10^{57}$ erg is instantaneously released at $t=0$ at $r=0$ (see equation (\[eq:Rs\])). For the initial distribution of the halo gas, we assume that $\omega=1.5$, $\rho_1=7.8\times
10^{-24}\rm\: g\: cm^{-3}$ and $r_1=0.1$ kpc to be consistent with gamma-ray observations (Section \[sec:result\]). The index $\omega=1.5$ is an approximation of the profile obtained by @guo12a. We solve equations (\[eq:diff\]) and (\[eq:psi\]) for $t\geq t_0=1\times 10^6$ yr. It is to be noted that if the SMBH at the GC injects energy at a rate of 16% of the Eddington luminosity ($\sim 5\times 10^{44}\rm\: erg\: s^{-1}$) for $1\times 10^6$ yr, the total energy is comparable to $E_{\rm tot}$. The energy that goes into CRs is $E_{\rm cr,tot}=0.2\: E_{\rm tot}$ in total. The spectral index of the accelerated CRs is assumed to be $q=4.1$. For the parameters we adopted, the maximum momentum at $t=t_0$ is $p_{\rm max}c=9\times
10^{14}$ eV. At $t=t_0$, we assume that the diffusion coefficient has typical Galactic values: $$\label{eq:kappa_i}
\kappa_i=10^{28}\left(\frac{E}{10\rm\: GeV}\right)^{0.5}
\left(\frac{B_0}{3\: \mu\rm G}\right)^{-0.5}\rm\: cm^2\: s^{-1}\:,$$ where $E$ is the energy of a CR proton [@gab09a]. From equation (\[eq:kappa\]), we obtain the initial wave energy density $\psi_i(r,p)=\psi(r,p,t_0)\propto \kappa_i^{-1}$. If the temperature of the halo gas is $T=2.4\times 10^6$ K [@guo12a], the Mach number of the shock at $t=3\times 10^6$ yr is ${\cal M}\sim 4$. Since it is generally believed that CR acceleration is ineffective at smaller Mach numbers [e.g. @gie00a], we assume that CR acceleration finishes at $t_f=3\times 10^6$ yr. The current age of the bubble is assumed to be $t_{\rm obs}=10^7$ yr and the bubble center (GC) is located at a distance of 8.5 kpc. The current bubble size is $R_{\rm sh}(t_{\rm
obs})=9.7$ kpc.
Other models are studied for comparison. Their parameters are the same as those for Model FD except for the followings. In Model NG, the wave growth is ignored, and we assume that $\kappa=\kappa_i$ for $r>R_{\rm
sh}$. In Model LA, acceleration of CRs starts later, and we adopt $t_0=4\times 10^6$ yr and $t_f=t_{\rm obs}=10^7$ yr. In Model CI, energy is continuously injected from the GC at a rate of $L_w=E_{\rm
tot}/t_{\rm obs}$ for $t<t_{\rm obs}$ (see equation (\[eq:Rsw\])), and we set $t_{\rm obs}=2\times 10^7$ yr so that the position of the peak of the surface brightness profiles is almost the same as that of Model FD.
Results {#sec:result}
=======
Figure \[fig:pro\] shows the surface brightness profiles of the bubble. They are calculated simply by projecting gamma-ray emissions on a plane at a distance of 8.5 kpc and we do not consider the detailed geometrical effects that come up when the distance to the Fermi bubbles is finite. Figure \[fig:pro\]a shows the results for Model FD, which are compared with the southern bubble data in Figure 9 in @su10a; one degree corresponds to $\pi/180\times 8.5$ kpc. Since our model is rather simple and we do not include background, we shift the observational data along the horizontal axis ($+5^\circ$) and the vertical axis ($-0.9\rm\: keV\: cm^{-2}\: s^{-1}\: sr^{-1}$ in the 1–5 GeV band and $-0.4\rm\: keV\: cm^{-2}\: s^{-1}\: sr^{-1}$ in the 5–20 band). At 2 and 10 GeV, the predicted profiles are fairly flat and have sharp edges at $r\sim 50^\circ$ as observations suggest (Figure \[fig:pro\]a). Significant gamma-ray emissions fill the bubble, because not all the gas is concentrated at the shock. (Model FD in Figure \[fig:sedov\]).
The sharp edges seen in Figure \[fig:pro\]a are created by the dense gas just behind the shock. Moreover, the wave amplification outside the shock also contributes to the formation of the sharp edges. Figure \[fig:amp\] shows the wave amplification $\psi/\psi_i$ at $r=R_{\rm sh,+}$ and $t=t_{\rm obs}$ for Model FD. The amplification leads to the decrease of the diffusion coefficient (equation (\[eq:kappa\])) and slows the CR diffusion out of the bubble. While the wave energy $\psi$ grows at a given radius outside the shock (equation (\[eq:psi\])), the wave energy at the expanding shock ($r=R_{\rm sh,+}(t)$) gradually decreases. While $\psi/\psi_i$ is not much dependent on CR momentum at $10^{10}\lesssim pc \lesssim
10^{14}$ eV (Figure \[fig:amp\]), $\psi_i\propto \kappa_i^{-1}$ is a decreasing function of CR momentum (equations (\[eq:kappa\]) and (\[eq:kappa\_i\])). This means that $\kappa\propto\psi^{-1}$ is an increasing function of CR momentum and CRs with larger energies diffuse faster. Assuming that CR acceleration stopped at $t=t_f < t_{\rm obs}$, CRs are left far behind the shock at $t=t_{\rm obs}$ if their diffusion is not much effective. This happens for GeV CRs in Model FD; most of them remain far behind the shock. However, this is not the case for CRs with much larger energies. At $t=t_{\rm obs}$ and $r=R_{\rm sh,+}$, the diffusion coefficient for CRs with $pc=10$ TeV is $\kappa=9.7\times
10^{28}\rm\: cm^2\: s^{-1}$. Thus, the diffusion scale-length is $l_{\rm
diff}\sim\sqrt{4\kappa (t_{\rm obs}-t_f)}\sim 3.0$ kpc. On the other hand, the shock velocity at $t=t_{\rm obs}$ is $V_{\rm sh}=540\rm\: km\:
s^{-1}$, and thus the advection scale-length is $l_{\rm adv}=V_{\rm
sh}(t_{\rm obs}-t_f)\sim 3.9$ kpc. Since $l_{\rm diff}\lesssim l_{\rm
adv}$, most CRs do not diffuse beyond the shock radius, although the diffusion cannot be ignored. This explains the profile of 1 TeV gamma-rays, which is created by CRs with $\sim 10$ TeV (Figure \[fig:pro\]a). The moderate diffusion enables some of the TeV CRs to reach the very high density region just behind the shock. This makes the surface brightness at 1 TeV a little brighter than those at smaller energies (Figure \[fig:pro\]a). We note that the slight increase of $\psi/\psi_i$ at $pc\sim 10^{14}$ eV in Figure \[fig:amp\] is caused by the higher-energy CRs that have arrived at $r\sim R_{\rm
sh}$. In Model NG, in which the wave growth is ignored, the diffusion for GeV CRs is the moderate one and the surface brightness in the GeV band is larger than that for Model FD (Figure \[fig:pro\]b). However, the diffusion of CRs with $\gtrsim$ TeV is much faster and the CRs diffuse beyond the shock radius (Figure \[fig:pro\]b). While the surface brightness profile for a given energy is relatively flat for Model NG, the spatial extension of CRs varies with their energies because of the fairly large and energy-dependent diffusion (equation (\[eq:kappa\_i\])).
In Model LA, the limb-brightening becomes more prominent because CRs do not have much time to diffuse out (Figure \[fig:pro\]b). Thus, $t_0$ must be much smaller than $t_{\rm obs}$, or CR acceleration must have started at the shock when the bubble is small. The observed flatness of the surface brightness profiles may also imply that the time-scale of the energy injection at the GC is much shorter than the age of the bubble ($t_{\rm inj}\ll t_{\rm obs}$). In Models FD, NG and LA, we implicitly assumed that $t_{\rm inj}\lesssim t_0$. On the other hand, in Model CI the energy has been continuously injected at the GC ($t_{\rm
inj}=t_{\rm obs}$). In this case, the halo gas inside the bubble ($r<R_{\rm sh}$) is compressed into a thin dense shell between the shock and the contact discontinuity at $r=0.86\: R_{\rm sh}$ (Figure \[fig:sedov\]). If the region behind the contact discontinuity ($r<0.86\: R_{\rm sh}$) is almost empty with gas, the gamma-ray image should have a shell-like structure (Model CI in Figure \[fig:pro\]b), because gamma-rays are created through the interaction between CRs and the gas of the thin shell. In Model CI, we assume that the gas density inside the contact discontinuity is $0.1\:\rho(R_{\rm sh,+})$ for a calculational purpose.
Figure \[fig:sp\] shows the gamma-ray spectrum of the bubble for Model FD. The observed hard gamma-ray spectrum is reproduced, which reflects that the spectral index of the CRs around the bubble is not much different from the original one ($q=4.1$). This is because of the decrease of the diffusion coefficient or the confinement of CRs around the bubble. For Model NG, the confinement depends on CR energies, and the original CR spectrum is not conserved [@ohi11b]. In Figure \[fig:sp\], the gamma-ray luminosity in the TeV band is slightly larger than that in the GeV band as was shown in Figure \[fig:pro\]a. We note that the TeV luminosity depends on $p_{\rm max}$. For example, larger $\eta_g$ makes $p_{\rm max}$ smaller (equation (\[eq:pmax\])). The dotted line in Figure \[fig:sp\] shows the spectrum when $\eta_g=100$; other parameters are the same as those for Model FD. As can be seen, the TeV luminosity is much reduced. In Figure \[fig:sp\], we also present the neutrino spectrum for Model FD. The flux is similar to the one predicted by @lun12a, and thus their discussion can be applied. Our results indicate that the neutrino flux is comparable to the background and it could be marginally detected [@lun12a].
Model FD indicates that the position of the shock is a few kpc outside the edge of the gamma-ray bubble. X-ray emission from the high-density gas just behind the shock could have been detected there [@sof00a; @bla03a]. In Model FD, in which CRs are accelerated to PeV, we also predict that the size of the bubble is larger in the TeV band (Figure \[fig:pro\]a), because of the faster diffusion of higher energy CRs. The difference of the size of the bubble between the GeV and TeV bands could be used to discriminate our hadronic model from other leptonic models. In the leptonic models, gamma-rays originate from electrons and the cooling time of the electrons with energies of $\gtrsim 30$ GeV is smaller than the age of the bubbles [Figure 28 in @su10a]. This means that the electrons must be being accelerated. Thus, the gamma-rays should be observed around the acceleration sites and the gamma-ray distribution should not depend on the energy band. In particular, since the cooling time of TeV electrons is very short ($\lesssim 10^6$ yr), the diffusion of the electrons can be ignored and the distribution of the gamma-rays from them should reflect the positions of the acceleration sites.
Summary and Discussion
======================
We solve a diffusion-advection equation to investigate the evolution of the distribution of CRs accelerated at the shock front of the Fermi bubbles. We found that the observed flat surface brightness profile with a sharp edge can be reproduced because of the gas inside the bubbles and the reduced diffusion coefficient owing to CR streaming instabilities. The latter also contributes to the hard spectrum of the bubbles. The CR acceleration must have started at the early stage of the bubble evolution and the time-scale of energy injection at the GC must be much smaller than the current age of the bubbles.
the hadronic model by @cro11a, the bubbles are long-lived ($\gtrsim 8$ Gyr) or steady. This is not likely in our model because the forward shocks rapidly cross the halo ($\sim 10^7$ yr), unless the background gas is rapidly falling toward the galactic plane. If the SMBH blows winds, reverse shocks may develop in the winds [@zub11a], and CRs may be accelerated there. Our model does not treat this type of acceleration. If the reverse shocks disappear in a short time, the situation may not be much different from the one we considered. Moreover, since the reverse shocks are located in the innermost region of the bubbles, it may take a longer time for the CRs accelerated there to diffuse out to the dense region at the bubble edge than those accelerated at the forward shocks. Thus, the contribution of the former to the gamma-ray emission may be less than that of the latter. @zub12a indicated that the activity of the SMBH lasted $\sim 10^6$ yr and the current age of the bubbles is $\sim 10^7$ yr, which are consistent with our model. They also indicated that buoyancy may deform the bubbles. Although it may somewhat change the height of the bubbles, our results may not be much affected as long as CRs generally move with the background gas.
This work was supported by KAKENHI (YF: 23540308, YO: No.24.8344). R. Y. was supported by the fund from Research Institute, Aoyama Gakuin University.
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---
abstract: |
CRTS J084133.15+200525.8 is an optically bright quasar at $z =
2.345$ that has shown extreme spectral variability over the past decade. Photometrically, the source had a visual magnitude of $V
\sim 17.3$ between 2002 and 2008. Then, over the following five years, the source slowly brightened by approximately one magnitude, to $V \sim 16.2$. Only $\sim 1$ in 10,000 quasars show such extreme variability, as quantified by the extreme parameters derived for this quasar assuming a damped random walk model. A combination of archival and newly acquired spectra reveal the source to be an iron low-ionization broad absorption line (FeLoBAL) quasar with extreme changes in its absorption spectrum. Some absorption features completely disappear over the 9 years of optical spectra, while other features remain essentially unchanged. We report the first definitive redshift for this source, based on the detection of broad H$\alpha$ in a Keck/MOSFIRE spectrum. Absorption systems separated by several 1000 km s$^{-1}$ in velocity show coordinated weakening in the depths of their troughs as the continuum flux increases. We interpret the broad absorption line variability to be due to changes in photoionization, rather than due to motion of material along our line of sight. This source highlights one sort of rare transition object that astronomy will now be finding through dedicated time-domain surveys.
author:
- 'Daniel Stern, Matthew J. Graham, Nahum Arav, S. G. Djorgovski, Carter Chamberlain, Aaron J. Barth, Ciro Donalek, Andrew J. Drake, Eilat Glikman, Hyunsung D. Jun, Ashish A. Mahabal, Charles. C. Steidel'
title: Extreme Variability in a Broad Absorption Line Quasar
---
Introduction
============
For galaxies hosting active galactic nuclei (AGNs), time-domain surveys have long proven to be fertile avenues of research. Indeed, optical continuum variability was recognized as a common feature of quasars shortly after their initial discovery [@Matthews:63], and has since been exploited for purposes ranging from identifying quasars [, @vandenBergh:73], to determining black hole masses through reverberation mapping [, @Blandford:82; @Bentz:09], to studying the inner circumnuclear environment [, @Risaliti:02]. Recent efforts using wide-area, time-domain surveys have vastly extended this avenue of research by exploring the optical variability of extremely large samples of quasars, numbering in the tens to hundreds of thousands [, @MacLeod:12; @Graham:14]. Besides determining the light curve properties of typical quasars, such work has identified interesting new phenomenology such as candidate periodic light curves suggestive of sub-parsec binary super-massive black hole systems [, @DOrazio:15a; @DOrazio:15b; @Graham:15; @Graham:15b; @Jun:15b; @Liu:15], AGN undergoing major flaring suggestive of microlensing or explosive activity in the accretion disk such as superluminous supernovae, mergers, or tidal disruption events [, @Drake:11; @Lawrence:16 Graham , submitted], and changing look AGN with the abrupt appearance or disappearance of broad emission lines [, @LaMassa:15; @Gezari:17].
One topic where quasar variability has received particular attention has been the temporal characteristics of broad absorption line (BAL) quasars. Specifically, over the past few years, several teams have reported on multi-epoch spectroscopic observations of BAL quasars [, @Barlow:92; @Lundgren:07; @Gibson:08; @Gibson:10; @Capellupo:11; @Capellupo:12; @Capellupo:13; @FilizAk:12; @FilizAk:13; @FilizAk:14; @Vivek:12; @He:14; @He:15; @Joshi:14; @Wildy:14; @Wildy:15; @Grier:15; @Zhang:15]. While variability in BAL trough strengths is relatively common, large ($> 50\%$) changes in the absorption equivalent width is quite rare [, @Hall:11]. A primary question in BAL variability studies has been whether observed changes in BAL trough strengths are primarily due to changes in the ionization state of the outflowing wind [, @Wang:15], or whether they are due to high column density BAL clouds moving through our line of sight [, @McGraw:15].
For example, @FilizAk:13 present a detailed analysis of $\approx 650$ BAL troughs identified in 291 quasars observed by the Sloan Digital Sky Survey (SDSS), sampling rest-frame timescales between 1 and 3.7 years. They estimate that the average lifetime of a BAL trough is a few thousand years, and that the emergence/disappearance of BAL features are extremes of general BAL variability. @FilizAk:13 also report coordinated BAL variability across multiple troughs at different velocities. They argue that changes in the opacity of the shielding gas producing changes in the ionizing radiation incident on the BAL material are the most probable cause for such coordinated variability.
@Grier:15 and @Wildy:15 reach similar conclusions based on the highly variable BAL lines seen in a spectroscopic monitoring campaigns. With variability seen on time-scales of just a few days, both authors conclude that the most likely cause of such rapid changes is the BAL gas responding to changes in the incident ionizing continuum.
Leading to an alternative explanation of BAL variability, @Capellupo:11 [@Capellupo:12] report on an ongoing monitoring campaign of a sample of 24 BAL quasars at $1.2 < z < 2.9$ on timescales ranging from $\sim 4$ months to $\sim 8$ years. Studying the BAL feature, @Capellupo:11 found variability in 40% of their sample on month-long timescales, and in 65% of their sample on year-long timescales. They find that higher-velocity BALs are more likely to vary than lower-velocity BALs, and that weaker BALs are more likely to vary than stronger BALs. They suggest that the observations are best understood as the movement of clouds within 6 pc of the central engine across the line of sight. In a detailed study of the first observation of $\lambda
\lambda 1118, 1128$ BAL variability in a quasar, @Capellupo:14 argue that the observations are best described by a BAL cloud at a distance of $\simlt 3.5$ pc moving across the line sight. The implied kinetic energy of the outflow would be $\sim 2\%$ of the quasar bolometric luminosity, which is sufficient to cause substantial feedback.
Also supporting this interpretation that BAL variability is not dominated by photoionization, @He:14 report on 18 epochs of SDSS/BOSS spectroscopy of a BAL quasar at $z = 2.72$. They find only a weak correlation between the BAL variability and the continuum luminosity, suggesting that continuum changes are not driving changes in the BAL trough amplitudes.
Here, we report on CRTS J084133.15+200525.8 (), an optically bright quasar that has shown extreme variability over the past decade (Figure \[fig:lc\]). The quasar transitioned from having a relatively stable visual magnitude of $V \sim 17.3$ between 2002 and 2008, to slowly brightening by a factor of $\sim 2.5$ over the course of 5 years and then plateauing at $V \sim 16.2$. As detailed below, a combination of archival and newly acquired spectroscopy reveal this source to be an iron low-ionization broad absorption line (FeLoBAL) quasar exhibiting extreme spectroscopic changes over the same time period, and the nature of these variations allow us to assess the likely cause of the BAL trough variability.
Independent of our own work on , @Rafiee:16 recently reported on this same source as part of a sample of three FeLoBAL quasars that have shown significant spectroscopic variability over the past decade. Interestingly, all three show decreasing strength of their low-ionization iron absorption. The current paper has several additions relative to that work. Specifically, we provide new data on , including a new epoch of optical spectroscopy which demonstrates continued spectral changes, and a near-infrared spectrum which provides the first precise redshift for the quasar as well as an estimate of its black hole mass. Finally, @Rafiee:16 remain agnostic as to whether absorber transverse motion or ionization variability is the more likely cause of the changes in the absorption troughs of this source. In contrast, the additional epoch of Palomar spectroscopy presented here allows us to argue that ionization variability is the more likely cause of the extreme absorption variability seen in .
Throughout this paper, we use Vega magnitudes unless otherwise indicated and we adopt the concordance cosmology, $\Omega_{\rm M}
= 0.3$, $\Omega_\Lambda = 0.7$ and $H_0 = 70\, \kmsMpc$.
Data and Results
================
Optical Light Curve
-------------------
The Catalina Real-time Transient Survey[^1] [CRTS; @Drake:09] leverages the Catalina Sky Survey, designed to search for near-Earth objects, as a probe of the time-variable universe. CRTS has used three telescopes for much of the past decade, two in the northern hemisphere and one in Australia, to cover up to $\sim 2500\, {\rm deg}^2$ per night. The filterless observations are broadly calibrated to Johnson $V$ [for details, see @Drake:13] with a nominal depth of $V
\sim 20$. The full CRTS data set contains time series for approximately 500 million sources.[^2]
CRTS represents the best data set currently available with which to systematically study quasar variability with large samples over a decade-length timescale. In an analysis of characteristic timescales of 240,000 known spectroscopically confirmed objects using Slepian wavelet variance, Graham et al. (submitted) originally identified as an extreme outlier in the plane defined by a linear trend (the Thiel-Sen statistic) and deviation from the median Slepian wavelet variance fit. In that analysis, has a characteristic timescale $\tau = 109.9$ days, which is significantly larger than expected for a quasar of its magnitude, $\tau = 48.0 \pm 5.9$ days.
If we instead characterize quasar light curves with a Gaussian process damped random walk model and only consider the subset of 79,749 quasars with at least 200 CRTS photometric measurements, again stands out. The two parameters from this model are the amplitude, $\sigma$, and the characteristic timescale, $\tau$, of the damped random walk [, @Kelly:09]. We use a kernel density estimator to determine the distribution of sources in the $\sigma-\tau$ plane, and we find that resides in an extreme location in this plane ($\log \Sigma = -7.8$, where $\Sigma$ is the density of sources in this plane). Only seven quasars stand out at this level or more from the population distribution, implying that only $\sim 1$ in 10,000 quasars show variability behavior as extreme as . Further inspection of the CRTS light curve of (Figure \[fig:lc\]) also indicates that the variable behavior is different from the expected stochastic damped random walk model that describes most quasars, and instead appears more consistent with a state change. Further support for this interpretation comes from earlier photometry of reported in @Rafiee:16 from the Palomar Sky Surveys (POSS-I, Palomar Quick V, and POSS-II), reaching back to the mid-1950s. @Rafiee:16 reports no evidence for a significant change in the optical brightness of prior to 2000.
SDSS imaged on UT 2004 December 12 (MJD = 53351), which is prior to the brightening episode. The source was unresolved, and based on its unusual and red colors, SDSS targeted for spectroscopic observations as a high-redshift quasar candidate.
Optical Spectroscopy
--------------------
was first observed spectroscopically by SDSS on UT 2005 December 1 [MJD = 53705; @Blanton:03] and was then re-observed by SDSS-III BOSS on UT 2011 January 4 [MJD = 55565; @Dawson:13]. The spectra, shown in Figure \[fig:optical\], show a source with many absorption features, making redshift identification challenging. Indeed, the SDSS data releases have reported a variety of redshifts for , always with warning flags, ranging from $z = 0.859$ (DR8; Warning = Many Outliers) to $z = 1.295$ (DR7; zStatus = Failed) to $z = 3.195$ (DR9; Warning = Negative Emission). Our visual inspection of the BOSS spectrum tentatively identified and blends in the region around 9400 Å, implying $z \sim 2.3$, consistent with both the visual inspection value of $z = 2.342$ in the SDSS DR12 quasar catalog [DR12Q; @Paris:14] and the results of our Keck infrared spectrum described in §2.4.
We obtained additional optical spectroscopy of using the Double Spectrograph on the Hale 200” Telescope at Palomar Observatory on UT 2014 April 22 (MJD = 56769). We obtained two 600 s exposures using the 1.0 slit in cloudy conditions. The data were reduced using standard procedures and relative spectrophotometric calibration was achieved using observations of standard stars obtained on the same night. Figure \[fig:optical\] presents the Palomar data, where we have scaled the spectra so that the long wavelength ($\simgt 5500$ Å) part of the spectra is of comparable flux density to the BOSS spectrum at the same wavelengths.
The multi-epoch spectra show the extreme variability exhibited by , as well as multiple strong absorption features, characteristic of an FeLoBAL quasar. FeLoBALs are notoriously challenging targets for redshift identification [, @Becker:97; @Brunner:03]. We see dramatic changes across the full spectrum, particularly in the spectral region between redshifted Ly$\alpha$ and . Some features do not change across the near-decade timescale of the spectroscopy, such as the saturated absorption at 5150 Å. Other features completely disappear, such as absorption lines at $\approx$ 5450 and 8500 Å. There is an overall uncovering of blue continuum emission, with the flux around Ly$\alpha$ increasing by an order of magnitude over the $>8$ years spanned by the spectroscopy. In addition, while the continuum between \] and is extremely choppy in the 2005 spectrum, by 2014 it is smoother, which is more typical of normal quasar spectra.
Imaging at Other Wavelengths
----------------------------
is a bright near- to mid-infrared source, well detected by both the Two Micron All Sky Survey [$K_s = 13.62 \pm 0.04$ — 2MASS; @Skrutskie:06] and the [*Wide-field Infrared Survey Explorer*]{} [$W3 = 9.58 \pm 0.06$ — ; @Wright:10]. With $W1 - W2 = 0.65$, is slightly bluer than the mid-infrared AGN selection criteria of @Stern:12, which are $W1 - W2 \geq
0.8$ and $W2 \leq 15.05$. However, as shown in @Assef:13, the AGN selection color can be relaxed for brighter sources.
There is little variability detected at longer wavelengths in this source. In AB magnitudes, the $z$-band magnitude recorded by SDSS was $z = 16.37 \pm 0.01$ on MJD 53351, closely matching the $z$-band magnitude of $Z = 16.34 \pm 0.01$ recorded by UKIRT Infrared Deep Sky Survey [UKIDSS; @Lawrence:07] on MJD 55141. In the near-infrared (in Vega magnitudes), 2MASS recorded $H = 14.41 \pm
0.05$ and $K_s = 13.62 \pm 0.04$ on MJD 51105, closely matching the UKIDSS values of $H = 14.32 \pm 0.02$ on MJD 54061 and $K = 13.57
\pm 0.02$ on both MJD 54061 and MJD 55238, where we have assumed a 2% floor on the UKIDSS photometric calibration [, @Hodgkin:09]. Similarly, the mid-infrared flux measured by [*WISE*]{} and [*NEOWISE*]{} [@Mainzer:14] varies by only $\sim 0.04$ mag, comparable to the typical uncertainty.
is not detected by [*ROSAT*]{}, nor was it (serendipitously) observed by either the [*Chandra X-Ray Observatory*]{} or [*XMM-Newton*]{}. is also not detected by the Faint Images of the Radio Sky at Twenty cm survey [FIRST; @Becker:95], implying $S_{\rm 1.4~GHz} \simlt 1$ mJy ($5 \sigma$). Finally, as expected, observations by the [*Galaxy Evolution Explorer*]{} [[*GALEX*]{}; @Martin:05], which sample below the Lyman limit for $z = 2.35$, do not detect .
Near-infrared Spectroscopy
--------------------------
We obtained a $K$-band (1.95-2.39 $\mu$m) spectrum of with the Multi-Object Spectrometer for InfraRed Exploration [MOSFIRE; @McLean:12; @Steidel:14] on UT 2014 May 5 (MJD=56782) in longslit mode. We obtained three dithered exposures of 180 s each through a 07 entrance slit under clear conditions with good seeing. The spectrum was reduced using a combination of the MOSFIRE data reduction pipeline (DRP) and custom routines [for details, see @Steidel:14]. Wavelength calibration was based on a combination of OH emission lines in the night sky and an internal Ne arc lamp. Flux calibration and telluric absorption removal was accomplished using spectra of an A0V star (Vega analog) observed at similar airmass. The final extracted spectrum (Figure \[fig:nearIR\]) shows strong continuum and a single broad emission line with a peak at 2.1956 $\mu$m, which we identify as H$\alpha$ at $z=2.3446$. The apparent asymmetry in the continuum straddling the line is well modeled by the @Boroson:92 template on the blue side of the line.
We use the broad H$\alpha$ emission line to estimate the mass of the black hole in . First, we apply a multiplicative correction to the $K$-band spectrum to match the $K$-band photometry from the UKIDSS observations. We then approximate the uncertainties in the spectrum by considering the standard deviation of the spectrum outside the strong emission line. We model the H$\alpha$ spectral region as the sum of two broad Gaussian lines, a single narrow Gaussian, an iron template (which elevates the continuum on the blue side of the emission line), and a power-law continuum. The full-width at half-maximum (FWHM) of the broad H$\alpha$ emission is $6086 \pm 42\, \kms$, and the combined luminosity of the broad H$\alpha$ components is $L_{\rm H\alpha} = (5.56 \pm 0.05) \times
10^{45} \ergs$. Modeling the broad-band (3000Å to 7$\mu$m) spectral energy distribution of the quasar as a sum of a power-law continuum, two blackbody thermal components (500 and 1250 K, to model the rest-frame IR emission), and line emission from H$\alpha$ and as determined from the Keck spectrum, we derive $L_{5100} = (1.24 \pm 0.02) \times 10^{47} \ergs$. Following @Jun:15a, we derive $\log(M_{\rm BH}/M_\odot) = 10.36 \pm
0.16$ using the $L_{5100}$ estimator and $\log(M_{\rm BH}/M_\odot)
= 10.29 \pm 0.17$ using the $L_{\rm H\alpha}$ estimator. We note that these statistical error bars underestimate the true uncertainty, both due to the non-simultaneity of the imaging and near-infrared spectroscopy and, more importantly, the systematic uncertainty in the virial scale factor, $f$, which is the typically the dominant source of uncertainty in black hole mass measurements; in this case, we adopt $f = 5.1 \pm 1.3$ from @Woo:13, as per @Jun:15a.
For comparison, without access to any well-detected emission features, @Rafiee:16 simply adopted a black hole mass of $M_{\rm BH}
= 6 \times 10^9\, M_\odot$ as a typical value. Adopting their value for the bolometric luminosity of , $L_{\rm bol} = (3.36 \pm
0.69) \times 10^{47}\, {\rm erg}\, {\rm s}^{-1}$ [based on the observed rest-frame 2900 Å flux density and a bolometric correction of ${\rm BC}_{2900} = 5 \pm 1$ from @Richards:06], we determine an Eddington ratio of $L_{\rm bol}/L_{\rm Edd} \sim 0.15$ (in comparison to their value of 0.45).
Discussion
==========
Figure \[fig:Fe\_Mg\] shows a comparison between the long wavelength portion of the three epochs of optical spectroscopy. In the spectral region beyond $\sim 7300$ Å there is much less blending of troughs from different ions, making the variability changes simpler to interpret. We identify two primary absorption systems. The first system, A, shows several troughs of UV absorption between 7500 and 8900 Å, as well as absorption at 9050 Å. The other system, B, shows absorption at 9300 Å. The two absorption systems are separated by 9000 , yet show coordinated reductions in the depth of their troughs as the quasar brightens. This is the expected behavior if the BAL spectral variability is driven by changes in the photoionization: as the ionizing continuum flux increases, the column densities of and decrease for all clouds along the line of sight. (Note that this expectation assumes that the ionizing continuum changes are correlated with the flux changes around 2500 Å). The scenario of clouds moving across our line of sight is hard pressed to explain both the coordinated changes of the trough depths as well as the observed trough weakening with increasing UV flux. A priori, there is no reason that troughs as widely separated in velocity as A and B would be correlated since they are different parcels of gas. Even more so, there is no reason in this scenario for changes in the trough depths to be correlated with flux changes. Therefore, we interpret the variability in the absorption troughs to be due to changes in photoionization, rather than motion of material into our line of sight. A follow-up paper will more carefully model the full multi-epoch spectroscopic data set, including additional spectroscopy from our continuing monitoring, with the goal of understanding the location and energetics of the outflow, and its impact on the host galaxy (Chamberlain , in preparation).
appears to be an FeLoBAL quasar in the process of transitioning to a more common low-ionization BAL (LoBAL) quasar, similar to FBQS J1408+3054 reported by @Hall:11. We note, however, that @Hall:11 interpreted the variability in that source as being related to structure in the BAL outflow moving out of our line of sight rather than being related to photoionzation changes.
highlights the sort of rare, extremely variable quasars that can be used to probe the physics of quasar outflows. We expect to find many more such examples with the new generation of wide-area, sensitive, high-cadence synoptic surveys. We were fortuitous in this case that multi-epoch archival spectroscopy was available for this source. In the future, it will be exciting to find similar major events in real time, allowing real-time multi-wavelength follow-up in order to more fully dissect the internal workings of AGN engines.
We thank the anonymous referee for a prompt and helpful referee report. CRTS was supported by the NSF grants AST-1313422, AST-1413600, and AST-1518308. The work of DS and HJ was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. DS also acknowledges support from NASA through ADAP award 12-ADAP12-0109. NA and CC acknowledge support from NSF through grant AST 1413319, and from NASA through STScI grants GO 11686 and GO 12022. Research by AJB was supported by NSF grant AST-1412693. EG acknowledges the generous support of the Cottrell College Award through the Research Corporation for Science Advancement. HJ is supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Universities Space Research Association under contract with NASA. The authors a grateful to the staff at the Palomar and Keck observatories, where some of the data presented here were obtained. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
, , , , ,
©2017. All rights reserved.
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[^2]: See [http://catalinadata.org/]{}.
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---
abstract: 'A class of network models with symmetry group $G$ that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising ‘triple-humped’ phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.'
address: 'AA, ST: Department of Mathematics, Imperial College, London SW7 2AZ, UK'
author:
- Alexis Arnaudon
- So Takao
bibliography:
- 'biblio.bib'
title: |
Networks of coadjoint orbits:\
from geometric to statistical mechanics\
---
Introduction
============
The main purpose of this paper is to provide a link between geometric mechanics and statistical mechanics following recent advances in stochastic geometric mechanics [@lazaro2008stochastic; @arnaudon2014stochastic; @holm2015variational; @arnaudon2016noise; @arnaudon2016stochastic], where the inclusion of a structure-preserving noise in Lagrangian or Hamiltonian systems have been considered. Geometric mechanics provides a mathematical framework for describing Hamiltonian mechanical systems, but with the addition of a structure-preserving noise and dissipation, a theory of statistical geometric mechanics becomes possible, as suggested in [@arnaudon2016noise]. Here, we take a step further to include lattices, or more generally, networks of these systems and study their stability and phase transitions, similar to the well-known Ising or Heisenberg models.
The Ising model is one the most well-studied systems in statistical mechanics, which is a simplified model of ferromagnetism. Its significance in statistical mechanics comes from the fact that it is the simplest model that exhibits [*phase transition*]{}, which is ubiquitous in the theory of matter, such as in the transition from liquids to gases or in the magnetisation of materials. See for example Chandler [@chandler1987introduction] for a classic reference in statistical mechanics. Relaxing the discrete nature of the Ising model to a system of interacting unit vectors in $\mathbb R^3$ on a lattice, we obtain the classical Heisenberg model, which, again is a simple model of ferromagnetism but now with a continuous symmetry group $SO(3)$, instead of the discrete $C_2$-symmetry of the Ising model. This model exhibits a phase transition, albeit one with a different universality class due to the difference in the symmetry group.
On the other hand, geometric mechanics is a mathematical discipline that deals with the dynamics of a ‘few-body’ mechanical system with symmetry and mainly concerned with issues such as symmetry reduction, integrability and stability of equilibria. For basic references in geometric mechanics, see for example Arnold [@arnold89mechanics], Marsden and Ratiu [@marsden1999book] and Holm [@holm2008geometric]. Although the two fields are very different, recent progress in the theory of stochastic geometric mechanics has shed some light to bridge the gap between the two. Initiated in the work by Bismut [@bismut1982mecanique] for canonical systems and extended to general Hamiltonian systems in Lázaro-Camí and Ortega [@lazaro2008stochastic], stochastic geometric mechanics have found applications in areas as vast as simple mechanical systems [@arnaudon2016noise2], fluid dynamics [@holm2015variational], geometric integrators [@bourabee2009stochastic], stability theory [@arnaudon2017stability] and shape analysis [@arnaudon2016stochastic]. In particular, Holm [@holm2015variational] introduced stochastic processes at the level of the variational principle for fluid systems in such a way that the momentum map is preserved, and this introduced the idea of [*structure preserving noise*]{}, where the added noise does not destroy the essential geometric feature of the system. In a similar spirit, adding a dissipative mechanism in Hamiltonian systems that preserve the basic geometric structure has been considered by Bloch et al. [@bloch1996euler], where a type of dissipative bracket that dissipates energy while preserving the coadjoint orbit is introduced. A similar type of bracket that dissipates a chosen conserved quantity while preserving another, called the [*selective decay mechanism*]{}, is considered in Gay-Balmaz and Holm [@gaybalmaz2013selective; @gaybalmaz2014geometric].
Our work is based on Arnaudon et al. [@arnaudon2016noise] where both noise and dissipation is introduced in a general finite-dimensional Hamiltonian system with configuration group $G$, phase space $T^*G$ and symmetry group $G$, in such a way that it preserves the coadjoint orbits, where the reduced dynamics take place. A remarkable consequence of adding noise and dissipation in this way is that, not only is the geometric structure of the equation preserved but also the invariant measure for the stochastic dissipative system is [*exactly given by the Gibbs measure*]{} restricted to the coadjoint orbit, provided that we choose an isotropic noise. The emergence of the Gibbs measure in this system is key to connecting our theory with statistical mechanics, where it arises naturally as the invariant measure in a canonical ensemble, which is a system of particles in statistical equilibrium that is coupled to an external heat bath. This strongly suggests that we can approach this system from the point of view of statistical mechanics and vice versa.
Our first goal of this paper is to construct a simple lattice model that has a general symmetry Lie group $G$, or more generally on a network, using symmetry reduction techniques in geometric mechanics and study their deterministic properties. By considering a $G$-invariant canonical Hamiltonian system on phase space $T^*G$ at each node of a given graph, we regard the momentum map $J: T^*G \rightarrow \mathfrak g^*$ as the [*spins*]{} in our model, which interact with other spins according to the structure of the graph, where the interaction takes place if they are connected by an edge. The coupling between the spins can be taken in two ways: (1) at the reduced space $\mathfrak g^*$ or (2) directly on the group $G$. In the former, which is the easier case, we introduce the coupling between two neighbouring spins in the [*reduced*]{} Hamiltonian [*after*]{} performing symmetry reduction. We call this approach [*momentum coupling*]{} since the variables that are coupled is the momentum map. We will see that this approach generalises the classical Heisenberg model to include general symmetry groups. For the latter, we consider a representation of $G$ on a vector space $V$ at each node and couple the neighbours in the [*unreduced*]{} Lagrangian that depend on vectors in $V^*$ that are acted on by $G$. Since the neighbours are coupled with the ‘positions’ of a given vector in $V^*$ that is rotated around by the $G$-action, we call this approach [*position coupling*]{}. The vector that is taken here for position coupling breaks the full $G$-symmetry of our system and hence we apply the semi-direct product reduction theorem given in Holm et al. [@holm1998euler] to obtain the corresponding Lie-Poisson system on the semi-direct product Lie algebra $\mathfrak g^* \,\circledS \, V^*$ at each node. This gives rise to a new system that finds no analogues in classical lattice models. In the special case where the Lie algebra $\mathfrak g = \text{Lie}(G)$ is compact and semi-simple, we were able to obtain the equilibrium solutions for both the momentum-coupled and position-coupled systems as the eigenvectors of a generalised graph Laplacian, which we construct from the underlying graph. Furthermore, we were able to classify these stationary eigenvectors into ferromagnetic and anti-ferromagnetic states, which is consistent with known equilibrium solutions of the Ising model or the Heisenberg model and we also investigated their corresponding stability properties using the Energy-Casimir method. Relaxing the condition that each spin must have unit length, many of the equilibrium solutions that we find here are new.
Our second goal is to investigate the statistical mechanical properties of the lattice model constructed above with noise and dissipation of the type considered in Arnaudon et al. [@arnaudon2016noise]. In particular, we study the phase transition exhibited in the two examples that we consider here: the rigid body lattice and the heavy top lattice which are the simplest systems that can be obtained via momentum-coupling and position-coupling respectively, by taking $G=SO(3)$. The invariant measure is found to be the Gibbs measure restricted to the coadjoint orbit for both of these systems, which allows us to introduce the notion of temperature and hence study their respective temperature phase transition behaviours. For the rigid body network, we observe a standard second-order phase transition in both the mean-field simulation and the direct simulation as expected. However, in the direct simulation of the heavy top network, we observe a new type of phase transition behaviour where the spins become aligned to two intermediate metastable states as we decrease the temperature before settling down to the lowest energy state, instead of jumping directly to the lowest energy state. This ‘triple-humped’ phase transition is not captured in our mean-field simulation, which follows a standard ’single-humped’ phase transition. In this study, we only looked at numerical simulations of the phase transition but much work needs to be done in the analysis to understand this phenomenon.
Structure of the paper
----------------------
In section \[background\], we begin with a quick review of statistical mechanics and stochastic geometric mechanics and study the statistics of a single coadjoint orbit system in section \[single\]. Then, we introduce the notion of momentum coupling in section \[SS-section\] to construct a lattice model with symmetry group $G$ and study in detail the equilibrium solutions and their stability properties. We will then add noise and dissipation to the system and show that the invariant measure is given by the Gibbs measure. In section \[section-RB\], we study an example of this system by taking $G=SO(3)$, which we call the rigid body network. In section \[SD-section\], we introduce the idea of position-coupling and derive the corresponding network Lie-Poisson model using semi-direct product reduction. Noise and dissipation are then added to the system in a similar way. This theory is then illustrated concretely in section \[section-HT\] by considering the case $G=SO(3)$, which we call the heavy top network. In section \[PT-section\], we study the mean-field approximations and the phase transition behaviours of the rigid body network and the heavy top network. Finally, we give a conclusion and discuss further work in section \[conclusion\].
List of symbols
---------------
This work contains notations from three topics: geometric mechanics, graph theory and stochastic analysis. For clarity, we summarize the main notations we will be using here.
- $G\qquad $ a Lie group
- $g\qquad $ an element in a Lie group
- $\mathfrak g\qquad $ Lie algebra corresponding to $G$
- $\mu\qquad $ an element in $\mathfrak g^*$
- $\xi\qquad $ an element in $\mathfrak g$
- $k\qquad $ dimension of $\mathfrak g$
- $\mathcal O\qquad $ coadjoint orbit of $G$ in $\mathfrak g^*$
- $\mathcal N\qquad $ connected, undirected graph
- $N\qquad $ number of vertices in $\mathcal N$
- $d_i\qquad $ number of edges stemming from vertex $i$ in $\mathcal N$
- $A\qquad $ adjacency matrix
- $D\qquad $ degree matrix
- $L_0\qquad $ the graph Laplacian
- $L\qquad $ the symmetric normalised graph Laplacian
- $\mathbb A\qquad $ the extended adjacency matrix
- $\mathbb D\qquad $ the extended degree matrix
- $\mathbb L\qquad $ the extended normalised Laplacian
- $\mathbb 1\qquad $ the identity matrix
- $\mathbb d\qquad $ stochastic time increment
- $\mathbb P\qquad $ probability distribution
- $\Braket {\cdot }\qquad $ averaging
- $dW_t\qquad\hspace{-0.17in} $ standard Wiener process
- $\sigma\qquad $ noise amplitude
- $\theta \qquad $ dissipation amplitude
- $\beta\qquad $ inverse temperature
Background
==========
Statistical mechanics
---------------------
Statistical mechanics is a physical theory used to describe the macroscopic properties, such as temperature and entropy of a many-particle system (e.g. gas or lattice) evolving as a canonical Hamiltonian system that fluctuates around a mean state. This theory is useful for systems with a large degree of freedom where first of all, solving the full system is computationally expensive and second of all, the information that we need is independent of the motion of individual particles in the system. In equilibrium statistical mechanics, one deals with a system that is in [*statistical equilibrium*]{}, that is, the particles are distributed according to an invariant measure of the underlying system. The system of particles are often assumed to be in statistical equilibrium according to the following three thermodynamic ensembles
1. [*Micro-canonical ensemble*]{}: the system is isolated and there is no energy exchange or particle exchange with the exterior. Each state is equally probable to occur.
2. [*Canonical ensemble*]{}: the system is coupled to a heat bath of fixed temperature and fixed average energy and there is no particle exchange. The states are distributed according to a Gibbs measure.
3. [*Grand-canonical ensemble*]{}: the system is coupled to a heat bath of a fixed temperature and is subject to particle exchange.
For systems with inter-particle interactions such as in lattice models, a phenomenon called [*phase transition*]{} is often observed as the temperature is varied, which describes an abrupt change of state in a system. A classic example of this is the transition from liquid to gas at the boiling point. Now, consider a simple 2-dimenstional lattice model with $N$ nodes that take the values [*up*]{} ($+1$) or [*down*]{} ($-1$) at each node. The energy of this system at a particular state is given by $$\begin{aligned}
H = - \sum_{i \sim j}^N J_{ij} s_i s_j, \quad s_i, s_j \in \{1, -1\}\, ,\end{aligned}$$ for constants $J_{ij}>0$ and $i \sim j$ means that nodes $i$ and $j$ are adjacent on the lattice. We denote by $\Omega := \{1, -1\}^N$ for the sample space of all possible configurations. Assuming that the system is in statistical equilibrium in a canonical ensemble with temperature $T$, the probability of selecting a configuration $\boldsymbol s = (s_1, \ldots, s_N) \in \Omega$ with a given energy level $H_0$ is given by the [*Gibbs distribution*]{} $$\begin{aligned}
\mathbb P(X = \boldsymbol s \,|\,T) = Z_\beta^{-1} e^{-\beta H_0}\, , \quad H(\boldsymbol s) = H_0\, ,\end{aligned}$$ where the normalising constant $Z_\beta$ is called the [*partition function*]{} and $\beta$ is the [*inverse temperature*]{}, defined by $\beta := \frac{1}{k_B T}$, where $k_B$ is the [*Boltzmann constant*]{}. It is well known that there exists some $T = T_c$, called the [*critical temperature*]{} such that the [*average magnetisation*]{} defined by $$\begin{aligned}
\langle M \rangle = \frac1N \sum_{\boldsymbol s \in \Omega} \sum_{i=1}^N s_i \, \mathbb P(X = \boldsymbol s \,|\,T)\end{aligned}$$ vanishes for $T > T_c$ and becomes strictly positive for $T < T_c$. This is an example of a so-called [*second-order phase transition*]{} and the model that we just considered is the [*2D Ising model*]{} which is the simplest known system that admits a phase transition. Now, instead of taking the values $\{1, -1\}$, one can generalise this system so that the spins $s_i$ are unit vectors in $\mathbb R^3$ and we obtain the [*classical Heisenberg model*]{} which also exhibits a similar second-order phase transition only if the interaction is anisotropic, in which case it is often called the $XYZ$, or $XXZ$-model.
The ideas that we discussed here are elementary in statistical mechanics and can be found in most textbooks on the topic. Here, we will refer to Chandler [@chandler1987introduction] for a more thorough physical exploration of this topic and to [@ruelle2004thermodynamic] for a rigorous mathematical formulation of statistical mechanics.
### Other developments in geometric statistical mechanics
Remarkably, in the early days of modern geometric mechanics, one of the founders of the field, Jean-Marie Souriau, had already attempted to apply ideas from geometric mechanics to statistical mechanics. His original idea was to impose natural symmetries on the Gibbs distribution and obtain a vector of inverse temperatures $\beta\in \mathfrak g$ in such a way that the symmetry Lie group $G$ corresponding to $\mathfrak g$ survives in the construction of statistical quantities, such as the entropy. Some of his works in this direction can be found in [@souriau1966definition; @souriau1974mecanique; @souriau1969structure]. However, this theory attracted very little attention at the time until only recently, where several authors modernized the original work of Souriau. We refer to [@barbaresco2014koszul; @barbaresco2015symplectic; @marle2016tools] as well as to [@barbaresco2016geometric] for a complete review on the history of Souriau’s theory.
In parallel to this, there has also been recent work by Gay-Balmaz and Yoshimura [@gay2017lagrangianI; @gay2017lagrangianII] based on infinite dimensional geometric mechanics and a certain class of dynamical constraints to describe various systems in the theory of non-equilibrium thermodynamics.
Stochastic geometric mechanics
------------------------------
A systematic theory for introducing stochastic processes into mechanics started with the work of Bismut [@bismut1982mecanique], where the noise was introduced at the level of the variational principle and the corresponding stochastic Euler-Lagrange equations were derived. This theory has seen a resurgence recently and has been extended by several authors such as Lázaro-Camí and Ortega [@lazaro2008stochastic] and Bou-Rabee and Owhadi [@bourabee2009stochastic] based on more modern geometric mechanical techniques. Also recently, Arnaudon, Chen and Cruzeiro [@arnaudon2014stochastic] and Holm [@holm2015variational] added noise in the variational principle that is compatible with symmetry reduction to obtain a corresponding stochastic Euler-Poincaré equation. In the former, the noise was introduced at the level of the Lie group and using symmetry reduction together with taking an expectation, they obtained a dissipative deterministic equation with applications in infinite dimensional systems, such as the Navier-Stokes equation in fluid dynamics. In the latter, which we will base our work on, the noise is introduced directly into the reconstruction relation in the variational principle and followed by symmetry reduction, a fully stochastic Euler-Poincaré equation is derived. Based on this work, Arnaudon, De Castro and Holm [@arnaudon2016stochastic] studied in detail the finite dimensional analogue of this construction together with double bracket dissipation, and the implication of noise on the nonlinear stability of relative equilibria was investigated in Arnaudon, Ganaba and Holm [@arnaudon2017stability]. We refer to [@arnaudon2016stochastic] and references therein for more details on other related works.
In the present text, we will use the finite dimensional equations derived in Arnaudon, De Castro and Holm [@arnaudon2016stochastic], hence we will refer to this work for more details about the derivations of the stochastic equations used here.
For a dynamical system with configuration group $G$, phase space $TG$ and a $G$-invariant Lagrangian $L(g, \dot g)$, where $(g, \dot g) \in TG$, we can apply the theory of reduction by symmetry to show that the Euler-Lagrange equation on $TG$ associated to this Lagrangian is equivalent to the Euler-Poincaré equation $$\begin{aligned}
\frac{d}{dt}\frac{\partial l}{\partial \xi} + \mathrm{ad}^*_\xi \frac{\partial l}{\partial \xi}=0\, , \end{aligned}$$ on $\mathfrak g$, where $\xi= g^{-1} \dot g\in \mathfrak g$, the reduced Lagrangian is $l(\xi) := L(g, \dot g)$ and $\mathrm{ad}^*$ is the dual of the adjoint action. From the definition of the reduced variable $\xi$, it is possible to reconstruct the solution on the original configuration manifold $G$, using the formula $$\begin{aligned}
\dot g = g\xi\, , \end{aligned}$$ called the reconstruction relation. The introduction of noise appears at this level in the theory of reduction by symmetry by replacing the above relation by a stochastic process $$\begin{aligned}
\mathbb d g = g\xi \, dt + \sum_{l=1}^K g \sigma_l \circ dW_t^l\, ,
\label{sto-rr}\end{aligned}$$ where $W_t^l$ are $K$ independent standard Wiener processes, $\mathbb d$ is the stochastic evolution operator and $\sigma_l\in \mathfrak g$ are given Lie algebra elements which represents the amplitude and direction of the noise. We interpret this as a Stratonovich SDE, represented by the symbol $\circ$ so that we can use the standard rules of calculus.
From the above stochastic reconstruction relation, one can compute the stochastic variations $$\begin{aligned}
\delta \xi = \mathbb d\eta + \mathrm{ad}_\xi\eta \,dt + \sum_{l=1}^K \mathrm{ad}_{\sigma_l}\eta \circ dW_t^l\, , \quad \eta := g^{-1} \delta g\, , \end{aligned}$$ and the stochastic Euler-Poincaré equation will follow from the same variational problem exactly as in the deterministic case.
As we mentioned in the introduction, the link between geometric mechanics and statistical mechanics requires a mechanism of dissipation to balance the energy input of the noise and hope for the system to reach a statistical equilibrium. As given in Arnaudon et al. [@arnaudon2016stochastic], we use the double bracket dissipation introduced by Bloch et al, [@bloch1996euler] and extended in Gay-Balmaz and Holm [@gaybalmaz2013selective; @gaybalmaz2014geometric] to obtain the following stochastic Lie-Poisson equation with dissipation $$\begin{aligned}
\mathbb d\mu &+ \mathrm{ad}^*_\frac{\partial h}{\partial \mu} \mu\, dt
+ \theta\, \mathrm{ad}^*_\frac{\partial C}{\partial \mu} \left [ \frac{\partial C}{\partial \mu}, \frac{\partial h}{\partial \mu} \right ]^\flat \, dt + \sum_l\mathrm{ad}^*_{\sigma_l} \mu \circ dW_t^l = 0 \,,
\label{SEP-Diss}\end{aligned}$$ where $h:\mathfrak g^*\to \mathbb R$ is the reduced Hamiltonian, $C$ is a given Casimir function of the Lie-Poisson structure and the momentum variable $\mu\in \mathfrak g^*$ is conjugate to the reduced velocity $\xi \in \mathfrak g$ via the Legendre transform. We denote by $\flat:\mathfrak g \to \mathfrak g^*$ to be the canonical isomorphism between $\mathfrak g$ and its dual given an inner product, and $\theta \in \mathbb R$ parametrizes the strength of dissipation. This equation can be derived from a variational principle where the dissipation is inserted as a force in the Lagrange-d’Alembert framework, see [@arnaudon2016stochastic] for more details.
One can see that the dissipative term and the noise preserves the coadjoint orbit $$\begin{aligned}
\mathcal O_\mu = \{ \mathrm{Ad}^*_g \mu, \quad \forall g \in G\} \, ,
\label{coadj-def}\end{aligned}$$ where $\mu= \mu(0)$. In the following, we will denote a generic coadjoint orbit by $\mathcal O$, discarding the foot point. In particular, the Casimir functions $C:\mathfrak g^*\to \mathbb R$ (in general, the system admits several Casimir functions, but we only select one here) is conserved whereas the energy decays due to the dissipation and fluctuate due to the noise.
The statistical information of the process can be captured via the Fokker-Planck equation, which describes the time evolution of the probability distribution of the process $\mathfrak \mu$. We refer to [@arnaudon2016stochastic] for the derivation of the Fokker-Planck equation for the process $\mathfrak \mu$ in geometric form, given by $$\begin{aligned}
\frac{d}{dt} \mathbb P(\mu) + \{h,\mathbb P\} +\theta\, \left \langle\left [\frac{\partial \mathbb P}{\partial \mu}, \frac{\partial C}{\partial \mu}\right], \left [ \frac{\partial h}{\partial \mu}, \frac{\partial C}{\partial \mu}\right]^{\flat} \right \rangle - \frac12 \sum_l \{\Phi_l,\{\Phi_l,\mathbb P\}\}=0\, ,
\label{FP-Diss}\end{aligned}$$ where $\langle \cdot, \cdot \rangle$ is the natural pairing on $\mathfrak g$, $\{f, h\}(\mu) =\left \langle \mu, \left [ \frac{\partial f}{\partial \mu}, \frac{\partial h}{\partial \mu}\right ]\right \rangle $ is the Lie-Poisson bracket and $\Phi_l= \langle \sigma_l, \mu\rangle$ are the stochastic potentials. We now state the result in [@arnaudon2016stochastic] that forms the basis of our present work.
\[FP-diss-thm\] The stationary distribution of the stochastic process with an isotropic noise, that is $\sigma_i = \sigma e_i$ (for $e_i$ a basis of $\mathfrak g$), is the Gibbs measure on the coadjoint orbit, that is $$\begin{aligned}
\mathbb P_\infty(\mu) = Z^{-1} e^{-\beta h(\mu)}\, ,
\label{Gibbs-def}
\end{aligned}$$ where $\beta= \frac{2\theta}{\sigma^2}= \frac{1}{k_B T}$ is the inverse temperature ($k_B$ is Boltzmann’s constant) and $Z$ is the normalisation constant, or partition function $$\begin{aligned}
Z = \int_{\mathcal O} e^{-\beta h(\mu)} d\mu \, .
\label{Z-def}
\end{aligned}$$
One can prove this by a direct substitution into the Fokker-Planck equation and obtain $\frac{d}{dt} \mathbb P_\infty= 0$.
We need to stress an important point here. As opposed to the standard theory of statistical mechanics where the partition function is an integral over the full phase space, our partition function is defined as an integral over the coadjoint orbit and not over the whole space $\mathfrak g^*$. This is a result of the nonlinear dissipation and the multiplicative noise and makes the effective phase space compact if the Lie group is compact. It also has the advantage of being an embedding in $\mathbb R^n$, rather than a general manifold where the definition of stochastic processes is more involved. Notice the requirement of isotropic noise, necessary to obtain the simple Gibbs form of the stationary distribution. In the non-isotropic case, the distribution would be close but different from the Gibbs distribution, see [@arnaudon2016stochastic] for more details.
Hereafter, we will not use the Fokker-Planck equation as we will assume implicitly that the system is at statistical equilibrium, that is, our system is distributed according to the Gibbs measure. We can therefore apply results from the theory of equilibrium statistical mechanics to study our system. One of the fundamental results is the fluctuation-dissipation theorem, which states that for a system to be at a statistical equilibrium, the diffusion coefficient must be a function of the dissipation coefficient or vice versa. In our case, this relation is given by $\theta = \frac12 \beta \sigma^2$, and is often called [*Einstein’s relation*]{}. For our system, statistical equilibrium states exist for all pairs $(\theta,\sigma)$ but when the temperature is fixed, one of the variables depends on the other. We will not go further into this issue as it is rather simple as we are dealing with a classical system, but this theorem is important especially for quantum systems. We refer to Kubo [@kubo1966fluctuation] for an early theoretical exposition of this theorem, and the many references therein for more details on this topic.
Statistics in a single coadjoint orbit {#single}
======================================
We illustrate our statistical analysis by first considering the statistics for a single coadjoint orbit. The key object in equilibrium statistical mechanics is the partition function since most statistical quantities can be derived from it. Unfortunately, the partition function cannot be integrated analytically in general, except for a few simple examples.
The average energy of the system is defined as $$\begin{aligned}
\Braket{E} := \int_{\mathcal O} h(\mu) \mathbb P_\infty (\mu) d\mu = Z^{-1}\int_\mathcal{O} h(\mu) e^{-\beta h(\mu)} d\mu\, ,\end{aligned}$$ but can be obtained directly from the partition function as $$\begin{aligned}
\Braket{E} = - \frac{\partial}{\partial \beta} \mathrm{log}(Z)\, . \end{aligned}$$
In the case of compact coadjoint orbits, the average energy is bounded from above and saturtes rapidly. Indeed, as $T\to \infty$, the Gibbs distribution $\mathbb P_\infty(\mu) \to 1$ for all $\mu\in \mathcal O$ and we have the inequality $$\begin{aligned}
\Braket{E}= \int_\mathcal{O} h(\mu) \mathbb P_\infty(\mu) d\mu \leq \int_\mathcal{O} h(\mu) \, d\mu < \infty\, . \end{aligned}$$
Similarly, the energy fluctuation around $\braket{E}$ is defined as $$\begin{aligned}
\Delta_2 \Braket{E} := \braket{E^2}- \braket{E}^2 = \frac{\partial^2}{\partial \beta^2} \mathrm{log} Z= - \frac{\partial \Braket{E}}{\partial \beta}\, . \end{aligned}$$ As for the mean energy, it saturates rapidly as $T\to \infty$.
We finally arrive at the definition of entropy $S$ on the coadjoint orbit, given by the famous relation $$\begin{aligned}
S := - k_B \int_\mathcal{O} \mathbb P_\infty(\mu) \mathrm{log} \left (\mathbb P_\infty(\mu)\right ) d\mu \, , \end{aligned}$$ or, in terms of the partition function, $$\begin{aligned}
S = k_B \frac{\partial}{\partial T} \left ( T\, \mathrm{log}(Z)\right )\, .
\label{S-Z}\end{aligned}$$ If the orbit is compact, the entropy saturates as well for large $T$.
The entropy is an important quantity as it is maximized by the Gibbs measure, under the constraint that the mean energy $\Braket{E}$ is fixed. This is, in a sense the definition of the canonical ensemble in statistical mechanics.
The Gibbs distribution is the distribution that maximizes the entropy of the system. \[S-thm\]
We enforce $\Braket{E}=E_0$ with a Lagrange multiplier $\beta$ (this choice will be clear later) and consider the functional $$\begin{aligned}
\mathbb S(\mathbb P) = \int_\mathcal{O} \left [ \mathbb P(\mu) \mathrm{log}( \mathbb P(\mu)) + \beta \left ( h(\mu ) \mathbb P(\mu) - E_0\right )\right ] d\mu\, . \end{aligned}$$ The variation with respect to $\mathbb P$ in the variational principle $\delta \mathbb S = 0$ gives the relation $$\begin{aligned}
\mathrm{log}\left (\mathbb P(\mu)\right) + 1 + \beta h(\mu) = 0 \, . \end{aligned}$$ By normalizing the solution $\mathbb P(\mu)$ to $1$, we obtain the Gibbs measure with inverse temperature $\beta$.
Network of coadjoint orbits I: Momentum coupling {#SS-section}
================================================
The two important systems that can be described using the theory of statistical mechanics are gas and lattices. The latter, which includes the Ising model and the Heisenberg model are well studied in statistical physics and exhibit many interesting properties such as phase transition. In this section, we will construct a lattice of interacting continuous spins, similar to the Heisenberg model, that take values on a coadjoint orbit, then study in details their deterministic properties and later introduce noise and dissipation to the system that preserve the coadjoint orbits. By coupling the neighbours in the [*momentum*]{}, we will recover the classical Heisenberg model as the simplest example using the coadjoint orbit of $SO(3)$, that is the sphere $S^2$. We will mostly restrict our exposition to compact semi-simple Lie algebras here, but extensions to more general Lie algebras should be possible in some cases.
Deterministic equations
-----------------------
Our aim here is to construct the equations of motion of spins interacting on a general undirected connected network $\mathcal N$. At each node, we will assign a coadjoint orbit $\mathcal O_i$ for $i=1,\ldots, N$, associated to the Lie group $G$, which is coupled to its neighbours according to the structure of the graph.
### Coupling in the momentum
Given an undirected, connected graph $\mathcal N$ and a left (or right) $G$-invariant canonical Hamiltonian system on $T^*G$, we introduce the notion of [*coupling in the momentum*]{} to construct a Lie-Poisson Hamiltonian system on $(\mathfrak g^*)^{\oplus N}$ such that at each node of $\mathcal N$, the system evolves as a Lie-Poisson system on $\mathfrak g^*$, with an additional interaction term arising from its neighbours. If we suppose for now that at each node, the system evolves independently as a canonical Hamiltonian system on $T^*G$ with left $G$-invariant Hamiltonians $H_i$ at node $i$, then according to the Lie-Poisson reduction theorem, we can collectivise the Hamiltonian as $H_i = h_i \circ \mathbf J_R$ where $\mathbf J_R : T^*G \rightarrow \mathfrak g^*$ is the momentum map corresponding to the right cotangent lift action of $G$ on $T^*G$ and we obtain a Lie-Poisson system on $\mathfrak g^*$ with reduced Hamiltonian $h_i$ at each node $i$. The idea of coupling in the momentum is to take the interaction between the neighbours at the level of the [*reduced space*]{} $\mathfrak g^*$, that is, we couple the momentum map $\mathbf J_R$ between neighbouring bodies. Introducing the symmetric and positive definite [*interaction tensor*]{} $$\begin{aligned}
\mathbb J_{ij} : \mathfrak g^* \to \mathfrak g\, , \quad \forall i,j = 1\, , \ldots, N\,,\end{aligned}$$ we take the [*momentum-coupled interaction potential energy*]{} to be $$\begin{aligned}
\label{int-h}
h^{\text{int}}_{ij}(\mu_i, \mu_j) = -\frac{1}{2\sqrt{d_id_j}} \langle \mu_i, \mathbb J_{ij} \mu_j \rangle_{\mathfrak g^* \times \mathfrak g}, \quad \mu_i, \mu_j \in \mathfrak g^*\,,\end{aligned}$$ where $d_i$ is the number of neighbours at node $i$ on graph $\mathcal N$ and the factor $\sqrt{d_id_j}^{-1}$ is taken to normalise the expression, which will be convenient later in our analysis. We take the total energy of the form $$\begin{aligned}
h(\boldsymbol \mu) = \sum_{i=1}^N h_i(\mu_i) + \sum_{i=1}^N \sum_{j \sim i} h^{int}_{ij}(\mu_i, \mu_j), \quad \boldsymbol \mu = (\mu_1, \ldots, \mu_N) \in (\mathfrak g^*)^{\oplus N}\, , \end{aligned}$$ where $i \sim j$ means that nodes $i$ and $j$ are adjacent on the graph $\mathcal N$. Taking the $(-)$ Lie- Poisson structure on $(\mathfrak g^*)^{\oplus N}$ which is given by the sum $$\begin{aligned}
\label{LP-full}
\{f, g\}^-_{LP}(\boldsymbol \mu) = - \sum_{i=1}^N \left<\mu_i\, , \left[\frac{\delta f}{\delta \mu_i}, \frac{\delta g}{\delta \mu_i} \right] \right>_{\mathfrak g^* \times \mathfrak g}\, ,\end{aligned}$$ we obtain the Lie-Poisson equation $$\begin{aligned}
\label{LP-eq-1}
\dot{\mu_i} = \text{ad}^*_{\delta h_i/\delta \mu_i} \mu_i - \sum_{j \sim i} \frac{1}{\sqrt{d_id_j}}\text{ad}^*_{\mathbb J_{ij}\mu_j} \mu_i, \quad \forall i = 1, \ldots, N\, .\end{aligned}$$
To be more precise, the interaction tensor $\mathbb J_{ij}$ is a map from $\mathfrak g^*$ at site $i$ to $\mathfrak g$ at site $j$, but since the Lie algebra $\mathfrak g$ is identical at each node, there is no need to distinguish between them.
If we consider a right $G$-invariant Hamiltonian instead, then the reduced variables are the left momentum map $\mathbf J_L$ and we take the $(+)$ Lie-Poisson structure. (See [@marsden1999book] for more details about the issue of left vs. right action.)
For the case where $\mathfrak{g} = \text{Lie}(G)$ is a compact, semi-simple Lie algebra, we can identify $\mathfrak g^*$ with $\mathfrak g$ using the inner product $\langle \cdot, \cdot \rangle = - \kappa(\cdot, \cdot)$ where $\kappa$ is the Killing form, which also satisfies the associativity property $\langle a, [b, c] \rangle = \langle [a, b], c \rangle$. From this, one can check that the quadratic functions $$\begin{aligned}
C_i (\mu_i) = \frac12 \langle \mu_i, \mu_i \rangle, \quad \forall i=1, \ldots, N,\end{aligned}$$ are Casimirs of the Lie-Poisson bracket and the coadjoint orbits of $G^{\times N}$ on $\mathfrak (g^*)^{\oplus N}$ are contained in their level sets, i.e. $$\begin{aligned}
\mathcal O \subset \left\{\boldsymbol \mu \in (\mathfrak g^*)^{\oplus N} : C_i(\mu_i) = c_i, \, i=1, \ldots, N\right\}\, ,\end{aligned}$$ where $c_1, \ldots, c_N$ are constants, which follows from the $\text{Ad}^*$-invariance of the Killing form. The coadjoint orbits are preserved by the dynamics due to the equivariance of the momentum map $\mathbf J_R$.
We also consider $$\begin{aligned}
C (\boldsymbol \mu) = \sum_i C_i(\mu_i ) = \frac12 \langle \boldsymbol \mu, \boldsymbol \mu \rangle\,,\end{aligned}$$ which is the sum of all the Casimirs. This is also a Casimir for this system and will be used in our analysis later. We will also use the following shorthand notation for a system of coadjoint orbits that are interconnected by a network.
Given a graph $\mathcal N$ with $N$ nodes, a Lie group $G$, and a coadjoint orbit $\mathcal O_{\boldsymbol \mu} = \left\{\text{\bf Ad}^*_{\boldsymbol g} {\boldsymbol \mu} : \boldsymbol g \in G^{\times N} \right\} \subset (\mathfrak g^*)^{\oplus N}$, where $\boldsymbol \mu \in (\mathfrak g^*)^{\oplus N}$ and $\text{\bf Ad}^*$ is the diagonal coadjoint action of $G^{\times N}$ on $(\mathfrak g^*)^{\oplus N}$, we call the triple $(\mathcal N, G, \mathcal O_{\boldsymbol \mu})$ a network of coadjoint orbits and equation a network Lie-Poisson equation on $(\mathcal N, G, \mathcal O_{\boldsymbol \mu})$.
Hereafter, we consider a simple mechanical system consisting only of a purely kinetic energy term $$\begin{aligned}
h^{KE}_i(\mu_i) = \frac12 \left \langle \mu_i, \mathbb I_i^{-1} \mu_i\right \rangle_{\mathfrak g^* \times \mathfrak g}\, , \qquad \forall i = 1, \ldots, N\,, \end{aligned}$$ where $\mathbb I_i:\mathfrak g\to \mathfrak g^*$ is the moment of inertia tensor assigned to node $i$, which is symmetric and positive definite, and an interaction potential energy term . We take the total Hamiltonian to be $$\begin{aligned}
\label{full-h-1}
h(\boldsymbol \mu) &= \sum_i h^{KE}_i(\mu_i) + \sum_i \sum_{j \sim i} h^{\text{int}}_{ij}(\mu_i, \mu_j) \nonumber \\
&= \frac12 \sum_i \left \langle \mu_i, \mathbb I_i^{-1} \mu_i\right \rangle - \frac12 \sum_i \sum_{j \sim i} \frac{1}{\sqrt{d_id_j}} \langle \mu_i, \mathbb J_{ij} \mu_j \rangle.\end{aligned}$$ In fact, this can be expressed in a more compact form using the language of graph theory, which we will review in the next section.
In the special case where the graph is regular, i.e. $d_1=\cdots=d_n=d$ and absorbing the $\frac1d$ facor in the interaction tensors $\mathbb J_{ij}$, we obtain the Hamiltonian $$\begin{aligned}
h_\mathrm{reg} (\boldsymbol \mu) = \frac12 \sum_i \left \langle \mu_i, \mathbb I_i^{-1} \mu_i \right \rangle - \frac12 \sum_i\sum_{i\sim j} \left \langle \mu_i , \mathbb J_{ij} \mu_j \right\rangle \, .\end{aligned}$$ which, ignoring the kinetic energy term, resembles the Hamiltonian for the Ising model or the classical Heisenberg model. Hence, the network Lie Poisson system we obtained by momentum coupling can be viewed as a generalisation of the Heisenberg model with spins taking values on a general coadjoint orbit and with an additional kinetic energy term at each node.
From this Hamiltonian, we see that the minimum energy configuration must be an aligned state, with all the $\mu_i$ pointing in the same direction, and the maximum energy state must be anti-aligned. In statistical mechanics, these two states are called ferromagnetic and anti-ferromagnetic states respectively and will be important in our discussion of equilibrium solutions later.
### Review of graph theory
The structure of a graph is described by the [*adjacency matrix*]{}, which for unweighted graphs, is given by $$\begin{aligned}
A_{ij}=
\begin{cases}
1 & \mathrm{if }\quad i\sim j\\
0 & \mathrm{otherwise}\, ,
\end{cases}\end{aligned}$$ where $i\sim j$ means that the node $i$ and $j$ are adjacent, or share an edge on the graph. This matrix is symmetric if the graph is undirected, which is the case here. Each node has $d_i = (A \boldsymbol 1_N)_i$ neighbours, where $\boldsymbol 1_N = (1,\ldots 1)\in \mathbb R^N$, and this is called the [*degree*]{} at node $i$. From this, we define the [*graph Laplacian*]{} $L_0 = D-A$, where $D= \mathrm{diag}(d_1, \ldots, d_N)$ is the [*degree matrix*]{}, and its normalised version $$\begin{aligned}
L= D^{-\frac12} L_0D^{-\frac12}= \mathrm{\mathbb 1}_{N} - D^{-\frac12} A D^{-\frac12} \,, \end{aligned}$$ which is a symmetric matrix. These Laplacians are usually used to define a random walk on a graph, where $\dot{\mathbf p} = \mathbf pL$ for a probability vector $\mathbf p\in \mathbb R^n$ of a random walker, which corresponds to a discrete version of the diffusion equation, where $L$ plays the role of the Laplace-de Rham operator.
### Network of coadjoint orbits
We will now extend this theory to write our system of interacting coadjoint orbits of dimension $k$ using the language of graph theory, in particular, using the graph Laplacian.
We extend our notion of the normalised Laplacian by weighting its components by the inertia tensor and the interaction tensor to get an [*extended normalised Laplacian*]{} $$\begin{aligned}
\mathbb L &:= \overline {\mathbb I}^{-1} -\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12}\,\label{ext-Lapl} \\
\mathrm{where }\quad \mathbb A_{ij} &= \mathbb J_{ij} A_{ij}, \quad \overline {\mathbb I}^{-1}= \mathrm{diag}(\mathbb I^{-1}_1, \ldots, \mathbb I^{-1}_n) \quad \mathrm{and} \quad \mathbb D = \mathrm{diag}(d_1 \mathbb 1_k, \ldots, d_n \mathbb{1}_k)\, . \nonumber \end{aligned}$$
The Hamiltonian of our system can then be expressed compactly as $$\begin{aligned}
\label{simplified-h}
h(\boldsymbol \mu ) = \frac12 \left \langle \boldsymbol \mu , \mathbb L \boldsymbol \mu \right \rangle_{(\mathfrak g^*)^{\oplus N}\times \mathfrak g^{\oplus N}}\, .\end{aligned}$$
Using this notation, the network Lie-Poisson equation with Hamiltonian can be written in the compact form $$\begin{aligned}
\dot{\boldsymbol \mu} = \mathbf{ad}^*_{\mathbb L \boldsymbol \mu} \, \boldsymbol \mu \label{LP-network}\,, \end{aligned}$$ where $\mathbf{ ad}^*: \mathfrak g^{\oplus N}\times (\mathfrak g^*)^{\oplus N}\to (\mathfrak g^*)^{\oplus n}$ is the diagonal coadjoint action $(\xi_1, \ldots, \xi_N) \times (\mu_1, \ldots, \mu_N) \mapsto (\text{ad}^*_{\xi_1} \mu_1, \ldots, \text{ad}^*_{\xi_N} \mu_N)$ where $\ad^*: \mathfrak g \times \mathfrak g^* \rightarrow \mathfrak g^*$ is the standard coadjoint action on $\mathfrak g^*$.
### Legendre transformation and Lagrangian formulation
We obtained equation by coupling the neighbours at the level of the reduced space [*after*]{} performing Lie-Poisson reduction at each node. It is then natural to ask whether we can recover the same equation by coupling the neighbours at the level of the group and then performing symmetry reduction. The answer is yes, except for a few cases where the graph Laplacian is not invertible on the full phase space.
For $\mathbb L$ to be invertible, one has to ensure that $\text{ker}(\mathbb L) = \emptyset$. For the simple case where $\mathbb I_i = \mathbb I $ and $\mathbb J_{ij} = \mathbb J$, this can be reduced to showing that $\mathbb I$ and $\mathbb J$ has no eigenvalues in common. Indeed, if they share an eigenvalue, then at least one of the ferromagnetic states will have a $0$ eigenvalue since the extended Laplacian reduces to the standard graph Laplacian $L = D- A$ which has a single $0$ eigenvalue. However, choosing $\mathbb J< \mathbb I$, the minimum eigenvalue of $\mathbb L$ becomes strictly positive since the reduced $L$ corresponding to ferromagnetic states become strictly diagonally dominant.
Assuming that the moment of inertia is invertible on the full space, we obtain the following reduced Lagrangian by the inverse Legendre transform $$\begin{aligned}
l(\boldsymbol \xi) = \frac12 \langle \boldsymbol \xi , \mathbb L^{-1} \boldsymbol \xi\rangle\, , \qquad \boldsymbol \xi = \mathbb L \boldsymbol \mu \in \mathfrak g^{\oplus N},\end{aligned}$$ which can also be obtained through the full Lagrangian on $TG^{\times N}$, given by $$\begin{aligned}
L(g, \dot{g}) = \frac12 \left< \dot{\boldsymbol g}, \mathbb L^{-1}_{\boldsymbol g} \, \dot{\boldsymbol g} \right>_{T_{\boldsymbol g}(G^{\times N}) \times T^*_{\boldsymbol g}(G^{\times N})}, \label{Lag-full}\end{aligned}$$ where we defined $\mathbb L^{-1}_{\boldsymbol g} := L^*_{{\boldsymbol g}^{-1}}\mathbb L^{-1}(L_{{\boldsymbol g}^{-1}})_* : T_{\boldsymbol g}(G^{\times N}) \to T^*_{\boldsymbol g}(G^{\times N})$ and $L_{\boldsymbol g}$ denotes left diagonal action of $\boldsymbol g$. We refer to appendix 2 of [@arnold89mechanics] for the reconstruction of the full Lagrangian from a reduced Lagrangian of this form. It is easy to check that the Lagrangian is left invariant under $G$ so we can apply Euler-Poincaré reduction on and perform Legendre transformation to obtain our network Lie-Poisson equation . However, notice that the matrix $\mathbb L^{-1}$ is far from being sparse, thus the interaction between sites take a complicated nonlocal (more than neighbours interactions) form on the Lagrangian side.
If one really wishes to interpret the system from the Lagrangian side with singular $\mathbb L$, the Lagrangian can be defined in terms of the Moore-Penrose pseudoinverse $\mathbb L^\dagger$. However by doing so, one has to restrict the reduced Lagrangian to the subspace $\text{ker}(\mathbb L^\dagger)^{\perp}$ and recover the reduced Hamiltonian on the subspace $\text{ker}(\mathbb L)^{\perp}$ by Legendre transformation. We refer to [@cendra1998maxwell] for more details on how to obtain an equivalent Lagrangian system for degenerate Hamiltonians, but within the context of plasma physics.
Nonlinear stability results
---------------------------
In this section, we will look for equilibrium solutions of the momentum-coupled network Lie-Poisson system on a compact, semi-simple Lie algebra and find their nonlinear stability properties. In particular, we will show that the eigenvectors of the extended graph Laplacian $\mathbb L$ correspond to equilibrium solutions of this system and they are either a ferromagnetic or an antiferromagnetic state. Furthermore, we will show that the eigenvectors with the lowest and highest eigenvalues correspond to nonlinearly stable equilibria.
### Equilibrium solutions
Since we are on a compact semi-simple Lie algebra $\mathfrak g$, we can take $\text{\bf ad}^* = -\text{\bf ad}$, so equation becomes $$\begin{aligned}
\dot{\boldsymbol \mu} = [\boldsymbol \mu, \mathbb L \boldsymbol \mu]\, .\end{aligned}$$ In general, the solutions to $\dot{\boldsymbol \mu} = 0$ are given by all $\boldsymbol \mu_e \in \mathfrak g^*$ such that $\mathbb L \boldsymbol \mu_e$ is contained in the centralizer $Z(\boldsymbol \mu_e)$ of $\boldsymbol \mu_e$. However, here we will only consider the case where $\mathbb L \boldsymbol \mu_e = \lambda_e \boldsymbol \mu_e$ for some $\lambda_e \in \mathbb R$, which is clearly contained in $Z(\boldsymbol \mu_e)$. So fixing $a > 0$, a regular value of $C:\mathfrak g^* \rightarrow \mathbb R$, we consider relative equilibrium configurations on the level set $C^{-1}(a)$ that are given by the $kN$ eigenvectors of $\mathbb L$, rescaled appropriately to satisfy $C(\boldsymbol \mu_e) = a$. Now, pairing both sides of $\mathbb L \boldsymbol \mu_e = \lambda_e \boldsymbol \mu_e$ with $\frac12 \boldsymbol \mu_e$, we get $$\begin{aligned}
\lambda_e = \frac{1}{C(\boldsymbol \mu_e)}h(\boldsymbol \mu_e)\, , \label{lambda-energy}\end{aligned}$$ so the eigenvalue of $\mathbb L$ is proportional to the total energy of the equilibrium state.
In the special case where $\mathbb J_{ij} = \mathbb J$ and $\mathbb I_i = \mathbb I$ for all $i,j=1, \ldots, N$, we see that these $kN$ eigenvectors can be categorised into two types: ferromagnetic or anti-ferromagnetic states, as given in the following proposition.
\[equib\] Consider a network Lie Poisson system on $(\mathcal N, G, \mathcal O)$ where $\mathfrak g = \text{Lie}(G)$ is compact, semi-simple and let $a > 0$ be a regular value of $C : \mathfrak g^* \rightarrow \mathbb R$. If the interaction tensor is the same for all edges, i.e. $\mathbb J_{ij} = \mathbb J$ and the moment of inertia tensor is the same at all nodes, i.e. $\mathbb I_i = \mathbb I$, then there exists $kN$ linearly independent relative equilibrium configurations $\boldsymbol \mu^e = (\mu_1^e, \cdots, \mu_N^e)$ on the level set $C^{-1}(a)$ such that
1. $k$ are ferromagnetic states, i.e. $\mu_i^e = \sqrt{d_i}\mu^e$ for all $i = 1, \ldots, N$, where $\mu^e$ is an eigenvector of the extended inertia matrix $\mathbb I_{\text{ext}} := \mathbb I^{-1} - \mathbb J$, and
2. the remaining $(N-1)k$ states are anti-ferromagnetic, i.e. $\sum_{i=1}^N\sqrt{d_i}\mu_i^e = 0 $.
Consider the orthogonal decomposition of $(\mathfrak g^*)^{\oplus N} $ into ferromagnetic and anti-ferromagnetic states, i.e. $(\mathfrak g^*)^{\oplus N} = V \oplus V^{\perp}$, where $$\begin{aligned}
&V = \left\{\boldsymbol \mu = (\mu_1, \cdots, \mu_N) \in (\mathfrak g^*)^{\oplus N} : \mu_i = \sqrt{d_i} \mu, \quad \mu \in \mathfrak{g}^*\right\} \\
&V^{\perp} = \left \{\boldsymbol \mu = (\mu_1, \cdots, \mu_N) \in (\mathfrak g^*)^{\oplus N} : \sum_i \sqrt{d_i} \mu_i = 0\right \}\, .\end{aligned}$$ It is easy to check that $V^{\perp}$ is the orthogonal complement of $V$ with $\text{dim}(V) = k$ and $\text{dim}(V^{\perp}) = (N-1)k$. We claim that $V$ and $V^{\perp}$ are invariant subspaces under the linear operation $\mathbb L$.
To verify the former, take $\boldsymbol \mu \in V$, so $ \mu_i = \sqrt{d_i} \mu$ for all $i = 1, \ldots, N$. Then, $$\begin{aligned}
(\mathbb L \boldsymbol \mu)_i = \sqrt{d_i} \,\mathbb I^{-1} \mu - \sum_j \frac{A_{ij}}{\sqrt{d_i}} \mathbb J \mu = \sqrt{d_i}\left(\mathbb I^{-1} \mu - \sum_j \frac{A_{ij}}{d_i} \mathbb J \mu \right) \nonumber = \sqrt{d_i} \,\widehat{\mu}\, ,\end{aligned}$$ where $\widehat{\mu} := \mathbb I_{\text{ext}}\,\mu \in \mathfrak{g} \cong \mathfrak{g}^*$ and we used the fact that $\sum_j A_{ij} = d_i$. So $\mathbb L \boldsymbol \mu \in V$ for all $\boldsymbol \mu \in V$ and therefore $V$ is invariant under $\mathbb L$.
For the other case, take $\boldsymbol \mu \in V^{\perp}$, so that $\sum_i \sqrt{d_i}\mu_i = 0$. Then, $$\begin{aligned}
\sum_{i=1}^N \sqrt{d_i} (\mathbb L \boldsymbol \mu)_i &= \sum_{i=1}^N \sqrt{d_i}\left(\mathbb I^{-1} \mu_i - \sum_j \frac{A_{ij}}{\sqrt{d_id_j}} \mathbb J \mu_j \right) \\
&= \mathbb I^{-1} \sum_i \sqrt{d_i} \mu_i - \sum_{i,j} \frac{A_{ij}}{d_j} \mathbb J \sqrt{d_j} \mu_j\\
&= \mathbb I_{\text{ext}} \sum_i \sqrt{d_i} \mu_i = 0\, , \end{aligned}$$ where we used the fact that $\sum_i A_{ij} = d_j$ and the fact that $\boldsymbol \mu \in V^{\perp}$ in the last line. Hence, $\mathbb L \boldsymbol \mu \in V^{\perp}$ for all $\boldsymbol \mu \in V^{\perp}$ as expected.
This implies that there is an appropriate change of basis such that $\mathbb{L}$ becomes block diagonal, $$\begin{aligned}
\mathbb{L}\rightarrow
\begin{pmatrix}
\mathbb{L}_1 & 0 \\
0 & \mathbb{L}_2
\end{pmatrix}\, ,\end{aligned}$$ where $\mathbb{L}_1 : V \rightarrow V^*$ and $\mathbb{L}_2: V^{\perp} \rightarrow (V^{\perp})^*$. It is easy to see that $\mathbb L_1 \equiv \mathbb I_{\text{ext}}$. Hence, $\mathbb{L}$ has $kN$ eigenvectors $\{\boldsymbol \mu^e_i\}_{i=1}^{kN}$, which are equilibrium solutions, where $\boldsymbol \mu^e_i= (\mathbf v_i, \mathbf 0_{V^{\perp}})$ for $i=1,\ldots k$ and $\boldsymbol \mu^e_i = (\mathbf 0_{V}, \mathbf w_i)$ for $i = k+1, \ldots, kN$, where $\mathbf v_i \in V$ are the $k$ eigenvectors of $\mathbb{L}_1\equiv \mathbb I_{\text{ext}}$ and $\mathbf w_i \in V^{\perp}$ are the $(N-1)k$ eigenvectors of $\mathbb{L}_2$ and this proves our result.
Rescaling appropriately, all of the ferromagnetic and anti-ferromagnetic equilibrium states given above exist on a level set of the summed Casimir $C$, but not all of them exist if we restrict to a single coadjoint orbit.
As we will see later with the $SO(3)$ example, most of the eigenvalues $\lambda_e$ of $\mathbb L$ have algebraic multiplicity $n_\lambda = \text{mult}(\lambda_e) > 1$, so if $\boldsymbol \mu^e_1$ and $\boldsymbol \mu^e_2$ are two eigenvectors of $\mathbb L$ sharing the same eigenvalue $\lambda_e$, then their linear combination is also an eigenvector and therefore an equilibrium solution. Hence, every point in the eigenspace $E(\lambda_e) := \text{ker}(\mathbb L - \lambda_e \mathbb 1)$ is an equilibrium solution with $\text{dim}(E(\lambda_e)) = n_\lambda$ (since $\mathbb L$ is symmetric, the algebraic and geometric multiplicity are equivalent). It is therefore possible to construct more complicated equilibrium states that are neither ferromagnetic nor anti-ferromagnetic by taking a linear combination of a ferromagnetic eigenvector and an anti-ferromagnetic eigenvector that share the same eigenvalue.
### Nonlinear stability
We now investigate the stability properties of the equilibrium solutions found above via the energy-Casimir method, as given in [@arnold1966priori; @holm1985nonlinear]. Our main result is stated as follows.
\[equib-config\] Fixing a level set $C^{-1}(a)$ for $a>0$, the equilibrium solutions $\boldsymbol \mu_e$ of the network Lie-Poisson equation corresponding to the highest and lowest eigenvalues $\lambda_e$ of $\mathbb L$ are nonlinearly stable provided $\text{mult}(\lambda_e) = 1$.
Consider an equilibrium solution $\boldsymbol \mu_e$, which is an eigenvector of $\mathbb L$. In accordance with the energy-Casimir method, we first consider the augmented Hamiltonian $$\begin{aligned}
h_\Phi = h+ \Phi\left (C\right )\, , \end{aligned}$$ where $\Phi$ is an arbitrary real-valued function such that $\boldsymbol \mu_e$ is a critical point of $h_\Phi$. That is, $$\begin{aligned}
D_{\boldsymbol \mu} h_\Phi(\boldsymbol \mu_e) \cdot \delta \boldsymbol \mu = \left(\mathbb L \boldsymbol \mu_e + \Phi'\left (a \right)\boldsymbol \mu_e\right) \cdot \delta \boldsymbol \mu = 0 \, ,\end{aligned}$$ This equation holds for any free variations $\delta \boldsymbol \mu \in \mathfrak{g}$, provided $ \Phi'(a) = -\lambda_e$, where $\lambda_e$ is the eigenvalue of $\mathbb L$ corresponding to the eigenvector $\boldsymbol \mu_e$.
Next, we compute the second variation of $h_\Phi$. $$\begin{aligned}
\left<\delta \boldsymbol \mu, D^2_{\boldsymbol \mu} h_\Phi(\boldsymbol \mu_e) \cdot \delta \boldsymbol \mu \right>&= \left<\delta \boldsymbol \mu, \left(\mathbb L + \Phi' (a)\right) \delta \boldsymbol \mu\right> + \Phi''\left (a\right ) (\boldsymbol \mu_e \cdot \delta \boldsymbol \mu)^2 \nonumber\\
&= \left<\delta \boldsymbol \mu, (\mathbb L- \lambda_e \mathbb{1}) \delta \boldsymbol \mu \right> + \Phi''(a) (\boldsymbol \mu_e \cdot \delta \boldsymbol \mu)^2 \, .
\label{hessian-RB}\end{aligned}$$ Since $\mathbb L$ is symmetric, we can change the basis such that $\mathbb L$ is diagonal with ordered eigenvalues, i.e. $\mathbb L \rightarrow \text{diag}(\lambda_1, \ldots, \lambda_{kN})$ with $\lambda_1 \geq \ldots \geq \lambda_{kN}$. Now, let $\boldsymbol \mu_e$ be an equilibrium solution with the highest eigenvalue, so that $\lambda_e = \lambda_1$ and assume that $\text{mult}(\lambda_e) = 1$. Then we have $\boldsymbol \mu_e = \sqrt{2a} \,\widehat{\boldsymbol e}_1$, where $\{\widehat{\boldsymbol e}_1, \ldots, \widehat{\boldsymbol e}_{kN}\}$ is the new basis and reduces to $$\begin{aligned}
\left< \delta \boldsymbol \mu, D^2_{\boldsymbol \mu} h_\Phi(\boldsymbol \mu_e)\delta \boldsymbol \mu \right> &= \sum_{i=1}^{kN} (\lambda_i - \lambda_1 ) \delta \hat \mu_i^2 + 2a\Phi''(a) \delta \hat{\mu}_1^2 \, ,\end{aligned}$$ where $\delta \hat{\mu}_i$ are the components of $\delta \boldsymbol \mu$ in the new basis. One can directly check that choosing $\Phi''(a) < 0$, the Hessian matrix $D^2_{\boldsymbol \mu} h_\Phi(\boldsymbol \mu_e)$ becomes strictly negative definite, so by the energy-Casimir method, the equilbrium configuration with the highest energy is nonlinearly stable. Similarly, choosing $\lambda_e = \lambda_{kN}$ (lowest energy) and $\Phi''(a) > 0$, the Hessian matrix $D^2_{\boldsymbol \mu} h_\Phi(\boldsymbol \mu_e)$ becomes strictly positive definite, so the equilibrium configuration with the lowest energy is also nonlinearly stable.
From , we see that this corresponds to the highest and lowest energy equilibrium states on the level set $C^{-1}(a)$.
We cannot deduce further about the nonlinear stability of the other equilibrium states, but as we will see with the $SO(3)$ example in section \[section-RB\], most of them are linearly unstable and the remaining few are linearly stable at best. To investigate the linear stability of the equilibrium solutions in the general setting, we first linearize the equation by setting $\boldsymbol \mu(t) = \boldsymbol \mu_e + \epsilon \, \delta \boldsymbol \mu(t)$ and dropping terms of $O(\epsilon^2)$ to get $$\begin{aligned}
\label{lin-eq}
\dot{\delta \boldsymbol \mu} = \mathrm{\bf ad}_{\boldsymbol \mu_e} \left( (\mathbb L - \lambda_e \mathbb{1}) \delta \boldsymbol \mu \right) \,, \end{aligned}$$ where we have used $\mathbb L \boldsymbol \mu_e = \lambda_e \boldsymbol \mu_e$. We can then investigate the linear instability of the equilibrium solution $\boldsymbol \mu_e$ by looking at whether the eigenvalues of the linear operator $\mathrm{\bf ad}_{\boldsymbol \mu_e} \left( (\mathbb L - \lambda_e \mathbb 1)\, \cdot \right)$ has a positive real part.
Noise and dissipation
---------------------
We now perturb our system with noise and dissipation that preserves the Casimir $C$, following [@arnaudon2016noise]. The equation is similar to the single coadjoint orbit system with equation , that is $$\begin{aligned}
\mathbb d\mu_i &+ \mathrm{ad}^*_\frac{\partial h}{\partial \mu_i} \mu_i\, dt
+ \theta\, \mathrm{ad}^*_\frac{\partial C}{\partial \mu_i} \left [ \frac{\partial C}{\partial \mu_i}, \frac{\partial h}{\partial \mu_i} \right ]^\flat \, dt + \sum_l\mathrm{ad}^*_{\sigma_{i,l}} \mu_i \circ dW_t^{i,l} = 0 \,,
\label{SEP-Diss-N}\end{aligned}$$ for $i=1, \ldots, N$, where $h = \frac12 \left<\boldsymbol \mu, \mathbb L \boldsymbol \mu \right>$ and $C$ is a given Casimir for the Lie-Poisson bracket. The noise term has $kN$ independent Lie algebra vectors $\sigma_{i,l}$ associated to independent Wiener processes $W_t^{i,l}$ and the dissipation considered here is a double bracket dissipation (See [@gaybalmaz2013selective; @bloch1996euler]). One could generalise this equation further by letting $\theta_i$ be node dependent, but we will not do this here to keep our equations simple and we also choose our noise to be [*isotropic*]{}, that is, $\sigma_{i,l} = \sigma \boldsymbol e_l$, where $\boldsymbol e_l$ is a basis vector of $\mathfrak g$, which are the same at every node.
Notice that without noise, the energy dissipates as $$\begin{aligned}
\frac{d}{dt} h_{\text{int}}(\boldsymbol \mu) = -\theta\, \sum_{i=1}^N \left [ (\mathbb L \boldsymbol \mu)_i , \mu_i \right ] ^2\, \end{aligned}$$ and will tend towards the equilibrium configuration with minimum energy on the coadjoint orbit. If the noise is isotropic, the stationary distribution can be computed in the same way as we did with the single orbit case.
The stationary distribution of with isotropic noise is the Gibbs distribution $$\begin{aligned}
\mathbb P_\infty(\boldsymbol \mu ) = \frac{1}{Z} e^{-\beta h(\boldsymbol \mu)} \, ,
\label{P_infty}\end{aligned}$$ where the inverse temperature is given by $$\begin{aligned}
\frac{1}{T} = \beta = \frac{\theta}{2\sigma^2} \, . \end{aligned}$$ The partition function is given by $$\begin{aligned}
Z= \int_{\boldsymbol {\mathcal O}} e^{-\beta h(\boldsymbol \mu)} d\boldsymbol \mu\, ,
\label{Z-int}\end{aligned}$$ where $\boldsymbol{ \mathcal O}= \mathcal O_1\times \ldots \times \mathcal O_{N}$ is the direct product of the coadjoint orbits at each lattice site, or the total coadjoint orbit of the lattice.
We refer to [@arnaudon2016noise] for the proof of this formula, which is done by direct substitution into the Fokker-Planck equation of the stochastic process .
The Gibbs distribution provides us with the notion of [*temperature*]{} in this system and as we will see in section \[PT-section\], we observe a phase transition for the case $G=SO(3)$ as we vary the temperature, similar to the classical Heisenberg model. The mean field approximation can also be derived using the partition function and will allow us to help detect the phase transition.
Example I: Networks of rigid bodies {#section-RB}
===================================
In this section, we study in more detail the case $G=SO(3)$, corresponding to a network of interacting rigid bodies. The corresponding Lie algebra $\mathfrak{so}(3)$ is compact and semi-simple so we are in the setting of the general theory studied in section \[SS-section\].
Equation of motion
------------------
Consider a single free rigid body with configuration group $SO(3)$. The reduced Hamiltonian is given by the kinetic energy $h(\mathbf \Pi) = \frac12 \mathbf \Pi \cdot \mathbb I^{-1} \mathbf \Pi$ where $\mathbf \Pi \in \mathfrak{so}^*(3) \cong \mathbb R^3$ is the [*angular momentum vector*]{} and the corresponding equation, given by the $\mathfrak{so}^*(3)$ Lie-Poisson structure, is $\dot{\mathbf \Pi} = \mathbf \Pi \times \mathbf \Omega$, where $\mathbf \Omega := \mathbb I^{-1} \mathbf \Pi$ is the [*angular velocity vector*]{}. We will not discuss the dynamics of a single body further here, as it is standard in geometric mechanics and instead refer to chapter 15 of Marsden and Ratiu [@marsden1999book] for a more detailed exposition of the system.
Extending this to a network $(\mathcal N, SO(3), \mathcal O)$ of interacting rigid bodies via momentum coupling with inertia tensor $\mathbb I_i$ and interaction tensor $\mathbb J_{ij} $, we get the Hamiltonian $$\begin{aligned}
h(\overline {\mathbf \Pi})= \frac12 \sum_i \mathbf \Pi_i\cdot \mathbf \Omega_i - \frac12 \sum_i \sum_{j \sim i} \frac{1}{\sqrt{d_id_j}}\mathbf \Pi_i \cdot \mathbb J_{ij} \mathbf \Pi_j\, ,
\label{RB-H-lattice}\end{aligned}$$ where $\overline {\mathbf \Pi} = (\mathbf \Pi_1, \ldots, \mathbf \Pi_N)$ is the network extended angular momentum vector and $\mathbf \Omega_i := \mathbb I_i^{-1} \mathbf \Pi_i$ is the angular velocity at site $i$. Taking the $(-)$ Lie-Poisson structure $$\begin{aligned}
\label{RB-bracket}
\{f, g\}_{LP}^-(\overline {\mathbf \Pi}) = -\sum_i \mathbf \Pi_i \cdot \frac{\partial f}{\partial \mathbf \Pi_i} \times \frac{\partial g}{\partial \mathbf \Pi_i},\end{aligned}$$ we obtain the corresponding network Lie-Poisson equation $$\begin{aligned}
\dot{ \mathbf{\Pi}}_i = \mathbf \Pi_i\times \mathbf \Omega_i - \sum_{j \sim i} \frac{1}{\sqrt{d_id_j}} \, \mathbf \Pi_i\times \mathbb J_{ij} \mathbf \Pi_j\quad \forall i = 1, \ldots, N\, .
\label{RB-Det-lattice}\end{aligned}$$ One can check that the Casimirs for the Lie-Poisson bracket are given by $$\begin{aligned}
C_i(\mathbf \Pi_i) = \frac12 \| \mathbf \Pi_i\|^2, \quad i=1, \ldots, N\, , \end{aligned}$$ and we denote the sum of the Casimirs by $$\begin{aligned}
C(\overline {\mathbf \Pi}) = \sum_{i=1}^N C_i(\mathbf \Pi_i) = \frac12 \|\overline{\mathbf \Pi}\|^2\, .\end{aligned}$$
The coadjoint orbit in this special case is given as follows.
The coadjoint orbit $\boldsymbol{\mathcal O} = \mathcal O_1 \times \cdots \times \mathcal O_N$ of $SO(3)^{\times N}$ on $\mathfrak{so}^*(3)^{\oplus N} \cong \mathbb R^{3N}$ is given by the level sets of the Casimirs $$\begin{aligned}
\mathcal O_i = \left\{\mathbf \Pi_i \in \mathbb R^{3} : C_i(\mathbf \Pi_i) = c_i \right\} \cong S^2\, ,\end{aligned}$$ for some constants $c_1, \ldots, c_N$.
Since the action of $SO(3)$ on the two-sphere $S^2$ (viewed as a submanifold in $\mathbb R^3$) is transitive, the coadjoint orbit is exactly given by the level sets of the Casimirs.
We observe that removing the kinetic energy term in and taking $d_i=d$ for all $i=1, \ldots, N$, we recover the Hamiltonian for the Heisenberg model. Hence, the rigid body network can be viewed as a Heisenberg model with mass-carrying spins.
Equilibrium solutions
---------------------
We now investigate the relative equilibrium configurations of the deterministic rigid body network and its corresponding stability properties. Since the Lie algebra $\mathfrak{so}(3)$ is compact and semi-simple, we can apply propositions \[equib\] and \[equib-config\] to obtain information about the equilibrium configurations of the rigid body network and its corresponding nonlinear stability properties, which we summarise in the following proposition.
\[stab-RB\] The relative equilibrium solutions $\overline {\mathbf \Pi}^e = (\mathbf \Pi^e_1, \ldots, \mathbf \Pi^e_N)$ of a rigid body network $(\mathcal N, SO(3), \mathcal O)$ correspond to the eigenvectors of $\mathbb L$ and those with the lowest and highest energy on a level set $C^{-1}(a)$ ($a > 0$) are nonlinearly stable. Furthermore, if $\mathbb J_{ij} = \mathbb J$ and $\mathbb I_i = \mathbb I$, then there exists $3N$ linearly independent relative equilibrium configurations such that
1. three are ferromagnetic states, i.e. $\mathbf \Pi_i^e = \sqrt{d_i} \mathbf \Pi^e$ for $i=1, \ldots, N$, where $\mathbf \Pi^e$ is an eigenvector of the extended inertia matrix $\mathbb I_{\text{ext}}:=\mathbb I^{-1} - \mathbb J$ and
2. the remaining $3N-3$ are anti-ferromagnetic states, i.e. $\sum_{i=1}^N \sqrt{d_i} \mathbf \Pi_i^e = \mathbf 0$
Now, from , the linearized equation for the rigid body network around an equilibrium configuration $\overline {\mathbf \Pi}^e$ is given by $$\begin{aligned}
\label{linearised-RB}
\frac{d}{dt}\delta \overline {\mathbf \Pi} = \widehat{\mathbf \Pi}^e(\mathbb L - \lambda_e \mathbb 1) \cdot \delta \overline {\mathbf \Pi}, \quad \widehat{\mathbf \Pi}^e := \text{diag}(\widehat{\mathbf \Pi}_1^e, \ldots, \widehat{\mathbf \Pi}_N^e)\, ,\end{aligned}$$ where $\,\widehat{ } : (\mathbb R^3, \times) \rightarrow \mathfrak{so}^*(3)$ is the standard Lie algebra isomorphism that takes a vector in $\mathbb R^3$ to a skew-symmetric matrix in $\mathfrak{so}^*(3)$.
In figure \[fig:RB-stability\], we computed the eigenvalues $\lambda_s$ of the linearised system plotted against its corresponding energy $\lambda_e$ for two cases: (a) $\mathbb I=\text{diag}(1,1,1)$, $\mathbb J=\text{diag}(1,2,3)$ and (b) $\mathbb I=\text{diag}(1,2,3)$, $\mathbb J=\text{diag}(1,1,1)$, where we took a $20 \times 20$ lattice with periodic boundary condition. As expected from proposition \[stab-RB\], we see that in both cases, the highest and lowest energy configurations are linearly stable (both have multiplicity 1), however the majority of the configurations that lie between these two states are unstable. Interestingly, we also see a few linearly stable states towards the high and low end of the energy, with significantly more of them in \[fig:RB-stability1\] than in \[fig:RB-stability2\].
We observed numerically that the linearly stable states (red dots) given in figure \[fig:RB-stability1\], are parallel to the $\Pi_3$-axis, which tend to be characterised by a regular ‘argyle’ pattern of aligned spins for low energy configurations (figure \[fig:RB-sol1a\]) and anti-aligned spins for high energy configurations (figure \[fig:RB-sol2b\]). As expected, the lowest energy state is a ferromagnetic state that is completely aligned with the $\Pi_3$-axis and the highest energy state is an anti-ferromagnetic state given by a fine checkerboard pattern. These last two configurations are furthermore nonlinearly stable by proposition \[stab-RB\].
The linearly stable states in figure \[fig:RB-stability2\] can be described as follows, from lowest to highest energy. The lowest energy configurations is a ferromagnetic state that is parallel to the $\Pi_3$-axis as expected, the next ($\lambda = -3.5$) is a four dimensional subspace consisiting of a similar ‘argyle’ pattern of aligned spins seen in the other case (figure \[fig:RB-sol2a\]), the next ($\lambda = -3$) is again a ferromagnetic state, along the $\Pi_1$-axis this time, the last few ($4 < \lambda < 5$) consist of ‘argyle’ patterns with anti-aligned spins parallel to the $\Pi_1$-axis (figure \[fig:RB-sol1b\]) and the highest energy state is a completely anti-ferromagnetic checkerboard patterned state as expected.
Network of coadjoint orbits II: Position coupling {#SD-section}
=================================================
In this section, we consider a Lie-Poisson network of coadjoint orbits $(\mathcal N, G, \mathcal O)$ that arise from a different approach to coupling the neighbours. We will call this [*position coupling*]{}. In contrast to momentum coupling, where coupling occurs at the level of the reduced space $\mathfrak g^*$, in position coupling, the coupling occurs at the level of the group by considering a representation of $G$, followed by symmetry reduction, which yields a Euler-Poincar[é]{}/Lie-Poisson equation. This introduces a semi-direct product group structure into our system, analogous to the heavy top. We saw earlier that the momentum-coupled equations arise somewhat unnaturally from the Euler-Poincar[é]{} framework, however, there is no such issue for the position coupling approach. Furthermore, the equations that we obtain by position coupling are completely new to our knowledge and further investigation is necessary to understand the new phase transition behaviour that we will describe in section \[PT-section\].
Deterministic equations
-----------------------
Consider a network $\mathcal N$, a group $G$ and a left representation of $G$ on a vector space $V$. Fixing a vector $\boldsymbol a_0 = (A_1, \ldots, A_N) \in (V^*)^{\oplus N}$ and introducing the interaction tensor $\mathbb J_{ij} : V^* \rightarrow V$, we define the [*position coupled interaction potential energy*]{} $$\begin{aligned}
\label{h-int-HT}
h^{\text{int}}_{ij}(g_i, g_j ; A_i, A_j) = -\frac{1}{2\sqrt{d_id_j}} \langle g_i^{-1} A_i, \mathbb J_{ij} \, g_j^{-1} A_j \rangle_{V^* \times V}\, , \quad g_i, g_j \in G,\end{aligned}$$ where nodes $i$ and $j$ are adjacent on the graph $\mathcal N$ and we denote by $gA$ to be the left action of $g \in G$ on $a \in V^*$. At node $i$, we assign a Lagrangian $L_i(g_i, \dot{g}_i; A_i)$ depending on the parameter $A_i$ that satisfies the following properties
1. $L_i(g_i, \dot{g}_i; A_i)$ is (left) invariant under $G_{A_i}$, namely the isotropy subgroup of $G$ which fixes the vector $A_i$.
2. The extended Lagrangian $L_i(g_i, \dot{g}_i, A_i)$ is invariant under the diagonal (left) action of $G$ on $TG \times V^*$.
We take our full Lagrangian to be $$\begin{aligned}
L(\boldsymbol g, \dot{\boldsymbol g}, \boldsymbol a_0) = \sum_i L_i(g_i, \dot{g}_i, A_i) - \sum_i \sum_{j \sim i} h^{\text{int}}_{ij}(g_i, g_j, A_i, A_j)\, ,\end{aligned}$$ where $\boldsymbol g = (g_1, \ldots, g_N) \in G^{\times N}$. We can check that this Lagrangian is invariant under the (left) diagonal action of $G^{\times N}$ on $TG^{\times N} \times (V^*)^{\times N}$, so we obtain the reduced Lagrangian $$\begin{aligned}
L(\boldsymbol g, \dot{\boldsymbol g}, \boldsymbol a_0) &= L(\boldsymbol e, \boldsymbol g^{-1}\dot{\boldsymbol g}, \boldsymbol g^{-1}\boldsymbol a_0) \nonumber \\
&:= l(\boldsymbol \xi, \boldsymbol a)=\sum_i l_i(\xi_i, a_i) + \frac12 \sum_{i} \sum_{j \sim i}\frac{1}{\sqrt{d_id_j}} \langle a_i, \mathbb J_{ij} a_j \rangle\, ,\end{aligned}$$ where $ \xi_i := g_i^{-1} \dot{g_i}, \quad a_i := g_i^{-1} A_i$, $\boldsymbol \xi = (\xi_1, \ldots, \xi_N)$ and $\boldsymbol a = (a_1, \ldots, a_N)$. This allows us to apply the semi-direct product reduction theorem given in Holm et al. [@holm1998euler] to obtain the following network Euler-Poincaré equations.
The following statements are equivalent.
1. For a fixed vector $\boldsymbol a_0 \in (V^*)^{\oplus N}$, Hamilton’s variational principle $$\begin{aligned}
\delta \int_{t_1}^{t_2} L(\boldsymbol g, \dot{\boldsymbol g}; \boldsymbol a_0) \, dt = 0
\end{aligned}$$ holds for variations $\delta \boldsymbol g$ vanishing at the endpoints.
2. $\boldsymbol g(t)$ satisfies the Euler-Lagrange equations for $L|_{\boldsymbol a_0}$ on $G^{\times N}$.
3. The constrained variational principle $$\begin{aligned}
\delta \int_{t_1}^{t_2} l(\boldsymbol \xi(t), \boldsymbol a(t)) \, dt = 0
\end{aligned}$$ holds on $(\mathfrak g \oplus V^*)^{\oplus N}$ with variations taking the form $$\begin{aligned}
\delta \xi_i = \dot{\eta_i} + [\xi_i, \eta_i], \quad \delta a_i = - \eta_i a_i\, ,
\end{aligned}$$ for all $i=1,\ldots,N$, where $\eta_i(t) := \delta g_i(t) \in \mathfrak g$ vanishes at the endpoints.
4. The following Euler-Poincar[é]{} equations hold on $(\mathfrak g \oplus V^*)^{\oplus N}$ $$\begin{aligned}
\frac{d}{dt} \frac{\delta l_i}{\delta \xi_i} &= \ad^*_{\xi_i} \frac{\delta l_i}{\delta \xi_i} + \frac{\delta l_i}{\delta a_i} \diamond a_i + \left(\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a\right)_i \diamond a_i, \label{EP-semidirect} \\
\frac{da_i}{dt} &= - \xi_i a_i
\end{aligned}$$ for all $i=1, \ldots, N$, where $\diamond: V^* \times V \rightarrow \mathfrak{g}^*$ is the momentum map defined by $$\begin{aligned}
\left< \left< q, \, \xi p \right> \right> = - \langle \xi, p \,\diamond \, q \rangle\, ,\end{aligned}$$ where $\langle \langle \cdot, \cdot \rangle \rangle$ is the natural pairing on $V^* \times V$.
### Legendre transformation and Lie-Poisson equation
We find the reduced Hamiltonian by taking the partial Legendre transform in the velocity variable, $$\begin{aligned}
h(\boldsymbol \mu, \boldsymbol a) &= \langle \boldsymbol \mu, \boldsymbol \xi \rangle - l(\boldsymbol \xi, \boldsymbol a) =: \sum_i h_i(\mu_i, a_i) - \frac12\left<\boldsymbol a, \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a \right>\, ,
\label{Legendre}\end{aligned}$$ where $\boldsymbol \mu = (\mu_1, \ldots, \mu_N)$ and $\quad \mu_i = \frac{\delta l}{\delta \xi_i}$. The Euler-Poincaré equation then becomes a Lie-Poisson equation $$\begin{aligned}
\begin{split}
\frac{d \mu_i}{dt} &= \ad^*_{\delta h_i / \delta \mu_i} \mu_i - \frac{\delta h_i}{\delta a_i} \diamond a_i - \left(\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a\right)_i \diamond a_i \\
\frac{d a_i}{dt} &= - \frac{\delta h_i}{\delta \mu_i} \cdot a_i\, ,
\end{split}\end{aligned}$$ which is indeed Lie-Poisson, with bracket given by the $(-)$ Lie-Poisson bracket on the semi-direct product algebra $(\mathfrak{g}^* \circledS V^*)^{\times N} $, which is given by $$\begin{aligned}
\{f,g\}^-_{LP}(\boldsymbol \mu, \boldsymbol a) &= -\sum_i \left(\left< \mu_i, \left[\frac{\delta f}{\delta \mu_i}, \frac{\delta g}{\delta \mu_i} \right] \right> + \left<a_i, \frac{\delta f}{\delta \mu_i}\frac{\delta g}{\delta a_i} - \frac{\delta g}{\delta \mu_i} \frac{\delta f}{\delta a_i} \right>\right)\, .
\label{LP-semidirect}\end{aligned}$$
Likewise, for a right invariant Lagrangian and right representation, we recover the $(+)$ Lie-Poisson bracket. Again, we refer the readers to Holm et al. [@holm1998euler] for more details about left vs. right group action.
For the special case where $\mathfrak g = \text{Lie}(G)$ is compact, semi-simple and given a coadjoint representation of $G$ on $V^* = \mathfrak g^*$, the Lie-Poisson structure becomes $$\begin{aligned}
\{f,g\}^-_{LP}(\boldsymbol \mu, \boldsymbol a) &= -\sum_i \left(\left< \mu_i, \left[\frac{\delta f}{\delta \mu_i}, \frac{\delta g}{\delta \mu_i} \right] \right> + \left<a_i, \left[\frac{\delta f}{\delta \mu_i},\frac{\delta g}{\delta a_i}\right] - \left[\frac{\delta g}{\delta \mu_i}, \frac{\delta f}{\delta a_i}\right] \right>\right)\, ,
\label{LP-semidirect-2}\end{aligned}$$ with the corresponding Lie-Poisson equation $$\begin{aligned}
\begin{split}
\frac{d \mu_i}{dt} &= \left[\mu_i, \frac{\delta h_i}{\delta \mu_i} \right] + \left[a_i, \frac{\delta h_i}{\delta a_i} - \left(\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a\right)_i\,\right] \\
\frac{d a_i}{dt} &= \left[a_i,\frac{\delta h_i}{\delta \mu_i} \right]\, .
\label{LP-eq-semidirect}
\end{split}\end{aligned}$$ We refer to Ratiu et al. [@ratiu1981euler; @ratiu1982lagrange] for more details on similar systems and in particular their complete integrability in the case of the Lagrange top (which will be lost here, due to the interaction terms).
One can check that the Casimirs for the bracket are given by $$\begin{aligned}
C_{i,1} = \langle \mu_i ,a_i\rangle \qquad \mathrm{and} \qquad C_{i,2} \langle a_i ,a_i\rangle\, , \end{aligned}$$ for $i=1, \ldots, N$, where we take $\langle \cdot, \cdot \rangle = -\kappa(\cdot, \cdot)$ and the coadjoint orbits of the semi-direct product group $(G \,\circledS \,\mathfrak g)^{\times N}$ are contained in the level sets of the Casimirs, i.e. $$\begin{aligned}
\mathcal O \subset \left\{(\boldsymbol \mu, \boldsymbol a) \in (\mathfrak g^* \,\circledS \,\mathfrak g^*)^{\times N} : C_{i,1}(\boldsymbol \mu, \boldsymbol a) = c_{i,1}, C_{i,2}(\boldsymbol \mu, \boldsymbol a) = c_{i,2}, \, \forall i = 1, \ldots, N \right\}\, ,\end{aligned}$$ for given $2N$ constants $c_{1,1}, \ldots, c_{N,1}$ and $c_{1,2}, \ldots, c_{N,2}$. Again, this follows from the $\text{Ad}^*$-invariance of the Killing form. The preservation of coadjoint orbits under the dynamics of the position coupled network Lie-Poisson system again follows from the equivariance of the momentum map $\mathbf J_R : T^*(G \,\circledS \,\mathfrak g)^{\times N} \rightarrow (\mathfrak g^* \,\circledS \,\mathfrak g^*)^{\times N}$ giving rise to the Lie-Poisson structure . We will also denote by $$\begin{aligned}
C_1(\boldsymbol \mu, \boldsymbol a) = \sum_{i=1}^N C_{i,1}(\mu_i, a_i)\qquad \mathrm{and} \qquad C_2( \boldsymbol a) = \sum_{i=1}^N C_{i,2}(a_i)\,, \end{aligned}$$ as the sum of the Casimirs.
Equilibrium positions {#HT-eq-general}
---------------------
We now seek for equilibrium solutions of that correspond to the critical points of the Hamiltonian restricted to the level sets of the summed Casimirs $C_1$ and $C_2$. Here, we specialise to Hamiltonians of the form $$\begin{aligned}
h(\boldsymbol \mu,\boldsymbol a) = h^{KE}(\boldsymbol \mu) + h^\mathrm{int}(\boldsymbol a)\, , \end{aligned}$$ where $$\begin{aligned}
h^{KE}(\boldsymbol \mu) = \frac12 \langle \boldsymbol \mu, \overline {\mathbb I}^{-1} \boldsymbol \mu \rangle, \quad \overline {\mathbb I}^{-1} := \text{diag}(\mathbb I^{-1}_1, \ldots, \mathbb I^{-1}_N)\, ,\end{aligned}$$ for inertia tensors $\mathbb I_i : \mathfrak g \rightarrow \mathfrak g^*$ at each node $i$ is the kinetic energy, and $$\begin{aligned}
h^{\text{int}}(\boldsymbol a) = - \frac12 \left< \boldsymbol a, \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \,\boldsymbol a \right>\, .\end{aligned}$$ is the interaction potential energy. Now consider the augmented Hamiltonian $$\begin{aligned}
h_{\phi,\psi}(\boldsymbol \mu,\boldsymbol a) = h(\boldsymbol \mu,\boldsymbol a) + \phi(C_1) + \psi(C_2)\, , \end{aligned}$$ where $\phi, \psi$ are arbitrary smooth functions and take its first variations, which set to $0$ gives the condition for $(\boldsymbol \mu_e, \boldsymbol a_e) $ to be an equilibrium solution. One could also choose a more general function of the two Casimirs, but this form turns out to be general enough for our purpose. Taking the variation, we get $$\begin{aligned}
\begin{split}
Dh_{\phi,\psi}(\boldsymbol \mu_e, \boldsymbol a_e) \cdot (\delta \boldsymbol \mu,\delta \boldsymbol a)^T &= \sum_i \left( \left<\mathbb I^{-1}_i \mu_i^e + \lambda_1 \, a_i^e, \, \delta \mu_i \right> \right.\\
&\left. + \left< -\left(\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a_e\right)_i+ \lambda_1 \, \mu_i^e + \lambda_2 \, a_i^e, \,\delta a_i \right> \right) = 0 \, ,
\end{split}\end{aligned}$$ where we defined $$\begin{aligned}
\lambda_1 &:= \phi'(c_1)\qquad \mathrm{and} \qquad \lambda_2 := 2\psi'(c_2)\, . \end{aligned}$$
The conditions for the equilibrium solutions are hence given by $$\begin{aligned}
&\delta \boldsymbol \mu: \overline {\mathbb I}^{-1} \boldsymbol \mu_e + \lambda_1 \boldsymbol a_e = 0 \label{mu-eq} \\
&\delta \boldsymbol a: -\mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a_e + \lambda_1 \boldsymbol \mu_e + \lambda_2 \,\boldsymbol a_e = 0\, ,
\label{a-eq}\end{aligned}$$ and pairing with $\boldsymbol \mu_e$ and with $\boldsymbol a_e$, we find $$\begin{aligned}
&\lambda_1 = -\frac{1}{c_1}\left<\boldsymbol \mu_e, \overline {\mathbb I}^{-1} \boldsymbol \mu_e \right> \label{lambda1-eq} \\
&\lambda_2 = \frac{1}{c_2}\left(\left<\boldsymbol \mu_e, \overline {\mathbb I}^{-1} \boldsymbol \mu_e \right> + \left<\boldsymbol a_e, \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a_e\right> \right)\, .
\label{lambda2-eq}\end{aligned}$$
Now, substituting into , we obtain the following equation $$\begin{aligned}
\mathbb L(\lambda_1) \,\boldsymbol a_e = -\lambda_2 \,\boldsymbol a_e\, ,
\label{L_l1l2}\end{aligned}$$ where $\mathbb L(\lambda_1) := -\lambda_1^2 \,\overline {\mathbb I} - \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12}$ is our new extended graph Laplacian. Hence, fixing $\lambda_1$, the equilibrium solutions $\boldsymbol a_e$ and $\boldsymbol \mu_e = -\lambda_1 \overline {\mathbb I} \,\boldsymbol a_e$ again correspond to the $kN$ eigenvectors of the graph Laplacian $\mathbb L(\lambda_1)$ with eigenvalue $-\lambda_2$. From this, we immediately deduce the following result.
Fixing $\lambda_1$ and one of the summed Casimirs $C_1$ or $C_2$, and furthermore, assuming that $\mathbb J_{ij} = \mathbb J$ and $\mathbb I_i = \mathbb I$ for all $i,j = 1, \ldots, N$, there exists $kN$ equilibrium solutions $\boldsymbol a_e$ and $\boldsymbol \mu_e = -\lambda_1 \overline {\mathbb I} \,\boldsymbol a_e$ such that
1. $k$ are ferromagnetic states, i.e. $a_i = \sqrt{d_i} a$ for all $i$, where $a$ is an eigenvector of the extended inertia tensor $\mathbb I_{\text{ext}}(\lambda_1) := -\lambda_1^2 \mathbb I - \mathbb J$,
2. The remaining $(N-1)k$ are anti-ferromagnetic states, i.e $\sum_{i=1}^N \sqrt{d_i} a_i = 0$.
The proof is similar to that of proposition \[equib\].
It is important to note that the ferromagnetic and anti-ferromagnetic equilbrium solutions given above are only found if we fix $\lambda_1$ and only one of the summed Casimirs $C_1$ or $C_2$. If instead, we want to find the equilibrium solutions on the level sets of both $C_1$ and $C_2$ while keeping the parameter $\lambda_1$ free, then one has to solve the full nonlinear equation , , , which in general is difficult to solve.
Nonlinear stability analysis
----------------------------
We saw that the eigenvectors of the graph Laplacian $\mathbb L(\lambda_1)$ correspond to equilibrium solutions of , similar to the momentum-coupled case. We will now show that the equilibrium configuration corresponding to the eigenvector of $\mathbb L(\lambda_1)$ with the [*lowest*]{} eigenvalue is nonlinearly stable. This differs slightly from the momentum coupled case where we were able to prove that both the lowest and highest eigenvalue configurations are stable.
Fixing $\lambda^e_1$, the equilibrium configuration corresponding to the eigenvector of $\mathbb L(\lambda_1^e)$ with the lowest eigenvalue $-\lambda_2^e$ is nonlinearly stable, provided $\text{mult}(-\lambda_2^e) = 1$.
We apply the energy-Casimir method to assess the stability of the lowest eigenvalue configuration. The Hessian $D^2 h_{\phi,\psi}(\boldsymbol \mu_e, \boldsymbol a_e)$ of the augmented Hamiltonian is a symmetric matrix given by $$\begin{aligned}
D^2 h_{\phi,\psi}(\boldsymbol \mu_e, \boldsymbol a_e) =
\begin{pmatrix}
\overline {\mathbb I}^{-1} + \hat \lambda_1 \,\boldsymbol a_e^T \boldsymbol a_e & \lambda^e_1 \,\mathbb 1 + \hat \lambda_1 \,\boldsymbol a_e^T \boldsymbol \mu_e \\
\lambda^e_1 \,\mathbb 1 + \hat \lambda_1 \,\boldsymbol \mu_e^T \boldsymbol a_e & \lambda^e_2 \,\mathbb 1 - \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} + \hat \lambda_2 \,\boldsymbol a_e^T \boldsymbol a_e + \hat \lambda_1 \,\boldsymbol \mu_e^T\boldsymbol \mu_e
\end{pmatrix}\, ,\end{aligned}$$ where $\hat{\lambda}_1 := \phi''(c_1)$ and $\hat{\lambda}_2 := 4\psi''(c_2)$. Setting $\hat{\lambda}_1 = 0$, this simplifies to $$\begin{aligned}
D^2 h_{\phi,\psi}(\boldsymbol \mu_e, \boldsymbol a_e) =
\begin{pmatrix}
\overline {\mathbb I}^{-1} & \lambda^e_1 \,\mathbb 1 \\
\lambda^e_1 \,\mathbb 1 & \lambda^e_2 \,\mathbb 1 - \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} + \hat \lambda_2 \,\boldsymbol a_e^T \boldsymbol a_e
\end{pmatrix}
=:
\begin{pmatrix}
X & Y \\
Y^T & Z
\end{pmatrix}\, .\end{aligned}$$ It is well-known that block matrices of this form are positive definite if and only if the upper left block $X$ and the Schur complement $B := Z - Y^T X^{-1} Y$ are both positive definite. Since we take $\mathbb I$ to be positive definite, it follows that $X$ is positive definite so we only need to show that $B$ is positive definite. Written in full, in terms of the graph Laplacian $\mathbb L(\lambda^e_1) = -(\lambda^e_1)^2 \,\overline {\mathbb I} - \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12}$, one can check that $$\begin{aligned}
\delta \boldsymbol a^T B \,\delta \boldsymbol a = \delta \boldsymbol a^T \left(\mathbb L(\lambda^e_1) + \lambda^e_2 \mathbb 1 \right) \delta \boldsymbol a + \hat \lambda_2 (\boldsymbol a_e \cdot \delta \boldsymbol a)^2\, .\end{aligned}$$ Since $\mathbb L(\lambda^e_1)$ is symmetric, we can diagonalise it so that $\mathbb L(\lambda^e_1) \rightarrow \text{diag}(\alpha_1, \ldots, \alpha_{kN})$ with $\alpha_1 \geq \ldots \geq \alpha_{kN}$. Recalling that the equilibrium solution satisfies the eigenvalue problem $\mathbb L(\lambda^e_1)\, \boldsymbol a_e = -\lambda_2^e \,\boldsymbol a_e$, we take $-\lambda^e_2 = \alpha_{kN}$ which is the lowest eigenvalue of $\mathbb L(\lambda^e_1)$ and we assume that $\text{mult}(-\lambda_2^e)=1$. We then have $\boldsymbol a_e = \sqrt{c_2}\,\widehat{\boldsymbol e}_{kN}$ and get $$\begin{aligned}
\delta \boldsymbol a^T B \,\delta \boldsymbol a = \sum_{i=1}^{kN} (\alpha_i - \alpha_{kN})\delta \hat{a}_i^2 + \hat{\lambda}_2 c_2 \delta \hat{a}_{kN}^2\, ,\end{aligned}$$ where $\delta \hat{a}_i$ for $i=1, \ldots,kN$ are the components of $\delta \boldsymbol a$ in this new-basis. From this, it is easy to see that the Schur complement $B$ is positive definite if we choose $\hat{\lambda}_2 > 0$. Hence, $D^2 h_{\phi,\psi}(\boldsymbol \mu_e, \boldsymbol a_e)$ becomes positive definite and this configuration is nonlinearly stable, by the energy-Casimir method.
Noise and dissipation
---------------------
Following Arnaudon et al. [@arnaudon2016noise], the general stochastic equations for this system is given by $$\begin{aligned}
\begin{split}
\mathbb d\mu_i &+ [\xi_i, \mu_i] +\left [\chi( \boldsymbol a)_i,a_i\right ]\, dt
+ \theta\, \left [\frac{\partial C}{\partial \mu_i}, \left [ \frac{\partial C}{\partial \mu_i}, \xi_i \right ]\right] dt
+\, \theta\, \left [ \frac{\partial C}{\partial a_i}, \left [ \frac{\partial C}{\partial \mu}, \chi(\boldsymbol a)_i \right ] +\left [\frac{\partial C}{\partial a_i} ,\xi_i\right ] \right ] dt \\
&\hspace{50mm}+ \sum_l [\sigma_l,\mu_i] \circ dW_t^{i,l} = 0\\
\mathbb da_i &+ [\xi_i, a_i]\, dt + \theta\,\left [\frac{\partial C}{\partial \mu_i}, \left [\frac{\partial C}{\partial \mu_i},\chi( \boldsymbol a)_i\right ]- \left [\frac{\partial C}{\partial a_i}, \xi_i\right ] \right ] dt + \sum_l [{\sigma_l},a_i] \circ dW_t^{i,l} = 0\,,
\end{split}
\label{SP-SD}\end{aligned}$$ where $$\begin{aligned}
\chi(\boldsymbol a)_i := \frac{\partial h}{\partial a_i} -\left ( \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \boldsymbol a\right )_i\, . \end{aligned}$$ The dissipative terms parametrized by $\theta$ are of double bracket form, and preserve the structure of the coadjoint orbit. Their complicated form is due to the semi-direct product structure, see [@gaybalmaz2013selective; @gaybalmaz2014geometric; @arnaudon2016noise] for more details. The important difference here is in the $\chi(\boldsymbol a)$ term which contains interactions between the neighbouring spins on the network and appears in the dissipative terms. These terms are crucial for the existence of the Gibbs distribution .
Example II: Networks of heavy tops {#section-HT}
==================================
We now consider the case $G=SO(3)$, and take the coadjoint representation of $SO(3)$ on $\mathfrak{so}^*(3) \cong \mathbb R^3$. The neighbours are coupled by the orientation of a fixed vector $\boldsymbol \Gamma_0 \in \mathbb R^3$ rotated around by the $SO(3)$ action. This breaks the symmetry in our Lagrangian so we extend our configuration manifold to $SO(3) \times \mathbb R^3$ such that the extended Lagrangian is invariant under the diagonal $SO(3)$ action on $SO(3) \times \mathbb R^3$. The Lie-Poisson equation that we obtain via symmetry reduction has the Lie-Poisson structure of the semi-direct product group $SE(3) \cong SO(3) \circledS \mathbb R^3$, hence we call this system the “heavy top network" as opposed to the rigid body network obtained via momentum coupling.
Equations of motion
-------------------
Starting with the full Lagrangian $$\begin{aligned}
L(\boldsymbol R, \dot{\boldsymbol R}, \mathbf \Gamma_0) = \frac12 \sum_i \langle \dot{R}_i, \mathbb I_i^{R_i} \,\dot{R}_i \rangle + \frac12 \sum_i \sum_{j \sim i} \frac{1}{\sqrt{d_id_j}} \langle R_i^{-1} \mathbf \Gamma^i_0, \mathbb J_{ij}R_j^{-1} \mathbf \Gamma^j_0 \rangle\, ,
\label{HT-Lagrangian-full}\end{aligned}$$ where $(R_i,\dot{R}_i) \in TSO(3), \boldsymbol \Gamma_0^i \in \mathbb R^3$ and $ \mathbb I_i^{R_i} = R_i \,\mathbb I_i R_i^{-1}$ (interpreted as matrix multiplication), we get the reduced Hamiltonian $$\begin{aligned}
\label{h-HT}
&h(\overline {\boldsymbol \Pi}, \overline {\boldsymbol \Gamma}) = \frac12 \left( \overline {\mathbf \Pi} \cdot \overline{\mathbb I}^{-1} \overline {\mathbf \Pi} - \overline {\mathbf \Gamma}\cdot \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \,\overline {\mathbf \Gamma} \right)\,,\end{aligned}$$ where $ \overline {\mathbf \Pi} = (\mathbf \Pi_1, \ldots, \mathbf \Pi_N)$, $ \overline {\mathbf \Gamma} = (\mathbf \Gamma_1, \ldots, \mathbf \Gamma_N)$, $\mathbf \Pi_i = R_i^{-1}\dot{R}_i$ and $\mathbf \Gamma_i = R_i^{-1} \mathbf \Gamma^i_0$. Taking the semi-direct product Lie-Poisson structure $$\begin{aligned}
\{f, g\}^-_{LP}(\overline {\mathbf \Pi}, \overline {\mathbf \Gamma}) = -\sum_i \left(\mathbf \Pi_i \cdot \frac{\delta f}{\delta \mathbf \Pi_i} \times \frac{\delta g}{\delta \mathbf \Pi_i} + \mathbf \Gamma_i \cdot \left(\frac{\delta f}{\delta \mathbf \Pi_i} \times \frac{\delta g}{\delta \mathbf \Gamma_i} - \frac{\delta g}{\delta \mathbf \Pi_i} \times \frac{\delta f}{\delta \mathbf \Gamma_i}\right)\right)\, ,\end{aligned}$$ we get the Lie-Poisson equation $$\begin{aligned}
\begin{split}
\dot{\mathbf \Pi}_i &= \mathbf \Pi_i\times \mathbf \Omega_i - \mathbf \Gamma_i \times \left( \mathbb D^{-\frac12} \mathbb A \mathbb D^{-\frac12} \,\overline {\mathbf \Gamma} \right)_i\\
\dot{\mathbf \Gamma}_i &= \mathbf \Gamma_i\times \mathbf \Omega_i \label{HT-eq}\, ,
\end{split}\end{aligned}$$ for $i=1, \ldots, N$, where $\mathbf \Omega_i := \mathbb I_i^{-1} \mathbf \Pi_i$ is the angular velocity at node $i$. One can check that the Casimirs for this bracket are given by $$\begin{aligned}
C_{i,1} = \mathbf \Pi_i \cdot \mathbf \Gamma_i\, ,\quad C_{i,2} = \|\mathbf \Gamma_i \|^2\, , \end{aligned}$$ so in particular, the sums $$\begin{aligned}
C_1 = \sum_i C_{i,1} = \sum_i \mathbf \Pi_i \cdot \mathbf \Gamma_i\, , \quad C_2 = \sum_i C_{i,2} = \sum_i \| \mathbf \Gamma_i\|^2\, , \end{aligned}$$ are conserved by the dynamics. Now, the coadjoint orbits of the heavy top network are given as follows.
The coadjoint orbits $\mathcal O = \mathcal O_1 \times \cdots \times \mathcal O_N$ of the heavy top network are given by, $$\begin{aligned}
\mathcal O_i = \left\{(\mathbf \Pi_i, \mathbf \Gamma_i) \in \mathbb R^{3}\times \mathbb R^{3} : C_{i,1} = c_{i,1}, \, C_{i,2} = c_{i,2} \right\} \cong TS^2\, ,\end{aligned}$$ if $c_{i,2} \neq 0$, which is a four-dimensional submanifold and $$\begin{aligned}
\mathcal O_i = \left\{(\mathbf \Pi_i, \boldsymbol 0) \in \mathbb R^{3}\times \mathbb R^{3} : \|\mathbf \Pi_i\|^2 = const \right\} \cong S^2\, ,\end{aligned}$$ if $c_{i,2} = 0$, which is a two-dimensional submanifold, unless $\|\mathbf \Pi_i\|^2 = 0$.
We refer to theorem [ 1.2]{} in [@ratiu1981euler] for the proof in the single body case. The general multi-body case considered here is an easy extension.
Noise and dissipation
---------------------
From the general equation , we use $C_{i,1} = \boldsymbol \Pi_i \cdot \boldsymbol \Gamma_i$ as the Casimir for the double bracket dissipation to obtain the stochastic equation $$\begin{aligned}
\begin{split}
\mathbb d\boldsymbol \Pi_i &+\left(\boldsymbol \Omega_i \, dt + \sum_l \boldsymbol{\sigma}_l\circ dW^l_t \right) \times\boldsymbol \Pi_i - \left(\boldsymbol \Gamma_i \times \boldsymbol \chi(\boldsymbol \Gamma)_i \right) dt \\
&+\theta\, \boldsymbol\Gamma_i\times(\boldsymbol\Omega_i\times\boldsymbol\Gamma_i)\, dt +\theta\, \left [ \boldsymbol \Pi_i\times ( \boldsymbol\chi(\boldsymbol \Gamma)_i\times \boldsymbol \Gamma_i) - \boldsymbol\Pi_i\times (\boldsymbol\Pi_i\times \boldsymbol\Omega_i)\right]\, dt= 0 \\
\mathbb d\boldsymbol \Gamma_i &+ \left(\boldsymbol \Omega_i \, dt + \sum_l\boldsymbol{\sigma}_l\circ dW^l_t \right) \times\boldsymbol \Gamma_i + \theta\, \left [ \boldsymbol\Gamma_i\times ( \boldsymbol \chi(\boldsymbol \Gamma)_i\times \boldsymbol\Gamma_i) - \boldsymbol \Gamma_i\times(\boldsymbol\Pi_i\times \boldsymbol\Omega_i)\right ]\, dt = 0 \,.
\end{split}\end{aligned}$$ where $$\begin{aligned}
\boldsymbol \chi(\boldsymbol \Gamma) = \chi+ \sum_{j\sim i}\frac{1}{\sqrt{d_i d_j}}\mathbb J_{ij} \boldsymbol \Gamma_j\, . \end{aligned}$$ Notice that the other Casimir $C_{i,2}$ is also preserved by this dissipation, thus we did not include it here. However, it is possible that including the Casimir $C_{i,2}$ would change the behaviour of the system, but we leave this investigation for future work.
Equilibrium solutions
---------------------
We now study the equilibrium solutions of the heavy top network and its corresponding stability properties. The classification of equilibrium solutions into ferromagnetic and anti-ferromagnetic states and its nonlinear stability property are summarised in the following proposition.
\[HT-equib\] The relative equilibrium solutions $\overline {\mathbf \Gamma}_e$ of a heavy top network correspond to the eigenvectors of $\mathbb L(\lambda_1^e)$ with $\overline {\mathbf \Pi}_e = -\lambda_1^e \,\overline {\mathbb I} \,\overline {\mathbf \Gamma}_e$ and the equilibrium solution corresponding to the lowest eigenvalue $-\lambda_2^e$ of $\mathbb L(\lambda_1^e)$ is nonlinearly stable. Furthermore, if $\mathbb J_{ij} = \mathbb J$ and $\mathbb I_i = \mathbb I$, then there exists $3N$ linearly independent equilibrium configurations such that
1. three are ferromagnetic, i.e. $\mathbf \Gamma_i^e = \sqrt{d_i} \mathbf \Gamma^e$ for $i=1, \ldots, N$, where $\mathbf \Gamma^e$ is an eigenvector of $\mathbb I_{\text{ext}} := -(\lambda_1^e)^2 \,\mathbb I^{-1} - \mathbb J$ and
2. the remaining $3N-3$ are anti-ferromagnetic, i.e. $\sum_{i=1}^N \sqrt{d_i} \mathbf \Gamma_i^e = 0$.
In order to investigate the stability of the other equilibrium configurations, we move to linear stability analysis.
### Linear stability
Linearising equations by taking $\overline {\mathbf \Pi}(t) = \overline {\mathbf \Pi}_e + \epsilon \,\delta \overline {\mathbf \Pi}(t)$ and $\overline{\mathbf \Gamma}(t) = \overline {\mathbf \Gamma}_e + \epsilon \,\delta \overline{\mathbf \Gamma}(t)$ and dropping terms of $O(\epsilon^2)$, we get in term of $\boldsymbol \Gamma_e$ only, $$\begin{aligned}
\frac{d}{dt}
\begin{pmatrix}
\delta \overline {\mathbf \Pi} \\
\delta \overline {\mathbf \Gamma}
\end{pmatrix}
=
\begin{pmatrix}
\lambda_1 \widehat {\mathbf \Gamma}^e -\lambda_1 \mathbb I \widehat{\mathbf \Gamma}^e \,\overline {\mathbb I}^{-1} & (-\lambda_1^2 \,\overline{\mathbb I} +\lambda_2 \mathbb 1) \widehat{\mathbf \Gamma}^e -\widehat{\mathbf \Gamma}^e \,\mathbb D^{-\frac12}\mathbb A \mathbb D^{-\frac12} \\
\widehat{\mathbf \Gamma}^e \,\overline {\mathbb I}^{-1} & \lambda_1 \, \widehat{\mathbf \Gamma}^e
\end{pmatrix}
\begin{pmatrix}
\delta \overline{\mathbf \Pi} \\
\delta \overline{\mathbf \Gamma}
\end{pmatrix}\, ,
\label{lin-stab-HT}\end{aligned}$$ where $\widehat{\mathbf \Pi}^e := \text{diag}(\widehat{\mathbf \Pi}^e_1, \ldots, \widehat{\mathbf \Pi}^e_N)$, $\widehat{\mathbf \Gamma}^e := \text{diag}(\widehat{\mathbf \Gamma}^e_1, \ldots, \widehat{\mathbf \Gamma}^e_N)$ and $\,\widehat{} : (\mathbb R^{3},\times) \rightarrow \mathfrak{so}(3)$ is the standard Lie algebra isomorphism that sends a vector in $\mathbb R^3$ to a skew symmetric matrix in $\mathfrak{so}(3)$. Hence, we can assess the linear stability of an equilibrium solution $(\overline {\mathbf \Pi}_e, \overline{\mathbf \Gamma}_e)$ by looking at the eigenvalues of the matrix on the right hand side of .
Determining the linear stability of all the equilibrium states of this system is impossible to do analytically, so we will only discuss the numerical results here. First, we solve to find all the equilibria of the lattice given a $\lambda_1$, and in figure \[fig:l1l2-HT\], we plotted the values of all possible $\lambda_2$ (i.e. minus the eigenvalues of $\mathbb L(\lambda_1)$) as a function of $\lambda_1$. In the case $\mathbb I= \mathrm{diag}(1,2,3)$ and $\mathbb J=\mathrm{diag}(1,1,1)$ (figure \[fig:l1l2-HT1\]), we see that as $\lambda_1\to 0$, all the solutions become degenerate with multiplicity $3$ (i.e. the eigenvalues $\lambda_2$ collapse) but this does not happen in the other case $\mathbb I= \mathrm{diag}(1,1,1)$ and $\mathbb J=\mathrm{diag}(1,2,3)$ (figure \[fig:l1l2-HT2\]). This can be explained by the fact that at $\lambda_1=0$, which corresponds to the zero momentum case $\bar{\mathbf \Pi} = 0$, the system is isotropic in case (a) and anisotropic in case (b).
In figure \[fig:stability-HT\], we display the largest, real part of the eigenvalues $\lambda_s$ of the linearised system plotted against $\lambda_2$ (minus the eigenvalue of $\mathbb L(\lambda_1)$) corresponding to equilibrium solutions $\mathbf \Gamma_e$ of for $\lambda_1=0.5$. As expected from proposition \[HT-equib\], the lowest eigenvalue state (or, the states with highest $\lambda_2$) in both cases \[stability\_HT1\] and \[stability\_HT2\] are linearly stable. This corresponds to a ferromagnetic equilibria along the $\Gamma_3$-axis in both cases. For $\mathbb I=\mathrm{diag}(1,2,3)$ and $\mathbb J=\mathrm{diag}(1,1,1)$ (figure \[stability\_HT1\]), the two states near this configuration with multiplicity 1 ($\lambda_2 \approx 4.2$ and $\lambda_2 \approx 4.5$) correspond to the other two ferromagnetic equilibria given in proposition \[HT-equib\], which we call states A and B and these are seen to be unstable. There also exists a subspace of anti-ferromagnetic equilibria between these two states ($\lambda_2 \approx 4.35$) with multiplicity $4$ that have a very small but positive $\lambda_s$. We call this state $C$. As we took $\lambda_1 \rightarrow 0$, we saw that state $C$ becomes linearly stable for some small $\lambda_1$, and states $A$ and $B$, while they were always found to be unstable, their corresponding values of $\lambda_s$ tended to $0$ (i.e. approaching a linearly stable state). In the other case $\mathbb I=\mathrm{diag}(1,1,1)$ and $\mathbb J=\mathrm{diag}(1,2,3)$ (figure \[stability\_HT2\]), the linearly stable equilibria excluding the nonlinearly stable state were found to be anti-ferromagnetic states with the ‘argyle’ pattern similar to that in figure \[fig:RB-sols\].
Numerical experiments {#HT-numerics}
---------------------
Before looking at phase transitions, we will present here an interesting phenomenon that arises in the heavy top lattice when $\mathbb I= \mathrm{diag}(1,2,3)$ and $\mathbb J = \mathrm{diag}(1,1,1)$. In figure \[fig:HT-stab-diss\], we show several simulations of the deterministic $20\times 20$ heavy top lattice starting from different initial conditions, with or without the double bracket dissipation. The initial conditions are taken to be nearly ferromagnetic, with spins $\mathbf \Gamma_i$ aligned to a fixed direction, except for a small perturbation at two nodes. This way, we can numerically assess the stability of the ferromagnetic equilibria. From proposition \[HT-equib\], we know that position $(0,0,1)$ (i.e. the $\Gamma_3$-axis) is nonlinearly stable, and we also observed this in our simulations. Hence, we will not display the corresponding plots here as it is not very interesting. Instead, we will display the simulations starting close to the two other ferromagnetic equilibria $(0,1,0)$ in figure \[g\_010\] and $(1,0,0)$ in figure \[g\_100\], which were seen to be unstable with or without dissipation.
There are two interesting behaviors of the system observed from the plots. The first, which is displayed in panel \[g\_100\], is that the solution starting close to the shortest axis $(1,0,0)$ with double bracket dissipation is stuck in this position for a while, then gets stuck in the middle axis position $(0,1,0)$ briefly, before relaxing to the lowest energy state $(0,0,1)$. This can also be seen in panel \[g\_010\] where the solution starting close to the middle axis $(0,1,0)$ is stuck there for a while before relaxing to the lowest energy state $(0,0,1)$. Although the double bracket dissipation can never stabilize an unstable equilibrium solution, as proven in Bloch et al. [@bloch1994dissipation], this indicates some kind of transient stability, or metastability of the shortest and middle axis equilibrium, which will be observed again in the phase transition plots in section \[PT-section\]. We note that this phenomenon is not apparent in the absence of dissipation and is only observed clearly in the presence of dissipation. Apart from these special cases, for instance when we start from $(1,1,1)$, which is not an equilibrium, the dissipation will always drive the system towards the lowest energy position without getting stuck (see figure \[g\_111\]).
The second observation that we made is the [*partial synchronisation*]{} phenomenon seen in the non-dissipative simulations (see the bold lines in \[g\_010\], \[g\_100\] and \[g\_111\]). The corresponding energy (total, kinetic and potential) plots are given by the bold lines in \[e\_010\], \[e\_100\] and \[e\_111\] to demonstrate the validity of the simulations. In all of the cases considered, we observe that after some time, the spins relax to a state where on average, it oscillates closely around the lowest energy configuration (i.e. ferromagnetic state along the $\Gamma_3$-axis), despite the absence of dissipation. We call this phenomenon “partial synchronisation”. This result is rather surprising as, by Liouville’s theorem, the volume of the phase space is conserved by the dynamics, so by the Poincaré recurrence theorem, any states starting near the equilibrium should return sufficiently close to it after a finite time. However, in our simulations, the trajectories after some time seem to get stuck in another part of the phase space without returning to a region close to the initial conditions. This is very counter-intuitive from the original dynamics of the rigid body, where an initial condition near the unstable (middle) axis will eventually come back near it, after traversing a long trajectory on the momentum sphere. One possible avenue of investigation of this phenomenon is to study the local dynamics of the lattice where almost periodic motions were observed (see the videos in <http://wwwf.imperial.ac.uk/~aa10213/>). This phenomenon may also be tied to the complex interaction between the $\mathbf \Gamma$ and $\mathbf \Pi$ variables and the non-compactness of the phase space. However, we will not study this further in the present work and instead leave it for future investigations.
Temperature phase transitions {#PT-section}
=============================
In this last section, we will study phase transitions in the two examples that we constructed above, namely, the rigid body network and the heavy top network. There exist various types of phase transitions, but here we will focus on second-order phase transitions that arise from varying the temperature of the system. This is characterized by a transition from an orderly state of the spins, measured by how much they are aligned with each another, to a completely disordered state. In order to detect a phase transition, one can either apply a mean field approximation of the model and try to solve it analytically or perform direct numerical simulations of the underlying dynamics of the lattice. Many other methods are available but are out of the scope of this first investigation.
Mean field approximation
------------------------
The mean field approximation relies on the assumptions that (1) the microscopic system is identical at each node and (2) the spins (momentum or position) of the rigid bodies in statistical equilibrium are close to its mean. This approximation is increasingly accurate if each site has more neighbours, which is the case for high dimensional lattices. In two dimensions, the approximation fails to properly assess the critical temperature and the corresponding critical exponents, but still, give a good indication of the presence of a phase transition.
### Mean field approximation of the rigid body network
We assume that the system is identitcal at each node, i.e. $\mathbb I_i = \mathbb I$ for all $i=1, \ldots, N$, and that the interactions between the neighbours are identical, that is, $\mathbb J_{ij} = \mathbb J$ for all $i,j = 1, \ldots, N$ for the mean field approximation to be valid. Now, define the [*averaged momentum*]{} $$\begin{aligned}
\langle \boldsymbol \Pi \rangle := \frac1N \sum_{i=1}^N \int_{\boldsymbol{\mathcal O}} \mathbf \Pi_i \, \mathbb P_{\infty} (\overline {\mathbf \Pi}) d \overline {\mathbf \Pi}, \quad \mathbb P_{\infty} (\overline {\mathbf \Pi}) = Z_{RB}^{-1} e^{-\beta h(\overline {\mathbf \Pi})},\end{aligned}$$ where $\beta = \frac{2 \theta}{\sigma^2} = T^{-1}$ is the inverse temperature, $\mathbb P_{\infty}(\cdot)\,d\overline {\mathbf \Pi}$ is the Gibbs measure and $\boldsymbol{\mathcal O} = \mathcal O_1 \times \cdots \times \mathcal O_N$ is the total coadjoint orbit. Since the system is assumed to be identical at each node, we have $\mathcal O_i = \mathcal O$ for all $i=1, \ldots, N$.
Since we assume that the spins $\mathbf \Pi_i = \Braket{\boldsymbol \pi} + \delta \mathbf \Pi_i$ are close to the mean $\langle \boldsymbol \pi \rangle$, we linearise the interaction Hamiltonian around $\Braket{\boldsymbol \Pi}$ to get $$\begin{aligned}
h^{\text{int}}(\overline {\mathbf \Pi})&= -\sum_{i} \sum_{j \sim i} \frac{1}{2d} ( \boldsymbol \Pi_i - \Braket{\boldsymbol \Pi} + \Braket{\boldsymbol \Pi}) \cdot \mathbb J (\boldsymbol \Pi_j-\Braket{\boldsymbol \Pi}+\Braket{\boldsymbol \Pi}) \nonumber\\
&= -\frac{1}{2d} \sum_{i} \sum_{j \sim i} \left(\delta \mathbf \Pi_i \cdot \mathbb J \, \delta \mathbf \Pi - \frac{1}{2d} \left ( \Braket{\boldsymbol \Pi } \cdot \mathbb J \boldsymbol\Pi_j +\boldsymbol\Pi_i \cdot \mathbb J \Braket{\boldsymbol\Pi}\right) \right) - \frac12 \sum_i \Braket{\boldsymbol\Pi}\cdot \mathbb J \Braket{\boldsymbol\Pi}\, \nonumber\\
&\approx -\sum_{i} \Braket{\boldsymbol \Pi } \cdot \mathbb J \boldsymbol\Pi_i, \nonumber\end{aligned}$$ where the first term in the second line is neglected since it is quadratic in $\delta \mathbf \Pi_i$ and is therefore very small, and the last term is also neglected since it is constant and does not contribute to the dynamics. We therefore obtain the complete mean field Hamiltonian $$\begin{aligned}
h_{\rm mf} =\sum_{i} \left( \frac12 \boldsymbol \Pi_i \cdot \mathbb I^{-1} \boldsymbol\Pi_i - \Braket{\boldsymbol \Pi } \cdot \mathbb J \boldsymbol\Pi_i \right) \, . \end{aligned}$$ Notice that the kinetic energy is still exact. From this Hamiltonian, the Gibbs distribution and the partition function take a simpler form, $$\begin{aligned}
\mathbb P_{\infty}^{\text{mf}} (\overline {\mathbf \Pi}) = \frac{1}{Z_{RB}^{\text{mf}}} e^{-\beta h_{\text{mf}}(\overline {\mathbf \Pi})}, \quad
Z^{\rm mf}_{RB}= \left (\int_{\mathcal O} e^{-\beta \left ( \frac12 \boldsymbol \Pi\cdot \mathbb I^{-1} \boldsymbol \Pi - \boldsymbol \Pi\cdot \mathbb J\Braket{\boldsymbol \Pi }\right )}d\boldsymbol \Pi\right)^N\,,\end{aligned}$$ and one can check that the average momentum $\Braket{\boldsymbol \Pi}$ simplifies to $$\begin{aligned}
\Braket{\boldsymbol \Pi} = \frac{\int_{\mathcal O} \mathbf \Pi \, e^{-\beta \left ( \frac12 \boldsymbol \Pi\cdot \mathbb I^{-1} \boldsymbol \Pi - \boldsymbol \Pi\cdot \mathbb J\Braket{\boldsymbol \Pi }\right )}d\boldsymbol \Pi}{\int_{\mathcal O} e^{-\beta \left ( \frac12 \boldsymbol \Pi\cdot \mathbb I^{-1} \boldsymbol \Pi - \boldsymbol \Pi\cdot \mathbb J\Braket{\boldsymbol \Pi }\right )}d\boldsymbol \Pi},
\label{mean-momentum}\end{aligned}$$ which is now an implicit equation for the order parameter $\Braket{\boldsymbol \Pi}$. This equation is difficult to solve analytically, as it involves integrals over the momentum sphere but can be numerically estimated using Monte-Carlo integration. We will display the numerical approximation of $\Braket{\boldsymbol \Pi}$ in the next section, compared to the full simulations.
### Mean field approximation of the heavy top network
In a similar fashion, we can derive the mean field approximation of the heavy top network. In this case, we take our order parameter to be the [*averaged position*]{}, defined by $$\begin{aligned}
\langle \boldsymbol \Gamma \rangle := \frac1N \sum_{i=1}^N \int_{\boldsymbol{\mathcal O}_1} \int_{\boldsymbol{\mathcal O}_2} \mathbf \Gamma_i \, \mathbb P_{\infty} (\overline {\mathbf \Pi}, \overline {\mathbf \Gamma}) d\overline {\mathbf \Pi} d \overline {\mathbf \Gamma}, \quad \mathbb P_{\infty} (\overline {\mathbf \Pi}, \overline {\mathbf \Gamma}) = Z_{HT}^{-1} e^{-\beta h(\overline {\mathbf \Pi}, \overline {\mathbf \Gamma})},\end{aligned}$$ where $\boldsymbol{\mathcal O}_1 = \mathcal O_{1,1} \times \cdots \times \mathcal O_{N,1}$ and $\boldsymbol{\mathcal O}_2 = \mathcal O_{1,2} \times \cdots \times \mathcal O_{N,2}$ are coadjoint orbits corresponding to level sets of the Casimirs $C_{i,1}$ and $C_{i,2}$ respectively. Again, since we assume the system to be identical at each node, we take $\mathcal O_{i,1} = \mathcal O_1$ and $\mathcal O_{i,2} = \mathcal O_2$ for all $i=1, \ldots, N$.
The partition function in the mean field approximation is found to be $$\begin{aligned}
Z_{HT}^{\rm mf} = \left( \int_{\mathcal O_1 \times \mathcal O_2} e^{-\beta \left(\frac12 \mathbf \Pi \cdot \mathbf \Omega - \mathbf \Gamma \cdot (\mathbb J \, \Braket{ \boldsymbol \gamma} )\right)} d \mathbf \Pi \, d \mathbf \Gamma \right)^N\, , \end{aligned}$$ and the equation for $\Braket{\boldsymbol \Gamma}$ simplifies to $$\begin{aligned}
\label{mean-position}
\langle \boldsymbol \Gamma \rangle &=
\frac{\int_{\mathcal O_2} \mathbf \Gamma \,e^{\beta \boldsymbol \Gamma \cdot (\mathbb J \, \Braket{\boldsymbol \gamma})} \left( \int_{\mathcal O_1} e^{-\beta \frac12 \mathbf \Pi \cdot \mathbf \Omega} \, d \mathbf \Pi \right) d \mathbf \Gamma
}
{\int_{\mathcal O_2} e^{\beta \mathbf \Gamma \cdot (\mathbb J \, \Braket{\boldsymbol \gamma})} \left( \int_{\mathcal O_1} e^{-\beta \frac12 \mathbf \Pi \cdot \mathbf \Omega} \, d \mathbf \Pi \right) d \mathbf \Gamma
}.\end{aligned}$$
\[mf-equivalence\] Notice that if one chooses $\mathbb I = \mathrm{diag}(1,1,1)$ so that $\boldsymbol \Omega = \boldsymbol \Pi$, the integral over the $\mathbf \Pi$-variable cancels out so equation becomes equivalent to .
Numerical simulations
---------------------
The simplest (but computationally expensive) way to detect phase transitions in lattices is by a direct simulation of the full stochastic equation, sampling from the Gibbs measure. This is equivalent to a classical Monte-Carlo simulation of the Ising model for example but in the more general setting of continuous spins on coadjoint orbits. The phase transitions we observe are all second-order and is detected as we increase the temperature $T = \frac{\sigma^2}{2 \theta}$. The order parameter for the rigid body and the heavy top network are given by the averaged momentum and position respectively.
We summarise here the phase transition behaviours of the four systems we simulated, displayed in figures \[fig:RB1\] and \[fig:HT-PT\].
1. [*Rigid body with $\mathbb I=\mathrm{diag}(1,1,1)$ and $\mathbb J=\mathrm{diag}(1,2,3)$ (Figure \[PT-RB1\])* ]{} This case corresponds to the classical Heisenberg model with anisotropic interactions (also known as $XYZ$-model). Notice that some simulations have anomalous low magnetisation at low temperatures. This can be explained by the fact that in some simulations, the system got stuck in a state consisting of large regions of opposite spins. A longer simulation would be necessary to see these two domains merge into one to reach the minimum energy state.
2. [*Rigid body with $\mathbb I=\mathrm{diag}(1,2,3)$ and $\mathbb J=\mathrm{diag}(1,1,1)$.(Figure \[PT-RB2\])* ]{} This case corresponds to a massive isotropic Heisenberg model with non-uniform mass. Our simulations did not get stuck in the state consisting of opposite spins, as seen in the previous case, and the mean field approximation is closer to the direct simulations. The reason that the mean field approximation is more precise in this case can be explained by the fact the interaction term, which is approximated, is isotropic but the kinetic term, which is exact, is anisotropic.
3. [*Heavy top with $\mathbb I=\mathrm{diag}(1,1,1)$ and $\mathbb J=\mathrm{diag}(1,2,3)$. (Figure \[PT-HT1\])* ]{} This case also corresponds to the classical Heisenberg model, at least from the mean field point of view (by remark \[mf-equivalence\]), even if the full dynamics is different. Again, we observe solutions getting stuck in the state consisting of opposite spins and hence the existence of anomalies.
4. [*Heavy top with $\mathbb I=\mathrm{diag}(1,2,3)$ and $\mathbb J=\mathrm{diag}(1,1,1)$. (Figures \[PT-HT2\],\[PT-HT3\] and \[PT-HT4\])*]{} This last case is the most interesting as it shows a more complex phase transition, which we will describe in details below.
### Triple-humped phase transition in the heavy top network
We now discuss the last case in more details. From the direct numerical simulations, we observed a phase transition from a strongly magnetised state (large value of $\Braket{\boldsymbol \Gamma}$) along the $\Gamma_3$-axis to a non-magnetized state (that is, $\Braket{\boldsymbol \Gamma} = 0$) as we increased the temperature. But between these two states, we also observed [*two intermediate phase transitions*]{}, where magnetisation along the other two axes also occur before becoming completely disordered. We call this a ‘trimple-humped’ phase transition. These intermediate phase transitions indicate that these unstable ferromagnetic equilibria along the $\Gamma_1$ and $\Gamma_2$ axes can still support magnetisation, which, in statistical physics is generically called a [*meta-stable state*]{}. We note that this phenomenon is not captured in our mean-field simulations.
From our linear stability analysis in section \[section-HT\], we observed that for small values of $\lambda_1$, equivalent to a small ratio $c_1/c_2$, the ferromagnetic equilibria along the $\Gamma_1$ and $\Gamma_2$ axes are unstable, but are close to being linearly stable. This is compatible with the observation that these intermediate phase transitions exist only for small values of $c_1/c_2$. For example, in panel \[PT-HT4\] we took $c_1/c_2 = 1.15$ and saw that the third phase transition is almost negligible. Increasing the value of this ratio further will also remove the other intermediate phase transition. On the other hand, decreasing $c_1/c_2$, the intermediate phase transitions persist and furthermore, will shift their respective critical temperature towards zero. In figure \[PT-HT3\], where we took $c_1/c_2 = 0.8$, we see that both critical temperatures $T_1$ and $T_2$ are smaller compared to those in figure \[PT-HT2\], where we took $c_1/c_2 = 1$.
We will not investigate this phase transition further here, and leave its mathematical understanding as a challenging open problem.
Conclusion and outlook {#conclusion}
======================
In this paper, we established a link between geometric mechanics and statistical mechanics by constructing a network of interacting Lie-Poisson system on $\mathfrak g^*$ with noise and dissipation that preserves the coadjoint orbits, which gave us a canonical ensemble for the system in statistical equilibrium. For the construction of the system, we considered two types of coupling, one where the neighbours are coupled in the reduced space and the other where the neighbours are coupled directly on the configuration group by considering a representation of the group on a given vector space. The first approach yielded a direct generalisation of the classical Heisenberg model to include general symmetry groups with an additional kinetic energy term and the second approach gave a system that is possibly new. In the special case where $\mathfrak g$ is compact and semi-simple with Hamiltonian of the form kinetic + potential energy, we were able to find the equilibrium solutions of the purely deterministic system as the eigenvectors of the underlying extended graph Laplacian and found that (1) for the momentum-coupled case, the equilibrium solution corresponding to the lowest and highest eigenvalue of the graph Laplacian are nonlinear stable and (2) for the position-coupled case, the equilibrium solution corresponding to the lowest eigenvalue of the graph Laplacian is nonlinearly stable. Furthermore, we showed that in both cases, these equilibrium solutions can be classified into ferromagnetic and anti-ferromagnetic states. In our numerical simulation of the rigid body lattice and the heavy top lattice, which are the simplest examples of momentum-coupled and position-coupled systems respectively, we observed a second order phase transition, similar to that in the Ising model or in the Heisenberg model. However, in the heavy top network, we also observed a ‘triple-humped’ phase transition, in which the system underwent two intermediate phase transitions before settling down to the lowest energy configuration as we decreased the temperature, which is unusual for simple lattice models.
In future work, we would like to investigate further this new type of phase transition behaviour that we observed for the heavy top network, which we believe is related to the metastability of the intermediate ferromagnetic states. However, even without noise, we numerically observed unusual behaviour of the heavy top network, such as when the spins start close to an arbitrary ferromagnetic state, there is an exchange between the kinetic and potential energy that causes the spins to relax to a state that oscillates closely around the stable ferromagnetic state with lowest potential energy, despite the absence of dissipation. So studying the deterministic heavy top network further could also be interesting and may help us understand this phase transition behaviour better. This partial synchronisation result deserves a more detailed study, in particular for more general networks, where synchronisation of oscillators are an active subject of research (see for example [@barahona2002synchronization] and the many subsequent works). Other interesting phenomena can be observed for Heisenberg models on certain types of networks, such as supra-oscillations (see for example[@expert2017graph] and references therein).
Regarding phase transitions, one could also compute the critical exponents and the corresponding universality class of the phase transition seen here, or even doing a more thorough analysis using Landau theory to better understand the dynamics near the critical temperature. We can also investigate different types of phase transitions, for instance by varying the external magnetic field instead of temperature, or even extending our domain from a simple lattice to a general network. One may also guess that certain topological phase transitions could even be observed in these systems, such as the Kosterlitz-Thouless transition [@kosterlitz1973ordering].
As we can see, there are many interesting questions that are open for further investigation which we hope to address in future works.
Acknowledgements {#acknowledgements .unnumbered}
----------------
[AA is grateful to John Gibbon for suggesting him to look further into this research direction. AA acknowledges EPSRC funding through award EP/N014529/1 via the EPSRC Centre for Mathematics of Precision Healthcare. ST acknowledges funding through Schrödinger scholarship scheme. ]{}
|
---
abstract: 'We propose an experimental arrangement to image, with attosecond resolution, transient surface plasmonic excitations. The required modifications to state-of-the-art setups used for attosecond streaking experiments from solid surfaces only involve available technology. [[Buildup and life times of surface plasmon polaritons can be extracted and local modulations of the exciting optical pulse can be diagnosed [*in situ*]{}.]{}]{}'
author:
- 'Mattia Lupetti$^a$, Julia Hengster$^b$, Thorsten Uphues$^b$, and Armin Scrinzi$^a$'
bibliography:
- 'library.bib'
title: Attosecond Photoscopy of Plasmonic Excitations
---
[[Surface plasmons]{}]{} are collective excitations of electrons that propagate along a metal-dielectric interface. Recently, plasmonics has gathered interest for the development of ultra-fast all-optical circuitry [@Ozbay2006], since it can combine the high operational speed of photonics (PHz scale) with the miniaturization provided by electronics (nm scale). For this purpose, it is important to understand the buildup dynamics and lifetime of the collective electronic excitation. Although the plasmon lifetime can be inferred from the plasmonic resonance width (of the transmission spectrum, see for instance [@ropers2005prl]), plasmon buildup is a process that cannot be addressed in terms of frequency analysis. In the present work, we propose an experimental setup to image the transient dynamics of a plasmonic mode, which can be realized as a modification of the so-called “attosecond streak camera” [@Kienberger2004], which has already been successfully applied to solid surfaces. The attosecond streak camera is a two-color pump-probe scheme, where a weak XUV attosecond pulse ionizes electrons from the solid, and a collinear, few-cycle ($\sim 5\,fs$ FWHM) NIR pulse serves as the probe, which accelerates the XUV photo-electrons after their escape from the solid. With this technique it was possible to resolve solid-state physics phenomena with resolution of a few attoseconds ($1\, \text{as} = 10^{-18}$ s) [@Cavalieri2007].
We benchmark our setup concept against the buildup of Surface Plasmon Polaritons (SPPs) excited by a NIR pulse on a grating surface. A time-delayed XUV pulse probes the SPPs during their evolution by detecting the effect of their field on XUV photoemission. In principle, pump and probe beams can be spatially separated, allowing to probe different surface regions. Thus, differently from atomic and surface streaking employed so far, the setup provides spatio-temporal information. To distinguish it from standard attosecond streaking experiments, we name our setup “attosecond photoscopy”.
A well established method for producing isolated attosecond pulses is the generation of high harmonic radiation (HHG) in noble gases [@Hentschel2001; @Cavalieri2007; @Corkum1993; @Agostini2004]. An intense few cycle NIR laser pulse is focused into a noble gas target and generates high harmonics of the fundamental radiation. The XUV radiation co-propagates with the driving laser pulse. Both pulses are focused onto a sample with a delayable two part mirror composed of an XUV multilayer mirror in the inner part and a broadband NIR mirror in the outer part. The multilayer mirror is designed as a high pass filter for the harmonics, which results in an isolated attosecond pulse. The pulse can be timed relative to the NIR with a precision of $\lesssim 10\,as$.
![Experimental setup of an attosecond photoscopy experiment. The XUV attosecond pulse liberates electrons in presence of the plasmonic field, which is excited by a short NIR pulse. Control of NIR-XUV time delay $\tau$ allows observation of the plasmon transient dynamics.[]{data-label="fig:setup"}](setup3D_mattia.eps){width="\linewidth"}
Figure \[fig:setup\] illustrates the setup discussed here. The NIR and XUV beams propagate in $y$-direction, at normal incidence onto the plane of the grating. Polarizations are in $x$-direction, perpendicular to the grooves. Using this arrangement, two counter-propagating plasmons are excited in the focus of the NIR pulse on the grating structure. [[A band gap at the zero crossing separates two plasmon branches [@barnes1996prb]. An optical pulse at normal incidence usually couples to only one of the branches, called the bright mode, but at tight focussing with about 5$^\circ$ angular dispersion also the second, “dark” mode is excited.]{}]{}
XUV photo-electrons are measured at perpendicular direction to the surface. As in [@Cavalieri2007], the final electron momenta are recorded as a function of the delay between the NIR and XUV beams. The electron *spectrogram* retrieved is a convolution of photoemission with acceleration in the plasmonic field at the location and time of the initial electron release.
Depending on the time delay between the NIR pulse and the probing attosecond pulse, the XUV generated photoelectrons experience a different plasmonic field amplitude and phase, leading to a modulation of the kinetic energy distribution by the emerging plasmonic field.
[[ The energy gap between dark and bright modes manifests itself in the spectrogram as a “transition” from the bright $\omega_b$ to the dark $\omega_d$ mode frequencies, which is measurable in our setup because of the attosecond resolution.]{}]{}
Below we analyze the photoscopic spectrogram using a basic analytical model as well as numerical solutions of the SPP propagation together with a Monte Carlo simulation of the electron streaking process. We will demonstrate that from the spectrograms one can recover the plasmonic field at the surface. The detailed analysis and interpretation will be discussed in the following.
Standard streaking experiments are based on electron sources that can be considered point-like with respect to the laser wavelength, such as atoms or molecules. For this reason the dipole approximation can be used: $\mathbf{A}(\mathbf{r},t)\simeq
\mathbf{A}(t)$. After emission, the electron canonical momentum is conserved: $ \mathbf{P}(t) = \mathbf{P}_i$, which translates into $\mathbf{p}(t) + \frac{e}{c}\mathbf{A}(t) =
\mathbf{p}_i + \frac{e}{c}\mathbf{A}(t_i)$, where $e$ denotes the electron charge and $|\mathbf{p}_i| = \sqrt{2m(E_{xuv}-W_f)}$ is the initial momentum of the electron released at time $t_i$ from a material with work function $W_f$. Assuming that $A(t\rightarrow\infty) = 0$, the final momentum recorded by the spectrometer is $$\label{eq:fin-mom}
\mathbf{p}_f = \mathbf{p}_i + \mathbf{a}(t_i),$$ where we defined $\mathbf{a}:= \frac{e}{c}\mathbf{A}$.
The spectral width of the XUV attosecond pulse is reflected in a momentum-broadening of the initial electron distribution $n_e =
n_e(\mathbf{p}_i, t_i)$. For simplicity we assume Gaussian distributions centered around momentum $\mathbf{p}_0$ and time $t_0$, respectively, where $t_0$ denotes the time of peak XUV intensity on target. With Eq. (\[eq:fin-mom\]) for the initial electron momentum, the time-integrated final momentum is $$\label{eq:mom-dis}
\sigma(\mathbf{p}_f) = \int_{-\infty}^{\infty}\!\!\! dt_i \, n_e(\mathbf{p}_f - \mathbf{a}(t_i), t_i).$$ The spectrogram for a series of delays $\tau$ becomes $$\label{eq:str-spe}
\sigma(\mathbf{p}_f, \tau) = \int_{-\infty}^{\infty} dt_i \, n_e(\mathbf{p}_f - \mathbf{a}(t_i), t_i-\tau).$$ From this, the NIR pulse can be reconstructed by analyzing the average momentum of the streaking spectrogram When applying the method to plasmonic excitations we have to consider that the SPP, acting as the streaking field, is spatially inhomogeneous and propagates on a surface. Previous work on streaking on nanoparticles [@Suessmann2011prb] clearly shows that spatial inhomogeneity of the streaking field leads to a smearing of the streaking trace obtained in a traditional setup. Thus, we need to include the position dependence into our initial electron distribution: $n_e(\mathbf{p}_i, t_i)\rightarrow n_e(\mathbf{r}_i, \mathbf{p}_i,t_i)$. The final momentum of the electrons accelerated in the plasmon field is then $$\label{eq:fin-mom-ext}
\mathbf{p}_f = \mathbf{p}_i - e \int_{-\infty}^{\infty}\mathbf{E}(\mathbf{r}(t'), t')\, dt'.$$ For a typical XUV photon energy of $80$ eV, the average initial speed of a photoelectron is $v_i = 5$ nm/fs. If the NIR pulse is $4$ fs short, it will give rise to a plasmonic field of a duration of few tens of femtoseconds. During this time, the electrons move by $\lesssim 100$ nm. The additional drift imparted by the plasmonic field is small compared to the initial velocity. As the plasmon evanescent field extends to about NIR wavelength ($800$ nm) beyond the surface, we can write $\mathbf{r}(t')\simeq
\mathbf{r}_i$ in Eq. (\[eq:fin-mom-ext\]). With this approximation, one obtains a position corrected analog of Eq.(\[eq:fin-mom\]): $$\label{eq:fin-mom-ext2}
\mathbf{p}_f = \mathbf{p}_i - \mathbf{a}(\mathbf{r}_i, t_i)$$ Since the photoelectron detector does not resolve the emission positions $\mathbf{r}_i$, the photoscopic spectrogram is the integral over time [*and the area covered by the XUV pulse*]{} $$\label{eq:pho-spe}
\sigma(\mathbf{p}_f, \tau) = \int_{\mathbb{R}^3}d^3r_i
\int_{-\infty}^{\infty} dt_i \, n_e(\mathbf{r}_i, \mathbf{p}_f - \mathbf{a}(\mathbf{r}_i, t_i), t_i-\tau).$$ [[The space-averaged momentum is independent of the time-delay, as the integral of a propagating pulse is negligible (exactly zero in free space). Thus]{}]{} for extracting time information from the photoscopic spectrogram, we use the delay-dependent momentum variance $$\label{eq:ps-mom2}
S(\tau) = \frac{\int d\mathbf{p}_f \, |\mathbf{p}_f|^2
\,\sigma(\mathbf{p}_f, \tau)}{\int d\mathbf{p}_f \,
\sigma(\mathbf{p}_f, \tau)} - |\langle \mathbf{p}_f
\rangle|^2.$$ As the XUV pulse duration is short compared to the NIR period, we treat photoemission as instantaneous. The distribution of the photoelectron yield along the surface is proportional to the XUV intensity profile. Furthermore, we neglect any transport effect in the solid and consider only the photoelectrons coming from the first few layers of material, as reported in [@Neppl2012b]. With these conditions one finds $$\nonumber n_e(\mathbf{r}_i, \mathbf{p}_i, t_i-\tau)\simeq g_{\text{x}}(x_i)n_e(\mathbf{p}_i)\delta(y_i - y_s)\delta(t_i-\tau-t_0),$$ where $y_s$ is the grating vertical position (we neglect any groove depth effect) and $g_{\text{x}}$ is a Gaussian function of width $w_\text{x}$, i.e. the XUV attosecond pulse focal spot.
As for the angular dependence of the photoemission we first restrict our discussion to the two extreme cases of 1) unidirectional emission with all initial momenta orthogonal to the grating plane, and 2) isotropic emission. For either distribution, the reconstructed times closely reproduce the actual dynamics. In reality, the XUV photoelectron distribution will be between these extreme cases and should be determined in a measurement without NIR field.
Unidirectional initial distributions can be written as $n_e(\mathbf{p}_i) = n_e(p_i\, \hat{\mathbf{n}}_s)$, where $p_i =
|\mathbf{p}_i|$ and $\hat{\mathbf{n}}_s$ is the direction orthogonal to the grating plane. Eq. (\[eq:pho-spe\]) now becomes $$\nonumber \sigma(p_f, \tau) = \int_{-\infty}^{\infty} dx_i\, g_\text{x}(x_i) n_e\left(p_f
- \hat{\mathbf{n}}_s\cdot\mathbf{a}(x_i, t_0-\tau)\right),$$ where $\hat{\mathbf{n}}_s$ denotes the surface normal. Near the surface, in the region that is probed by the electrons, the plasmonic field is predominantly perpendicular to the surface. Therefore, we can approximate $\hat{\mathbf{n}}_s\cdot\mathbf{a} = \mathsf{a}_y
\simeq \mathsf{a}_{\text{spp}}$. Computing the variance Eq. (\[eq:ps-mom2\]) for a Gaussian distribution of the initial electron momenta, we obtain
$$\begin{aligned}
S(\tau) &= \Delta p^2 + \int_{-\infty}^{\infty} dx_i\,
g_\text{x}(x_i)\mathsf{a}_\text{spp}^2(x_i,
t_0-\tau).\label{eq:ps-mom2-uni}\end{aligned}$$
For isotropic XUV photo-electron emission, the initial distribution can be written as: $n_e(\mathbf{p}_i) = \frac{1}{\pi} n_e(p_i)= \frac{1}{\pi}n_e(|\mathbf{p}_f - \mathbf{a}|)$, where we employed $ p_i = |\mathbf{p}_i|$. We use $|\mathbf{a}|\ll|\mathbf{p}_{f}|$ to approximate $|\mathbf{p}_f -
\mathbf{a}| \simeq p_{f} - \,\mathbf{a}\cdot\hat{\theta}$, where $\theta$ is the angle between the final momentum and the surface normal. The spectrogram then reads $$\label{eq:pho-spe-iso}
\sigma(p_f, \tau) = \frac{1}{\pi}\int_{-\infty}^{\infty} dx_i\,
g_\text{x}(x_i)\, n_e(p_f - \mathbf{a}\cdot\hat{\theta}).$$ A straightforward calculation for the angular integrations leads to the expression of the variance $$\begin{aligned}
S(\tau) &= \Delta p^2 + \frac{1}{\pi}\int_{-\infty}^{\infty} dx_i\,
g_\text{x}(x_i)|\mathbf{a}(x_i, \tau)|^2.
\label{eq:ps-mom2-iso}\end{aligned}$$ In either case, by Eqs. (\[eq:ps-mom2-uni\]) and (\[eq:ps-mom2-iso\]), measuring the variance of the photo-emission spectrogram provides direct access to the space-averaged vector potential $\mathbf{a}^2$ at the surface in the direction of photo-detection. The surface vector potential $|\mathbf{a}|^2 = \mathsf{a}_x^2 + \mathsf{a}_\text{spp}^2$ also includes $a_x$, the NIR field at the grating surface. Modifications of the surface field compared to the incident beam can be measured [*in situ*]{} (see below). Simulations of the plasmonic field were performed with the finite-difference time-domain (FDTD) method [@Taflove1995], using a freely available software package [@Oskooi2010]. Material properties were included through the appropriate model of gold dielectric function [@Rakic98ao]. We assume a Gaussian 4 fs FWHM pulse at a central wave length of 800 nm. The grating parameters are optimized for maximal absorption from the NIR pulse, assuming a gold surface. Beam waists of NIR and XUV were 5 and 10 $\mu$m, respectively.
The XUV photoemission process is approximated as a sudden ejection of electrons from the surface boundary, with the appropriate unidirectional and isotropic initial momentum distribution, respectively. The electron trajectories and final momenta are computed by solving the Lorentz equation for each photoelectron in the previously simulated electromagnetic field.
The spectrogram variance obtained by Monte Carlo simulation is compared in Fig. \[fig:comp\_iso\_sim\] with the space integral of the squared vector potential along $y$ from the FDTD simulation. We assume isotropic initial momentum distribution and a TOF detector of 5$^\circ$ acceptance centered around the perpendicular direction.
![Comparison between variance of photoscopic spectrogram in the “filtered isotropic” case (red) and $\int |\mathsf{a}_y|^2 dx$ computed in the FDTD (blue). The offset of the filtered isotropic case is due to the XUV pulse energy width.[]{data-label="fig:comp_iso_sim"}](comp_iso_sim.eps){width="1.0\linewidth"}
Note that the variance directly images the integral of the surface plasmonic field squared without further assumptions or input from theory. The agreement is robust w.r.t. to the angular distribution of photo-electron momenta: one obtains analogous results for unidirectional emission.
![Photoscopic spectrograms at perpendicular (left) and grazing (right) electron emission. The measurements retrieve plasmonic and NIR field, respectively. Solid lines are the momentum variances. []{data-label="fig:pho-spe-iso"}](nor_vs_gra.eps){width="1.05\linewidth"}
The detailed image of the fields provides for an [*in situ*]{} diagnosis both, of the plasmon field and exciting NIR source, including possible distortions due to the NIR reflection on the grating. In Figure \[fig:pho-spe-iso\] spectrograms observed in the perpendicular and grazing direction are shown, which reflect the two contributions.
From the plasmonic (perpendicular) component, we extract buildup and life-times, as well as contributions of the bright and dark modes to the spectrograms. We parametrize the field as follows: we assume plasmonic fields with a Gaussian envelope $\mathsf{a}_{\text{spp}} = \exp[i\varphi]\exp[-\varphi^{2}/2\omega_\text{spp}^{2}T^{2}]$, with $ \varphi = k_{\text{spp}}x - \omega_\text{spp} t$. There are two counter-propagating SPP wave-packets, each containing a bright $\omega_b$ and and a dark $\omega_d$ frequency. These terms are multiplied by a “buildup” and “decay” function $f(t) =
\exp(-t/2\tau_m)\times(1-\text{erf}((\sigma_m^2-2\tau_m t)/(2\sqrt{2}\sigma_m\tau_m)))$, which is the convolution of a Gaussian excitation profile with exponential decay. Source duration and plasmon mode decay rate are denoted by $\sigma_m$ and $\tau_m$, respectively, for $m = b,d$. When $f(t)$ multiplies the plasmonic term, the respective $\tau_m$ parametrizes the lifetime, while the Gaussian half-width half-maximum in intensity $\xi_m = \sigma_m\sqrt{\ln 2}$ parametrizes the buildup time.
The remaining fit parameters are the amplitudes of the respective plasmon modes. The explicit form of the parametrization is given in the supplementary materials. The relevant free parameters in this model are the excitation buildup times $\xi_b,\xi_d$, the plasmon decay times $\tau_b,\tau_d$ and the plasmon frequencies $\omega_b,\omega_d$ for the bright and dark modes, respectively.
Fitting to the simulated variance, we find plasmon frequencies are $\hbar\omega_b = 1.65$ eV and $\hbar\omega_d = 1.62$ eV, consistent with the plasmonic band gap of 14 nm given in Ref. [@ropers2007njp]. Results for the buildup- and life-times are reported in Table \[tab:fit-res\]. Because of spatial integration, the plasmon pulse extension $T$ has little influence on the variance. The values in the table were obtained with $T=15$ fs (FWHM). A conservative lower bound of $T$ is given by the diameter of the NIR spot size, an upper bound by that size plus plasmon propagation during excitation. [[Variation in the range of $T=10$ and $20$ fs has only a small effect on buildup and decay times. Due to the superposition of bright mode decay with dark mode buildup, variation is largest for these parameters with about 0.7 fs.]{}]{} For any given value of $T$ in this interval, the buildup and decay extracted from the FDTD surface field and from the spectrogram variance are in good agreement.
$\quad$ Filtered Isotropic Unidirectional FDTD
------------ --------- -------------------- ---------------- ------ --
$\xi_b$ $\quad$ 2.07 2.06 2.01
$\tau_b$ $\quad$ 3.0 3.1 2.96
$\xi_d$ $\quad$ 6.6 6.2 5.3
$\tau_d$ $\quad$ 32.5 33.3 34.6
$\omega_b$ $\quad$ 1.61 1.62 1.62
$\omega_d$ $\quad$ 1.65 1.65 1.65
: \[tab:fit-res\] Carrier frequency $\omega_m$, buildup time $\xi_m$ and lifetime $\tau_m$ resulting from fits of the theoretical model to the numerically simulated data. The cases isotropic emission with perpendicular detection (“filtered”), unidirectional emission, as well as values extracted directly from the FDTD calculation are shown. (Times in fs. Frequencies in eV)
A comparison of the two spectrograms in Figure \[fig:pho-spe-iso\] of the NIR vs. the plasmonic field allows the evaluation of the field enhancement, which is in the present case $\sim$ 1. >From the spectrogram at grazing direction, we get a NIR pulse duration of $\Delta t_\text{fwhm} = 4.5$ fs, in good agreement with the $4.6$ fs from the FDTD code. Such a measurement provides an independent *in situ* diagnosis of the field distortions of the NIR field caused by the interaction with the grating.
In conclusion, we have shown how to obtain, with existing experimental instrumentation, direct, time-resolved images of the SPP surface field. Time resolution is determined by controlling the relative pulse delay. This allows the extraction of basic parameters such as SPP buildup and life times. Attosecond resolution, in our example, provides for the distinction of bright and dark mode oscillations. The same setup also provides [*in situ*]{} diagnostics of the NIR pulse.
Once spatially separated XUV attosecond and NIR pulses become available, one may resolve in space and time also other surface phenomena: by letting the NIR field excite a surface mode in some region, one can image SPP propagation along complex plasmonic waveguides or plasmonic switches by simply pointing the attosecond XUV pulse on the region of interest.
We are grateful to C. Ropers for useful discussions. We acknowledge support by the DFG, by the excellence cluster “Munich Center for Advanced Photonics (MAP)”, by the Austrian Science Foundation project ViCoM (F41), by the Landesexzellenzcluster “Frontiers in Quantum Photon Science” and the Joachim Herz Stiftung.
|
---
author:
- |
C. Done\
Department of Physics, University of Durham, South Road, Durham, DH1 3LE:\
chris.done@durham.ac.uk
title: Observational Characteristics of Accretion onto Black Holes
---
Observational Characteristics of Accretion onto Black Holes
===========================================================
Abstract
--------
These notes resulted from a series of lectures at the IAC winter school. They are designed to help students, especially those just starting in subject, to get hold of the fundamental tools used to study accretion powered sources. As such, the references give a place to start reading, rather than representing a complete survey of work done in the field.
I outline Compton scattering and blackbody radiation as the two predominant radiation mechanisms for accreting black holes, producing the hard X-ray tail and disc spectral components, respectively. The interaction of this radiation with matter can result in photo-electric absorption and/or reflection. While the basic processes can be found in any textbook, here I focus on how these can be used as a toolkit to interpret the spectra and variability of black hole binaries (hereafter BHB) and Active Galactic Nuclei (AGN). I also discuss how to use these to physically interpret real data using the publicly available [xspec]{} spectral fitting package (Arnaud et al 1996), and how this has led to current models (and controversies) of the accretion flow in both BHB and AGN.
Fundamentals of accretion flows: observation and theory
-------------------------------------------------------
### Plotting Spectra {#s:plot}
Spectra can often be (roughly) represented as a power law. This can be written as a differential photon number density (photons per second per square cm per energy band) as $N(E)=N_0 E^{-\Gamma}$ where $\Gamma$ is photon index. The energy flux is then simply $F(E)=EN(E) =
N_0 E^{-(\Gamma-1)}= N_0 E^{-\alpha}$ where $\alpha=\Gamma-1$ is energy index.
Power law spectra are broad band, i.e. the emission spans many decades in energy. Thus in general we plot logarithmically, in $\log E$, with $d\log E$ rather than $dE$ as the constant. The number of photons per bin is $N(E)dE =N(E)E dE/E=EN(E)d\log E=F(E)d\log E$. Thus, somewhat counter-intuitively, plotting $F(E)$ on a logarithmic energy scale shows the [*number*]{} of photons rather than flux.
Similarly, energy per bin is $F(E) dE =F(E)E dE/E = EF(E) d\log E$. Thus to see the energy at which the source luminosity peaks on a logarithmic frequency scale means we have to plot $\log E F(E)$ ($=\nu F(\nu)$) versus $\log E$. In these units, hard spectra have $\Gamma<2$ so peak at high energies. Soft spectra have $\Gamma > 2$ and peak at low energies. Flat spectra with $\Gamma=2$ means equal power per decade. These are illustrated in Fig. \[f:pl\].
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One other very common way for spectra to be plotted is in counts per second per cm$^2$, $C(E)$. This is [*not*]{} the same as $N(E)$, the counts refer to number of photons [*detected*]{} not emitted. Such spectra show more about the detector than the intrinsic spectrum as these are convolved with the detector response giving $C(E)=\int
N(E_0) R(E,E_0) dE_0$ where $R(E,E_0)$ is the detector response i.e. the probability that a photon of input energy $E_0$ is detected at energy $E$. Instrument responses for X-rays are generally complex so the spectra are generally analysed in counts space, by convolving models of the intrinsic spectrum through the detector response and minimizing the difference between the predicted and detected counts to derive the best fit model parameters. While such ’counts spectra’ constitute the basic observational data, they do not give a great deal of physical insight. Deconvolving these data using a model to plot them in $\nu f(\nu)$ space is strongly recommended i.e. in [xspec]{} using the command [iplot eeuf]{}.
### Plotting variability {#s:pow}
Plotting variability is analogous to plotting spectra. A light-curve, $I(t)$, spanning time $T$, with points every $\Delta t$ can be decomposed into a sum of sinusoids: $$I(t)=I_0+\Sigma_{i=1}^N A_i\sin (2\pi \nu_i t + \phi_i)$$ where $I_0$ is the average flux over that timescale, $\nu_i=i/T$ with $i=1,2...N$ and $N=T/(2\Delta t)$. This is more useful when normalized to the average, giving the fractional change in intensity $I(t)/I_0= 1+ \Sigma (A_i/I_0)\sin (2\pi \nu_i t + \phi_i)$. The power spectrum $P(\nu )$ is $(A_i/I_0)^2$ versus $\nu_i$, and the integral $\int P(\nu )d\nu = (\sigma/I)^2$ i.e. the squared total r.m.s. variability of the lightcurve.
Similar to the energy spectra, the power spectrum is generally broad band so is plotted logarithmically. Then the total variability power in a bin is $P(\nu ) d\nu = P(\nu ) \nu d\nu/\nu = \nu P(\nu )
d\log\nu$. Thus, similarly to spectra, a peak in $\nu P(\nu )$ versus $\log\nu$ shows the frequency at which the variability power peaks, so this is the more physical way to plot power spectra.
[c]{}
The data often show power-law power spectra, with $P(\nu ) \propto \nu^0$ on long timescales breaking to $\nu^{-1}$ at a low frequency break $\nu_b$ and then breaking again to $\nu^{-2}$ at a high frequency break $\nu_h$. This is termed band limited noise or flat-top noise, where the ’flat-top’ has $P(\nu ) \propto \nu^{-1}$ as this has equal variability power per decade.
Fig.\[f:pow\] illustrates this for data from a low/hard state (see section \[s:states\]) in Cyg X-1. The light-curves are shown over different timescales, $T$. These are all normalized to their mean, so it is easy to see from the light-curves that the fractional variability is very low on timescales shorter than 0.01 s. At longer timescales the fractional variability increases, then remains constant for 1-10 s and then drops again.
### Spectra and Variability of the Shakura-Sunyaev disc {#s:ss}
The underlying physics of a Shakura-Sunyaev accretion disc can be illustrated in a very simple derivation just conserving energy (rather than the proper derivation which conserves energy and angular momentum).
A mass accretion rate $\dot{M}$ spiraling inwards from $R$ to $R-dR$ liberates potential energy at a rate $dE/dt=L_{pot}=(GM\dot{M}/R^2)\times dR$. The virial theorem says that only half of this can be radiated, so $dL_{rad}=GM\dot{M} dR/(2R^2) $. If this thermalises to a blackbody then $dL=dA \sigma_{SB} T^4$ where $\sigma_{SB}$ is the Stephan-Boltzman constant and area of the annulus $dA=2 \times 2
\pi R \times dR$ (where the factor 2 comes from the fact that there is a top and bottom face of the ring). Then the luminosity from the annulus $dL_{rad}=GM\dot{M}dR/(2R^2) = 4 \pi
R \times dR \sigma_{SB} T^4$ or $\sigma_{SB}T^4(R)= GM\dot{M}/8 \pi R^3 $. This is only out by a factor $3(1-(R_{in}/R)^{1/2})$ which comes from a full analysis including angular momentum (see H. Spruit in these proceedings).
Thus the spectrum from a disc is a sum of blackbody components, with increasing temperature and luminosity emitted from a decreasing area as the radius decreases. The peak luminosity and temperature then comes from $R_{in}$ (modulo the corrections for the inner boundary condition). Using the very approximate treatment above, the total total luminosity of the disc $ L_{disc} = GM\dot{M}/(2R_{in})$ so substituting for $GM\dot{M}$ gives $\sigma_{SB} T^4(R)= R_{in} L_{disc} /4 \pi R^3 \propto
L_{disc}\times (R_{in}/R) \times 1/(4 \pi R^2)$. So $\sigma_{SB}
T^4_{max}=\sigma_{SB} T(R_{in})= L_{disc}/(4 \pi R_{in}^2)$, giving an observational constraint on $R_{in}$ if we can measure $T_{max}$ and $L_{disc}$. This is important as $R_{in}$ is set by General Relativity at the last stable orbit around the black hole, which is itself dependent on spin. Angular momentum, $J$, is typically a mass, times a velocity, times a size scale. The smallest size scale for a black hole is that of the event horizon, which is always larger than $R_g=GM/c^2$, and the fastest velocity is the speed of light. Thus $|J| < M c GM/c^2$, or spin-per-unit mass, $|J|/M = a_* GM/c$, where $a_*\le 1$. The last stable orbit is at $6~R_g$ for a zero spin ($a_*=0$: Schwarzschild) black hole, decreasing to $1~R_g$ for a maximally rotating Kerr black hole for the disc co-rotating with the black hole spin ($a_*=1$), or $9~R_g$ for a counter-rotating disc ($a_*=-1$). Thus to convert the observed emission area to spin we need to know the mass of the black hole so we can put the observed inner radius into gravitational radii $r_{in}=R_{in}/R_g$, giving us a way to observationally measure black hole spin.
In [xspec]{}, a commonly used model for the disc is [diskbb]{}. This assumes that $T^4\propto r^{-3}$ i.e. has no inner boundary condition. It is adequate to fit the high energy part of the disc spectrum i.e. the peak and Wien tail, but the derived normalization needs to be corrected for the lack of boundary condition. It also assumes that each radius emits as a true blackbody, which is only true if the disk is effectively optically thick to absorption at all frequencies. Free-free (continuum) absorption drops as a function of frequency, so the highest energy photons from each radii are unlikely to thermalize. This forms instead a modified (or diluted) blackbody, with effective temperature which is a factor $f_{\rm col}$ (termed a colour temperature correction) higher than for complete thermalization. The full disk spectrum is then a sum of these modified blackbodies, but this can likewise be approximately described by a single colour temperature correction to a ’sum of blackbodies’ disk spectrum (Shimura & Takahara 1995), giving rise to a further correction to the [diskbb]{} normalization. The final factor is that the emission from each radius is smeared out by the combination of special and general relativistic effects which arise from the rapid rotation of the emitting material in a strong gravitational field (Cunningham 1975, see section \[s:rel\]). Again, these corrections can be applied to the [diskbb]{} model (e.g. Kubota et al 2001; Gierlinski & Done 2004a), but are only easily available as tabulated values for spin $0$ and $0.998$ in Zhang et al (1997). Hence a better approach is to use [kerrbb]{}, which incorporates the stress-free boundary condition and relativistic smearing for any spin (Li et al 2005) for a given colour temperature correction factor. An even better approach is to use [bhspec]{}, which calculates the intrinsic spectrum from each radius using full radiative transfer through the disc atmosphere, including partially ionized metal opacities, rather than assuming a colour temperature corrected blackbody form (Davis et al 2005). This imprints atomic features onto the emission from each radius, distorting the spectrum from the smooth continuum as produced by [kerrbb]{} (Done & Davis 2008, Kubota et al 2010).
To zeroth order, the emitted spectrum does not require any assumptions about the nature of the viscosity, parameterized by $\alpha$ by Shakura & Sunyaev (1973). However, variability is dependent on this, as variability in the emitted spectrum requires that the mass accretion rate through the disc changes. Material can only fall in if its angular momentum is transported outwards via ’viscous’ stresses, now known to be due to the magneto-rotational instability (MRI, see J. Hawley lecture notes in this proceedings). The viscous timescale, $t_{visc}\approx \alpha^{-1} (H/R)^{-2} t_{dyn}$ where $H$ is the vertical scale height of the disc and $t_{dyn}= 2\pi R_g(r^{3/2}+a_*)/c$ is the dynamical (orbital) timescale which is $\sim 5$ ms for a Schwarzschild black hole of $10M_\odot$.
Modeling the observed variability of the disc gives an estimate for $\alpha=0.1$ (King, Pringle & Livio 2007), though current simulations of the MRI gives stresses which are an order of magnitude lower than this (see J. Hawley lecture notes in this proceedings).
The disc models give a geometrically thin solution $H/R \sim 0.01$, so the very fastest variability from changes in mass accretion rate at the innermost edge are 100,000 times longer than the dynamical timescale. Thus accretion discs in black hole binaries (hereafter BHB) should only vary on timescales longer than a few hundred seconds.
### Observed Spectra and Power Spectra of Black Hole Binaries {#s:states}
#### Disc dominated states {#s:lt}
The mass accretion rate through the entire disc in BHB can vary over weeks-months-years, triggered by the disk instability (see R. Hynes in these proceedings or Done, Gierlinski & Kubota 2007, hereafter DGK07). This means that a single object (with constant distance, inclination and spin) can map out how the spectrum changes as a function of luminosity as shown in Fig. \[f:lc\].
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Fig. \[f:gx339\]a shows that these can indeed show spectra which look very like the simple accretion disc models described above (section \[s:ss\]). This disc emission also shows very little rapid variability on timescales of less than a few hundred seconds, as expected (Churazov et al 2001). Collating disc spectra on longer timescales at different $\dot{m}$ gives $L_{disc} \propto T^4_{max}$ (Fig. \[f:gx339\]b), clear observational evidence for a constant size-scale inner radius to the disc despite the large change in mass accretion rate. This is exactly as predicted for the behaviour at the last stable orbit, and is a key test of Einstein’s gravity in the strong field limit. Indeed, given how close the last stable orbit is to the event horizon ($r=6$ compared to the horizon at $r=2$ for a Schwarzschild black hole), this represents almost the strongest gravitational field we could ever observe.
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Folding in all the required corrections (see above) allows us to measure this fixed size-scale and translate it into a measure of spin if there are good system parameter estimates (see R. Hynes in these proceedings). To date, all objects which show convincing $L\propto T^4$ tracks give moderate spins, as opposed to extreme Kerr (Davis et al 2006; Shafee et al 2006; Middleton et al 2006; Gou et al 2010). This is as expected from current (probably quite uncertain) supernovae collapse models (see P. Podsiadlowski, these proceedings, also Gammie, Shapiro & McKinney 2004; Kolehmainen & Done 2010) and BHB in low mass X-ray binaries should have a spin distribution which accurately reflects their birth spin. This is because the black hole mass must approximately double in order to significantly change the spin, which is not possible in an LMXB even if the black hole (which must be more than $3~M_\odot$) completely accretes the entire low mass ($\la 1~M_\odot$) companion star (King & Kolb 1999). However, high spins are derived for two objects which do not have good $L\propto T^4$ tracks, namely GRS1915+105 (McClintock et al 2006, but see Middleton et al 2006) and LMC X-1 (Gou et al 2009)
#### The high energy tail
However, even the most disc dominated (also termed high/soft state) spectra also have a tail extending out to higher energies with $\Gamma
\sim 2$ (see Fig. \[f:gx339\]a). This tail carries only a very small fraction of the power in the data discussed above, but it can be much stronger and even dominate the energetics. Where the tail coexists with a strong disc component (very high/intermediate or steep power law state) it has $\Gamma> 2$, so the spectra are soft. But where the disc is weak the tail can dominate the total energy, with $\Gamma< 2$ so the spectra are hard (low/hard state), peaking above 100 keV. All these very different spectra are shown in Fig. \[f:states\]a, colour coded in the same way as the long term light-curve (Fig. \[f:lc\]). The combination of these two plots shows that typically, hard spectra are only seen at low fractions of Eddington, while disc or disc-plus-soft-tail spectra are seen only at high fractions of Eddington. This is actually very surprising as the classic Shakura-Sunyaev disc is unstable to a radiation pressure instability above $\sim 0.05 L_{Edd}$. Thus we might expect that the disc is disrupted into some other type of flow at high fractions of Eddington, and that we see clean disc spectra only at low fractions of Eddington. This is entirely the reverse of what is seen (as first recognized by Nowak 1995, see also Gierlinski & Done 2004a; DGK07).
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The tail has more rapid variability than the disc component (Churazov et al 2001; Gierlinski & Zdziarski 2005; though the disc variability is enhanced on timescales longer than 1 sec in the low/hard state: Wilkinson & Uttley 2009), typically with large fluctuations on timescales from 100-0.05 seconds. The dynamical (Keplarian orbit) timescale for a $10~M_\odot$ black hole at $6R_g$ is 0.005s, so these timescales are at least $10
\times$ longer than Keplarian. If the variability is driven by changes in mass accretion rate then the expected timescale is a factor $\alpha (H/R)^2$ longer, so this requires $H/R \sim 1$ i.e. a geometrically thick flow.
### Theory of geometrically thick flows: ADAFs
Pressure forces must be important in a geometrically thick flow, by contrast to a Shakura-Sunyaev disc. If these are from gas pressure then the flow must be hot, with protons close to the virial temperature of $10^{12}$ K. The data require that the electrons are hot in the low/hard state in order to produce the high energy tail via Compton scattering (see section \[s:comp\]), but generally only out to 100 keV i.e. $10^9$ K. These two very different requirements on the temperature can both be satisfied in a two temperature plasma, where the ion temperature is much hotter than the electron temperature. This happens in plasmas which are not very dense, as the electrons can radiate much more efficiently than the protons, so the electrons lose energy much more rapidly than the protons. Even if the electrons and protons are heated at the same rate then the proton temperature will be hotter than the electrons if the protons and electrons do not interact enough to equilibriate their temperatures. This gives a further requirement that the optical depth of the flow needs to be low.
This leads to the idea of a hot, geometrically thick, optically thin flow replacing the cool, geometrically thin, optically thick Shakura-Sunyaev disc. The exact structure of this flow is not well known at present - Advection Dominated Accretion Flows (ADAFs) are the most well known, but there can also be additional effects from convection, winds and the jet (DGK07). Ultimately, magneto-hydrodynamic simulations in full general relativity including radiative cooling are probably needed to fully explore the complex properties of these flows (J. Hawley, this volume).
When ADAFs (Narayan & Yi 1995) were first proposed, a key issue was how to produce such flow from an originally geometrically thin cool disc (to the extent that one theoretician said ’turbulence generated by theorists waving their hands’). However, there is now a mechanism to do this via an evaporation instability. If the cool disc is in thermal contact with the hot flow then there is heat conduction between the two, which can lead to either evaporation of the disc into the hot flow, or condensation of the hot flow onto the disc. At low mass accretion rates, evaporation predominates in the inner disc, giving rise to a radially truncated disc/hot inner flow geometry (Liu et al 1999; Rozanska & Czerny 2000; Mayer & Pringle 2007). This is exactly the geometry required in the phenomenological truncated disc/hot inner flow models described in the next section
The hot flow can only exist if the electrons and protons do not interact often enough to thermalise their energy. This depends on optical depth, and the flow collapses when $\tau \ga 2-3$ which occurs at $\dot{m}= 1.3\alpha^2\sim 0.01$ for $\alpha=0.1$ (Esin et al 1997). This is very close to the luminosity of the transition from soft to hard spectra seen on the outburst decline (Maccarone 2003), though more complicated behaviour is seen on the hard to soft transition on the rise (hysteresis: Miyamoto et al 1995; Yu & Yan 2009), plausibly due to the rapid accretion rate changes pushing the system into non-equilibriuim states (Gladstone, Done & Gierlinski 2007).
But there are issues. I stress again that the flow should be more complex than an ADAF as these do not include other pieces of physics which are known to be present (convection, winds, the jet, changing advected fraction with radius: see e.g. DGK07). Indeed, standard ADAF solutions are somewhat too optically thin and hot to match the observed Compton spectra (Malzac & Belmont 2009), and this is especially an issue at the lowest $L/L_{Edd}$ where the observed X-ray spectra are far too smooth to be produced by the predicted very optically thin flow (Pszota et al 2008).
### Truncated disc/hot inner flow models
These two very different types of solution for the accretion flow can be put together into the truncated disc/hot inner flow model. At high fractions of Eddington we typically see strong evidence for a disc down to the last stable orbit (see Fig. \[f:gx339\]). At low fractions of Eddington, we can have one of these hot, optically thin geometrically thick flows. The only way to go from one to the other is for the disc to move inwards. As it penetrates further into the flow then more seed photons from the disc are intercepted by the flow so the Compton spectrum softens (see Section \[s:comp\]).
The MRI turbulence in the hot inner flow generates the rapid variability at each radius, modulating the mass accretion rate to the next radius. Thus the total variability is the product (not the sum!) of variability from all radii within the hot flow (Lyubarskii 1997; Kotov, Churazov & Gilfanov 2001; Aravelo & Uttley 2006). This rather natually gives rise to a key observational requirement that the r.m.s variability $\sigma$ (see section \[s:pow\]) in the lightcurve, as measured over a fixed set of frequencies (duration $T$ and sampling $\Delta t$), is proportional to the mean intensity $I_0$. This is the r.m.s.-flux relation and cannot be produced by a superposition (addition) of uncorrelated events such as the phenomenological ’shot noise’ models. Instead this observation [*requires*]{} that the fluctuations are multiplicative (Uttley & McHardy 2001; Uttley, McHardy & Vaughan 2005), as sketched in Fig. \[f:fluc\].
[c]{}
As the disc extends progressively inwards for softer spectra, the flow at larger radii cannot fluctuate on such large amplitudes as the disc is underneath it. The large amplitude fluctuations can only be produced from radii inwards of the truncated disc. This gets progressively smaller as the disc comes in, so the longer timescale (lower frequency) fluctuations are progressively lost, so the power spectrum narrows, with $\nu_b$ increasing while the amount of high frequency power stays approximately the same, as seen in the data (see Fig. \[f:pdf\_trans\]a) These models can be made quantatative, and can match the major features of the correlated changes in both the energy spectra and the power spectra as the source makes a transition from the low/hard to high/soft states (Fig. \[f:pdf\_trans\]b, Ingram & Done 2010).
However, the power spectrum also contains a characteristic low frequency QPO which also moves along with $\nu_b$ (Fig. \[f:pdf\_trans\]a). This can be very successfully modeled in both frequency and spectrum as Lense-Thirring (vertical) precession of the entire hot inner flow (Ingram, Done & Fragile 2009), using the same transition radius as required for the low frequency break in the broad band power spectrum (Ingram & Done 2010, see Fig. \[f:pdf\_trans\]c). Since this is a model involving vertical precession, the QPO should be strongest for more highly inclined sources, as observed (Schnittman, Homan & Miller 2006).
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The hot flow is also a good candidate for the base of the jet, in which case the collapse of the hot flow seen in the high/soft state spectra also triggers the collapse of the jet, as observed (see R. Fender, this volume).
### Scale up to AGN {#s:agn}
These models can be easily scaled up to AGN, keeping the same geometry as a function of $L/L_{Edd}$ but changing the disc temperature as expected for a super-massive BH. The luminosity scales as M for mass accretion at the same fraction of Eddington, and the inner radius scales as M. Thus the emitting area scales as $M^2$ so $T^4 \propto
L/A \propto 1/M$ so disc temperature [*decreases*]{}, peaking in the UV rather than soft X-rays. Interstellar absorption in the host galaxy and in our galaxy effectively screens this emission (see section \[s:nabs\]), so the disc peak cannot be directly observed in the same way as in BHB.
The strong UV flux from the disc also excites multiple UV line transitions (see section \[s:iabs\]) from any material around the nucleus. This environment is much less clean than in LMXRB as there are many more sources of gas to be illuminated in the rich environment of a galaxy centre (molecular clouds, the obscuring torus...), giving strong line emission from the broad line region (BLR) and narrow line region (NLR).
Apart from these differences, we should otherwise see the same behaviour in terms of the spectral and variability changes as the mass accretion rate changes, and in the jet power. Thus these models predict intrinsic changes in the ionizing nuclear spectrum as a function of mass accretion rate as well as changes in the observed spectrum due to obscuration. This is in contrast to the original ’unification models’ of AGN in which Seyfert 1 and 2 nuclei were intrinsically the same, but viewed at different orientations to a molecular torus.
There is growing evidence for intrinsic differences in nuclear spectra. The optical emission line ratios can be quite different in [*unobscured*]{} AGN of similar mass e.g. LINERS show different line ratios to Seyfert 1s which are different to Narrow Line Seyfert 1s. This is a clear indication that the ionizing spectrum is intrinsically different, as expected from their very different $L_{bol}/L_{Edd}$ This can be seen directly from compilations of the spectral energy distributions (SED) of these different types of AGN. The fraction of power carried by the X-rays drops for increasing $L_{bol}/L_{Edd}$ in much the same way as for BHB. The soft tail at high $L/L_{Edd}$ carries a smaller fraction of bolometric luminosity so $L_x$ has to be multiplied by a larger factor to get $L_{bol}$ (Vasudevan & Fabian 2007).
The jet should also change with state as in BHB. This gives a clear potential explanation for the origin of radio loud/radio quiet dichotomy. This matches quite well to the observed radio populations (Koerding, Jester & Fender 2006), but there is a persistent suggestion that this is not all that is required, with the most powerful radio jets being found in the most massive AGN (e.g. Dunlop et al 2003). Incorporating super-massive black hole growth and its feedback onto galaxy formation into the semi-analytic codes to model the growth of structures in the Universe may give the answer to this. These show a correlation between super-massive black hole mass and mass accretion rate such that largest black holes in massive ellipticals are now all accreting in gas poor environments so accrete via a hot flow with correspondingly strong radio jet (Fanidakis et al 2010).
Compton scattering to make the high energy tail {#s:comp}
-----------------------------------------------
Compton scattering is just an energy exchange process between the photon and electron and the energy exchange is completely analytic. The output photon energy, $\epsilon_{out}$, is given by $$\epsilon_{out} = {\epsilon_{in} (1- \beta \cos \theta_{ei}) \over 1-\beta
\cos \theta_{eo} +(\epsilon_{in}/ \gamma)(1-\cos \theta_{io}) }$$ where $\theta_{ei}$, $\theta_{eo}$ and $\theta_{io}$ are the angles between the electron and input photon, electron and output photon, and input and output photon, respectively, $\gamma=(1-\beta^2)^{-1/2}$ is the electron Lorentz factor so that its kinetic energy is $E=(\gamma^2-1)^{1/2} m_e c^2$ and $\epsilon=h \nu
/m_ec^2$ is the photon energy relative to the rest mass energy of the electron. In general, this simply says whichever of the input electron and photon has the most energy shares some of this with the other.
An electron at rest has $E=0 < \epsilon$. The photon hits the electron and momentum conservation means that the electron recoils from the collision, so the photon loses energy in Compton downscattering. If the photons and electrons are isotropic and $\epsilon_{in}<<1$ then the angle averaged energy loss is $\epsilon_{out}=\epsilon_{in}/(1+\epsilon_{in}) \approx
\epsilon_{in}(1-\epsilon_{in})$. Thus the change in energy $\epsilon_{out}-\epsilon_{in}=\Delta \epsilon =
-\epsilon_{in}^2$. Alternatively, for $\epsilon_{in}>>1$ then the photon loses almost all its energy in the collision.
### Thermal Compton Upscattering: Theory {#s:thermal}
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Since Comptonization conserves photon number, it is easiest to draw in $\log F(E)$ versus $\log E$ as $F(E) d\log E =$ photon number per bin on a logarithmic energy scale (see section \[s:plot\]). See also the review by Gilfanov (2010).
In a thermal distribution of electrons, the typical random velocity is set by the electron temperature $\Theta=kT_e/m_e c^2$ as $v^2 \sim 3
kT_e/m_e$ so $\beta^2=3 \Theta$. Again, for isotropic electon and photon distributions this can be averaged over angle to give $\epsilon_{out}=(1+4\Theta+16\Theta^2+...)\epsilon_{in} \approx
(1+4\Theta)\epsilon_{in}$ for $\Theta <<1$. So, in scattering we change energy by $\epsilon_{out}-\epsilon_{in} = \Delta \epsilon = 4 \Theta
\epsilon_{in}$ and photons are Compton upscattered. Obviously there is a limit to this, since the photon cannot gain more energy than the electron started out with, so this approximation only holds for $\epsilon_{out} \la 3\Theta$.
Photons can only interact with electrons if they collide. The probability a photon will meet an electron can be calculated from the optical depth. An electron has a (Thomson) cross-section $\sigma_T$ for interaction with a photon. Thus it is probable that the photon will interact if there is one electron within the volume swept out by the photon where the length of the volume is simply the path length, $R$, and the cross-sectional area is the cross-section for interaction of the photon with an electron, $\sigma_T$. Optical depth, $\tau$, is defined as the number of electrons within this volume, so $\tau=n R
\sigma_T$ where $n$ is the electron number density. The scattering probability is $e^{-\tau} \approx 1-\tau$ for $\tau <<1$.
The seed photons are initially at some energy, $\epsilon_{in}$, so only a fraction, $\tau$, of these are scattered in optically thin material to energy $\epsilon_{out,1}=
(1+4\Theta)\epsilon_{in}$. But these scattered photons themselves also can be scattered to $\epsilon_{out,2}=(1+4\Theta)\epsilon_{out,1}=(1+4\Theta)^2
\epsilon_{in}$. These photons can be scattered again to $\epsilon_{out,3}$ etc until they reach the limit of the electron energy after N scatterings where $\epsilon_{out,N}=(1+4\Theta)^N
\epsilon_{in} \sim 3 \Theta$. Since the energy boost and fraction of photons scattered is constant then this gives a power law of slope $\log f(\epsilon) \propto \ln (1/\tau)/\ln (1+4\Theta)$ i.e. $f(\epsilon ) \propto \epsilon^{-\alpha}$ with $\alpha=\ln
\tau/\ln (1+4 \Theta)$. This is a power law from the seed photon energy at $\epsilon_{in}$ up to $3\Theta$. Thus the power law index is determined by [*both*]{} the temperature and optical depth of the electrons. These cannot be determined independently without observations at high energy to constrain $\Theta$ as the same spectral index could be produced by making $\tau$ smaller while increasing $\Theta$ (see Fig. \[f:thermal\]a and b). However, there are some constraints as the spectrum is only a smooth power law in the limit where the orders overlap i.e. $\tau$ not too small and $\Theta$ not too big and the energy bandpass is not close to either the electron temperature or seed photon energy (see Fig. \[f:thermal\]a and b).
This list of caveats means that often a power law is [*not*]{} a good approximation for a Comptonized spectrum. If the temperatures are non- relativistic and the optical depth not too small then [comptt]{} (Titarchuk 1994) or [nthcomp]{} (Zdziarski Johnson & Magdziarz 1996, where the ’n’ in front of thcomp denotes that it does not have the reflected component of Zycki, Done & Smith 1999) can be used as these both include the downturn in the Compton emission close to the seed photon energy which affects the derived disc properties (Kubota & Done 2004), as well as the rollover at the electron temperature.
However, for temperatures much above $\sim 100$ keV with good high energy data then relativistic corrections become important and [compps]{} (Poutanen & Svensson 1996) or [eqpair]{} (Coppi 1999) should be used. The Compton rollover at the electron temperature is rather sharper than an exponential, so using an exponentially cutoff power law is not a good approxmation (see Fig. \[f:xspec\_comp\]a), and will distort the derived reflected fraction (see Section \[s:refl\]).
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In optically thick Comptonization, with $\tau\gg 1$, almost all the photons are scattered each time so almost all of them end up at the electron temperature of $3\Theta$ forming a Wien peak (Fig. \[f:optthick\]a). The average distance a photon travels before scattering is $\tau=1$ i.e. a mean free path of $\lambda=1/(n
\sigma_T)$. Thus after 1 scattering, the distance travelled is $d_1^2=\lambda^2 +\lambda^2 - 2 \lambda^2 \cos \theta_{12}$ while after 2 scatterings this is $d_2^2=d_1^2 + \lambda^2 - 2 \lambda d_1
\cos \theta_{2,3}$ and after N scatterings $d_N^2=d_{N-1}^2 +
\lambda^2-2 \lambda d_{N-1} \cos \theta_{N,N+1}$. Since the scattering randomizes the direction then the angles average out, leaving $d_N^2=N \lambda^2= N /(n \sigma_T)^2$. The photon can escape when $d_N=R$, so the average number of scatterings before escape is $N\sim \tau^2$ (see Fig. \[f:optthick\]b).
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The amount of energy exchange from the electrons to photons in Comptonization can be roughly characterized by the Compton $y$ parameter. The fractional change in energy of the photon distribution, $y$, is the average number of scatterings times average fractional energy boost per scattering such that $y \approx (4\Theta + 16
\Theta^2)(\tau+\tau^2) \approx 4\Theta \tau^2$ in the optically thick, low temperature limit. If $y\ll 1$ then the electrons make very little difference to the spectrum, while for $y \ga 1$, Comptonization is very important in determining the emergent spectrum.
### Comptonization via energetics
Describing the spectrum in terms of $\tau$ and $\Theta$ is the ’classical’ way to talk about Compton scattering. But the physical situation is better described by $\tau$ and energetics. There is some electron region with optical depth $\tau$, heated by a power input $\ell_h$, making a Comptonized spectrum from some seed photon luminosity $\ell_s$. Now we are doing this by luminosity instead of photon number we should plot $\nu F(\nu)$ rather than $F(\nu )$. The seed photons peak at $\sim 3kT_{seed}$, with $\nu F(\nu)=\ell_s$ The ’power law’ Compton spectrum always points back to this point, forming a power law of energy index $\alpha\sim \log\tau/\log(1+4\Theta)$ which extends from here to $\sim 3kT_e$, and has total power $\ell_h$. The resulting equation can then be solved for $\Theta$ (see Haardt & Maraschi 1993). This is shown in Fig. \[f:xspec\_comp\]b, using the [eqpair]{} model. The seed photons (red) are Comptonised by hot electrons which have $10\times$ as much power as in the seed photons. For a large optical depth (blue: $\tau=10$) this energy is shared between many particles, so the electron temperature is lower than for a smaller optical depth (green: $\tau=1$). The spectrum cannot extend out to high energies, so has to be harder in order to contain the requisite amount of power.
This energetic approach gives more physical insight when we come to consider seed photons produced by reprocessing of the hard X-ray photons illuminating the disc (see section \[s:rep\]).
### Thermal Compton Scattering: Observations of Low/Hard State {#s:1118}
Fig. \[f:1753\] shows two examples of low/hard state spectra from a BHB, one which rolls over at $\sim 90$ keV, and one which extends to $\sim 200$ keV. Both rollovers look like thermal Comptonization, but with different temperatures ($\sim 30$ keV, and $\sim 100$ keV, respectively, i.e. $\Theta \sim 0.06$ and $0.2$). The optical depth can then be derived from the spectral index but not quite as easily as described above as $\tau$ is of order unity rather than $\ll 1$ as required for the analytic expression. A more proper treatment gives $\tau \sim 0.6$. Then the fraction of unscattered seed photons should be only $1-e^{-\tau}$ but we actually see more than this in Fig. \[f:1753\]a. Thus we require that not all the seed photons go through the hot electron region i.e. the geometry is either a truncated disc or the electron regions are small compared to the disc - either localised magnetic reconnection regions above the disc or a jet. See Section \[s:rep\] for how the energetics of Compton scattering strongly favour the truncated disc.
The spectra shown above are fit assuming that the observed soft X-ray component provides the seed photons for the Compton upscattering. This can be seen explicitly where there are multiwavelength observations, extending the bandpass down to the optical/UV where the outer parts of the disc can dominate the emission. Fig. \[f:1753\]a shows this for a bright low/hard state in the transient BHB XTE J1753.5-0127 (Chiang et al 2009). It is clear that extrapolating the hard X-ray power law down to the optical/UV will produce far more emission than observed. Thus the hard X-ray power law must break between the UV and soft X-ray, i.e. the seed photons for the Compton upscattering should be somewhere in this range. Since there is an obvious soft X-ray thermal component, this is the obvious seed photon identification. Fitting the soft X-rays with a disc component slightly underproduces the optical/UV emission, but this can be enhanced by reprocessing of hard X-rays illuminating the outer disc (van Paradijs 1996).
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However, for dimmer low/hard states, the hard X-rays extrapolate directly onto the optical/UV emission (Fig. \[f:1753\]b: Chiang et al 2009; Motch et al 1985). This looks much more like the seed photons are at lower energies than the optical. In the truncated disc picture, the disc can be so far away that it subtends a very small solid angle to the hot electron region which is concentrated at small radii. Thus the amount of seed photons from the disc illuminating the hot electrons can be very small, and can be less important than seed photons produced by the hot flow itself. The same thermal electrons as make the Compton spectrum can make both bremsstrahlung (from interactions with protons) and cyclo-synchrotron (from interactions with any magnetic field such as the tangled field produced by the MRI). The bremsstrahlung spectrum will peak at $kT_e$, so these seed photons have similar energies to the electrons so cannot gain much from Compton scattering. However, the cyclo-synchroton typically peak in the IR/optical region so these can be the seed photons for a power law which extends from the optical to the hard X-ray region (Narayan & Yi 1995; Di Matteo, Celotti & Fabian 1997; Wardzinski & Zdziarski 2000; Malzac & Belmont 2009).
Evidence for a change in seed photons is also seen in the variability (see R. Hynes, this volume). In bright states (both bright low/hard and high/soft/very high states) the optical variability is a lagged and smoothed version of the X-ray variability, showing that it is from reprocessed hard X-ray illumination of the outer accretion disc. However, this changes in the dim low/hard state, with the optical having more rapid variability than the hard X-rays, and often [*leading*]{} the hard X-ray variability (Kanbach et al 2001; Gandhi et al 2008; Durant et al 2008; Hynes et al 2009). This completely rules out a reprocessing origin, clearly showing the change in the optical emission mechanism.
### Non-Thermal Compton Scattering: High/Soft State {#s:nonthermal}
While the low/hard state can be fairly well described by thermal Compton scattering, the same is not true for the tail seen in the high soft state. This has $\Gamma \sim 2$ and clearly extends out past 1 MeV, and probably past 10 MeV for Cyg X-1 (Gierlinski et al 1999; McConnell et al 2002) , as shown in Fig. \[f:cygx1\_soft\]a. If this were thermal Compton scattering then the electron temperature must be $\Theta \ga 1$, requiring optical depth $\tau <<1$ in order to produce this photon index. The separate Compton orders are then well separated and the spectrum should be bumpy (see Fig. \[f:thermal\]b) rather than the smooth power law seen in the data (the bump in the data is from reflection: see Section \[s:refl\]). Thus this tail cannot be produced by thermal Comptonization.
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Instead it can be non-thermal Compton scattering, where the electron number density has a power law distribution rather than a Maxwellian i.e. $n(\gamma) \propto
\gamma^{-p}$ from $\gamma=1$ to $\gamma_{max}$. Going back to the original equation for the Compton scattered energy boost, for $\gamma>>1$ the output photon is beamed into a cone of angle $1/\gamma$ along the input electron direction. This gives an angle averaged output photon energy of $\epsilon_{out}= (4/3 \gamma^2
-1)\epsilon_{in} \approx \gamma^2 \epsilon_{in}$ for an isotropic distribution of input photons and electrons. Thus the Compton scattered spectrum extends from $\epsilon_{in}$ to $\gamma^2_{max}
\epsilon_{in}$, forming a power law from a single scattering order.
The power law index of the resulting photon spectrum can be calculated from an energetic argument. The rate at which the electrons lose energy is the rate at which the photons gain energy, giving $F(\epsilon ) d\epsilon \propto \dot{\gamma} n(\gamma) d\gamma$ where $\dot{\gamma}\propto \gamma^2$ is the rate at which a single electron of energy $\gamma$ loses energy and $n(\gamma)$ is the number of electrons at that energy. Thus $F(\epsilon) \propto \gamma^2
\gamma^{-p} d\gamma/d\epsilon$. Since $\epsilon\sim
\gamma^2\epsilon_i$ then $d\epsilon/d\gamma=2\gamma$ so $F(\epsilon)
\propto \gamma^{-(p-1)} \propto \epsilon^{-(p-1)/2}$ i.e. an energy spectral index of $\alpha=(p-1)/2$ (G. Ghisellini, private communication).
For an optically thin electron region, the electrons intercept only a fraction $\tau$ of the seed photons, and scatter them to $\gamma^2_{max} \epsilon_{in}$ with an energy spectral index of $(p-1)/2$. These can themselves be scattered into a second order Compton spectrum to $\gamma^2_{max} (\gamma^2 \epsilon_{in}) = \gamma^4
\epsilon_{in}$ but very soon the large energy boost means that these hit the limit of the electron energy of $\epsilon_{out}=\gamma_{max}$. The resulting spectrum is shown schematically in Fig. \[f:nonthermal\].
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[l]{}
Thus the high/soft state of Cyg X-1 requires $\gamma_{max} > 30$ to get seed photons from the disc at 1 keV upscattered to 1 MeV, or $\gamma_{max} > 100$ to get to 10 MeV, while the energy spectral index of $1.2 =
(p-1)/2$ implies $p\sim 3$. Such non-thermal Compton spectra can be modelled using either [compps]{} or [eqpair]{}.
Fig. \[f:cygx1\_soft\]a shows that the seed photons from the disc are clearly seen as distinct from the tail. This requires that either the optical depth is very low, or the electron acceleration region does not intercept many of the seed photons from the disc i.e. localized acceleration regions as shown schematically in Fig. \[f:cygx1\_soft\]b.
### Thermal - Nonthermal (Hybrid) Compton Scattering {#s:hybrid}
The high/soft states can transition smoothly into the very high or intermediate state spectra, with the tail becoming softer and carrying a larger fraction of the total power (Fig. \[f:vhs\]a). The disc then merges smoothly into the tail, showing that the hot electron region completely covers the inner disc emission and it is optically thick. The tail still extends up to 1 MeV, so clearly also contains non-thermal electrons, and is rather soft but has a complex curvature (see Fig. \[f:vhs\]b).
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The extent of the tail shows that there must be non-thermal Compton scattering, as in the high/soft state. However, the tail is softer so the electron index must be more negative than in the high/soft state spectrum, so the mean electron energy is lower. Yet the lack of direct disc emission requires an optical depth of $ \ga 1$. This means that there are multiple Compton scattering orders forming the spectrum in a similar way to thermal Compton scattering, but from a non-thermal distribution. The energetic limit to which photons can be scattered is $\gamma_{max}$ but because the energy boost on each scattering is small, the spectrum actually rolls over at $m_ec^2=511$ keV as the cross-section for scattering drops at this point where $\gamma\epsilon\sim 1$ (as the cross-section transitions to Klein-Nishina rather than the constant Thomson cross-section seen at lower collision energies). Thus optically thick, nonthermal Comptonized spectra with a steep power law electron distribution does not produce a power law spectrum. Instead there is a break at 511 keV (Ghisellini 1989), as shown schematically in Fig. \[f:steep\]a, which means that this cannot fit the observed tail at high energies seen in the very high state spectrum, as shown in Fig. \[f:steep\]b (Gierlinski & Done 2003)
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Thus neither thermal nor non-thermal Compton scattering can produce the tail seen in these very high state data. Instead, the spectra require both thermal and non-thermal electrons to be present. This could be produced in a single acceleration region, where the initial acceleration process makes a non-thermal distribution but where the resulting electrons have a hybrid distribution due to lower energy electrons predominantly cooling through Coulomb collisions (which thermalize) while the higher energy electrons maintain a power law shape by cooling via Compton scattering (Coppi 1999). Such hybrid thermal/non-thermal spectra can be modeled in [xspec]{} using either [compps]{} or [eqpair]{}. Alternatively, this could indicate that there are two separate acceleration regions, one with thermal electrons, perhaps the remnant of the hot inner flow, and one with non-thermal, perhaps magnetic reconnection regions above the disc or the jet (DGK07). This could be modeled by two separate [compps]{} or [eqpair]{} components, one of which is set to be thermal, and the other set to be non-thermal.
Whatever the electrons are doing, they are optically thick and cover the inner disc. Hence it is very difficult to reconstruct the intrinsic disc spectrum in these states. The derived temperature and luminosity of the disc depend on how the tail is modeled. A simple power law model for the tail means that it extends below the putative seed photons from the disc. Instead, in Compton upscattering, the continuum rolls over at the seed photon energy, so there are fewer photons at low energies from the tail so the disc has to be more luminous and/or hotter in order to match the data. Compton scattering conserves photon number, so all the photons in the tail were initially part of the disc emission, so the intrinsic disc emission is brighter than that observed by this (geometry dependent) factor. This means that the intrinsic disc luminosity and temperature cannot be unambiguously recovered from the data when $\tau\ga 1$, where the majority of photons from the disc are scattered into the tail (Kubota et al 2001; Kubota & Done 2004; Done & Kubota 2006; Steiner et al 2009).
For more observational details of spectral states see Remillard & McClintock (2006) and Belloni (2009).
Atomic Absorption {#s:abs}
-----------------
The intrinsic continuum is modified by absorption by material along the line of sight. This can be the interstellar medium in our galaxy the host galaxy of the X-ray source, or material associated with the source. For AGN this can be the molecular torus, the NLR or BLR clouds, or an accretion disc wind. For BHB this is just an accretion disc wind, and any wind from the companion star. For magnetically truncated accretion discs such as seen in the intermediate polars (white dwarf) and accretion powered millisecond pulsars (neutron stars) there is an accretion curtain which can be in the line of sight (de Martino et al 2004), while for extreme magnetic fields the disc is completely truncated and there is only an accretion column (polars in white dwarfs) but this overlays the X-ray hot shock, giving complex absorption (Done & Magdziarz 1998).
Again the key concept is optical depth, $\tau = \sigma(E) n R$, only now the cross-section, $\sigma(E)$, has a complex dependence on energy (rather than the constant electron scattering cross-section, $\sigma_T$, for energies below 511 keV). We can combine $nR =N_H$ as the number of Hydrogen atoms along a line of sight volume with cross-sectional area of 1 cm$^{-2}$, so the optical depth is simply related to column density by $\tau(E) = \sigma(E) N_H$.
### Neutral Absorption {#s:nabs}
The photo-electric absorption cross-section of neutral hydrogen is zero below the threshold energy of 13.6 eV, below which the photons do not have enough energy to eject the electron from the atom. It peaks at this threshold edge energy at a value of $6\times
10^{-18}$ cm$^{-2}$ and then declines as $ \approx
(E/E_{edge})^{-3}$. Thus the optical depth is unity for a Hydrogen column of $1.6\times 10^{17}$ cm$^{-2}$ at 13.6 eV while a typical column through our galaxy is $>10^{20}$ cm$^{-2}$, showing how effectively the UV emission is attenuated.
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The drop in cross-section with energy means that an H column of $10^{20}$ cm$^{-2}$ has $\tau=1$ at an energy of $\sim 0.2$ keV, allowing the soft X-rays to be observed. Fig. \[f:col\]a shows how a hydrogen column of $\log N_H=19,20,21,22,23$ progressively absorbs higher energy X-rays. However, the column is not made up soley of H. There are other, heavier, elements as well. These have more bound electrons, but the highest edge energy will be from the inner $n=1$ shell (also termed the K shell) electrons as these are the closest to the nuclear charge. Since this charge is higher, then the $n=1$ electrons are more tightly bound than those of H e.g. for He it is 0.024 keV, C, N, O is 0.28, 0.40 and 0.53 keV. These elements are less abundant than H so they form small increases in the total cross-section. Fe is the last astrophysically abundant element, and this has a K edge energy of 7.1 keV. Fig. \[f:col\]b shows this for column of $10^{22}$ cm$^{-2}$ for progressively adding higher atomic number elements assuming solar abundances. Helium has an impact on the total cross-section, but additional edges from heavier elements are important contributions to the total X-ray absorption, especially Oxygen.
In [xspec]{}, this can be modeled using [tbabs]{}/[zthabs]{} (Wilms, Allen and McCray 2000, where the latter has redshift as a free parameter) or [phabs]{}/[zphabs]{} (Balucinska-Church & McCammon 1992) if the abundances are assumed to be solar, or [tbvarabs]{}/[zvphabs]{} if the data are good enough for the individual element abundances to be constrained via their edges such as in GRS 1915+104 (Lee et al 2002). However, with excellent spectral resolution data from gratings then the line absorption (especially from neutral Oxygen) becomes important (see section \[s:lines\]), and [tbnew]{} (see J. Wilms web page at http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/) should be used (Juett, Schulz & Chakrabarty 2004).
### Ionised Absorption {#s:iabs}
Photo-electric absorption leaves an ion i.e. the nuclear charge is not balanced. Thus all the remaining electrons are slightly more tightly bound, so all the energy levels increase. The ion can recombine with any free electrons, but if the X-ray irradiation is intense then the ion can meet an X-ray photon before it recombines, so that the absorption is dominated by photo-ionized ions. For H, this means there is no photo-electric absorption, since it has no bound electrons after an ionization event, so some fraction of the total cross-section disappears. Helium may then have 1 electron left, so its edge moves to 0.052 keV. At higher ionizations, Helium is completely ionized, so its contribution to the total cross-section is lost and there are only edges from (ionized) C, N, O and higher atomic number elements. Thus the effect of going to higher ionization states is to reduce the overall cross-section as the numbers of bound electrons are lower. In the limit where all the elements are completely ionized, there is no photo-electric absorption at all.
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The ion populations are determined by the balance between photo-ionization and electron recombination. For a given element $X$, the ratio between $X^{+i}$ and the next ion stage up, $X^{i+1}$ is given by the equilibrium reaction $$X^{+i} + h \nu \rightleftharpoons X^{+(i+1)} + e^-$$ The photo-ionization rate depends on the number density of the ion $N_X^{+i}$ and of the number density of photons, $n_\gamma$, above the threshold energy $\nu_{edge}$ for ionization for that species. This number density can be approximated by $n_\gamma \sim L /(h \nu 4 \pi r^2 c)$ where $L$ is the source luminosity, $h \nu$ is the typical photon energy and $4 \pi
r^2 c $ is the volume swept out by the photons in 1 second. Actually, what really matters is the number of photons past the threshold energy which photo-ionize the ion, so this depends on spectral shape as well.
Equilibrium is where the photo-ionization rate, $N_X^{+i} n_{\gamma} \sigma(X^{+i})
c$ balances the recombination rate $N_X^{+(i+1)} n_e
\alpha(X^{+(i+1)}, T)$ where $\sigma(X^{+i})$ is the photoelectric absorption cross-section for $X^{+i}$ and $\alpha(X^{+(i+1)}, T)$ is the recombination coefficient for ion $X^{+(i+1)}$ at temperature $T$. Hence the ratio of the abundance of the ion to the next stage down is given by $${N_X^{+(i+1)} \over N_X^{+i}} = {n_{\gamma} \sigma(X^{+i}) c \over
n_e \alpha(X^{+(i+1)}, T)} \propto {n_\gamma\over n_e}$$ Thus the ratio of photon density to electron density determines the ion state. If the photon density is highest, the ion meets a photon first, so is ionized to the next stage. Conversely, if the electron density is higher, then the ion meets an electron first and recombines to the lower ion stage. The ratio of photon to electron number density can be written as $n_\gamma/n_e = \xi /(4 \pi h \nu c)$ where $\xi=L/nr^2$ is the photo-ionization parameter. There are other ways to define this such as $\Xi =P_{rad}/P_{gas}=L/(4 \pi r^2 c n_e k T) = \xi /(4 \pi c
k T)$ but whichever description is used, the higher the ionization parameter, the higher the typical ionization state of each element.
In general, the equilibrium reaction means that there are at least two fairly abundant ionization stages for each element, so as long as the higher ionization stage is not completely ionized then there are multiple edges from each element as each higher ion stage has a higher edge energy as the electrons are more tightly bound. This can be clearly seen in Fig. \[f:ionise\]a, where the H-like Oxygen ($O^{+7}$) edge at 0.87 keV is accompanied by the He-like ($O^{+6}$) edge at 0.76 keV for $\xi=100$ and 10.
The [absori]{} (Done et al 1992) model in [xspec]{} calculates the ion balance for a given (rather than self-consistently computed) temperature and hence gives the photo-electric absorption opacity from the edges. However, this can be very misleading as it neglects line opacities (see below, and Fig. \[f:ionise\]b)
### Absorption Lines {#s:lines}
There are also line (bound-bound) transitions as well as edges (bound-free transitions). These can occur whenever the higher shells are not completely full. Hence Oxygen can show absorption at the n=1 to n=2 (1s to 2p) shell transition even in neutral material whereas elements higher than Ne need to be ionized before this transition can occur. The cross-section in the line depends on the line width. This is described by a Voigt profile. The ’natural’ line width is set by the Heisenburg uncertainty relation between the lifetime of the transition $\Delta t \Delta \nu \la \hbar $. This forms a Lorentzian profile, with broad wings. However, the ions also have some velocity due to the temperature of the material $v_{thermal}^2\sim
kT_{ion}/m_{ion}$. Any additional velocity structure such as turbulence adds in quadrature so $v^2=v_{thermal}^2+v_{turb}^2$. These velocities Doppler shift the transition, giving a Gaussian core to the line. This combination of Gaussian core, with Lorentzian wings, is termed a Voigt profile (see Fig. \[f:profile\]a)
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The line equivalent width is the width of a rectangular notch (down to zero) in the spectrum which contains the same number of photons as in the line profile, as shown in Fig. \[f:profile\]b. The equivalent width of the line grows linearly with the column density of the ion when the line is optically thin in its core, as the line gets deeper linearly with column (called unsaturated). However, this linear behaviour stops when the core of the line becomes optically thick i.e. none of the photons at the line core can escape at the rest energy of the transition. Increasing the column cannot lead to much more absorption as there are no more photons at the line center to remove. The wings of the line can become optically thick but Doppler wings are very steep so the line equivalent width does not increase much as the column increases (called saturated). Eventually, the Lorentzian wings start to become important, and then the line equivalent width increases as the square root of the column density (called heavily saturated). This relation between column and line equivalent width is termed a ’curve of growth’. While the linear section of this is unique, the point at which the line becomes saturated depends on the Doppler width of the line, i.e. on the velocity structure of the material as shown in Fig. \[f:cog\].
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For ’reasonable’ velocities, the line absorption equivalent width can be larger than the equivalent width of the edges, so that line transitions dominate the absorption spectrum. Fig \[f:ionise\]b shows the total absorption (line plus edges computed using [warmabs]{}, a model based on the XSTAR photo-ionization code (Kallman & Bautista 2001; Kallman et al 2004) compared to the more approximate ionization code which just used edge opacities ([absori]{} in [xspec]{}). The differences are obvious. There are multiple ionized absorption lines which dominate the spectrum as well as photo-electric edges.
Soft X-ray absorption features, especially from H and He like Oxygen, are seen in half of type 1 Seyferts (Reynolds 1997). High resolution grating spectra from Chandra and XMM-Newton observations have shown these in some detail. Typically, the best datasets require several different column densities, each with different ionization parameter, in order to fit the data. By contrast, the BHB show little in the way of soft X-ray absorption (GROJ1655-40 is an extreme exception: Miller et al 2006a), but highly inclined systems do show H and He–like Fe K$\alpha$ absorption lines in bright states (Ueda et al. 1998; Yamaoka et al. 2001; Kotani et al. 2000; Lee et al. 2002; Kubota et al 2007).
In each case, the origin of this ionized material can be constrained by measuring the velocity width (or an upper limit on the velocity width) of a line to get the column density of that ion via a curve of growth. This can be done separately for each transition, but a better way is to take a photo-ionization code and use this to calculate the (range of) column density and ionization state required to produce all the observed transitions assuming solar abundance ratios. The spectrum gives an estimate for $L$ so the equation for the ionization parameter $\xi=L/nR^2$ can be inverted by writing $N_H=n\Delta R$ to give $R = L/(N_H \xi )\times (\Delta R/R)$. Since $\Delta R/R\la 1$, then the distance of the material from the source is $R\la L/(N_H \xi
)$. A small radius means that the material is most probably launched from the accretion disc itself, probably as a wind.
### Winds {#s:winds}
Wherever X-rays illuminate material they can photo-ionize it. They also interact with the electrons by Compton scattering. Electrons are heated by Compton upscattering when they interact with photons of energy $\epsilon \ga \Theta$ but are cooled by Compton scattering by photons with energy $\epsilon \la \Theta$. Since the illuminating spectrum is a broadband continuum, then the spectrum both heats and cools the electrons. The Compton temperature is the equilibrium temperature where heating of the electrons by Compton downscattering equals cooling by Compton upscattering. Section \[s:comp\] shows that the mean energy shift is $\Delta \epsilon \sim 4 \Theta \epsilon -
\epsilon^2$, so integrating this over the photon spectrum gives the net heating which is zero at the equilibrium Compton temperature, $\Theta_{ic}$, so $0=\int
N(\epsilon) \Delta \epsilon d\epsilon = \int N(\epsilon) (4
\Theta_{ic}\epsilon - \epsilon^2) d\epsilon$. Hence $$\Theta_{ic}= {\int N(\epsilon) \epsilon^2 d \epsilon \over 4 \int
N(\epsilon) \epsilon d\epsilon}$$ For a photon spectrum with $\Gamma
= 2.5$ between $\epsilon_i$ to $\epsilon_{max}$ this gives $\Theta_{ic}\approx \frac{1}{4}\sqrt{\epsilon_{max}\epsilon_i}$ $\sim
2.5~keV$ for energies spanning 1-100 keV. Alternatively, for a hard spectrum with $\Gamma =1.5$ this is $\approx \epsilon_{max}/12$ or 8 keV. The effective upper limit to $\epsilon_{max}$ is around 100 keV as the reduction in Klein-Nishina cross-section means that higher energy photons do not interact very efficiently with the electrons.
Thus the irradiated face of the material can be heated up to this Compton temperature, giving typical velocities in the plasma of $v^2_{ic}=3 k T_{ic}/m_p$. This is constant with distance from the source as the Compton temperature depends only on spectral shape (though the depth of the heated layer will decrease as illumination becomes weaker). This velocity is comparable to the escape velocity from the central object when $v^2_{ic} \sim GM/R_{ic}$, defining a radius, $R_{ic}$, at which the Compton heated material will escape as a wind (Begelman, McKee & Shields 1983). This is driven by the pressure gradient, so has typical velocity at infinity of the sound speed $c_s^2 = kT_{ic}/m_p \sim
v^2_{ic} \sim GM/R_{ic}$. Thus the typical velocity of this thermally driven wind is the escape velocity from where it was launched.
As the source approaches Eddington, the effective gravity is reduced by a factor $(1- L/L_{Edd})$, so the thermal wind can be launched from progressively smaller radii as the continuum radiation pressure enhances the outflow, forming a radiation driven wind.
The Eddington limit assumes that the cross-section for interaction between photons and electrons is only due to electron scattering. However, where the material is not strongly ionized, there are multiple UV transitions, both photo-electric absorption edges and lines. This reduces the ’Eddington’ limit by a factor $\sigma_{abs}/\sigma_T$ which can be as large as 4000. This opacity is mainly in the lines, and the outflowing wind has line transitions which are progressively shifted from the rest energy to $\Delta \nu/\nu \approx
v_{\infty}/c$. This large velocity width to the line means that its equivalent width is high, so it can absorb momentum from the line transition over a wide range in energy. This gives a UV line driven wind.
The final way to power a wind is via magnetic driving, but this difficult to constrain as it depends on the magnetic field configuration, so it is invoked only as a last resort.
Much of the ’warm absorber’ systems seen in AGN have typical velocities, columns and ionization states which imply they are launched from size scales typical of the molecular torus (e.g. Blustin et al 2005). The much faster velocities of $\sim 0.1-0.2$ c implied by the broad absorption lines (BAL’s) seen bluewards of the corresponding emission lines in the optical/UV spectra of some Quasars are probably a UV line driven wind from the accretion disc. However, the similarly fast but much more highly ionized (H and He-like Fe) absorption systems seen in the X-ray spectra of some AGN (see ahead in Fig \[f:mkn766\]c and d) probably require either continuum driving with $L \sim L_{Edd}$ or magnetic driving (as the ionization state is so high that the line opacity is negligible with respect to $\sigma_T$).
By contrast, the BHB typically show fairly low outflow velocity of the highly ionized Fe K$\alpha$, mostly consistent with a thermally driven wind from the outer accretion disc (e.g. Kubota et al 2007, DGK07), although the extreme absorber seen in one observation from GRO J1655-40 may require magnetic driving (Miller et al 2006a, Kallman et al 2009). However, this does assume that the observed luminosity, $L_{obs}$ measures the intrinsic luminosity, $L_{int}$. If there is electron scattering in optically thick, completely ionized material along the line of sight then $L_{obs}=e^{-\tau} L_{int} \ll L_{int}$ (Done & Davies 2008). Such scattering would strongly suppress the rapid variability power (Zdziarski et al 2010) and indeed the variability power spectra of these data lack all high frequency power above 0.3-1 Hz, rather than extending to the $\sim 10$ Hz seen normally (see Fig. \[f:states\]). Thus thermal winds potentially explain all of what we see in terms of absorption from BHB.
Reflection {#s:refl}
----------
Whereever X-rays illuminate optically thick material such as the accretion disc the photons have some probability to scatter off an electron, and so bounce back into the line of sight. This reflection probability is set by the relative importance of scattering versus photo-electric absorption. For neutral material, photo-electric absorption dominates at low energies so the reflected fraction is very small. However, the photo-electric cross-section decreases with energy so the reflected fraction increases. Iron is the last astrophysically abundant element (due to element synthesis in stars as released in supernovae - see P. Podsiadlowski, this volume), so after 7.1 keV there are no more significant additional sources of opacity. The cross-section decreases as $E^{-3}$, becoming equal to $\sigma_T$ at around 10 keV for solar abundance material (Fig. \[f:refl\]a). Above this, scattering dominates, leading to a more constant reflected fraction, but at higher energies, the photon energy is such that Compton downscattering is important, so the reflection is no longer elastic. Photons at high energy do reflect, but do not emerge at the same energy as they are incident. This gives a break at high energies as Compton scattering conserves photon [*number*]{}, and the number of photons is much less at higher energies. Thus neutral reflection gives rise to a very characteristic peak between 20-50 keV, termed the reflection hump, where lower energy photons are photo-electrically absorbed and higher photons are (predominantly) downscattered (Fig. \[f:refl\]b: George & Fabian 1991; Matt, Perola & Piro 1991)
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The dependence on photo-electric absorption at low energies means that the spectrum is also accompanied by the associated emission lines as the excited ion with a gap in the K (n=1) shell decays to its ground state. This excess energy can be emitted as a fluorescence line (K$\alpha$ if it is the n=2 to n=1 transition, K$\beta$ for n=3 to n=1 etc). However, at low energies, the reflected emission forms only a very small contribution to the total spectrum, so any emission lines emitted below a few keV are strongly diluted by the incident continuum. These lines are also additionally suppressed as low atomic number elements have a higher probability to de-excite via Auger ionization, ejecting an outer electron rather than emitting the excess energy as a fluorescence line. This means that iron is the element which has most impact on the observed emission, as this is emitted where the fraction of reflected to incident spectrum is large, and has the highest fluorescence probability. All this combines to make a reflection spectrum which contains the imprint of the iron K edge and line features, as well as the characteristic continuum peak between 20-50 keV (Fig. \[f:refl\]b: George & Fabian 1991; Matt, Perola & Piro 1991).
### Ionised Reflection
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The dependence on photo-electric absorption at low energies means that reflection is sensitive to the ionization state of the reflecting material. Fig. \[f:ionref\]a shows how the absorption cross-section changes as a function of ionization state using a very simple model for photoelectric absorption which considers only the photoelectric edge opacity. The progressive decrease in opacity at low energies for increased ionization state means an increase in reflectivity at these energies as shown in Fig. \[f:ionref\]b for simple models of the reflected continuum ([pexriv]{} model in [xspec]{}).
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However, this continuum is accompanied by emission lines and bound-free (recombination) continua and the lines can be more important for ionized material. This is due to both an increase in emissivity (He-like lines especially have a high oscillator strength) and to the fact that the increased reflected fraction at low energies mean that these lines are not so diluted by the incident continuum. Better models of ionized reflection are shown in Fig. \[f:reflion\]. These include both the self consistent emission lines and recombination continua, and the effects of Compton scattering within the disc. By definition, we only see down to a depth of $\tau(E)=1$. Thus the reflected continuum only escapes from above a depth of $\tau(E) \sim 1$. Fig. \[f:refl\]a shows that the iron line is produced in a region with $\tau_T\sim 0.5$, so a fraction $e^{-\tau_T} \sim 1/3$ of the line is scattered. For neutral material, this forms a Compton downscattering shoulder to the line, but for ionized material the disc is heated by the strong irradiation up to the Compton temperature. Compton upscattering can be important as well as downscattering, so the line and edge features are broadened (Young et al 2001, see Fig. \[f:reflion\]b). Models including these effects are publically available as the [xspec]{} [atable]{} model, [reflionx.mod]{}, and this should be used rather than [pexriv]{} for ionized reflection. However, the incident continuum for this model is an exponential power law, so this can have problems with the continuum form at high energies (see Fig. \[f:xspec\_comp\]a).
### Ionization Instability: Vertical Structure of the disc {#s:instability}
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The slab models described above assume that the irradiated material has constant density. Yet if this material is a disc, then it should be in hydrostatic equilibrium, so the density responds to the irradiation heating. The top of the disc is heated to the Compton temperature, so expands, so its density drops, so its ionization state is very high. Further into the disc, hydrostatic equilibrium means that the pressure has to increase in order to hold up the weight of the layers above. The Compton temperature remains the same, so the density has to increase, but an increase in density means an increase in importance of bremsstrahlung cooling. This pulls the temperature down further, but the pressure must increase so the density has to increase further still, so the cooling increases. Eventually the temperature/ionization state drops to low enough levels that not all the material is completely ionized. Bound electrons means that line cooling can contribute, pulling the temperature down even faster, with a corresponding increase in density. The material thus makes a very rapid transition from almost completely ionized to almost completely neutral. Thus the reflection spectrum is a composite of many different ionization parameters, with some contribution from the almost completely ionized skin, and some from the almost completely neutral material underlying the instability point, but with very little reflection at intermediate ionization states (Nayakshin, Kazanas & Kallman 2000; Done & Nayakshin 2007).
The difference in Compton temperature for hard spectral illumination and soft spectral illumination only changes the ionization state of the skin. For hard illumination, the high Compton temperature gives a very low density and high ionization state so the skin is almost completely ionized as described above. For softer illumination, the Compton temperature is lower, so the density is higher and the ionization state of the skin is lower, making it highly ionised rather than completely ionized. However, there is still the very rapid transition from highly ionized to almost neutral due to the extremely rapid increase in cooling from partially ionized material (Fig. \[f:instability\]; see also Done & Nayakshin 2007). Fig. \[f:xion\]a and b shows how this very different vertical structure for temperature-density-ionization affects the expected reflection signature.
The same underlying ionization instability, but for X-ray illumination of dense, cool clouds in pressure balance with a hotter, more diffuse medium (Krolik, McKee & Tarter 1981), may well be the origin of the multiple phases of ionization state seen in the AGN ’warm absorbers’ (e.g. Netzer et al 2003; Chevallier et al 2006).
### Radial Structure
This vertical structure of a disc should change with radius, giving rise to a different characteristic depth of the transition and hence a different balance between highly ionized reflection from the skin and neutral reflection from the underlying material. This does depend on the unknown source geometry as well as the initial density structure of the disc, and how it depends on radius, which in turn depends on the (poorly understood) energy release in the disc (see e.g. Nayakshin & Kallman 2001). The [xion]{} model (Nayakshin) incorporates both the vertical structure from the ionisation instability, and its radial dependence for some assumed X-ray geometry and underlying disc properties.
However, if the material is mostly neutral, then neither vertical nor radial structure gives rise to a change in the reflected spectrum with radius. Neutral material remains neutral as the illumination gets weaker, so these models are very robust.
### Relativistic Broadening {#s:rel}
The reflected emission from each radius has to propagate to the observer but it is emitted from material which is rapidly rotating in a strong gravitational field. There is a combination of effects which broaden the spectrum. Firstly, the line of sight velocity gives a different Doppler shift from each azimuth, with maximum blueshift from the tangent point of the disc coming towards the observer, and maximum redshift from the tangent point on the receding side. Length contraction along the direction of motion means that the emission is beamed forward, so the blueshifted material is also brightened while the redshifted side of the disc is suppressed. These effects are determined only by the component of the velocity in the line of sight, so are not important for face on discs. However, the material is intrinsically moving fast, in Keplerian rotation, so there is always time dilation (fast clocks run slow, also sometimes termed transverse redshift as it occurs even if the velocity is completely transverse) and gravitational redshift.
All these effects decrease with increasing radius. The smaller Keplerian velocity means smaller Doppler shifts and lower boosting giving less difference between the red and blue sides of the line. The lower velocity also means less time dilation while the larger radius means less gravitational redshift. Thus larger radii give narrower lines, so the line profile is the inverse of the radial profile, with material furthest out giving the core of the line and material at the innermost orbit giving the outermost wings of the line (Fabian et al 1989; Fabian et al 2000). The relative weighting between the inner and outer parts of the line are given by the radial emissivity, where the line strength $\propto r^{-\beta}$. This gives $\beta=3$ for either an emissivity which follows the illumination pattern from a gravitationally powered corona, or ’lamppost’ point source illumination. This characteristic line profile is given by the [diskline]{} (Fabian et al 1989) and [laor]{} (Laor 1991) models for Schwarzschild and extreme Kerr spacetimes, respectively.
However, these relativistic effects should be applied to the entire reflected continuum, not just the line. This can be modeled with the [kdblur]{} model (a re-coding of the [laor]{} model for convolution), or the newer [ky]{} models which work for any spin (Dovciak, Karas & Yaqoob 2004). Fig. \[f:rel\]a shows the [reflionx.mod]{} reflection [atable]{} convolved with [kdblur]{} for $r_{in}=30, 6$ and $1.23 R_g$ for $i=60^\circ$. Blueshifts slightly predominate over redshifts, with the ’edge’ energy (actually predominantly set by the blue wing of the line) at 7.8 keV for $r_{in}$ =6 and 1.23 (green and blue, respectively) compared to 7.1 keV for $r_{in}$ = 30 (red) and in the intrinsic slab spectrum (grey). Redshifts are more important for lower inclinations. Fig. \[f:rel\]b shows a comparison of the iron line region for $i=60^\circ$ (dotted lines) with $i=30^\circ$ (solid lines). The ’edge’ energy is now $\sim 6.7$ keV for both $r_{in}$ =1.235 and 6.
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Relativistic smearing is harder to disentangle for ionized material as the iron features are broadened by Compton scattering so there are no intrinsically sharp features to track the relativistic effects (see Fig. \[f:rel\]c and d). Nonetheless, the energies at which the line/edge features are seen are clearly shifted.
### Observations of reflected emission in AGN: iron line and soft X-ray excess
Reflection is seen in AGN. There is a narrow, neutral iron line and reflection continuum from illumination of the torus, and there is also a broad component from the disc. This broad component is often consistent with neutral reflection from material within $50 R_g$ produced from $r^{-3}$ illumination (Nandra et al 2007). However, a small but significant fraction of objects require much more extreme line parameters, with $r_{in}< 3 R_g$, and much more centrally concentrated illumination $\propto r^{-\beta}$ where $\beta=5-6$!!(MCG-6-30-15: Wilms et al 2001; Fabian & Vaughan 2003; 1H0707-495: Fabian et al 2002; Fabian et al 2004; NGC4051: Ponti et al 2006). These all generally high mass accretion rate objects, predominantly Narrow Line Seyfert 1s.
However, the X-ray spectrum is [*not*]{} simply made up of the power law and its reflection. There is also a ’soft X-ray excess’, clearly seen in the spectra of most high mass accretion rate AGN below 1 keV. Fig. \[f:mkn766\]a shows the changing shape of this soft excess for different intensity sorted spectra of Mkn766. This is rather smooth, so looks like a separate continuum component. However, fitting this for a large sample of AGN gives a typical ’temperature’ of this component that is remarkably constant irrespective of mass of the black hole (Czerney et al 2003; Gierlinski & Done 2004b). This is very unlike the behaviour of a disc or any component connected to a disc, requiring some (currently unknown) ’thermostat’ to maintain this temperature. Another, more subtle, problem is that ’normal’ BHB do not show such a component in their spectra, so these AGN spectra do not exactly correspond to a scaled up version of the BHB high/very high states. However, there [*is*]{} such a separate component in the most luminous state of the brightest BHB GRS 1915+105 which may scale up to produce the soft excess in the most extreme AGN (Middleton et al 2009; see Fig. \[f:agn\_variety\]c).
More clues to the nature of the soft excess can come from its variability. These objects are typically highly variable, and the spectrum changes as a function of intensity in a very characteristic way. At the highest X-ray luminosities, the 2-10 keV spectrum can often be well described by a $\Gamma\sim 2.1$ power law, with resolved iron emission line, and a soft excess which is a factor $\sim 2$ above the extrapolated 2-10 keV power law at 0.5 keV. Conversely, at the lowest luminosities, the apparent 2-10 keV power law index is much harder (and there are absorption systems from highly ionized iron), and the soft excess above this extrapolated emission is much larger. Fig. \[f:mkn766\]a shows this spectral variability for Mkn 766.
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The entire spectrum at the lowest luminosity looks like moderately ionized reflection (with the iron K alpha He and H like lines absorption lines superimposed), but the lack of soft X-ray lines (as well as the lack of a resolved iron emission line) means it would have to be extremely strongly distorted by relativistic effects. The classic extreme iron line source, MCG-6-30-15 has similar spectral variability but with stronger ’warm absorption’ features around 0.7-1 keV and stronger ionized iron absorption lines (see Fig. \[f:mkn766\]b). In both these sources, most of the spectral variability can be modeled if there is an extremely relativistically smeared reflection component which remains constant, while the $\Gamma=2.1$ power law varies, giving increasing dilution of the reflected component at high fluxes (Fabian & Vaughan 2003).
The obvious way for the reflected emission to remain constant is if it is produced by far off material, but the extreme smearing requirement conflicts with this. Instead, lightbending from a source very close to the event horizon could give both the required central concentration of the illumination pattern and apparent constancy of reflection if the variability is dominated by changes in source position giving changes in lightbending (Miniutti et al 2003; Fabian et al 2004; Miniutti & Fabian 2004). As the X-ray source gets closer to the black hole, the X-ray emission is increasingly focused onto the inner disc, and so the amount of direct emission seen drops.
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However, there are some physical issues with this solution. Firstly, unlike the iron line, the soft excess requires extreme smearing (small inner radius, strongly centrally concentrated emissivity) in almost all the objects in order to smooth out the strong soft X-ray line emission predicted by reflection (Crummy et al 2006). Secondly, to produce the soft excess from reflection requires that the reflecting material is moderately ionized over much of the disc photosphere. Yet the ionisation instability for material in pressure balance means that only a very small fraction of the disc photosphere can be in such a partially ionised state (see section \[s:instability\]). Either the disc is not in hydrostatic equilibrium (held up by magnetic fields?) or the soft excess is not formed from reflection (Done & Nayakshin 2007).
Instead, it is possible that the soft excess is formed by partially ionised material seen in absorption through optically thin material rather than reflection from optically thick material. However, there is still the issue of the lack of the expected partially ionised lines - in absorption this time rather than from emission (see Fig. \[f:ionise\]b). These could be smeared out in a similar way to reflection if the wind is outflowing (Gierlinski & Done 2004b; Middleton et al 2007), but the outflow velocities required are similarly extreme to the rotation velocities (Schurch & Done 2008; Schurch et al 2009). Even including scattering in a much more sophisticated wind outflow model shows that ’reasonable’ outflow velocities are insufficient to blend the atomic features at low energies into a pseudo-continuum, though this can explain the broad iron line shape without requiring extreme relativistic smearing (Sim et al 2010).
Potentially a more plausible geometry is if the absorber is clumpy, and only partially covers the source, then there are multiple lines of sight through different columns. Any mostly neutral material gives curvature underneath the iron line, making an alternative to extreme reflection for the origin of the red wing (L. Miller et al 2007; 2008; Turner et al 2008). Such neutral material will produce an iron fluorescence line, but this line can also be absorbed, so current observations cannot yet distinguish between these two models (Yaqoob et al 2009). This much more messy geometry perhaps gives more potential to explain the larger range of complex variability seen in some of the other high mass accretion rate AGN, as shown in Fig. \[f:agn\_variety\]a and b.
It is clearly important to find out which one of these geometries we are looking at. Either we somehow have a clean line of sight down to the very innermost regions of the disc despite it being an intense UV source which should be powering a strong wind, or we are looking through a material in strong, clumpy, wind, which has important implications for AGN feedback models, giving another way to strongly suppress nuclear star formation. These questions are currently an area of intense controversy and active research.
### Observations of Reflected Emission and Relativistic smearing in BHB {#s:bhobs}
Reflection is also seen in BHB, and here the controversy over its interpretation occurs at both low and high mass accretion rates.
The amount of reflection is determined by the solid angle subtended by the disc to the hard X-ray source i.e. the fraction of the sky that is covered by optically thick material as viewed from the hard X-ray emission region. The truncated disc models for low mass accretion rates predict that this should increase as the disc moves in towards the last stable orbit, identified with the source making a transition from the low/hard to high/soft state, while the decrease in inner disc radius means that this should also be more strongly smeared by relativistic effects. With RXTE data, the solid angle of reflection increases as expected, but the poor spectral resolution means that it is very difficult to constrain the relativistic smearing. Nonetheless, these do appear more smeared in the RXTE data (Gilfanov, Churazov & Revnivtsev 1999; Zdziarski, Lubinski & Smith 1999; Ibragimov et al 2005, Gilfanov 2010).
These results were derived assuming neutral reflection, whereas the reflected spectrum is plainly ionised in the high/soft states (e.g. Gierlinski et al 1999). While some of this ionization can be from photo-ionization by the illuminating flux, at least part of it must be due to the high disc temperature (i.e. collisional ionization: Ross & Fabian 2007). Thus the high/soft and very high state require fitting with complex ionised reflection models in order to disentangle the relativistic smearing and solid angle from the ionization state. Nonetheless, attempts at this using simplisitic models of ionized reflection ([pexriv]{}) with the RXTE data gave fairly consistent answers. The high/soft data seemed to show that the solid angle subtended by the disc is of order unity, and the (poorly constrained) smearing gives $r_{in}\approx 6$ for emissivity fixed at $\beta=3$, as expected from the potential geometries sketched in Fig. \[f:cygx1\_soft\]b (Gierlinski et al 1999, Zycki, Done & Smith 1998). By contrast, the very high state geometry discussed in section \[s:hybrid\] required that the inner disc is covered by an optically thick corona, predicting a smaller amount of reflection and smearing, again consistent with the RXTE observations (Done & Kubota 2006).
However, the first moderate and good spectral resolution results appeared to conflict with the neat picture described above. The extent of this is best seen in Miller et al (2009), who compiled some XMM-Newton spectra of BHB (plus a few datasets from other satellites) and fit with the best current relativistically smeared, ionised reflection models (together with a power law and disc spectrum). Their Table 3 shows that all the very high state spectra ($\Gamma>2.4$), require a large solid angle of reflection from the very inner regions of the disc, at odds with the geometry proposed from the continuum shape where an optically thick corona completely covers the cool material (Done & Kubota 2006).
There are even worse conflicts with the models for the low/hard state. The truncated disc/hot inner flow models clearly predict that the amount of relativistic smearing should be lower than in the high/soft state. However, XMM- Newton data from low/hard state observations also show a line which is so broad that the disc is required to extend down to the last stable orbit of a high spin black hole. The most famous of these is GX339-4 (Miller et al 2006b; Reis et al 2008), though this one is probably an artifact of instrumental distortion of the data due to pileup (Fig. \[f:lowhard\]a, Done & Diaz-Trigo 2010). However, there are other data which also show a line in the low/hard state which is so broad that the disc is required to extend down to the last stable orbit of a rapidly spinning black hole (SAX J1711.6-3808: Miller et al 2009; GRO J1655-40 and XTE J1650-500: Reis et al 2009, see also Reis et al 2010). While these are not so compelling as the (piled up) data from GX339-4, they still rule out the truncated disc/hot inner flow models for the low/hard state if the extreme broad line is the only interpretation of the spectral shape.
The nature of the low/hard state in BHB and AGN
-----------------------------------------------
This conflict motivates us to look again at the low/hard state in particular, especially as there are other observations which also challenge the hot inner flow/truncated disc geometry. Again, this is currently an area of intense controversy and active research.
### Intrinsic disc emission close to the transition
The high/soft state disc dominated spectra trace out $L \propto T^4$ giving strong evidence for a constant size scale inner radius (see section \[s:lt\]). After the transition to the low/hard state there is still a (weak) soft X-ray component which can be seen in CCD spectra (but not in RXTE due to its low energy bandpass limit of 3 keV). This has temperature and luminosity which is consistent with the same radius as seen in the high/soft state data, implying that the disc does not truncate (Rykoff et al 2007, Reis et al 2010).
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Fig. \[f:lowhard\]b illustrates this with both RXTE (green) and SWIFT (black) data from the transient XTE J1817-330 (Rykoff et al 2007, Gierlinski, Done & Page 2008). The disc goes below the RXTE bandpass just as the source makes a transition from the high/soft state, but its evolution can be followed by the lower energy data from SWIFT. Plainly the SWIFT data show that the disc [*does*]{} recede during (i.e triggering!) the transition, but then in the low/hard state proper, it apparently bounces back to give the same radius as in the high/soft state. However, just after the transition, the disc is not too far recessed, so it can be strongly illuminated by the energetically dominant hard X-ray source. This changes the temperature/luminosity relation from that expected from just gravitational energy release, as shown by the red points in Fig. \[f:lowhard\]. Additionally, the difference in inner boundary condition (going from the stress-free last stable orbit to a continuous stress across the truncated inner disc radius) means that the same temperature/luminosity relation implies the disc is bigger, which would move the two low/hard state points up to 250-300 in these units! This is even before taking into account that some of the disc photons are lost to our line of sight through Compton scattering if the disc underlies some of the hot inner flow. Putting these photons back into the disc gives higher luminosity/larger radii. (Makishima et al 2008).
These data show the difficulty in unambiguously reconstructing the inner radius of the disc when the disc component does not dominate the spectrum. They can be consistent with the disc down at the last stable orbit (Rykoff et al 2007; Reis et al 2010), but they can equally well be consistent with a truncated disc (Gierlinski et al 2008). However, the data during the transition are fairly clear that the disc starts to recede, so it seems most likely to me that that this continues into the low/hard state.
### Intrinsic disc emission at very low luminosities {#s:rep}
The transition is going to be complicated. The disc surely does not truncate in a smooth way, so there can be turbulent clumps, as well as issues with the overlap region suppressing the observed disc while also giving rise to strong illumination as discussed above. Instead, a much cleaner picture should emerge instead from a dimmer low/hard state if the disc truncates, as here it should be far from the X-ray source, so irradiation and overlap effects should be negligible. Yet there is still a weak soft X-ray component, with temperature and luminosity such that the emitting area implied is very small, of order the size scale of the last stable orbit. This has now been seen in several different CCD observations of low $L/L_{Edd}$ sources so is clearly a robust result (Reis, Miller & Fabian 2009; Chiang et al 2009; Wilkinson & Uttley 2009)
However, putting the disc down to the last stable orbit is not a solution to these data. It is then co-spatial with the hard X-ray emission, as this must be produced on small size scales. This then runs into difficulties with reprocessing. If the hard X-ray corona overlies the optically thick disc and emits isotropically then half of the hard X-ray emission illuminates the disc. Some fraction, $a$, (the albedo) of this is reflected, but the remainder is thermalized in the disc and adds to its disc luminosity. The minimum disc emission is where there is no intrinsic gravitational energy release in the disc, only this reprocessed flux. This is $L_{rep}=(1-a) L_h/2 \sim L_h/3$ as the reflection albedo cannot be high for hard spectral illumination as high energy X-rays cannot be reflected elastically. Instead they deposit their energy in the disc via Compton downscattering. Yet we see $L_{soft} \sim L_h/20$. Thus the geometry must be wrong! Either the hard X-ray source is not isotropic, perhaps beamed away from the disc as part of the jet emission, or the material is a small ring rather than a disc so that its solid angle is much less than $2\pi$ for a full disc (Chiang et al 2009; Done & Diaz-Trigo 2010).
However, for one source, XTE J1118+480, the galactic column density is so low that there are simultaneous UV and even EUVE constraints on the spectrum (Esin et al 2001). These show that this soft X-ray component co-exists with a much more luminous, cooler UV/EUV component. If the soft X-rays are the disc, what is the UV/EUV component? Alternatively, since the UV/EUV component looks like a truncated disc, what is the soft X-ray component? It must come from a much smaller emission area than the UV/EUVE emission, and one potential origin is the inner edge (rather than top and bottom face) of the truncated disc (Chiang et al 2009). The truncation region is probably highly turbulent, so there can be intrinsic variability produced by clumps forming and shreding, as well as them reprocessing the hard X-ray irradiation. This may also explain the variability seen in this component (Wilkinson & Uttley 2009).
Thus in my opinion, none of the current observations require that there is a disc down to the last stable orbit in the low/hard state. More fundamentally, it is very difficult to make such a model not conflict with other observations. Reprocessing limits on the hardness of the spectrum requires that the X-ray source is either patchy or beamed away from the disc (Stern et al 1995; Beloborodov 1999). A patchy corona would give a reflection fraction close to unity, which is not observed even considering complex ionization of the reflected emission (Barrio et al 2003; Malzac et al 2005). Beaming naturally associates the X-ray source with the jet, but is unlikely to be able to simultaneously explain the extreme broad line parameters derived for some low/hard state sources since the illumination pattern becomes much less centrally concentrated by the beaming. I suspect that more complex continuum modelling may make these lines less extreme, but this then removes a challenge to the truncated disc as well! And unlike the beaming models, the truncated disc/hot inner flow can additionally give a mechanism for the major state tranistions, and the variability.
Conclusions
-----------
The intrinsic radiation processes of blackbody radiation and Comptonization go a long way to explaining the underlying optical-to-X-ray continuum seen in both BHB and AGN. These, together with the atomic processes of absorption and reflection, and relativistic effects in strong gravity give us a toolkit with which to understand and interpret the spectra of the black hole accretion flows. This is currently an area of intense and exciting research, to try to understand accretion in strong gravity. If you got this far, congratulations, and come and join us: we get to play around black holes!
Acknowledgements
----------------
I would like to thank the organizers of the IAC winter school for inviting me to give this series of lectures, finally giving me the motivation to write these things down. But I only know these things because of the many people who have given me their physical insight on radiation processes, especially Andy Fabian and Gabriele Ghisellini. I also thank ISAS and RIKEN for visits during which I developed some of these lectures. It also could not have been written without the 8 hours spent at Tenerife South Airport where their lack of baggage check facility put paid to my plan to spend the day at a nearby surfing beach!
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---
abstract: 'X-ray charge-coupled devices (CCDs) are the workhorse detectors of modern X-ray astronomy. Typically covering the 0.3–10.0 keV energy range, CCDs are able to detect photoelectric absorption edges and K shell lines from most abundant metals. New CCDs also offer resolutions of 30–50 (E/$\Delta$E), which is sufficient to detect lines in hot plasmas and to resolve many lines shaped by dynamical processes in accretion flows. The spectral capabilities of X-ray CCDs have been particularly important in detecting relativistic emission lines from the inner disks around accreting neutron stars and black holes. One drawback of X-ray CCDs is that spectra can be distorted by photon “pile-up”, wherein two or more photons may be registered as a single event during one frame time. We have conducted a large number of simulations using a statistical model of photon pile-up to assess its impacts on relativistic disk line and continuum spectra from stellar-mass black holes and neutron stars. The simulations cover the range of current X-ray CCD spectrometers and operational modes typically used to observe neutron stars and black holes in X-ray binaries. Our results suggest that severe photon pile-up acts to falsely narrow emission lines, leading to falsely large disk radii and falsely low spin values. In contrast, our simulations suggest that disk continua affected by severe pile-up are measured to have falsely low flux values, leading to falsely small radii and falsely high spin values. The results of these simulations and existing data appear to suggest that relativistic disk spectroscopy is generally robust against pile-up when this effect is modest.'
author:
- 'J. M. Miller, A. D’Aì, M. W. Bautz, S. Bhattacharyya, D. N. Burrows, E. M. Cackett, A. C. Fabian, M. J. Freyberg, F. Haberl, J. Kennea, M. A. Nowak, R. C. Reis, T. E. Strohmayer, M. Tsujimoto'
title: 'On Relativistic Disk Spectroscopy in Compact Objects with X-ray CCD Cameras'
---
Introduction
============
X-ray observations of accreting neutron stars and black holes necessarily probe regions close to the compact object. The long–held promise of these observations is that aspects of the compact object itself, and the innermost accretion flow, may be revealed. Notable early attempts to realize this promise took several forms. For instance, many attempts were made to use the blackbody spectra of Type-I X-ray bursts in accreting neutron star systems to infer the stellar radius; however, derived radii were implausibly small, likely due to scattering effects (see London, Taam, & Howard 1986). In the case of black holes, careful efforts to measure the inner radius of the accretion disk (see, e.g., Makishima et al. 1986) were similarly complicated by various observational uncertainties such as mass and distance, and also by scattering effects (Shimura & Takahara 1995; Merloni, Fabian, & Ross 2000).
At the most basic level, measurements of stellar or disk radii using blackbody or modified blackbody continua amount to counting photons under a curve to measure an absolute flux. This is extremely difficult (though not impossible), and easily complicated by additional continuum components and detector flux calibration uncertainties. Partially owing to difficulties associated with continuum spectroscopy, and partially owing to the detection of broad, possibly relativistic Fe K emission lines in Cygnus X-1 (Barr, White, & Page 1985) and 4U 1543$-$475 (van der Woerd, White, & Kahn 1989; also see Park et al. 2004), parallel efforts to measure fundamental parameters focused on line spectroscopy.
Theoretical line models for non-spinning (Schwarzschild; $a = 0$, where $a = cJ / GM^{2}$) black holes and maximal-spin (Kerr $a =
0.998$; Thorne et al. 1974) black holes have been calculated for some time (Fabian et al. 1989; Laor 1991). Self-consistent models for hard X-ray illumination of an accretion disk that could give rise to such emission lines also have a long history (e.g. George & Fabian 1991). These models predict that lines arising in the inner disk should have skewed, asymmetric shapes, including a strong blue wing that is enhanced by special relativistic beaming and a long red wing that arises through Doppler shifts and gravitational red-shifts. Line shapes are very difficult to discern at the resolution afforded by gas proportial counter spectrometers (e.g. $E/\Delta E \simeq 6$). Confirmation of this line shape was first achieved in an [*ASCA*]{} observation of the Seyfert-1 galaxy MCG-6-30-15 (Tanaka et al. 1995). It was facilitated by the X-ray charge-coupled devices (CCDs) aboard the mission, which delivered a resolution of $\sim 12$ in the Fe K band. Relativistic iron lines are merely the most prominent part of the broad-band response of an accretion disk to hard X-rays, known as the disk reflection spectrum (see, e.g., George & Fabian 1991; Magdziarz & Zdziarski 1995; Dovciak, Karas, & Yaqoob 2004; Ross & Fabian 2005).
X-ray CCDs and relativistic spectroscopy of black holes and neutron stars are thus intimately linked. The advanced CCD spectrometers aboard [*Chandra*]{}, [*XMM-Newton*]{}, and [*Suzaku*]{} have confirmed relativistic lines in a number of Seyfert-1 spectra (for a review, see Miller 2007). Just as importantly, the ability of these spectrometers to handle high flux levels has made it possible to clearly detect asymmetric lines in the spectra of stellar-mass black holes, and to reject the possibility that broad lines are actually a collection of narrow lines (see, e.g., Miller et al. 2001). If the inner disk is assumed to be truncated at the innermost stable circular orbit (ISCO), which is set by the spin of the black hole (see Bardeen, Press, & Teukolsky 1972), relativistic lines can be used to infer black hole spin parameters. In a number of stellar-mass black holes, the lines observed are sufficiently broad that varying degrees of black hole spin may be required (e.g. Miller et al. 2002, 2004, 2008; Reis et al. 2009; Hiemstra et al. 2010; for a self-consistent analysis of eight systems, see Miller et al. 2009).
Very recently, X-ray CCD spectroscopy of transient and persistent neutron star X-ray binaries has revealed skewed, asymmetric disk lines in these systems (see, e.g., Bhattacharyya & Strohmayer 2007, Cackett et al. 2008, 2009, Di Salvo et al. 2009, D’Aì et al. 2009, D’Aì et al. 2010; for a self-consistent analysis of 10 systems, see Cackett et al. 2010.) In these sources, relativistic disk lines can be exploited to constrain the radius of the star, since the stellar surface (if nothing else) must truncate the disk wherein the lines arise. The inner extent of the disk is also related to the Alfven radius, and so can provide a constraint on stellar magnetic fields. In the case of the relativistic line observed in the millisecond X-ray pulsar SAX J1808.4$-$3658 (Cackett et al. 2009, Papitto et al.2009), the resulting field limits are commensurate with those derived from X-ray timing (Cackett et al. 2009).
New, parallel efforts to derive fundamental properties of compact objects and inner accretion flows have been fueled by new disk models. New prescriptions explicitly incorporate inner torque conditions, radiative transfer through the disk atmosphere, and even black hole spin parameters (Zimmerman et al. 2005, Li et al. 2005, Davis & Hubeny 2006). Using these models, black hole spin parameters have been constrained in a number of systems (see, e.g., McClintock et al.2006, Shafee et al. 2006; also see Zhang, Cui, & Chen 1997). The improved low-energy range of X-ray CCD detectors (often extending down to 0.3 keV) relative to current gas detectors (coverage below $\sim3$ keV is not possible with [*RXTE*]{} nor [*INTEGRAL*]{}) allows for improved measurements of the disk flux, and enables observers to separate disk emission from line-of-sight absorption in the interstellar medium (though in many cases dispersive spectroscopy may be required for this purpose; Miller, Reis, & Cackett 2009). Thus, while the improved energy resolution of X-ray CCD spectrometers is not necessarily critical for continuum spectroscopy, their energy range and spectral resolution are beneficial.
While measuring the width of lines is often easier than measuring an absolute flux, it still requires an accurate characterization of the underlying continuum flux, and an accurate knowledge of how the spectrometer reacts to high flux levels. The latter issue is typically unimportant for gas proportional counter spectrometers, but it can be important for X-ray CCD spectrometers. Indeed, the reaction of X-ray CCDs is important for both forms of relativistic spectroscopy of compact objects: an instrumental failure to accurately record the flux level is problematic for disk continua, and distortions to the continuum shape and energy response are problematic for line spectroscopy. Photon pile-up occurs when two or more photons land within a detection cell within a single CCD frame time. This causes a degeneracy between detecting ${\rm N}$ photons with energy ${\rm
E}_{i}$ and a single photon with energy of $\Sigma_{\rm i}^{\rm
N}~{\rm E}_{\rm i}$. Thus when pile-up occurs, energy and flux information are lost.
It has recently been suggested that ineffective pile-up mitigation could produce spurious results concerning black hole spin, neutron star radii, and the radial extent of accretion disks as a function of the mass accretion rate (e.g. Yamada et al. 2009; Done & Diaz Trigo 2010, Ng et al. 2010). Advanced statistical descriptions of photon pile-up in X-ray CCD detectors have been developed (Davis 2001; also see Ballet 1999), and implemented into fitting packages such as XSPEC and ISIS. The Davis et al. (2001) model has been applied in a number of regimes to correct for photon pile-up distortions, including spectra from: isolated neutron stars (e.g. van Kerkwijk et al. 2004), ultra-luminous X-ray sources (e.g. Roberts et al. 2004), transient Galactic black hole candidates (e.g. Jonker et al. 2004), low-mass X-ray binaries (e.g. Heinke et al. 2006), and even AGN (e.g. Wang et al. 2010). Most notably, Nowak et al. (2008) employed the basic elements of the Davis model to correct distortions to the [*Chandra*]{} spectra of the black hole candidate 4U 1957$+$11. Inner disk parameters consistent with those derived using spectra from other missions, including [*XMM-Newton*]{} and [*RXTE*]{}, were then recovered. A similar treatment of [*Chandra*]{} spectra of Cygnus X-1 by Hanke et al. (2009) was able to bring different spectra into close agreement.
With the aim of developing a rigorous understanding of how pile-up may affect relativistic spectroscopy, we have conducted extensive simulations based on a range of assumed input spectra, flux levels, and detector properties. This effort is timely as X-ray CCD spectrometers will be the standard in the field for the foreseeable future, at least for moderate resolution spectroscopy of bright targets. In the following sections, we briefly review the operation of X-ray CCD detectors, review the pile-up model used in our simulations, describe our simulations and results, and discuss the impacts of our findings.
The primary goal of this analysis is not to understand how to best avoid photon pile-up, though the results offer some insights on this point. Rather, the goal of this exercise is to understand the nature and magnitude of distortions to relativistic disk parameters when photon pile-up is present but unknown to an observer, poorly quantified, or difficult to mitigate in a rigorous manner. In a subset of our simulated spectra, then, photon pile-up is quite severe. When faced with similar real data, an observer would likely be motivated to extract an annulus or to otherwise at least partially mitigate the effects of pile-up.
The Basics of X-ray CCD Spectrometers
=====================================
At the most basic level, a CCD is an array of coupled capacitors. When an incident photon interacts in the semiconductor substrate, electrons are liberated. The charge is collected and stored in pixels, and it can be transfered to neighboring pixels – and eventually into an amplifier and read-out system – by virtue of the coupling.
In many respects, the operation of X-ray CCDs is similar to that of optical CCDs. One important difference is that optical photons only liberate a small number of electrons (typically zero or one) in an interaction. X-ray photons have much more energy, of course, and liberate many electrons. Indeed, the number of electrons liberated in an interaction is proportional to the energy of the incident X-ray. This fact means that X-ray CCDs are not merely imaging instruments, but medium-resolution spectrometers as well. The gas proportional counters aboard [*RXTE*]{} have an energy resolution of $\sim6$; current X-ray CCDs (e.g. [*Chandra*]{}/ACIS) achieve an energy resolution of $\sim30$ at 5.9 keV.
Most X-ray CCDs cover the 0.3–10.0 keV band. These limits are set by a combination of factors, including the efficiency of capturing and clocking charge clouds generated by low energy X-rays (this is partially set by the electrodes), the depletion layer thickness, the ability of current X-ray mirrors to focus hard X-rays, and the ability of the instrument to distinguish hard X-rays from non-X-ray events. The detailed performance characteristics of a given X-ray CCD depend on several variables, including the nature of the device structure (n- or p- substrate, MOS or pn junction); whether charge is stored and read-out close to the X-ray-illuminated face of the CCD, or far from the illuminated face; the temperature at which the CCD is being operated (this affects dark currents and noise), and a number of more subtle factors.
Photons are detected and elementary screening of desirable (e.g. X-ray) versus undesirable (e.g. cosmic ray) events is achieved in part by assigning event “grades”. Interactions are characterized using event “boxes”, which are usually 3x3 cells of pixels. The pattern of pulseheights that are consistent with photon interactions can be defined and differentiated from “hot” pixel and cosmic ray patterns.
Efficient spectroscopy with an X-ray CCD depends on each event box recording zero or one photons per CCD frame time. When the X-ray flux incident on a CCD is too high, multiple photons can be read as a single high energy photon (see above). The resulting flux is thus registered as falsely low, and the resulting spectrum as falsely hard. Grade migration provides a potential means of recognizing and diagnosing the degree of photon pile-up suffered in a given observation. When pile-up is important, charge patterns consistent with single photon interactions are reduced and patterns consistent with higher energy photon interactions or multiple photon interactions are increased.
Various strategies can be deployed to deliver nominal spectroscopic response even at high X-ray flux levels. The nominal CCD frame time can be greatly reduced, thus lowering the probability of two photons registering in a given event box. This often comes at the expense of imaging information, observing efficiency, or both. Over-sampling a broad telescope PSF with a high number of event boxes is another viable means of limiting the number of photons that may land in a given event box. The expense of a broad PSF is, of course, the loss of fine image quality. Observers can try to further prevent or limit photon pile-up by only extracting events from the wings of the PSF in annuli, by placing a source off of the optical axis of the telescope to blur its flux over more pixels, or by adopting a rigid grade selection that permits only single-photon events. In practice, however, some spectra obtained with X-ray CCDs will suffer from photon pile-up.
The Davis Photon Pile-up Model
==============================
Pile-up is a complex but well-understood statistical process: At a given flux level, a given event box has a fixed probability of getting 0, 1, or N photons. The probability of a given event grade for each outcome is also well-determined. If the input spectrum is known, or if it can be estimated, the distribution of “good” event grades (those likely to originate from actual X-ray photons) can be used to infer the extent and severity of photon pile-up. On this basis, Davis (2001) has developed a model of photon pile-up. The model not only accounts for the energy shifts due to photon pile-up, but also for flux decrements due to grade migration.
The crux of the Davis model is a new integral equation for the number of counts detected from a point source. The conventional equation for the number of counts detected in an X-ray spectrometer is linear, and discussed in detail in Gorenstein, Gursky, & Garmire (1968). However, pile-up is a nonlinear process, and requires a more advanced treatment. Davis (2001) presents an alternative integral equation that accurately characterizes the counts observed in the presence of photon pile-up. A fundamental assumption of the pile-up model is that a charge cloud in the CCD arising from N photons can be treated as the linear superposition of N individual charge clouds. Since the drift time for charge clouds in a CCD is typically much less than a microsecond, which is much smaller than typical readout times, and since a CCD amplifier is highly linear at X-ray signal levels, this assumption is valid for all of the CCD modes that will be considered in this work.
In practice, the chief uncertainty in correcting for pile-up using this model is that the details of the input spectrum are usually not known a priori. Exceptions might include instances where different missions are simultaneously observing the same source, or an instance wherein a given CCD spectrometer alternates between different operational modes. [*This uncertainty does not enter into our analysis.*]{} Simulating the effects of photon pile-up by convolving different input spectra with the pile-up model of Davis (2001) is a well-controlled experiment – the nature of the input spectrum is known perfectly. The results of this procedure can be thought of as characterizing the nature and severity of systematic errors on relativistic disk parameters arising due to photon pile-up.
It is worth noting that although the pile-up model was developed for [*Chandra*]{}, it is easily applicable to other missions. The crucial modification in describing pile-up in other CCD spectrometers is to accurately account for the fraction of the telesope point spread function (PSF) that is enclosed in each event box. This can differ considerably from mission to mission. Whereas a single event box captures a high fraction of [*Chandra’s*]{} PSF, a single event box captures a very small fraction of [*Suzaku’s*]{} PSF. In the latter case, to evaluate pile-up in a reasonable extraction region, many event boxes must be considered jointly.
As implemented in the XSPEC spectral fitting package (Arnaud & Dorman 2000), the Davis pile-up model has six parameters: the CCD frame time, the maximum number of photons to pile-up, the grade correction for a single photon, the grade morphing parameter (the grade migration function is a probability assumed to be proportional to $\alpha^{p-1}$, $0 \leq \alpha \leq 1$, where $\alpha$ is the grade morphing parameter and $p$ is the number of piled photons), the PSF fraction considered, and the number of event regions. This description of grade migration is based on the premise that photon arrival times in a given cell obey Poission statistics. It correctly captures the fact that the probability of recording an unwanted event grade increases with the number of piled photons.
The Simulated Spectra
=====================
To understand the specific effects of photon pile-up on disk continua and reflection parameters, we created a large number (325) of simulated spectra using the Davis pile-up model. Realistic, multi-component spectral forms based on published results were employed to make the simulations as realistic as possible. The spectra were generated using the “fakeit” command in XSPEC version 12. For simplicity, and because this exercise is necessarily concerned with bright sources, no backgrounds were used in making the source spectra. In all cases, an exposure time of 100,000 seconds was used when creating simulated spectra. This exposure is longer than a typical observation, but allows for excellent photon statistics.
In all of the simulations made in this work, default values were assumed for the maximum number of photons to pile-up (5) and the grade correction for a single photon (1). The frame time, PSF fraction, and number of event boxes were set according to the telescope and CCD combination used in each simulation. With these parameters fixed, the grade morphing parameter $\alpha$ effectively sets the severity of photon pile-up (when the incident source flux in each event box leads to pile-up).
As implemented in XSPEC, the pile-up model is a convolution model. Input spectra were convolved with the pile-up kernel to produce a resultant spectrum that is distorted by photon pile-up effects. In practice, for a given detector and mode, the degree of pile-up is set by the incident flux level. In our simulations, the degree of photon pile-up was controlled by adjusting the grade morphing parameter $\alpha$. Steps of 0.1 were used for $0.1 \leq \alpha \leq 0.8$, and steps of 0.04 were used for $0.8 \leq \alpha \leq 0.99$. The step size was reduced for high values of $\alpha$ in order to provide better resolution when pile-up typically starts to introduce strong distortions. For each combination of input spectrum and detector mode, then, 13 simulated spectra were generated.
Input Spectral Forms
====================
X-ray binaries display a wide variety of phenomena. Periods of correlated multi-wavelength behaviors can be classified into “states” that may correspond to distinct changes in the accetion flow (for a review, see, e.g. Remillard & McClintock 2006; also see Belloni et al. 2005). In this work, we consider three commonly-recognized “states” for black hole X-ray binaries, and a single input spectral form for neutron star low-mass X-ray binaries (the so-called “Z” and “atoll” sources). In the interest of simplicity and reproduciblity, we have adopted phenomenological spectral models for each state with values consistent with those reported in the literature. Table 1 lists values for all of the input spectra considered in this work. The paragraphs below offer some context and motivation for why certain models and values were used to generate the spectra.
[*The Very High State*]{}: This state can be extremely bright, and it has seldom been observed using CCD spectrometers. Continua observed in this state often consist of a hot disk and steep power-law component. We assumed the continuum parameters measured in the very high state of GX 339$-$4 using the [*XMM-Newton*]{}/EPIC-pn in “burst” mode (Miller et al. 2004). This mode has a frametime of just 7 $\mu$sec and a deadtime of 97%, but it is effective at preventing photon pile-up for fluxes up to 6 Crab. The particular spectrum chosen as a template is broadly consistent with very high state spectra obtained from [*RXTE*]{} monitoring of other transients (e.g. 4U 1543$-$475, Park et al. 2004; GRO J1655$-$40, Sobczak et al. 1999; XTE J1550$-$564, Sobczak et al. 2000, Miller et al.2003).
The “intermediate” state is similar to the “very high” state in terms of its spectral form (both thermal disk and non-thermal power-law-like emission are important), but typically has a lower flux. The effects of photon pile-up in this state are likely to be similar to the effects on very high state spectra at low values of the grade migration parameter $\alpha$. The “intermediate” state is not treated separately in this work.
[*The High/Soft State*]{}: This state can also be quite bright, but it typically persists for a longer period than the “very high state”. As a result, it is more commonly observed with CCD spectrometers. Continua observed in this state are strongly dominated by a hot disk component, accompanied by a weak, steep power-law. To model the high/soft state, we selected one of the spectrally softest observations of 4U 1543$-$475 obtained using [*RXTE*]{} (Park et al. 2004). This observation occurred on MJD 52475, and is consistent with the spectra selected for relativistic disk spectroscopy by Shafee et al. (2006). Here again, this high/soft state is typical of spectra of other black hole transients in the same state (see, e.g., Sobczak et al. 1999, 2000).
[*The Low/Hard State*]{}: The low/hard state has been a frequent target with CCD spectrometers recently, in an effort to understand the accretion flow at low mass accretion rates (for a recent survey, see Reis, Fabian, & Miller 2010, also see Tomisck et al. 2009). Spectral continua in this state are dominated by a hard power-law, sometimes with weak emission from the accretion disk. The continuum spectrum assumed in our simulations is a composite of the parameters reported in observations of GX 339$-$4 by Miller et al. (2006), Tomsick et al. (2008), and Wilkinson & Uttley (2009).
[*A Typical Z/Atoll Spectrum*]{}: Unlike most black hole X-ray binaries with low-mass companions, neutron stars with low magnetic fields are typically persistent (but variable) sources. Whereas black hole continua can often be well-characterized in terms of thermal disk emission and a power-law, neutron star spectra often require a third component (Lin, Remillard, & Homan 2007; Cackett et al. 2008, Cackett et al. 2010, D’Aì et al. 2010). A simple blackbody likely corresponds to emission from the boundary layer between the accretion disk and stellar surface (Revnivtsev & Gilfanov 2006). Emission from this region may be Comptonized, but blackbody emission that is Compton-upscattered in a region with high $\tau$ and low $kT_{e}$ can be modeled with a hotter, smaller blackbody (London, Taam, & Howard 1986). We used the spectral continuum parameters reported by Cackett et al. (2008) for Serpens X-1.
Relativistic Line Properties
----------------------------
Relativistic line components were added to all continuum spectra, apart from those for the “high/soft” state. The “Laor” relativistic emission line model (Laor 1991) was assumed to simulate emission lines from the inner disk. In all cases, the input inner radius was fixed at $6~GM/c^{2}$, the emissivity index was fixed at $q=3$ (where $J(r) \propto r^{-q}$), and the inclination was fixed at $30^{\circ}$. This radius is substantially larger than the innermost stable circular orbit for a maximally-spinning Kerr black hole ($1.24~GM/c^{2}$), and significantly larger than measured in a number of stellar-mass black holes (Miller et al. 2009). However, one purpose of this investigation is to see if photon pile-up can artificially [ *broaden*]{} relativistic lines, giving falsely small inner radii and falsely high spin values. Setting the inner radius at $6~GM/c^{2}$, then, permits distortions due to photon pile-up ample opportunity to produce falsely broad lines. For all simulated black hole spectra, the input flux of the relativistic “Laor” line was normalized to give an equivalent width of $300$ eV as per Miller et al. (2004).
Similarly, $6~GM/c^{2}$ is approximately 12.4 km for a 1.4 $M_{\odot}$ neutron star – only slightly larger than the canonical stellar radius of 10 km employed in many circumstances. Assuming the stellar surface (if not the boundary layer) must truncate the accretion disk, a relativistic line suggesting a smaller radius might be taken as evidence of a “strange” star. In the case of neutron stars, then, fixing the input radius at $6~GM/c^{2}$ serves to assess the ability of photon pile-up to give false evidence of “strange” stars through falsely small disk radii. For all simulated neutron star spectra, the input flux of the “Laor” line was normalized to given an equivalent width of $150$ eV as per Cackett et al. (2008).
Iron emission lines in accreting black holes and neutron stars are merely the most prominent part of the disk reflection spectrum (e.g. Dovciak, Karas, & Yaqoob 2004; Ross & Fabian 2005). Self-consistent modeling of real spectra requires modeling of the entire reflection spectrum, not just the broadened emission line. For simplicity, and because the line drives constraints on inner disk radii (and therefore spin) in fits to real data, disk reflection models are not treated in this analysis.
The Thermal Disk Continuum
--------------------------
In all spectra, thermal continuum emission from the accretion disk was simulated using the simple “diskbb” model (Mitsuda et al. 1984) within XSPEC. Using this model, the inner disk radius can be derived via: $r = (d / 10 {\rm kpc}) \times (K/{\rm cos}\theta)^{1/2}~{\rm km}$, where $K$ is the model flux normalization, $d$ is the distance to the source, and $\theta$ is the inclination of the inner disk. If the mass of the source is known, this radius can be converted to gravitational units.
This is merely a “color” radius, however, and does not account for the effects of spectral hardening due to radiative transfer through the disk atmosphere (Shimura & Takahara 1995; Merloni, Fabian, & Ross 2000). It is well-known that this model is overly simple in other ways. For instance, it does not include a zero-torque condition at the inner edge, whereas including this condition can reduce implied radii by a factor of $\sim$2 (Zimmerman et al. 2005). Moreover, a number of new disk models have been developed in which black hole spin is an explicit parameter (Davis & Hubeny 2006).
Deriving spins using these newer, more physical models requires extremely accurate knowledge of the absolute disk flux, however, which can be complicated by a number of effects (e.g. the flux calibration of the detector, the accuracy to which line of sight absorption is known, the nature of the hard component, etc.). And whereas new models can have as many as 10 free parameters, the “diskbb” model is able to accurately characterize the thermal continuum with only two (temperature and flux normalization). Owing to its simplicity and prior application to many neutron star and black hole spectra over numerous years and X-ray missions, then, all thermal disk continua were simulated using the “diskbb” model, and our analysis is primarily concerned with changes in the flux normalization parameter.
In the Z/atoll spectra, a simple (single-temperature) blackbody function was used to describe emission from the stellar surface, independent of the thermal emission from the disk.
Interstellar Absorption
-----------------------
Each input spectrum was modified by interstellar absorption using the “tbabs” model (Wilms, Allen, & McCray 2000). In all cases, the equivalent neutral hydrogen column density was fixed at ${\rm N}_{\rm
H} = 5.0 \times 10^{21}~ {\rm cm}^{-2}$. This is a moderate value, consistent with columns that have facilitated studies of the thermal disk continuum in a number of sources and at different mass accretion rates.
Simulations with Gaussian Lines
-------------------------------
Although a number observations made with the [*Chandra*]{}/HETGS (e.g. Di Salvo et al. 2005) and with [*Suzaku*]{} (Cackett et al.2008, Reis, Fabian, & Young 2009, Cackett et al. 2010) find broad, relativistic line shapes in the spectra of neutron stars, such lines are less established in neutron star spectra than in black hole spectra. Ng et al. (2010) suggest that neutron star lines may actually be narrower ($\sigma = 0.33$ keV, on average) and symmetric. Therefore, we also simulated neutron star spectra using the same neutron star continuum described above and in Table 1, but with Gaussian model with ${\rm E} = 6.7$ keV, $\sigma = 0.33$ keV, and an equivalent width of 150 eV, rather than a relativistic line.
Detectors and Modes
===================
The CCD spectrometers and modes considered in our simulations are detailed in Table 2. The modes selected reflect those that users might realistically select when observing bright sources. The “burst” mode of the EPIC-pn camera aboard [*XMM-Newton*]{} is not considered, as it is capable of handling exceptionally high flux levels without pile-up distortions. Common imaging modes available on the ACIS and XRT instruments aboard [*Chandra*]{} and [*Swift*]{}, respectively, are also not considered; these are known to readily suffer pile-up for even moderately bright sources and are typically avoided by users. Finally, with [*Chandra*]{}, pile-up can be largely mitigated while obtaining a high-resolution spectrum with the HETGS; however, assessing pile-up in this circumstance is especially complex and beyond the scope of this investigation.
An “effective” frame time is given in Table 2. This is the frame time that is important to consider when assessing the extent and impacts of photon pile-up. It multiplies the time required to extract one element of charge (typically a pixel, but sometimes a larger “macropixel”) by the number of elements necessary to define an event box. (Some missions – but not all – redefine event boxes for different modes; see the discussion below.) If one could know a priori that a sequence of events were all “singles” wherein charge from photon events was contained in only one pixel, then the effective frame time would equal the time required to clock just one row of pixels. However, it is not possible to know this a priori, and [ *in practice one only knows that an event was a “single” if the other rows in an event box are free of charge*]{}. Thus, for pile-up, the time scale of concern is the time required to transfer a full event box.
A significant simplification in our simulations is that the grade morphing parameter $\alpha$ is assumed to be constant in each extraction region. In practice, this is not true; $\alpha$ would depend on radius as an actual telescope PSF focuses successively less energy into successively larger annuli. Our simulations therefore capture the strongest impacts of photon pile-up distortions, and ignore how less distorted spectra from large annuli might affect results when summed with spectra from smaller annuli. In a partial effort to minimize the impact of the assumption of a constant $\alpha$, the extraction regions assumed in our simulations do not encircle the same fraction of the total incident energy, but rather attempt to sample the part of the PSF where the encircled energy changes only linearly with radius. This is not practical in the case of [*Chandra*]{}, however, which has a very tight PSF. A secondary simplification is that the pile-up model and our simulations do not consider how charge from event boxes might bleed into adjacent boxes in the case of severe pile-up.
Additional details concerning the individual spectrometers and modes considered, and how they are treated in our simulations, are given below:
Chandra/ACIS Continuous Clocking Mode
-------------------------------------
ACIS event boxes consist of a 3x3 box of pixels, each approximately 0.5” on a side. In continuous clocking or “CC” mode, charge is continuously transferred from the CCD (usually ACIS-S3) at a rate of 2.85 msec per row. Two-dimensional imaging is sacrificed in favor of fast read-out. The definition of ACIS event boxes is unchanged between standard imaging modes and CC mode. In our simulations, then, we have assumed an effective frametime of 8.55 msec, since this is the time require to clock a full event box.
As noted above, given the tight PSF of [*Chandra*]{}, it is impractical to extract photons from a region that encircles less than 90% of the incident energy. In our simulations, we therefore assumed an extraction radius of 2.5”, which corresponds to 90% of the incident energy. In the one-dimensional regime of CC mode, the extraction region is then a bar 5” in length. Note that this means that only 3.3 event boxes tile the extraction region.
Where possible, generic, readily-available response matrices were used in order to facilitate checks on this work by other teams. To simulate ACIS CC mode spectra, then, we used the ACIS response matrices made available by the mission for simulations supporting Cycle 11 proposals: “aciss\_aimpt\_cy11.rmf” and “aciss\_aimpt\_cy11.arf”. For more information about the ACIS spectrometer and CC mode, please see Garmire et al. (2003) and the [*Chandra*]{} Proper’s Observatory Guide (http://cxc.harvard.edu/proposer/POG/).
Swift XRT Windowed Timing Mode
------------------------------
In “windowed timing” or “WT” mode, charge rows are grouped by 10 in the read-out direction (effectively creating “macropixels”) and rapidly tranferred from the CCD. In the standard photon counting mode, a 3x3 grouping of pixels (each pixel is 2.36” on a side) is used to define an event box. However, in windowed timing mode, the event box is altered to be 10x7 pixels: 1 macropixel in the clocking direction, and 7 pixels in the orthogonal direction. This change improves the ability of the detector to identify and screen events in windowed timing mode. Note that because the new event box is only 1 macropixel in the clocking direction, the nominal and effective frame times are the same in windowed timing mode (see Table 2). Like [*Chandra*]{}/ACIS “continuous clocking” mode, “windowed timing” mode has a livetime fraction of 1.0.
In our simulations, we assumed an extraction radius of 9.0”, which roughly corresponds to the half power radius for the [*Swift*]{}/XRT. The extraction region is tiled by only 1.1 event boxes. For simplicity, we have also assumed an effective frame time of 1.77 msec (see Table 2). In practice, depending on where a photon strikes, it may take up to two read-out times of 1.77 msec to clock the charge through the extraction region (which roughly matches the half-power diameter of the PSF). In this sense, our simulations slightly under-estimate the severity of pile-up distortions to [*Swift*]{}/XRT spectra obtained in “windowed timing” mode.
To simulate “windowed timing” mode spectra, we used current, standardized response functions available through the HEASARC calibration database, “swxwt0to2s6\_20070901v011.rmf” and “swxwt0to2s6\_200101 01v011.arf”. These responses have been developed specifically for WT mode, and differ from the responses appropriate for normal imaging modes.
For more information about the [*Swift*]{}/XRT, please see Hill et al. (2004), Burrows et al. (2005), the [*Swift*]{} Technical Handbook (http://swift.gsfc.nasa.gov/docs/swift/proposals/appendix\_f. html) and the XRT User’s Guide (http://swift.gsfc.nasa.gov/docs/ swift/analysis/xrt\_swguide\_v1.2.pdf).
XMM-Newton EPIC-pn “Timing” Mode
--------------------------------
The “timing” mode of the EPIC-pn camera is similar to the “windowed timing” mode of the [*Swift*]{}/XRT in some respects. To achieve a high time resolution, normal CCD pixels are grouped into “macropixels”. Each macropixel is 10 pixels in the read-out direction and 1 pixel in the orthogonal direction. Since each pixel is 4.1” on a side, this means that a single macropixel is actually 41” in the read-out direction. In its long dimension, then, a single macropixel is equivalent to the 70% encircled energy diameter of the telescope. In standard imaging modes, [*XMM-Newton*]{} event boxes are the standard 3x3 pixel element, as per other X-ray CCD cameras. In timing mode, however, an event box is 3 macropixels by 3 macropixels. This means that a single event box is 123” long (equivalent to the 95% encircled energy diameter) and 12.3” wide (appoximately equal to the half-power diameter). This event box is extremely large, and roughly equivalent to the extraction region that is often used by observers using this mode.
Creating macropixels enables a short read-out time of 0.03 msec [ *per macropixel*]{}. This translates to an effective frame time of 0.09 msec [*per event box*]{}. Again, this is a simplification that may serve to under-estimate the severity of photon pile-up distortions to “timing” mode spectra. In practice, each macropixel takes at least three read-out times to be completely clocked, and it may take four timescales depending on where the photon strikes. Thus, each macropixel may contain more charge than anticipated in these simplified simulations.
The short frame time achieved in “timing” mode is impressive, especially considering that “timing” mode operates with a live-time fraction of 0.99. Unlike its cousin, “burst” mode (live-time fraction: 0.03), “timing” does not handle high fluxes by making exceptionally short exposures while discarding the bulk of the incident flux. These abilities come at the expense of tiling the PSF with many event boxes, however, which is another important means of reducing the severity of photon pile-up distortions.
A distinctive feature of the [*XMM-Newton*]{}/EPIC-pn camera is that diagonally adjacent pixels are treated as two single pixel events. Differences between detectors at this level are assumed to be small.
Generic and separate redistribution and ancillary matrix files are not readily available to [*XMM-Newton*]{} users. Timing mode response matrices were generated using the “rmfgen” and “arfgen” tools in SAS version 9.0.0. These matrices where then used to simulate all timing mode spectra.
For more information about [*XMM-Newton*]{} and the EPIC-pn camera, please see Str[ü]{}der et al. (2001), Haberl et al. (2004), and the User’s Handbook (http://xmm.esac.esa.int/external/xmm\_user\_support/ documentation/index.shtml).
XMM-Newton EPIC MOS “Full Frame” Mode
-------------------------------------
“Full frame” mode is a standard imaging mode for the MOS 1 and MOS 2 cameras aboard [*XMM-Newton*]{}, not a specialized timing mode like those described above. There are two reasons for considering it in this analysis. First, the small, 1.1” pixels of the MOS cameras allow for a large number of event boxes to tile an extraction region, which is one viable strategy for reducing the effects of photon pile-up. Second, by extracting annuli that excise central regions within a piled-up source image, observers are often able to make use of data obtained in this and similar operating modes. Whether or not pile-up is totally mitigated through this procedure is sometimes in doubt, and this analysis can help to clarify the nature of any distortions imposed on the extracted spectrum by residual pile-up.
In simulating MOS “full frame” spectra, circular extraction regions with 20” radii were assumed. Within this range, the encircled energy fraction of the PSF changes almost linearly with radius. The regions assumed would encircle approximately 70% of the total incident energy. As with pn “timing” mode, responses were generated using the “rmfgen” and “arfgen” functions within SAS version 9.0. These responses were then used to create all simulated “full frame” spectra.
For additional information on the [*XMM-Newton*]{}/EPIC-MOS cameras, please see Turner et al. (2001).
Suzaku Windowed Burst Mode
--------------------------
The XIS spectrometer aboard [*Suzaku*]{} provides many different operational modes that can be used to optimize science returns and to reduce or eliminate photon pile-up. Each XIS CCD has a nominal frame time of 8.0 seconds, but this can be reduced by choosing a “window” option. For instance, using a 1/4 window means that only 1/4 of the CCD will be exposed, reducing the frame time to 2.0 seconds. (Smaller windows are possible but cannot be used owing to drift in the spacecraft pointing over the course of an observation.) The frame time can be further reduced by selecting a “burst” option; in this case, the CCD only exposes for a fraction of the nominal frame time. The remainder of the frame time is then dead time, reducing the overall observing efficiency and requiring longer exposures. For this analysis, the second-most conservative combination of window and burst option has been considered. For all simulated spectra, we assumed a 1/4 window and 0.3 second burst option.
It should be noted that the XIS can be operated in “PSUM” mode, wherein the full height of the CCD is read-out in the time normally required to clock one row of charge. In some respects, this mode is similar to the “timing” and “burst” modes available on the [*XMM-Newton*]{}/EPIC-pn camera, and for similar reasons it is actually less effective at mitigating pile-up than the most conservative combinations of XIS window and burst options.
Each simulated spectrum assumed a circular extraction region with a radius of 60”, which encircles approximately half of the incident energy for the [*Suzaku*]{} PSF. Within this radius, the encircled energy fraction changes in a roughly linear way with radius. Each XIS pixel is 1” on a side, and standard 3 pixel by 3 pixel boxes are used to define event grades. Within the extraction region, then, there are 1256.0 event boxes. Tiling a broad PSF with many event boxes acts to reduce the flux per event box.
For more information on the XIS, please see the [*Suzaku*]{} ABC Guide (http://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/) and the [*Suzaku*]{} Technical Description (http://heasarc.gsfc.nasa. gov/docs/suzaku/aehp\_prop\_tools.html).
Analysis and Results
====================
All simulated spectra were fit using the same model components that were employed to generate the spectrum, but without using the pile-up kernel. This procedure permits an examination of how spectral fitting results are distorted by photon pile-up effects, when such effects have not been mitigated entirely (e.g. by extracting events in an annulus, by selecting a shorter frame time, etc.). For the purpose of illustrating how parameters are distorted even in the event of severe photon pile-up, fits were made even when the $\chi^{2}$ goodness-of-fit statistic was not acceptable. All errors reported in this analysis are $1\sigma$ confidence errors, and were determined using the “error” command within XSPEC. In the cases where the fits were unacceptable, the errors are not true $1\sigma$ errors.
A few simplifying assumptions were made in fitting the simulated spectra. The equivalent neutral hydrogen column density was held fixed at the input value, ${\rm N}_{\rm H} = 5.0\times 10^{21}~ {\rm
cm}^{-2}$. In practice, this might be equivalent to assuming the column density measured via a sensitive [*Chandra*]{}/HETGS or [*XMM-Newton*]{}/RGS spectrum in which individual neutral photoelectric absorption edges were detected (e.g. Miller, Cackett, & Reis 2010). Some of the relativistic iron line parameters were constrained to ranges and/or fractional uncertainties that are common in the literature. Specifically, all fits to the simulated line spectra assumed $6.70~ {\rm keV} \leq {\rm E} \leq 6.97~ {\rm keV}$, $3 \leq q
\leq 5$, and $10^{\circ} \leq \theta \leq 50^{\circ}$. Fits to the simulated spectra were otherwise unconstrained; line and continuum model parameters were free to take on whatever value best described the distorted spectra.
The goal of this analysis is simply to understand how disk reflection and disk continua are affected by pile-up. A detailed treatment of distortions to (more physical) continuum spectra (e.g. Comptonization spectra, and/or coupled disk plus corona models such as “eqpair”) might also be timely, but it is beyond the scope of this paper. The sections that follow are narrowly focused on the results of fits to relativistic lines and simple disk continua.
Pile-up and Relativistic Lines
------------------------------
The results of these limited exercises are clear: when photon pile-up is severe, it causes relativistic emission lines in black hole spectra to be measured as falsely narrow. In fits to the simulated spectra with the Laor line model, measured radii were all consistent with or significantly larger than the true value of $6~{\rm GM}/{\rm c}^{2}$. Figures 1 and 2 trace the evolution of measured radius as a function of pile-up severity (set by the grade migration parameter $\alpha$). Figure 3 shows how the entire spectrum is affected by pile-up. Figure 4 shows a sequence of line profiles from simulations with increasing photon pile-up. The same results hold in the case of our simulated neutron star spectra, with one modest exception (see below).
Even the baseline flux level assumed in our “very high” state simulations (corresponding to low values of $\alpha$) is found to completely overwhelm [*Chandra*]{}/ACIS “continuous clocking” mode and [*XMM-Newton*]{}/EPIC-pn “timing” mode. The emission line could not be detected nor reliably fit except at low values of the grade migration parameter $\alpha$. In any real observation at an even higher flux level (corresponding to a higher value of $\alpha$), then, a line may not be clearly detected. Similarly, for the [*XMM-Newton*]{}/EPIC MOS in “full frame” mode, event loss dominates over grade migration in the simulated spectra, and so the [*shape*]{} of the spectrum – including the line – is preserved though the line and continuum flux are a small fraction of their input values. The results shown in Figure 1 do not indicate that the MOS “full frame” mode is equipped to deal with typical “very high” state flux levels.
The [*Swift*]{}/XRT in “WT” mode and [*Suzaku*]{}/XIS (with 1/4 window and 0.3 second burst option) are less overwhelmed by the high flux associated with our baseline “very high” state spectrum. In fits to those simulated spectra, the inner disk radius correlates with the severity of pile-up as traced by the grade migration parameter $\alpha$ (again, $\alpha$ serves as a proxy for increasing the flux over the baseline levels given in Table 1). The results shown in Figure 1 indicate that the [*Suzaku*]{}/XIS is actually best-equipped to deal with such high flux levels, although even in this case fits to the iron line fail to recover the full line width and give falsely large inner disk radii. The apparent fluctuations in inner disk radius at high values of $\alpha$ seen in the XRT trend in Figure 1 are likely the result of errors that are under-estimated because the overall fit is poor.
In the “low/hard” state, the degree of spectral distortions is generally reduced with respect to the “very high” state (see Figure 2). The most accurate inner disk radii were recovered in the simulated [*Suzaku*]{}/XIS spectra. A 1/4 window option and 0.3 seconds burst option appear to suffer minimal disortion due to photon pile-up, at least for the flux levels that would generate the values of $\alpha$ that were simulated. Photon pile-up distortions are more pronounced in the other spectra: measured inner disk radii are found to depart more strongly from the true value with increasing $\alpha$. The [*XMM-Newton*]{}/EPIC-pn “timing” mode and [*Swift*]{}/XRT “windowed timing” mode appear to only suffer modest distortions as pile-up becomes more severe: in each case, lines are measured to be falsely narrow and to give inner disk radii that are too large by 2–3 ${\rm GM}/{\rm c}^{2}$. Fits to the simulated [*XMM-Newton*]{}/EPIC MOS spectra in “full frame” mode show the most marked trend: even for the lowest flux levels (lowest values of $\alpha$), pile-up is so severe that measured inner radii depart from the input value by factors of a few. The [*Chandra*]{}/ACIS “continuous clocking” mode is still overwhelmed at the baseline flux level give in Table 2, and lines cannot be required in spectral fits for $\alpha > 0.4$.
As with the “low/hard” state, the simulated [*Chandra*]{}/ACIS “CC” mode neutron star spectra were severely distorted by photon pile-up, and other modes were less affected. It was only possible to detect and fit the iron line reliably for $\alpha = 0.1$ for [*Chandra*]{}/ACIS “continuous clocking” mode (see Figure 5). In the case of the other simulated neutron star spectra, a limited number of measured radii were 10% smaller (e.g. $0.6~GM/c^{2}$) than the input value. This effect was only seen in simulations of spectra obtained with the EPIC-pn camera in “timing” mode. If these results are an accurate characterization of the systematic errors incurred when fitting mildly piled-up spectra with the EPIC-pn camera, it is a modest systematic error. Systematic uncertainties in the flux calibration between different cameras on the same observatory are approximately 7% ([*XMM-Newton*]{} calibration document XMM-SOC-CAL-TN-0083) and uncertainties between cameras on different missions can easily exceed 10% or more.
The neutron star spectra that were simulated assuming a narrower and symmetric Gaussian emission line with E$=$6.7 keV and $\sigma =
0.33$ keV (as per Ng et al. 2010) also give clear results. The Gaussian line width, line centroid energy, and the normalization of the disk blackbody component are plotted versus $\alpha$ in Figure 6. (The disk blackbody component is the lowest-energy flux component in the spectral model, and changes to its flux trace the degree of flux redistribution due to photon pile-up.) When photon pile-up is modest, the line width and line energy are not strongly affected. At the flux levels simulated, the [*XMM-Newton*]{}/EPIC MOS “full frame” mode suffers more severe pile-up. The line becomes [*narrower*]{} as photon pile-up (traced by the grade migration parameter $\alpha$) becomes more severe (see Figure 6). In no case do fits to the simulated spectra measure a line that is falsely broad, asymmetric, nor displaced to a significanly lower centroid energy. In short, narrower and symmetric lines are not observed to take on a relativistic shape due to photon pile-up distortions.
The result that pile-up generally tends to produce [*falsely narrow*]{} lines can largely be understood in simple terms. At high flux levels, photon pile-up will cause a CCD to register some low energy events as having a higher energy. [*Photon pile-up adds to the high energy continuum*]{}. Adding extra flux in the Fe K band has the effect of hiding the true profile of a relativistic line: the red wing blends with the continuum and the (relatively) narrow blue wing is all that remains. The details of the input spectrum and the nature of a given CCD camera (its effective area curve, its effective frame time, how many event boxes tile an extraction region) are likely important. This may partially explain why slightly different results are obtained from simulated EPIC-pn “timing mode” spectra of neutron stars.
The trends seen in Figure 4 are generally observed whenever pile-up is important. A broad range of incident spectra peak at approximately 1.0–1.5 keV, as do many detector effective area curves. Pile-up of photons to twice that energy, combined with a typical drop in effective area at about 2 keV due abrupt changes in the mirror reflectivity and also Si absorption in the CCD, causes a false excess in the 2–3 keV band. Depending on the specifics of the detector area curve, this excess may be structured. Pile-up also adds appreciably through and above the Fe K band, causing the power-law to be falsely hard and falsely high in flux. This is seen as a steadily increasing excess above 7 keV in the data/model ratio shown in Figure 4. The flux excess is greater when pile-up is more severe. When fit with typical spectral models, the flux excess at 2–3 keV and above 7 keV act to create a false flux deficit just below the line energy, which is again more severe as pile-up becomes more severe. Owing to the fact that more flux redistribution occurs when pile-up is more severe, the full width of relativistic lines becomes increasingly difficult to measure accurately.
Pile-up and Disk Continua
-------------------------
As shown in Figure 7, severe photon pile-up can have a strong impact on efforts to study thermal emission from the accretion disk. Depending on the specific detector and the input flux level (again, traced by $\alpha$), pile-up can falsely reduce the flux in the disk continuum by a factor of a few. The radius inferred from fits with the “diskbb” model depends on the square root of the flux normalization. More recent and physical disk models have more parameters, but still measure the flux to determine an innermost emission radius, and can be expected to be affected in the same degree. The results shown in Figure 7 depict the results of fits to simulated “high/soft” state spectra, but consistent results are also obtained in fits to the continuum in simulated spectra of other states (see the bottom panel in Figure 6).
This effect may also be understood relatively simply. Spectra of bright X-ray binaries tend to peak at or near to the peak of the effective area curve for X-ray telescopes with CCD spectrometers. This is typically also the energy range in which the thermal disk continuum dominates. When pile-up is important, then, low energy flux is preferentially lost from the accretion disk component due to grade migration and flux redistribution.
It should be noted that even the baseline “high/soft” state flux levels are actually quite high (see Table 1). Simulations with successively higher values of the grade migration parameter $\alpha$ trace successively higher flux levels. Only the [*Suzaku*]{}/XIS with a 1/4 window and 0.3 second burst option, and the [*Swift*]{}/XRT in “windowed timing” mode, are even marginally able to cope with this flux level. Even for these detectors and modes, strong mitigations (e.g. annular extraction regions with large inner radii) would be required to recover accurate spectral parameters. The non-response of the other detectors and modes to increases in $\alpha$ (see Figure 7) merely indicates that the limits of the simulations are reached. The flux decrements indicated in Figure 7 for the [*XMM-Newton*]{}/EPIC MOS in “full frame” mode, EPIC-pn in “timing mode”, and the [*Chandra*]{}/ACIS in “continuous clocking” mode, should not be taken as realistic estimates of observed flux decrements.
Discussion
==========
The results of the numerous simulations and fitting exercises detailed above suggest that severe photon pile-up affects relativistic disk spectra in clear and predictable ways:
Redistributing the low energy continuum to the high energy portion of the spectrum causes relativistic emission lines to become [*falsely narrow*]{}, giving a [*falsely high*]{} value for the inner disk radius. In turn, this means that estimates of black holes spin parameters based on the inner disk radius would be [*falsely low*]{}. In spectra where pile-up may be important and the success of mitigation is uncertain, emission lines give [*upper limits*]{} on the radius of the accretion disk, and [*lower limits*]{} on the value of the black hole spin parameter. The same trends are also observed in the case of neutron stars, both when relativistic and simple narrow Gaussian line functions are assumed.
Redistributing the low energy continuum to the high energy portion of the spectrum has exactly the opposite impact on disk continua. The modified disk continuum gives [*falsely low*]{} values of the inner disk radius, equating to [*falsely high*]{} values of black hole spin based on that radius. Therefore, when pile-up may be important, and/or when efforts to mitigate pile-up may not have been entirely successful, radii inferred from continuum fits are [*lower limits*]{} and inferred spin parameters are [*upper limits*]{}.
When two diagnostics are skewed in the same sense, it is difficult to detect a bias, and a potential check on derived quantities is lost. The results of the simple exercise undertaken in this paper are therefore fortuitous: photon pile-up distorts relativistic disk lines and the disk continuum in [*opposing*]{} ways. Fitting both lines and the disk continuum with relativistic spectral models and either checking that the radii agree, or explicitly requiring agreement in the fit (see Miller et al. 2009), will help to derive results that are robust against pile-up distortions. This procedure may be more difficult in the case of neutron stars, however, because the disk continuum is typically less distinct in neutron star spectra.
Efforts to understand the inner accretion flow geometry through correlations between the flux in a disk line and the ionizing continuum are thus also impacted by photon pile-up. A particularly interesting explanation of the flux trends seen in Seyfert-1 AGN such as MCG-6-30-15 (Miniutti & Fabian 2004) and NGC 4051 (Ponti et al.2006) is that gravitational light bending is altering the flux that impinges on the disk from a power-law source of ionizing radiation. In observations of such sources, photon pile up [*is not*]{} a concern, but it might be a concern if CCD spectometers were to monitor a stellar-mass accretor intensively. It is worth noting that current evidence of gravitational light bending in stellar-mass black holes is drawn from gas spectrometer data that is unaffected by pile-up (Miniutti, Fabian, & Miller 2004; Rossi et al. 2005).
Two papers have recently commented on the possibility that photon pile-up might create falsely broad emission lines in black hole spectra. In the first example, [*Suzaku*]{}/XIS spectra of GX 339$-$4 in an “intermediate” state were extracted from different annuli and compared (Yamada et al. 2009). Spectra from larger annuli (presumably suffering from less pile-up) are not found to strongly require black hole spin. Yamada et al. (2009) suggest that pile-up may influence the line shape, but fitting results are consistent with a very broad line and a spinning black hole at the 90% level of confidence (see Miller et al. 2008). The relatively small number of counts in the wings of the PSF can partially account for the lack of a strong spin requirement and lower statistical certainty. A separate but equally important issue is that the disk reflection model used by Yamada et al. (2009) was not convolved with the line element expected for emission from the inner disk; this is physically inconsistent in that it implies a stationary, non-orbiting reflector. The immediate effect is to falsely add to the continuum in the vicinity of the emission line, falsely narrowing the line.
Recent work by Done & Diaz Trigo (2010) examined [*XMM-Newton*]{} spectra of GX 339$-$4 in a “low/hard” spectral state. Spectra from the MOS cameras were found to suffer from pile-up, even when extracting counts in annuli, whereas the EPIC-pn “timing” mode spectra were claimed to be free of distortions from photon pile-up. The pn spectra were also found to deliver narrower line profiles than the MOS spectra, implying a larger inner disk radius compared to values reported in prior work (e.g. Miller et al. 2004; Reis et al.2008; also see Wilkinson & Uttley 2009). Repeating the MOS extraction exactly as detailed in Reis et al. (2008), it is apparent that the disk line profile does not vary with the inner radius of annular extraction regions (see Figure 8). The line profile in the annulus identified by Done & Diaz Trigo (2009) as being largely free of pile-up, closely matches the line profiles seen in spectra extracted from smaller annuli, when each spectrum is allowed to have its own continuum. This means that the radii measured in the MOS spectra are not distorted by photon pile-up, but that the radii derived in fits to the pn spectra are distorted.
This conclusion is echoed by our simulations. The results detailed above strongly suggest that the fact of a narrower line profile in the pn means that it is actually piled-up and does not measure the true line width. Indeed, the departure of 2–3 ${\rm GM}/{\rm c}^{2}$ shown in Figure 2 is commensurate with the radius difference between fits to the MOS spectra reported by Miller et al. (2006) and Reis et al. (2008), and the pn spectrum as fit by Wilkinson & Uttley (2009). The data/model ratio in Figure 7 of Done & Diaz Trigo (2009) shows a flux excess at 2 keV and a flux decrement between 3 keV and the Fe K line, very similar to the trends shown in Figure 4 in this paper. The observed spectra and our simulation results both show that photon pile-up makes lines artificially narrow.
Following Done & Diaz Trigo (2010), Ng et al. (2010) have analyzed a number of spectra of neutron star low-mass X-ray binaries, observed with the [*XMM-Newton*]{}/EPIC-pn camera in “timing” mode. Their analysis suggests that the spectra suffer from a degree of photon pile-up. When the center of the PSF is excluded, the line profiles are found to be more consistent with Gaussian profiles. Ng et al.(2010) conclude that pile-up acted to falsely create skewed, relativistic line profiles. This conclusion runs counter to the results of our simulations, which show that (1) severe pile-up acts to falsely narrow both relativistic and simple Gaussian lines, not to falsely broaden lines; and (2) pile-up distortions to typical neutron star line spectra are not expected to be extreme (see Figure 5) for sources with flux levels similar to Serpens X-1, except for [*Chandra*]{}/ACIS “continuous clocking” mode.
Independent [*Suzaku*]{} and [*XMM-Newton*]{} observations of Serpens X-1 reveal relativistic Fe lines with a clear red wing (see Figure 9; also see Bhattacharyya & Strohmayer 2007, Cackett et al. 2008, Cackett et al. 2010). It is already clear that [*Suzaku*]{} line profiles in this source and similar sources do not depend on the inner radius of event extraction annuli (Cackett et al. 2010; see Figure 11) – the relativistic profiles do not result from photon pile-up. The nature of the line seen with [*Suzaku*]{} strongly suggests that the red wing seen in the [*XMM-Newton*]{} spectrum is due to dynamical broadening and red-shifting, not photon pile-up.
We re-reduced the [*XMM-Newton*]{} pn observation of Serpens X-1 exactly as detailed in Bhattacharyya & Strohmayer (2007), and extracted spectra including and excluding the center of the PSF. The primary effect of excluding the center of the PSF is merely to reduce the number of photons in the resultant spectrum, thereby lowering the sensitivity to the point that the red wing cannot be detected. The line is neither narrower nor weaker when the center of the PSF is excluded – it is simply less defined (see Figure 10). Ng et al.(2010) do not present direct comparisons of line profiles in spectra obtained in dfferent extraction regions.
Given that: (1) [*Suzaku*]{} spectra of neutron stars reveal relativistic lines that are clearly not due to photon pile-up distortions, (2) a number of [*Suzaku*]{} and [*XMM-Newton*]{} line profiles are remarkably similar (e.g. SAX J1808.6$-$3808, see Figure 3 in Cackett et al. 2010; GX 349$+$2, see Figure 9 in Cackett et al.2010), (3) the effect of excluding the center of the PSF in the [*XMM-Newton*]{} spectrum of Serpens X-1 is merely to reduce definition in the Fe line (not its width nor its strength), and (4) our simulations show that severe pile-up acts to artificially [*narrow*]{} both relativistic and simple Gaussian lines, it is likely that dramatic reductions in sensitivity drove the results obtained by Ng et al.(2010) and led to faulty conclusions regarding the ability of pile-up to create false relativistic line profiles.
The fact that sensitive spectra are required to detect the red wings of relativistic line profiles is not a revelation. This fact fueled the requirement of minimum sensitivity thresholds in recent surveys of relativistic lines in Seyfert AGN (see Nandra et al. 2007). Recent work on neutron star spectra makes the necessity of sensitive spectra even more clear. Deep observations of a source such as GX 349$+$2 with [*Suzaku*]{} reveal a relativistic line profile, whereas short observations with the [*Chandra*]{}/HETGS – which has a lower collecting area – only recovers the narrower blue wing of the line profile (see Figure 3 in Cackett et al. 2009b). The [*Chandra*]{} spectrum is fully consistent with the relativistic line profile found using [*Suzaku*]{}, it simply does not have the sensitivity needed to detect the red wing against the continuum.
The above discussion is mostly focused on [*XMM-Newton*]{} and [*Suzaku*]{} spectroscopy, because the large effective area of these missions ensures that they are well-suited to relativistic spectroscopy. [*Swift*]{} is highly flexible, but its smaller collecting area and short observations mean that typical spectra lack the sensitivity to detect and measure relativistic disk lines well. However, [*Swift*]{} is very well-suited to black hole spin measurements using the disk continuum, and our results suggest that even in “windowed timing” mode, annular extraction regions are needed to avoid spectral distortions due to photon pile-up (see Figure 6). Brief discussions of pile-up and excluding the central portion of the PSF in “windowed timing” mode are given in Rykoff et al. (2007) and Rykoff, Cackett, & Miller (2010).
Two combinations of detectors and modes are not treated in this work [*because*]{} they represent effective means of preventing photon pile-up. As noted previously, the “burst” mode of the EPIC-pn camera aboard [*XMM-Newton*]{} may provide the [*best*]{} means of preventing photon pile-up, as its short read-out time is suited to extremely bright sources. This comes at a cost in sensitivity, however, as “burst” mode has a livetime fraction of just 0.03. Pile-up can also be avoided by observing with the [*Chandra*]{}/HETGS, which disperses a spectrum onto the ACIS spectrometer. Pile-up can be further avoided by reading-out the dispersed spectrum in “continuous clocking” mode.
[*Chandra*]{}/HETGS observations of the stellar-mass black hole GX 339$-$4 in an “intermediate” state measured an inner disk radius of $1.3_{-0.1}^{+1.7}~{\rm GM}/{\rm c}^{2}$ (Miller et al. 2004b). This analysis used the same relativistically-blurred reflection model applied to a later observation in the same state using [*Suzaku*]{} (Miller et al. 2008). Analysis of the later observation found a spin parameter of $a = 0.89\pm 0.04$, correspinding to ${\rm r} = 2.0\pm
0.2~{\rm GM}/{\rm c}^{2}$ (Bardeen, Press, & Tuekolsky 1972). The radius values measured in GX 339$-$4 are consistent in the two separate observations, using very different detectors. Similarly, [*Chandra*]{}/HETGS observations of the neutron star X-ray binary 4U 1705$-$44 find a very broad line, with a Gaussian width (FWHM) of $1.2\pm
0.2$ keV; when fit with a relativistic diskline model, a radius of ${\rm r} = 7^{+4}_{-1}~{\rm GM}/{\rm c}^{2}$ is measured (Di Salvo et al. 2005). Here again, the line properties found using grating spectra – which avoid severe pile-up – are consistent with those found using [*Suzaku*]{} and [*XMM-Newton*]{} (Reis, Fabian, & Young 2009, Di Salvo et al. 2009, Cackett et al. 2010), and inconsistent with the much narrower lines found by Ng et al. (2001) when extracting only the wings of [*XMM-Newton*]{}/EPIC-pn “timing” mode spectra.
An [*XMM-Newton*]{}/EPIC-pn “burst” mode observation of GRS 1915$+$105 in a “plateau” state (similar to the low/hard state in most black holes) may also validate the results of our simulations and the discussion above. Martocchia et al. (2006) observed GRS 1915$+$105 twice in the “plateau” state – once in “timing” mode and once in “burst mode” – at consistent flux levels. The “timing” mode spectrum may suffer from modest distortions due to photon pile-up. The power-law index was measured to be harder than in the “burst” mode observation ($\Gamma = 1.686^{+0.008}_{-0.012}$ versus $\Gamma = 2.04^{+0.01}_{-0.02}$), indicative of possible pile-up. The iron line is found to be fairly narrow in the “timing” mode observation: the line is only visible above $\sim$6.2 keV when the spectrum is fit with a power-law, the inner radius is constrained to be greater than 240 ${\rm GM}/{\rm c}^{2}$, and a reflection fraction of ${\rm R} = 0.35^{+0.02}_{-0.02}$ is measured. In contrast, when observed using “burst” mode, the line is visible down to 5 keV (or lower) relative to a simple continuum, a radius of less than $20~{\rm GM}/{\rm c}^{2}$ is required, and a reflection fraction of $R=1.69^{+0.16}_{-0.04}$ is measured (Martocchia et al. 2006). (This [*XMM-Newton*]{}/EPIC-pn “burst” mode spectrum is broadly conistent with a [*Suzaku*]{} observation of GRS 1915$+$105 in the “plateau” state. In that spectrum, the line is also broad, and consistent with the ISCO; see Blum et al. 2009.) These results square with a central result of the simulations presented in this paper: photon pile-up acts to make relativistic lines appear to be falsely narrow.
Though an investigation is beyond the scope of this paper, it should be noted that efforts to mitigate pile-up can also distort spectra. The PSF of an X-ray telescope is energy-dependent. Extracting events in annuli may avoid the piled-up core of a given PSF, but the spectrum is then derived from a portion of the PSF that is not calibrated as well as the core. Moreover, depending on the PSF, annular regions may only extract a vanishing fraction of the incident photon flux, complicating the detection of weak spectral lines. Relativistic disk lines are often 10–20% features above the continuum; extracting spectra from regions of the PSF from which the energy and flux calibration are not known to much better than 10% could easily make it difficult to recover the details of a given line profile. Depending on how much flux is excluded in the annulus, limited photon statistics will also serve to complicate the detection of line asymmetry.
Finally, it is worth emphasizing that our results depend on a specific combination of mirror technology and detector technology. Presently, CCD spectrometers sit at the focus of gold foil or gold-coated mirors; this partially accounts for an overall telescope efficiency curve that is highest below 2 keV. New mirror technology, such as Si pore optics (e.g. Beijersbergen, M., et al., 2004), may produce different CCD photon pile-up effects, and distortions to disk continua and disk lines may not longer skew in the opposite sense.
Conclusions
===========
Extensive simulations of photon pile-up for a number of spectral forms and detector parameters suggest that severe pile-up can distort spectroscopic signatures of the inner accretion disk, in largely predictable ways. The results of our work can be summarized as follows:\
[$\bullet$ The degree of photon pile-up in a spectrum depends on the input flux level from a source and the effective area of the telesecope, but it is also depends on the number of event boxes tiling the PSF and the frame time of the CCD.]{}
[$\bullet$ The relevant time scale for pile-up considerations is the time required to clock a full event box, not merely one row of charge.]{}
[$\bullet$ Tiling the PSF with many event boxes and short frame times are two independent means of reducing photon pile-up. However, when short frame times are achieved by changing the size of an event box and under-sampling the PSF, pile-up mitigation is partially compromised.]{}
[$\bullet$ Flux redistribution due to photon pile-up causes emission lines – whether relativistic or symmetric and intrinsically narrow – to be observed as falsely narrow. Measured inner disk radii are then falsely large, and inferred black hole spin parameters are falsely low. Relevent observed spectra, though small in number, support this finding.]{}
[$\bullet$ Grade migration due to photon pile-up causes the low energy spectrum – principally the disk component – to have a falsely low flux. Measured inner disk radii are therefore falsely small and inferred spin parameters are falsely high.]{}
[$\bullet$ Deriving inner disk radii and/or black hole spin parameters by linking that parameter to a joint value in both disk continuum and disk reflection models may yield more robust results when photon pile-up cannot be avoided or successfully mitigated.]{}
We wish to acknowledge the referee, Keith Jahoda, for a thoughtful review of this work and for comments that improved the paper. We thank John Davis, Matthias Ehle, Lothar Str[ü]{}der, J[ö]{}rn Wilms, and Maria Diaz-Trigo, Mark Reynolds, Dipankar Maitra, and Tiziana Di Salvo, and Giorgio Matt for helpful discussions.
Arnaud, K. A., and Dorman, B., 2000, XSPEC is available via the HEASARC on-line service, provided by NASA/GSFC
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[lllll]{} Parameter & Very High / Intm. & Low/Hard & High/Soft & Z/Atoll\
${\rm N}_{\rm H}~(10^{22}~{\rm cm}^{-2})$ & 0.5 & 0.5 & 0.5 & 0.5\
${\rm kT}_{\rm disk}$ (keV) & 0.76 & 0.18 & 0.97 & 1.21\
${\rm K}_{\rm disk}$ & 2300.0 & 160000.0 & 7050.0 & 103.0\
${\rm kT}_{\rm bbody}$ (keV) & – & – & – & 2.28\
${\rm K}_{\rm bbody}$ ($10^{-2}$) & – & – & – & 4.9\
$\Gamma$ & 2.60 & 1.60 & 2.48 & 3.60\
${\rm K}_{\rm pl}$ & 2.20 & 0.50 & 3.38 & 0.18\
${\rm E}_{Laor}$ (keV) & 6.7 & 6.7 & – & 6.7\
$q$ & 3.0 & 3.0 & – & 3.0\
${\rm r}_{in} (\rm GM/c^{2})$ & 6.0 & 6.0 & – & 6.0\
$i$ (deg.) & 30.0 & 30.0 & – & 30.0\
${\rm K}_{\rm line} (10^{-3})$ & 7.7 & 8.5 & – & 8.0\
Flux ($10^{-9}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}$) & 36.9 & 2.9 & 86.2 & 6.0\
\
[llllllllll]{} & ${\rm T}_{\rm R}^{a}$ & Radius$^{b}$ & Pixel Size$^{c}$ & Event Box$^{d}$ & ${\rm T}_{\rm eff.}^{g}$ & Regions$^{h}$\
& ($10^{-3}$ s) & (arcsec) & (arcsec) & (pixels) & ($10^{-3}$ s) & \
& 2.85 & 2.5 & 0.5 & 3x3 & 8.55 & 3.3\
& 1.77 & 9.0 & 2.36 & 10x7 & 1.77 & 1.1\
& 0.03 & 120 & 4.1 & 30x3 & 0.09 & 1.0\
& 2600 & 20 & 1.1 & 3x3 & 2600 & 115.3\
& 300 & 60.0 & 1.0 & 3x3 & 300.0 & 1256.0\
\
|
---
abstract: |
We explore the combined pressure and finite-size effects on the in-plane penetration depth $\lambda _{ab}$ in YBa$_{2}$Cu$_{4}$O$_{8}$. Even though this cuprate is stoichiometric the finite-size scaling analysis of $
1/\lambda_{ab}^{2}\left( T\right)$ uncovers the granular nature and reveals domains with nanoscale size $L_{c}$ along the $c$-axis. $L_{c}$ ranges from 33.2 Å to 28.9 Å at pressures from 0.5 to 11.5 kbar. These observations raise serious doubts on the existence of a phase coherent macroscopic superconducting state in cuprate superconductors.
author:
- 'R. Khasanov'
- 'T. Schneider'
- 'J. Karpinski'
- 'H. Keller'
title: 'Finite-size and pressure effects in YBa$_{2}$Cu$_{4}$O$_{8}$ probed by magnetic field penetration depth measurements'
---
Introduction
============
Since the discovery of superconductivity in cuprates by Bednorz and Müller[@Bednorz86] a tremendous amount of work has been devoted to their characterization. Indeed, the issue of inhomogeneities and their characterization is essential for applications and the interpretation of experimental data. Furthermore, there is even increasing evidence that inhomogeneities are an intrinsic property of cuprates. [@Mesot93; @Furrer94; @Alekseevskii88; @Liu91; @Chang92; @Cren00; @Lang02] Studies of different cuprate families revealed the segregation of the material in superconducting and non-superconducting regions. In particular, neutron scattering experiments provide evidence for nanoscale cluster formation and percolative superconductivity in various cuprates.[@Mesot93; @Furrer94] Electron paramagnetic resonance (EPR) studies reveal nanoscale phase separation in superconducting and dielectric regions of YBa$_2$Cu$_3$O$_{7-\delta}$ with $0.15\leq\delta\leq0.5$. [@Alekseevskii88] Nanoscale spatial variations in the electronic characteristics have also been observed in underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ with scanning tunnelling microscopy (STM),[@Liu91; @Chang92; @Cren00; @Lang02] while x-ray diffraction in oxygen doped La$_{2}$CuO$_{4}$ single crystals [@DiCastro00] provide evidence for superconducting domains with spatial extent $L_{ab}\approx 300$ Å in the $ab$-plane. Accordingly, there is considerable evidence for granular superconductivity in the cuprates. Although crystals of the cuprates are not granular in a structural sense, it occurs when microscopic superconducting domains are separated by non-superconducting regions through which they communicate for instance by Josephson tunnelling to establish the macroscopic superconducting state.
On the other hand, there is evidence for nearly isolated, homogeneous and superconducting domains of nanoscale extent embedded in a non-superconducting matrix. [@Schneider02; @Schneider03; @Schneider03a; @Schneider03b] It stems from a finite-size scaling analysis of the thermal fluctuation contributions to the specific heat [@Schneider02] and magnetic penetration depth data. [@Schneider03; @Schneider03a; @Schneider03b] In an isolated domain there is a finite-size effect because the correlation length cannot grow beyond the length of the superconducting domain in direction $i$. Accordingly there is no sharp phase transition and the specific heat coefficient will exhibit a blurred peak with a maximum at $T_{p}<T_{c}$, where $T_{c}$ is the transition temperature of the homogeneous bulk system. At $T_{p}$ the correlation length $\xi \left( T\right) $ reaches the limiting length $L$ of the domain. Similarly, $1/\lambda^{2}$, where $\lambda$ is the magnetic field penetration depth, does not vanish at $T_{c}$, but exhibits a tail with an inflection point at $T_{p}$. This raises serious doubts on the existence of macroscopic phase coherent superconductivity, suggesting that bulk superconductivity is achieved by a percolative process. Therefore, superconducting properties and the spatial extent of the domains can be probed by thermal fluctuations and the finite-size effects. This includes the effects of oxygen isotope exchange and pressure on the domain size. Recently a significant change of spatial extent of the superconducting domains upon oxygen isotope exchange has been demonstrated in Y$_{1-x}$Pr$_{x}
$Ba$_{2}$Cu$_{3}$O$_{7-\delta }$.[@Schneider03] It revealed the relevance of local lattice distortions in occurrence of superconductivity.
This paper addresses the pressure studies of the finite-size effect in YBa$_2$Cu$_4$O$_8$, which exhibits a rather large and positive pressure effect (PE) on $T_{c}$, with $dT_{c}/dp=0.59$ K/kbar.[@Scholtz92; @Bucher89; @VanEinige90] Even though this cuprate is [*stoichiometric*]{} the finite-size scaling analysis uncovers the existence of [*nanoscale domains*]{} with a spatial extent $L_{c}$ along the crystallographic $c$-axis. The value of $L_c$ decreases from $33.2$ to $28.9$ Å with increasing pressure from $0.5$ to $11.5$ kbar. Accordingly, $T_{c}$ [*increases*]{} with [*reduced thickness*]{} $L_{c}$ of the domains.
The paper is organized as follows. In Sec. \[seq:Theoretical\_background\] we sketch the finite-size scaling theory adapted for the analysis of penetration depth data. In Sec. \[seq:Experimental\] we describe sample preparation procedure and the experimental technique, adopted to deduce the in-plane magnetic field penetration depth $\lambda _{ab}$ from the Meissner fraction measurements. In Sec. \[seq:The\_finite-size\_analysis\] we perform the finite-size analysis of the data for the in-plane magnetic penetration depth $\lambda _{ab}\left( T\right) $ taken at different pressures. Sec. \[seq:pressure\_dependence\_domain\_size\] comprises the analysis of the pressure dependence of the domain lengths $L_{c}$ along the crystallographic $c$-axis.
Theoretical background {#seq:Theoretical_background}
======================
In a homogeneous bulk system, undergoing a fluctuation dominated continuous phase transition at $T_{c}$, the correlation length diverges as [@Shan-Keng76] $$\xi (T)=\xi _{0}^{\pm }|T/T_{c}-1|^{-\nu }=\xi _{0}^{\pm
}|t|^{-\nu }
\label{eq:critical_exponent}$$ ($\pm $ refers to $T>T_{c}$ and $T<T_{c}$, respectively, $\xi
_{0}^{\pm }$ is the critical amplitude and $\nu $ is the associated critical exponent). There is mounting evidence that in the experimentally accessible critical regime cuprates belong to the 3D-XY universality class (like superfluid He$^{4}$) with $\nu
\approx 2/3$.[@Schneider00] Since the order parameter $\Psi$ is a complex scalar it corresponds to a vector with two components. For this reason below $T_{c}$ there are two correlation lengths. The longitudinal one, $\xi ^{l}$, is associated with the correlation function $\left\langle Re\Psi
\left( R\right) Re\Psi \left( 0\right) \right\rangle -\left\langle
Re\Psi \left( 0\right) \right\rangle ^{2}$, while the transverse one, $\xi ^{t}$, measures the decay of $\left\langle Im\Psi \left(
R\right) Im\Psi \left( 0\right) \right\rangle $. In the long wavelength limit considered here, the total correlation function is dominated by transverse fluctuations so that $\xi ^{t}$ is the relevant length scale.[@Schneider00; @Hohenberg76] Suppose that the cuprates are granular, consisting of superconducting domains embedded in a non-superconducting matrix. Denoting the spatial extent of the domains along the crystallographic $a$, $b$ and $c$ -axis with $L_{a}$, $L_{b}$ and $L_{c}$, the transverse correlation lengths $ \xi _{i}^{t}$ cannot diverge according to Eq. \[eq:critical\_exponent\] but are limited by $$\xi _{i}^{t}\xi _{j}^{t}\leq L_{k}^{2},\ i\neq j\neq k.
\label{eq:correlation_limit}$$ Consequently, for finite superconducting domains, the thermodynamic quantities, like the specific heat and penetration depth, are smooth functions of temperature. As a remnant of the singularity at $T_{c}$ these quantities exhibit a so called finite-size effect,[@Fisher72; @Cardy88] namely a maximum or an inflection point at $T_{p_{i}}$ $$\xi _{i}^{t}(T_{p_{i}})\xi _{j}^{t}(T_{p_{i}})=L_{k}^{2},\ i\neq
j\neq k. \label{eq:correlation_limit_inf_point}$$
Close to criticality of the infinite system, the thermodynamic properties of its finite counterpart are well described by the finite-size scaling theory.[@Fisher72; @Cardy88] In particular, the thermodynamic observable $Q$ adopts the scaling form[@Schultka95] $$\frac{Q\left( t,L\right) }{Q\left( t,L=\infty \right) }=f\left(
x\right) ,\ x=\frac{L}{\xi \left( t,L=\infty \right) }. \label{O}$$ The scaling function $f$ depends only on the dimensionless ratio $L/\xi \left( t,L=\infty \right) $ and does not depend on microscopic details of the system. It does, however, depend on the boundary conditions and the geometry of the system. As an example, for $Q(t,L)=\xi _{i}^{t}\xi_{j}^{t}$ we obtain from Eqs. (\[eq:correlation\_limit\_inf\_point\]) and (\[O\]) the finite-size scaling relation $$\frac{\xi _{i}^{t}\xi _{j}^{t}}{\xi _{0i}^{t}\xi
_{0j}^{t}}|t|^{2\nu
}=f\left( \frac{sign(t)|t|^{\nu }L_{k}}{\sqrt{\xi _{0i}^{t}\xi _{0j}^{t}}}\right) .
\label{scaling-finction_coherence_length}$$ To relate this combination of transverse correlation lengths to an experimentally accessible quantity we invoke the universal relation [@Schneider00; @Schneider98] $$\frac{1}{\lambda _{i}^{2}(T)}=\frac{16\pi ^{3}k_{B}T}{\Phi
_{0}^{2}\xi _{i}^{t}(T)},
\label{eq:universal relation}$$ which holds in the 3D-XY universality class ($\Phi _{0}$ is the flux quantum, $k_{B}$ is the Boltzmann’s constant and $\lambda_{i}$ is the London penetration depth). In this case the Eq. (\[scaling-finction\_coherence\_length\]) reduces to $$\frac{\lambda _{0i}\lambda _{0j}}{\lambda _{i}\lambda
_{j}}|t|^{-\nu }=\left( \frac{\xi _{i}^{t}\xi _{j}^{t}}{\xi
_{0i}^{t}\xi _{0j}^{t}}\right) ^{-\frac{1}{2}}|t|^{-\nu }=g(y),
\label{eq:scaling-finction_lambda_1}$$ where $y$ is equal to $$y=sign\left( t\right) \left| t\right| \left(
\frac{L_{k}}{\sqrt{\xi
_{0i}^{t}\xi _{0j}^{t}}}\right) ^{1/\nu }=sign\left( t\right) \left| \frac{t}{t_{p_{k}}}\right|
\nonumber$$
For the homogenous system ($L_{k}\rightarrow \infty $) and $t\neq
0$ ($y\neq 0$), $g(y)$ corresponds to the stepwise function $$g_{\infty }\left( y<0\right) =1,\ g_{\infty }\left( y>0\right) =0.
\label{eq10c}$$ While for the system confined by the finite geometry ($L_{k}\neq
0$) the scaling function $g(y)$ diverges at $t\rightarrow 0$ ($y\rightarrow 0$) as $$g\left( y\rightarrow 0\right) =g_{0k}y^{-\nu }=g_{0k}\left( \left| \frac{t}{t_{p_{k}}}\right| \right) ^{-\nu }.
\label{eq:10d}$$
In order to obtain the absolute values of the inflection temperature $T_{p_i}$ and the superconducting domain size $L_i$ one can use the following procedure. Combining Eqs. (\[eq:correlation\_limit\_inf\_point\]) and (\[eq:universal relation\]) one obtains at the inflection point $$\left. \frac{1}{\lambda _{i}(T)\lambda _{j}(T)}\right| _{T=T_{p_{k}}}=\frac{16\pi ^{3}k_{B}T_{p_{k}}}{\Phi _{0}^{2}}\frac{1}{L_{k}}.
\label{eq4}$$ For an infinite and homogeneous system $1/\left( \lambda
_{i}\left( T\right) \lambda _{j}\left( T\right) \right) $ decreases continuously with increasing temperature and vanishes at $T_{c}$, while for finite domains it does not vanish and exhibits an inflection point at $T_{p_{k}}<T_{c}$, so that $$\left. d\left( \frac{1}{\lambda _{i}\left( T\right) \lambda
_{j}\left( T\right) }\right) /dT\right| _{T=T_{p_{k}}}=extremum
\label{eq5}$$ Note, that in this paper we analyze experimental data for the temperature dependence of the in-plane penetration depth $\lambda_{ab}$ taken at various applied pressures. In this case the domain size along $c$-axis ($L_{c}$) can be estimated according to Eq. (\[eq4\]) as $$L_{c}=\frac{16\pi ^{3}k_{B}T_{p_{c}}\left( \lambda _{a}\left(
T\right) \lambda _{b}\left( T\right) \right) _{T=T_{p_{c}}}}{\Phi
_{0}^{2}},
\label{eq:Lc_one}$$ that for $\lambda_a\simeq\lambda_b$ reduces to $$L_{c}\simeq \frac{16\pi ^{3}k_{B}T_{p_{c}}\lambda _{ab}^{2}\left(
T_{p_{c}}\right) }{\Phi _{0}^{2}}.
\label{eq:Lc_two}$$
To summarize, the main signatures for the existence of finite-size behavior appearing in the temperature dependence of $\lambda
_{ab}^{2}$ are:
- The scaling function $g(y)$ diverges at $T=T_{c}$ $(t=0,$ $y=0)$ \[Eq. (\[eq:10d\])\].
- $\lambda ^{-2}(T)$ has an inflection point at $T_{p}<T_{c}$ \[Eq.(\[eq4\])\].
- The first derivative of $\lambda ^{-2}(T)$ at $T=T_{p}$ has an extremum \[Eq.(\[eq5\])\].
Experimental Details {#seq:Experimental}
====================
Sample preparation and characterization
---------------------------------------
The polycrystalline YBa$_{2}$Cu$_{4}$O$_{8}$ samples were synthesized by solid-state reactions using high-purity Y$_{2}$O$_{3}$, BaCO$_{3}$ and CuO. The samples were calcinated at $880-935$ $^{o}$C in air for $110$ hours with several intermediate grindings. The phase-purity of the material was examined using a powder x-ray diffractometer. Only YBa$_{2}$Cu$_{3}$O$_{7-x}$ and CuO phases were revealed. The synthesis was continued at high oxygen pressure of $500$ bar, at $1000$ $^{o}$C during 30 hours. The x-ray diffraction measurements performed after the final stage of the synthesis revealed 95 % of YBa$_{2}$Cu$_{4}$O$_{8}$ phase. The sample was then regrounded in a mortar for about $60$ min in order to obtain sufficiently small grains, as required for the determination of $\lambda$ from Meissner fraction measurements. The field-cooled (FC) magnetization ($M$) measurements were performed with a Quantum Design SQUID magnetometer in a field of $
0.5$ mT for temperatures ranging from $5$ K to $100$ K. The absence of weak links between grains has been confirmed by the linear magnetic field dependence of the FC magnetization, measured at $0.5$ mT, $1$ mT and $1.5$ mT for each pressure at $T=10$ K.
The hydrostatic pressure was generated in a copper-beryllium piston cylinder clamp that was especially designed for magnetization under pressure measurements (see Ref. \[\]). The sample was mounted in a led container filled with Fluorient FC77 as a pressure transmitting medium with a sample to liquid volume ratio approximately $1/8$. The pressure was measured in situ at 7 K by using the $T_{c}$ shift of the led container.
Determination of the temperature dependence of $\lambda$ from Meissner fraction measurements
--------------------------------------------------------------------------------------------
The temperature dependence of $\lambda ^{-2}$ was extracted from the Meissner fraction $f$ deduced from low-field (0.5 mT FC) magnetization data using the relation:[@Blundel01] $$f(T)=\left( \frac{H}{M(T)}-N\right) ^{-1},$$ where $H$ denotes the external magnetic field and $N$ is the demagnetization factor. $N=1/3$ was taken assuming that the sample grains are spherical. Fig. \[Magnetization\] shows the temperature dependence of the Meissner fraction close to $T_{c}$ for different pressures. Three important features emerge: (i) The transition temperature $ T_{c}$ increases with increasing pressure. The value of $dT_c/dp= 0.59$ K/kbar is found, which is in good agreement with the literature data. [@Scholtz92; @Bucher89; @VanEinige90] (ii) The value of $f$ (Fig. \[Magnetization\]) is much smaller than 1, confirming that the average grain size of the sample is compatible with $\lambda$. The reduction of $f$ is caused by the field penetration at the surface of each individual grain for distances of the order of $\lambda $.[@Zhao97] (iii) The absolute value of the Meissner fraction increases with pressure. Since the average grain size does not change under pressure and the grains are decoupled from each other, the rise of $f$ must be attributed to a decrease of the magnetic penetration depth $\lambda $.
![The temperature dependence of the Meissner fraction $f$ obtained from low-filed (0.5 mT, FC ) magnetization measurements for various pressures.[]{data-label="Magnetization"}](Fig1BW.EPS){width="1.0\linewidth"}
The temperature dependence of $\lambda$ was analyzed on the basis of model suggested by Shoenberg.[@Shoenberg40] According to Ref. \[\] the temperature dependence of the Meissner fraction is given by $$f(T)=1-3\left( \frac{\lambda (T)}{R}\right) \coth {\left(
\frac{R}{\lambda (T)}\right) }+3\left( \frac{\lambda
(T)}{R}\right) ^{2},
\label{eq:Shoenberg}$$ where $2R$ is the average grain diameter. By solving this nonlinear equation, $\lambda$ for each value of $f$ was extracted, and with that the whole temperature dependencies of $\lambda$ was reconstructed. Since the sample consists of an anisotropic non-oriented powder, the extracted $\lambda$ is the so called effective penetration depth $\lambda _{eff}$ (powder average). However, for sufficiently anisotropic extreme type II superconductors, including YBa$_{2}$Cu$_{4}$O$_{8}$, $\lambda
_{eff}$ is proportional to the in-plane penetration depth in terms of $\lambda _{eff}=1.31\lambda _{ab}$.[@Fesenko91] The resulting temperature dependencies of $\lambda _{ab}^{-2}$ evaluated at different pressures are depicted in Fig. \[fig:lambda\]. Due to the unknown average grain size the data in Fig. \[fig:lambda\] are normalized to the $\lambda_{ab}^{-2}(0)$, taken from $\mu $SR measurements. [@Khasanov04] The solid lines indicate the leading critical behavior of $\lambda _{ab}^{-2}(T)$ for homogenous and infinite domains $$\lambda _{ab}^{-2}(T)=\lambda _{0ab}^{-2}|t|^{\nu },\ \ \nu =2/3
\label{eq:lambda_leading_critical_behavior}$$ This equation is a consequence of Eqs. (\[eq:critical\_exponent\]) and (\[eq:universal relation\]) with critical amplitude $$\lambda _{0ab}^{-2}=\frac{16\pi ^{3}k_{B}T_{c}}{\Phi _{0}^{2}\xi
_{0ab}^{t}}.$$ The values of $\lambda _{0ab}$ and $T_{c}$ obtained from the fit of Eq. (\[eq:lambda\_leading\_critical\_behavior\]) to the experimental data presented in Fig. \[fig:lambda\] are summarized in Table \[Table1\]. Since Eq. (\[eq:lambda\_leading\_critical\_behavior\]) is valid in the vicinity of $T_{c}$ only, we restricted the fit to the interval from $T_{c}$ to $T_{c}-3$ K.
![The temperature dependence of $\lambda_{ab}^{-2}$ for various pressures obtained from $f(T)$ data (see Fig. \[Magnetization\]) by using Eq. (\[eq:Shoenberg\]). The solid lines indicate the leading critical behavior of a homogenous bulk system according to Eq. (\[eq:lambda\_leading\_critical\_behavior\]) with the parameters listed in Table \[Table1\]. The inset shows $\lambda_{ab}^{-2}(T)$ in the vicinity of $T_c$ for $p=0.5$ kbar. The deviation of data points from theoretical curves clearly indicates the finite-size behavior of the system. []{data-label="fig:lambda"}](Fig2BW.EPS){width="1.0\linewidth"}
The finite-size analysis {#seq:The_finite-size_analysis}
========================
The essential characteristic of a homogeneous bulk cuprate superconductor is a sharp superconductor to normal state transition. A glance to the inset of Fig. \[fig:lambda\] shows that in the samples considered here that is not the case. The transition occurs smoothly and there is a tail pointing to a finite-size effect associated with an inflection point at some characteristic temperature $T_{p}$. As outlined above \[Eq. (\[eq:correlation\_limit\_inf\_point\])\] at $T_{p}$ the transverse correlation length $\xi _{ab}^{t}$ attains the limiting length along the $c$-axis. To substantiate the occurrence of an inflection point we show in Fig. \[fig:derivative\_lambda\] $d\lambda _{ab}^{-2}(T)/dT$ versus $T$ for YBa$_{2}$Cu$_{4}$O$_{8}$ samples at different pressures. The extreme in the first derivative of $\lambda _{ab}^{-2}(T)$ clearly reveals the existence of an inflection point at $T_{p_{c}}<T_{c}$. The absolute values of $T_{p_c}$ obtained from a parabolic fit to experimental data around maximum point are summarized in Table \[Table1\].
![$d\lambda_{ab}^{-2}(T)/dT$ vs. $T$ at different pressures. The maximum of $d\lambda_{ab}^{-2}(T)/dT$ is at a $T=T_{p_c}$.[]{data-label="fig:derivative_lambda"}](Fig3BW.EPS){width="1.0\linewidth"}
To substantiate the finite-size scenario further, we explore the scaling properties of the data with respect to the consistency with the finite-size scaling function. Noting that, $\lambda
_{a}\simeq \lambda _{b}$ and $\nu \approx 2/3$ in the 3D-XY universality class, Eq.(\[eq:scaling-finction\_lambda\_1\]) can be rewritten as $$\frac{\lambda _{0ab}^{2}}{\lambda _{ab}^{2}}|t|^{-2/3}=g\left( \frac{t}{|t_{p_{c}}|}\right) .
\label{eq:scaling-finction_lambda}$$ Figure \[fig:scaling\_function\](a) shows the resulting scaling function. For comparison, we included the limiting behavior of the finite-size scaling function $g_\infty$ \[see Eq. (\[eq10c\])\] for a homogeneous ($L_c\rightarrow \infty $) system \[the solid line in Fig. \[fig:scaling\_function\](a)\]. As it is seen, there is quite a good agreement between the experimental data and $g_\infty$ function for $t/t_{p_c}$ far from 0 ($T$ far from $T_c$), whereas in the vicinity of 0 ($T\sim T_c$) the data are completely inconsistent with such a stepwise behavior. The experimental data have to diverge at $t/t_{p_c}\rightarrow 0$. As outlined in Sec. \[seq:Theoretical\_background\] divergence of the scaling function $g(t/|t_{p_c}|)$ at $t/t_{p_c}\rightarrow 0$ implies that the system is confined by a finite geometry. To strengthen this point we display in Fig. \[fig:scaling\_function\](b) the comparison with the leading finite-size behavior $g_{c}(t\rightarrow
0)=g_{0c}|t/t_{p_{c}}|^{-2/3}$, as it is follows from Eq. (\[eq:10d\]). Noting that the amplitude $g_{0c}$ depends on the shape of the domains and on the boundary conditions, [@Schneider03b] it is remarkable that the experimental data collapses on two branches. Accordingly, the shape of the domains and the boundary conditions on the interface do not change significantly under applied pressure. The straight line in Fig. \[fig:scaling\_function\] (b) corresponds to $g_{0c}$ $\simeq 0.25$. This value is compatible with $g_{0_{c}}\simeq 0.5$ found for YBa$_{2}$Cu$_{3}$O$_{6.7}$ oriented powder and for Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ thin films. [@Schneider03b] Much larger values, $g_{0c}\approx 1.1-1.6$, have been found for Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ single crystals [@Schneider03b] and for Y$_{1-x}$Pr$_{x}$Ba$_{2}$Cu$_{3}$O$_{7-\delta }$ powders with $0.0\leq x\leq0.3$.[@Schneider03]
![Finite size scaling function $g$ \[Eq. (\[eq:scaling-finction\_lambda\])\] versus $t/\left|
t_{p_{c}}\right| $ (a) and versus $\left|t/ t_{p_{c}}\right|$ in a logarithmic scale (b) for different pressures. The solid stepwise line in (a) is the $g_\infty$ function for the homogenous ($L_c\rightarrow\infty$) system \[see Eq. (\[eq10c\])\]. The solid line in (b) is the leading finite-size behavior $g_{0c}|t/t_{p_c}|^{-2/3}$ at $t\rightarrow 0$ with $g_{0c}=0.25$ as it follows from Eq. (\[eq:10d\]). []{data-label="fig:scaling_function"}](Fig4BW.EPS){width="1.0\linewidth"}
----------- ---------- ----------- ---------------------- -------------------------------- --------- -------------------------- ---------------------------------- ----------------------------------------------------------------------- -------------------------- -- --
$T_c$ $T_{p_c}$ $\lambda_{0ab}^{-2}$ $\lambda_{ab}^{-2}$($T_{p_c}$) $L_c$ $\frac{\Delta T_c}{T_c}$ $\frac{\Delta T_{p_c}}{T_{p_c}}$ $\frac{\Delta\lambda_{ab}^{-2}(T_{p_c})}{\lambda_{ab}^{-2}(T_{p_c})}$ $\frac{\Delta L_c}{L_c}$
(K) (K) ($\mu$m$^{-2}$) ($\mu$m$^{-2}$) (Å) (%) (%) (%) (%)
0.5 kbar 80.58(4) 79.85(3) 93.8(1.5) 3.85(12) 33.2(9) - - - -
3.0 kbar 82.26(3) 81.45(3) 100.6(1.5) 4.29(11) 30.5(6) 2.08(6) 2.00(5) 11.3(3.7) -8.2(3.7)
8.9 kbar 85.34(3) 84.55(3) 108.0(1.4) 4.60(13) 29.3(8) 5.91(6) 5.89(5) 19.7(4.0) -11.7(4.0)
11.5 kbar 86.49(3) 85.65(3) 109.8(1.5) 4.74(13) 28.9(6) 7.33(6) 7.26(5) 23.3(3.8) -12.8(3.7)
----------- ---------- ----------- ---------------------- -------------------------------- --------- -------------------------- ---------------------------------- ----------------------------------------------------------------------- -------------------------- -- --
To summarize, the finite-size scaling analysis of the in-plane penetration depth data for YBa$_{2}$Cu$_{4}$O$_{8}$ is fully consistent with a finite-size effect. Indeed we established the consistency with all three characteristics of a finite-size effect (see Sec. \[seq:Theoretical\_background\]). The finite-size estimates for $\lambda_{0ab}^{-2}$ and $\lambda_{ab}^{-2}$($T_{p_c}$) at different pressures are summarized in Table \[Table1\].
The pressure dependence of the domain size {#seq:pressure_dependence_domain_size}
===========================================
Using Eq. (\[eq:Lc\_two\]) and the estimates for $T_{p_{c}}$ and $\lambda _{ab}^{-2}(T_{p_{c}})$ listed in Table \[Table1\], the domain size $L_{c} $ along the $c$-axis is readily calculated. In Fig. \[fig:Lc\_vs\_pressure\] we display the pressure dependence of $L_{c}$, $T_{c}(p)$ and $T_{p_{c}}(p) $. It is seen that $L_{c}$ decreases with pressure, whereas $T_{c}(p)$ and $T_{p_{c}}(p)$ increase almost linearly. On the other hand $T_{c}$ and $T_{p_{c}}$ increase with decreasing $L_{c}$. This agrees with the behavior found in Y$_{1-x}$Pr$_{x}$Ba$_{2}$Cu$_{3}$O$_{7-\delta }$, where $T_{c}$ and $ T_{p_{c}}$ were found to increase with reduced $L_{c}$. [@Schneider03] Here $T_{c}$ and $T_{p_{c}}$ have been reduced by increasing the Pr content $x$. In this context it is interesting to note that in granular aluminium $T_{c}$ was found to increase with reduced grain size.[@deutscher2] Another striking feature is the nanoscale magnitude of $L_{c}$, even though YBa$_{2}$Cu$_{4}$O$_{8}$ is stoichiometric.
![The pressure dependence of the domain size along the $c$-axis $L_c$ ($\blacksquare$), transition temperature $T_c$ ($\bullet$), and inflection temperature $T_{p_c}$ ($\circ$) in [YBa$_2$Cu$_4$O$_{8}$ ]{}. []{data-label="fig:Lc_vs_pressure"}](Fig5BW.EPS){width="1.0\linewidth"}
To check the consistency of our estimates we plot in Fig. \[fig:finite-size\_shifts\] the relative shifts of $T_{p_{c}}$, $\lambda _{ab}^{2}\left( T_{p_{c}}\right) $ and $L_{c}$ versus the relative shift of $T_{c}$. According to Eq.(\[eq4\]) these shifts are not independent but related by $$\frac{\Delta L_{c}}{L_{c}}=\frac{\Delta
T_{p_{c}}}{T_{p_{c}}}-\frac{\Delta \lambda _{ab}^{-2}\left(
T_{p_{c}}\right) }{\lambda _{ab}^{-2}\left( T_{p_{c}}\right) }.
\label{eq:relative_shifts}$$ The straight lines correspond to the linear fits with $\Delta
L_{c}/L_{c}= -2.18(26)\cdot\Delta T_{c}/T_{c}$, $\Delta
T_{p_{c}}/T_{p_{c}}=0.99(1)\cdot \Delta T_{c}/T_{c}$ and $ \Delta
\lambda _{ab}^{-2}\left( T_{p_{c}}\right) /\lambda
_{ab}^{-2}\left( T_{p_{c}}\right) = 3.34(27)\cdot\Delta
T_{c}/T_{c}$, revealing that Eq. (\[eq:relative\_shifts\]) is well satisfied. Furthermore, these estimates show that the reduction of $L_{c}$ with pressure reflects the fact that the pressure effect on $\lambda _{ab}^{-2}(T_{p_{c}})$ exceeds the effect on $ T_{c}$ considerably. Having established the consistency of our estimates it is essential to recognize that the occurrence of nanoscale superconducting domains is not an artefact of YBa$_{2}$Cu$_{4}$O$_{8}$ and our samples. Indeed the existence of nanoscale domains has been established in a variety of cuprates, including films, powders and single crystals.[@Schneider03b]
![The relative shifts of $\Delta \protect\lambda
_{ab}^{-2}(T_{p_{c}})/\protect \lambda _{ab}^{-2}(T_{p_{c}})$ ($\blacktriangledown$); $\Delta L_{c}/L_{c}$ ($ \blacksquare $), and $\Delta T_{p_{c}}/T_{p_{c}}$ ($ \bigcirc $) vs the relative shift of $\Delta T_{c}/T_c$. The solid lines represent the linear fits with $\Delta L_{c}/L_{c}= -2.18(26)\cdot\Delta T_{c}/T_{c}$, $\Delta T_{p_{c}}/T_{p_{c}}=0.99(1)\cdot \Delta T_{c}/T_{c}$ and $
\Delta \lambda _{ab}^{-2}\left( T_{p_{c}}\right) /\lambda
_{ab}^{-2}\left( T_{p_{c}}\right) = 3.34(27)\cdot\Delta
T_{c}/T_{c}$. []{data-label="fig:finite-size_shifts"}](Fig6BW.EPS){width="1.0\linewidth"}
Conclusion
==========
To summarize, we report the first observation of combined finite-size and pressure effects on the lengths $L_{c}$ of the superconducting domains along the $c$-axis and the in-plane penetration depth $\lambda _{ab}$ in a cuprate superconductor. The evidence for a finite-size behavior of the system arises from the tail in $\lambda_{ab}^{-2}\left( T\right)$ observed in the vicinity of $T_{c}$. We have shown that the scaling properties of the tail are fully consistent with a finite-size effect, arising from domains with nanoscale size along the $c$-axis. Indeed the essential characteristics of a finite-size effect, as (i) the limiting properties of the scaling function $ g(t/|t_{p_{c}}|)$, (ii) the existence of an inflection point in $\lambda ^{-2}(T)$ and (iii) the extremum in $d\lambda ^{-2}(T)/dT$ at $T_{p_{c}}$ have been verified. Even though YBa$_{2}$Cu$_{4}$O$_{8}$ is [*stoichiometric*]{} we have shown that the size of the domains along the $c$-axis is of nanoscale only, ranging from 33.2 Å to 28.9 Å at pressures from 0.5 to 11.5 kbar. This raises serious doubts on the existence of macroscopic phase coherent superconductivity. Contrariwise this does not exclude a percolative resistive superconductor to normal state transition, when the superconducting domains percolate. Indeed, a granular superconductor is usually characterized by two parameters:[@deutscher] the first is the domain’s size and the second the coupling between them. The coupling is accomplished randomly with a temperature dependent probability. Such a mechanism is a percolation process; at the temperature where the coupling probability is equal to the percolation threshold, an infinite cluster of coupled superconducting grains is formed. This suggests that resistive bulk superconductivity is achieved by a percolative process, while the phase coherent superconducting properties and the spatial extent of the domains can be probed by thermal fluctuations and finite-size effects.
The authors are grateful to J. Roos for stimulating discussions, K. Conder for help during sample preparation and T. Strässle for providing pressure cell for magnetization measurements. This work was supported by the Swiss National Science Foundation and by the NCCR program *Materials with Novel Electronic Properties* (MaNEP) sponsored by the Swiss National Science Foundation.
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[On Random Walks and Diffusions Related to Parrondo’s Games]{}
.5in
[Ronald Pyke]{}\
[University of Washington]{}\
\
.5in
.5in
[*AMS 1991 Subject Classification.*]{} 60J10, 60J15, 60J60
[*Key words and phrases.*]{} Parrondo games, simple random walk, shift diffusions, stationary probabilities, $\mod{m}$ random walk.
Introduction
============
The purpose of this paper is to study a family of random walks that include those arising in the games devised by J. M. R. Parrondo in 1997 to illustrate the apparent paradox that two ‘losing’ games can result in a ‘winning’ game when one alternates between them. We refer the reader to Harmer and Abbott (1999a,b), Harmer, Abbott and Taylor(2000) and Harmer, Abbott, Taylor and Parrondo(2000) in which Parrondo’s paradox is discussed, large simulations of specific Parrondo games and mixtures thereof are presented and certain theoretical results are given. These authors also give a heuristic explanation of the paradox in terms of the Brownian ratchet, the original motivation for the suggestion of these games. Other references to the general subject are included in the above mentioned papers by Harmer and Abbott. The reader may also note the reference Durrett, Kesten and Lawler(1991) which also deals with the general question of showing that winning games can be formed by mixing fair ones.
The suggested paradox may be visualized as follows. You are about to play a two-armed slot machine. The casino that owns this two-armed bandit advertises that both arms on their two-armed machines are “fair” in the sense that any player who plays either of the arms is assured that the average cost per play approaches zero as the number of plays increase. However, the casino does not constrain you to stay with one arm; you are allowed to use either arm on every play. You just tell the machine before beginning how many plays you wish to make. At the end of that number of plays, the machine displays the total amount won or lost. The question of interest in this context would be whether it is possible for the casino to still make money using only “fair” games.
In this paper a random walk will refer to a Markov chain $\{S_n:n=0,1,2,
\ldots\}$ taking values in the integers, ${\mathbb{Z}}$, which satisfies the discrete continuity condition $$|S_n-S_{n-1}|=1\quad\hbox{a.s.\ for each}\quad n\ge 1.$$ Let the transition probabilities for the random walk be denoted by $$p_j=P(S_{n+1}-S_n=1|S_n=j),\qquad q_j=P(S_{n+1}-S_n=-1|S_n=j)$$ and $$r_j=1-p_j-q_j=P(S_{n+1}=S_n|S_n=j)$$ for $j\in{\mathbb{Z}}$. Assume that $p_jq_j\ne 0$ for all $j$. For fixed integer $m\ge 1$, define a *mod m random walk* to be a random walk in which the transition probabilities $p_i$, $r_i$, $q_i$ depend only upon the congruence class $\mod{m}$ of the state $i$. Thus, these [*lattice regular*]{} or [*periodic*]{} random walks are such that for some specified integer $m>1$, $p_j=p_{j+m}$ and $q_j=q_{j+m}$ for all $j\in {\mathbb{Z}}$. More generally, define a *mod m Markov chain* on the integers to be one whose parameters depend only upon the congruence classes $\mod{m}$ of the states, namely, $p_{ij} = p_{i+m,j}$ for all integers $i,j$. This paper is concerned with the case of random walks, but places where the approach applies more generally are pointed out.
A $\mod{m}$ random walk is determined by the $2m$ parameters $p_j, q_j;0\le j <m$. Write $\bp=(p_0,p_1,\cdot\cdot \cdot,p_{m-1})$ with an analogous use of $\bq$ to specify the walk’s parameters. Observe that when $m=1$ the walk is classical simple random walk, so our main interest is in the cases of $m>1$.
These random walks are viewed as games with the increment $X_n=S_n-S_{n-1}(n\ge 1)$ denoting the gain at the $n$-th play. We say that the game is a winning/losing/fair game according as the almost sure limit of $S_n/n$ is positive/negative/zero.
For given $m$, write ${{\mathbb{Z}}_m}:=m{\mathbb{Z}}=\{km:k\in{\mathbb{Z}}\}$ for the integer lattice of span $m$. In the games introduced by Parrondo, it is assumed that the transition probabilities depend on the state only to the extent that the state is or is not in ${{\mathbb{Z}}_m}$. Thus, Parrondo’s games are characterized by $$\label{eqn1.1}
P(X_{n+1}=1\mid S_0,S_1,\ldots, S_n) = p'1_{[S_n\in{{\mathbb{Z}}_m}]}+p1_{[S_n\not\in{{\mathbb{Z}}_m}]}$$ for some $p, p'\in[0,1]$ and all $n\ge 0$. Write $q=1-p$ and $q'=1-p'$. We may also write $k\equiv j\mod{m}$ when $k\in j+{{\mathbb{Z}}_m}$.
For simplicity, we write ${G(m, \bp,\bq)}$ to denote a general $\mod{m}$ random walk or game, but write ${G(m, \bp)}$ for the game when each $q_j=1-p_j$ (i.e. each $r_j=0$) and write ${G(m, p, p')}$ for the special Parrondo random walk or game satisfying (\[eqn1.1\]).
The required notation and preliminary structure are introduced in the following section, in which the limiting results for ${G(m, p, p')}$ games are given for illustration. The general case is covered in Section 3, while in Section 4 we resolve the central question about whether random mixtures of losing Parrondo’s games can be winning ones. The asymptotic gain is derived in Section 5 while in Section 6 a certain expected interoccurrence time that appears in the previously obtained expression for this is also dreived. The method used to solve the recursion equations in these sections makes use of an extension of results of Mihoc and Fréchet (cf. Fréchet(1952)) that are provided in the Appendix to this paper. Continuous analogues to the random walks considered here are introduced in Section 7. These $\mod{m}$ diffusions have drift functions that are periodic step functions so that their embedded walks on the integers are ${G(m, \bp,\bq)}$ walks. In Theorem 7.2 the drift rates under which the embedded walk has specified transition probabilities is determined.
Preliminaries and Parrondo’s Examples
======================================
In the games suggested by Parrondo, the transition probabilities depend on the state only to the extent that it is or is not in ${{\mathbb{Z}}_m}$; see (\[eqn1.1\]) above. The asymptotic behavior of these games, as for any $\mod{m}$ random walk is determined by that of its embedded walk on the lattice ${{\mathbb{Z}}_m}$. Since this embedded walk is equivalent to simple random walk, its asymptotics are well known and dependent solely upon a single parameter, the walk’s probability of ’success’. In this section we introduce the notation required for the general case in Section 3 below, and illustrate the approach in the special case of a Parrondo ${G(m, p, p')}$ walk by substituting in known results for simple random walk.
Let $T_1<T_2<\cdots$ be the successive transition times of the embedded walk on ${{\mathbb{Z}}_m}$. That is $T_1 =\min\{n\ge 0:S_n\in {{\mathbb{Z}}_m}\}$ and, for $k>1$, $$T_{k+1}=\min\{n>T_k:S_n-S_{T_k}=\pm m\}$$ with the minimum of a null set being defined to equal $+\infty$. Set $T_0=0$. Write $$J_n=m^{-1}S_{T_{n+1}},\ \ \xi_n=T_{n+1}-T_n$$ for $n\ge 0$ so that $\{(J_n, \xi_n):n\ge 0\}$ is a (possibly delayed) Markov renewal process (MRP) in which the embedded random walk $\{J_n\}$ is simply a classical random walk with constant probability of ‘ success’, $$\label{eqn2.1}
p^\ast_m:=P(J_{n+1}-J_n=1\mid J_n).$$ Hence, once $p_m^{\ast}$ is known, the winning/losing/fair nature of the walk is easily determined.
In general, for $n$ satisfying $T_k<n\le T_{k+1}$, $$\frac{mJ_{k+1}-2m}{T_{k+1}} = \frac{S_{T_{k+1}}-2m}{T_{k+1}} \le
\frac{S_n}{n} \le \frac{S_{T_k}+m}{T_k} = \frac{mJ_k+m}{T_k}.$$ Thus, $$\label{eqn2.7}
m\left(\frac{mJ_{k+1}}{k+1} - \frac{2}{k+1}\right)/\frac{T_{k+1}}{k+1} \le
\frac{S_n}{n} \le m\left(\frac{J_k}{k} + \frac{1}{k} \right)/\frac{T_k}{k}.$$ It is known for the classical random walk $\{J_n\}$ that $J_k/k$ converges a.s. as $k\to\infty$ to $p^\ast_m-q^\ast_m$, with $q^\ast_m = 1-p^\ast_m$. Moreover, the stopping times $\{T_k\}$ are partial sums of iid r.v.’s having finite expectations so that $$T_k/k \mathop{\longrightarrow}\limits^{\mathrm{a.s.}} E(T_2-T_1)<\infty.$$ Upon taking limits in (\[eqn2.7\]) one obtains that with probability one, $$\label{eqn2.8}
\lim_{n\to\infty} \frac{S_n}{n} = \frac{m(p^\ast_m-q^\ast_m)}{E(T_2-T_1)}.$$ Clearly then, this limit is $0$, $>0$ or $<0$ according as $p^\ast_m=$, $>$ or $<q^\ast_m$.
The quantity $p^\ast_m$ is evaluated for the general ${G(m, \bp,\bq)}$ walk in Lemma \[lem6.2\] below. However, for the special Parrondo ${G(m, p, p')}$ random walk, the evaluation is immediate once we introduce the notation and approach that is needed for the general case, and so we give it separately here as
\[lem2.1\] The ${{\mathbb{Z}}_m}$-embedded MRP of the ${G(m, p, p')}$ random walk has transition probabilities determined by the ‘success’ probability $$\label{eqn2.2}
p^\ast_m=\frac{p'p^{m-1}}{p'p^{m-1}+q'q^{m-1}}$$ for all $p,p'\in[0,1]$ satisfying $|p-p'| < 1$.
The first part of this proof, through (\[eqn2.4\]) below, is general and will be needed in Section 3. The rest is substitution of known results.
Suppose $J_n=k$. That is, for the original walk suppose $S_{T_{n+1}}=km$. Since $T_{n+1}$ is a stopping time, $P(J_{n+1}-J_n=1\mid J_n=k
)$ is just the probability that starting at $S_0=0$, the random walk $\{S_n\}$ reaches $m$ before it reaches $-m$. But $S_1$ equals 1 or $-1$ with probability $p_0$ or $q_0$, respectively. Thus if we let $A$ denote the event that $\{S_n:n>1\}$ reaches 0 before it reaches $mS_1$ then the Markov property implies that $p^\ast_m$, the success probability for the embedded walk, satisfies the following recursion relation, in which we partition the event according to whether the original walk hits zero before $m$ or not: $$\label{eqn2.3}
p^\ast_m=P(A)p^\ast_m+p_0\{1-P(A\mid S_1=1)\}.$$ Hence $$\label{eqn2.4}
p^\ast_m=p_0P(A^c\mid S_1=1)/P(A^c).$$ Since for the special case of this lemma, the conditional probabilities given $S_1$ are just those that arise in the classical gambler’s ruin problem, (cf Feller (1968, Chap. XIV) it is known that $$\label{eqn2.5}
P(A^c \mid S_1=1) = \left\{\begin{array}{ll}
(qp^{m-1}-p^m)/(q^m-p^m) & \hbox{if }p\ne q \\
1/m & \hbox{if }p=\frac{1}{2}\end{array}\right.$$ and $P(A^c\mid S_1=-1)$ is similar but with $p$ and $q$ interchanged. Substitution of (\[eqn2.5\]) into (\[eqn2.4\]) now gives, when $p\ne q$, $$\label{eqn2.6}
P(A^c)
= (q'q^{m-1}+p'p^{m-1})(q-p)/(q^m-p^m)$$ and, therefore, $p^\ast_m$ is as required by (\[eqn2.2\]). When $p=\frac{1}{2}$, the substitution of (\[eqn2.5\]) yields $p^\ast_m=p'$ to complete the proof.
Note that by (\[eqn2.2\]), $p^\ast_m$ is the conditional probability that $S_n$ reaches $m$ before $-m$ given that $S_0=0$ and that the first $m$ steps of $S_n$ are monotone. This structure is more readily seen in the general case of Lemma \[lem6.2\] below.
For the special ${G(m, p, p')}$ case the above result yields
\[cor2.2\] (Harmer and Abbott(2000a)) When $|p-p'| < 1$, the game ${G(m, p, p')}$ is a fair, winning or losing game according as $$p'p^{m-1}-q'q^{m-1}=0,\qquad >0\qquad \hbox{or}\qquad <0.$$
The condition in Corollary \[cor2.2\] is more clearly expressed in terms of new variables $x=p/q$ and $y=p'/q'$, namely, the game ${G(m, p, p')}$ is a fair, winning or losing one according as $$\label{eqn2.9}
y-x^{-(m-1)}=0,\qquad >0\qquad\hbox{or}\qquad <0.$$ Recall that the degenerate case $q'=0=p$ has been excluded. Since the inverse relationships are $p=x/(1+x)$ and $p'=y/(1+y)$, it follows from (\[eqn2.9\]) that ${G(m, p, p')}$ is fair if for some $x\ge 0$, $p$ and $p'$ are related as $$q=1-p=\frac{1}{1+x}\qquad\hbox{ and }\qquad
p'=\frac{x^{-m+1}}{1+x^{-m+1}}=\frac{1}{1+x^{m-1}}.$$
Here are some examples. For $x=1$, $G(m,\frac{1}{2}, \frac{1}{2})$ is fair for every $m\ge 1$. For $m=4$ and $x=2$, the game $G(4, \frac{4}{5},
\frac{1}{65})$ is seen to be fair, and for $m=5$ and $x=2$, $G(5, \frac{2}{3},
\frac{1}{17})$ is fair. When $m=3$ and one chooses $x=3$, one obtains the fair game $G(3, \frac{3}{4}, \frac{1}{10})$. The associated games $G(3,
\frac{3}{4}-{\varepsilon}, \frac{1}{10}-{\varepsilon})$ for a range of ${\varepsilon}>0$ are the losing games used in the simulation study of Harmer and Abbott(1999a). The fact that these are losing games as indicated there is immediate from the following observation: If $G(m, p_0, p'_0)$ is a fair game, then ${G(m, p, p')}$ is a losing game whenever $0\le p'\le p'_0$ and $0\le p\le p_0$ with $p+p'<p_0+p'_0$; simply observe that $p'/q'$ and $p/q$ are increasing functions of $p'$ and $p$, respectively, so that $p'_0/q'_0=(p_0/q_0)^{-m+1}$ implies $p'/q'\le (p/q)^{-m+1}$ whenever $p'\le p'_0$ and $p\le p_0$. Since $G(3, \frac{3}{4}, \frac{1}{10})$ is a fair game the result follows.
General Mod m Random Walks
==========================
Let $\{S_n:n\ge 0\}$ be a general (discretely continuous) random walk on the integers ${\mathbb{Z}}$ in the sense described in the Introduction above. The asymptotic behavior of $\{S_n\}$ can be described in terms of the two associated reflecting random walks on the negative and positive integers. The latter is obtained, for example, by replacing $r_0$ and $q_0$ by $\bar r_0=1-p_0$ and $\bar q_0=0$. It is known (cf. Feller (1968), Chap. XV.8 or Chung (1967), Sect. I.12) that the corresponding reflecting random walk on ${\mathbb{Z}}^+=\{0,1,2,\ldots\}$ is recurrent or transient according to $$\label{eqn6.1}
\sum^\infty_{i=1}\frac{q_1q_2\cdots q_i}{p_1p_2\cdots p_i}=\infty$$ or not. When one looks similarly at the reflecting random walk on ${\mathbb{Z}}^-=\{0, -1, -2,\break
\ldots\}$, the roles of the $p$’s and $q$’s are interchanged so that recurrence in this case holds if and only if $$\label{eqn6.2}
\sum^\infty_{i=1} \frac{p_{-1}p_{-2}\cdots p_{-i}}{q_{-1}q_{-2}\cdots
q_{-i}}=\infty.$$ Now return to the original walk on ${\mathbb{Z}}$. The positive part of this walk, $\{S^+_n\}$, is a Markov renewal process in which all sojourn times are equal to one except those between successive visits to state $0$. The distribution of these latter sojourn times is a possibly deficient mixture that includes with probability $q_0$ the distribution of the first passage time from state $-1$ to state $0$. The latter passage time is finite with probability one only if the reflecting random walk on ${\mathbb{Z}}^-$ is recurrent. Hence $\{S_n\}$ is a recurrent random walk if and only if both reflecting random walks are recurrent, or equivalently, if and only if both (\[eqn6.1\]) and (\[eqn6.2\]) hold. Consequently, the walk is transient if and only if at least one of these series converges. Accordingly, the boundary of a transient random walk may consist of either or both of $+\infty$ and $-\infty$, depending upon which one or both of the series converge. (Cf. Karlin and McGregor (1959), Section 4 where the integral representations of the transition probabilities of the doubly infinite random walk are expressed in terms of those of the two corresponding reflecting walks.)
Consider now, for fixed integer $m\ge 1$, a *mod m random walk* as defined in Section 1. random walk in which the transition probabilities $p_i$, $r_i$, $q_i$ depend only upon the congruence class $\mod{m}$ of the state $i$. (Note that when $m=1$, the $\mod{1}$ random walk is just the classical random walk with constant transition probabilities.) Thus, for $i=sm+l$ for some $s\in{\mathbb{Z}}$ and $l=0,1,\ldots, m-1$, we know that $(p_i, r_i, q_i)=(p_l, r_l, q_l)$. Moreover, for $s\ge 0$, the summand in (\[eqn6.1\]) becomes $$\label{eqn6.3}
\frac{q_1q_2\cdots q_i}{p_1p_2\cdots p_i} =
\frac{p_0}{q_0}\left\{\frac{q_0q_1\cdots q_{m-1}}{p_0p_1\cdots
p_{m-1}}\right\}^s \frac{q_0q_1\cdots q_l}{p_0p_1\cdots p_l}$$ while for $s<0$, a similar representation holds with the $p$’s and $q$’s interchanged. If we define $$\label{eqn6.4}
\rho_m:=\frac{p_0p_1\cdots p_{m-1}}{q_0q_1\cdots q_{m-1}}$$ then the divergence of (\[eqn6.1\]) holds if and only if $\rho_m\le 1$ while (\[eqn6.2\]) holds if and only if $\rho_m\ge 1$. By the above discussion, the walk is then recurrent, transient to $+\infty$ or transient to $-\infty$ according as $\rho_m$ is equal to, greater than or less than one. This then proves
\[lem6.1\] For $m\ge 1$, a $\mod{m}$ random walk is recurrent, transient toward $+\infty$ or transient toward $-\infty$ according as $$\label{eqn6.5}
p_0p_1\cdots p_{m-1}-q_0q_1\cdots q_{m-1}=0,\qquad >0 \quad \hbox{ or }<0.$$
It remains to evaluate $p^{\ast}_m$, the probability of ’success’, $p^{\ast}_m$, for the embedded walk on ${{\mathbb{Z}}_m}$.
\[lem6.2\] For $m\ge 1$, and a $\mod{m}$ random walk ${G(m, \bp,\bq)}$ with parameters ${\mathbf{p}}=(p_0,p_1,
\ldots, p_{m-1})$ and ${\mathbf{q}}=(q_0, q_1,\ldots, q_{m-1})$ satisfying $p_iq_i\ne 0$ for $i=0,1,\ldots, m-1$, one has $$\label{eqn6.6}
p^\ast_m = \frac{p_0p_1\cdots p_{m-1}}{p_0p_1\cdots p_{m-1}+q_0q_1\cdots
q_{m-1}} = \frac{\rho_m}{1+\rho_m}.$$
For this general case, set $$\label{eqn6.7}
v_m=P(A^c|S_1=1)\qquad\hbox{and}\qquad \bar v_m=P(A^c|S_1=-1).$$ so that the expression for $p^{\ast}_m$ in (\[eqn2.4\]) becomes $$p^\ast_m=p_0v_m(p_0v_m+q_0\bar v_m)^{-1}$$ Thus (\[eqn6.6\]) will be proved once it is established that $$\label{eqn6.8}
\frac{v_m}{\bar v_m}=\frac{p_1p_2\cdots p_{m-1}}{q_1q_2\cdots q_{m-1}}.$$ By definition, $v_m$ $(\bar v_m)$ is the probability (of ’ruin’) that starting at $1$ $(-1)$ the random walk reaches $m$ $(-m)$ before it reaches $0$. Moreover, by the modulo structure of the walk, $\bar v_m$ is the same as the probability that starting at $m-1$, the random walk reaches $0$ before $m$. Thus, $v_m$, for example is the same as $\ _0f_{1m}$ in the usual notation for these taboo probabilities; cf. Chung (1967, Sect. I.12) where these are derived for the random walk. Direct substitution of these exact values would then justify (\[eqn6.6\]). Since we only require the ratio of these two taboo probabilities, the following mapping approach suffices, and may be of separate interest.
We first construct a $1-1$ correspondence between the set, $\Gamma_k$, of paths that go from $1$ to $m$ without hitting $0$ and the set, $G_k$, of paths that go from $m-1$ to $0$ without hitting $m$. Thi s correspondence is a simple reversal: If ${\mathbf{s}}_k=(s_1, s_2, \ldots, s_k,
m)$ denotes a path in $\Gamma_k$ so that $s_1=1$, $s_k=m-1$ and $1\le s_i\le
m-1$ for $1\le i\le k$, the corresponding reversed path in $G_k$ is $${\mathbf{t}}_k({\mathbf{s}}_k)={\mathbf{t}}_k=(t_1, t_2, \ldots, t_k, 0) \equiv (s_k, s_{k-1},
\ldots, s_1, 0).$$ (The reader can visualize the reversal of a path in the illustration of Figure 1. In fact, the result becomes fairly transparent once one recognizes the effect on paths of flipping the time axis.)
{width="6in"}
For a given path ${\mathbf{s}}_k$, let $\nu^+_i$ ($\nu^-_i$) equal the number of transitions from $i$ to $i+1$ ($i$ to $i-1$). Then $$\begin{aligned}
\label{eqn6.10}
P\left((S_1, S_2, \ldots, S_{k+1}) = {\mathbf{s}}_k|S_1=1\right) & = &
\prod^{m-1}_{i=1}p^{\nu^+_i}_i q^{\nu^-_i}_i \nonumber \\
& = & \left(\prod^{m-1}_{i=1}p_i\right)\prod^{m-2}_{j=1}
(p_jq_{j+1})^{\nu^+_j-1},\end{aligned}$$ with the last step following since $\nu^-_1=0$, $\nu^+_{m-1}=1$ and $\nu^-_{i+1}=\nu^+_i-1\ge 0$ for $1\le i\le m-2$. For the reversed path ${\mathbf{t}}_k({\mathbf{s}}_k)$, where an ‘up’ transition of $s_j$ to $s_{j+1}$ in ${\mathbf{s}}_k$ becomes a ‘down’ transition of $s_{j+1}$ to $s_j$. Write $\widehat\nu^+_i$ and $\widehat\nu^-_i$ for the corresponding numbers for ${\mathbf{t}}_k$ so that $$\label{eqn6.11}
P\left((S_1, S_2, \ldots, S_{k+1})={\mathbf{t}}_k|S_1=m-1\right) =
\left(\prod^{m-1}_{i=1}q_i\right)\prod^{m-1}_{j=2}(q_jp_{j-1})^{\widehat\nu^-_j-
1}.$$ But it is clear from the correspondence that $\widehat\nu^-_j=\nu^+_{j-1}$. Thus for every $k\ge m-1$ and every path ${\mathbf{s}}_k\in\Gamma_k$ the ratio of (\[eqn6.10\]) over (\[eqn6.11\]), namely $p_1p_2
\cdots
p_{m-1}/q_1q_2\cdots q_{m-1}$, is constant. It now follows immediately that (\[eqn6.8\]) holds, thereby completing the proof.
Random Mixtures of Parrondo Games ${G(m, p, p')}$
=================================================
The main question of interest for these games concerns what happens to a player’s fortune when two or more games are played in some alternating fashion. For example, if two different games are known to be fair, can a player create a winning game by randomly choosing between the two at each play? Observe first of all that for $\pi\in[0,1]$, the random mixture of two games, ${G(m, \bp,\bq)}$ and $G(,m,\bP,\bQ)$, in which at each play the former is chosen with probability $\pi$,is also a $\mod{m}$ game, namely, $G(m,\pi\bp+(1-\pi)\bP,\pi\bq+(1-\pi)\bQ)$. Since Lemma 3.1 characterizes the winning or losing nature of any such game, the question of whether the random mixture of two fair games is a winning game or not has been theoretically answered. By the way, the criterion in Lemma 3.1 implies that if $\bp$ and $\bq$ are interchanged in a fair game ${G(m, \bp,\bq)}$, it remains fair, whereas a losing game would be turned into a winning game. Moreover, the nature of the criterion is such that it should be the exception rather than the rule for a random mixture of fair games to remain fair. Thus at this stage, the existence of fair games whose mixture is winning (or losing) would appear to be less paradoxical.
A couple of general questions of interest are as follows. Suppose we say that two fair games, A and B, are [*mutually supportive*]{} if any other game consisting of a sequence of plays of game A or B is not a losing game whenever the game choices are made independently of previous outcomes. Do mutually supportive pairs of distinct games exist? Is it true that if a non-trivial random mixture (in which game A is chosen independently at each stage with constant probability) does not result in a losing game, then the two games are mutually supportive?
In this section, we give a complete answer to the structure of random mixtures in the special case of the Parrondo game, ${G(m, p, p')}$. Although this is done by rather elementary methods, more general questions involving mixtures appear to be quite difficult.
Consider the random mixture, $G(m, \pi p+(1-\pi)\beta, \pi p'+(1-\pi)\beta')$, of the two Parrondo games $G(m, p, p')$ and $G(m, \beta,\beta')$, in which the mixing probability is $\pi \in (0,1)$. Set $$\label{eqn3.1}
x=p/q,\ \ y=p'/q',\ \ \widehat x=\frac{\beta}{1-\beta},\ \ \widehat
y=\frac{\beta'}{1-\beta'}$$ and $$\label{eqn3.2}
\overline x=\frac{\pi p+(1-\pi)\beta}{1-\{\pi p+(1-\pi)\beta\}}=
\frac{p+\lambda\beta}{q+\lambda(1-\beta)},\qquad
\overline y=\frac{p'+\lambda\beta'}{q'+\lambda(1-\beta')},$$ where $\lambda=(1-\pi)/\pi$. Assume without loss of generality that $\beta <p$, or equivalently, $\widehat x <x$.
The question to consider is whether the random mixture of two losing games can be a winning game. Suppose first that the two given games are fair. That is, by Corollary \[cor2.2\] in the form (\[eqn2.9\]), our question is whether it is possible to have $$\label{eqn3.3}
y=x^{-m+1},\qquad \widehat y=\widehat x^{-m+1}\qquad\hbox{and}\qquad
\overline y>\overline x^{-m+1}.$$ For simplicity, write $m-1=r$ so that $r=1,2,\cdots$. Simple algebra leads to $$\label{eqn3.4}
\overline x=\frac{x(1+\widehat x)+\lambda\widehat x(1+x)}{1+\lambda+\widehat x+\lambda x},\qquad
\overline y=\frac{y(1+\widehat y)+\lambda\widehat y(1+y)}{1+\lambda+\widehat y+\lambda y}.$$
Substitution of the first two equations of (\[eqn3.3\]) into $\overline y$ permits the inequality $\overline y>\overline x^{-r}$ to be written after simplification as $$\label{eqn3.5}
\frac{1+\lambda+\widehat x^r+\lambda x^r}{(1+\lambda)(\widehat x x)^r+\lambda\widehat x^r+x^r} >
\frac{(1+\lambda+\widehat x+\lambda x)^r}{((1+\lambda)(\widehat x x)+\lambda\widehat x+x)^r}.$$ Clearly, this can never hold if $r=1$, (i.e $m=2$). We assume, therefore, that $m >2$ in the remainder of this section.
If one introduces functions $f(a)=a^r$ and $g(a,b)=(1+\lambda +a+\lambda b)/((1+\lambda)ab+\lambda a+b)$, then (\[eqn3.5\]) involves a form of inverse composition, namely, $$g(f(\widehat x), f(x)) > f(g(\widehat x,x)).$$ On the other hand, (\[eqn3.5\]) may be written equivalently in terms of $\pi$ as $$\label{eqn3.6}
\frac{1+\pi\widehat x^r+(1-\pi)x^r}{(1+\pi\widehat x+(1-\pi)x)^r}
> \frac{1
+\pi\widehat x^{-r}+(1-\pi)x^{-r}}{1+\pi\widehat x^{-1}+(1-\pi)x^{-1})^r}.$$ Thus, this inequality is one about norms on the simplex as may be seen as follows: If we set ${\mathbf{u}}=(1,
\widehat x,x)$ and ${\mathbf{v}}=(1,1/\widehat x, 1/x)$, (\[eqn3.5\]) is equivalent to $$\|{\mathbf{u}}\|_{r,\mu}/\|{\mathbf{u}}\|_{1,\mu} > \|{\mathbf{v}}\|_{r,\mu}/\|{\mathbf{v}}\|_{1.\mu},$$ where the norms are with respect to the measure $\mu$ that assigns masses $1,\pi,1-\pi$ to the coordinates $1,2,3$, respectively. \[In this context, the special case of $\widehat x=1$, in which the first game is the classical fair random walk, (and which is the case relevant to the examples in Harmer and Abbott (1999a)), is describable as a comparison between the $r$-norms of the ray projection onto the unit simplex of the vectors $(1,1,x)$ and $(1,1,x^{-1})$ (or equivalently, $(1,x,x)$. Moreover, in the case of purely random mixing $(\pi=1/2)$, the inequality is more enticing in that it may be stated as above but for vectors $(1,1,1,x)$ and $(1,1,1,x^{-1})$ under counting measure on the coordinates.\]
Fix $\widehat x=a\ge 1$. By cross multiplying in (\[eqn3.6\]), the inequality is equivalent to the positivity of the polynomial $$\label{eqn3.7}
\begin{array}{rcl}
Q(x):&=&(1+\lambda+a^r+\lambda x^r)((1+\lambda)ax+\lambda a+x)^r \nonumber\\
&&\quad \quad -((1+\lambda)a^rx^r+\lambda a^r+x^r)(1+\lambda+a+\lambda x)^r \\
&=&(1+\lambda+a^r)((1+\lambda)ax+x+\lambda a)^r-\lambda a^r(1+\lambda +a+\lambda x)^r\\
&&\quad +x^r\{\lambda((1+\lambda)ax+x+\lambda a)^r-((1+\lambda)a^r+1)(1+\lambda +a+\lambda x)^r\} \\
&=&\D{\sum^r_{j=0}{r\choose j}\{(1+\lambda+a^r)((1+\lambda)a+1)^j(\lambda a)^{r-j}-\lambda a^r(1+\lambda +a)^{r-j}\lambda^j\} x^j}
\\
&&\D{\quad +
\sum^r_{k=0}{r\choose k}\{\lambda((1+\lambda)a+1)^k(\lambda a)^{r-k}-((1+\lambda)a^r+1)(1+\lambda +a)^{r-k}\lambda^k\}x^{r+k}.}
\end{array}$$ Upon writing $Q(x)=\sum^{2r}_{j=0}q_jx^j$, it follows that the coefficients are $$\label{eqn3.8}
q_j = \left\{ \begin{array}{ll}
{r\choose j}a^r\{(1+\lambda+a^r)(1+\lambda+a^{-1})^j\lambda^{r-j} - (1+\lambda+a)^{r-j}\lambda^{j+1}\} & \hbox{for }0\le j<r,
\\
(1+\lambda+a^r)((1+\lambda)a+1)^r - ((1+\lambda)a^r+1)(1+\lambda +a)^r & \hbox{for }j=r, \\
{r\choose j-r}a^r\{(1+\lambda+a^{-1})^{j-r}\lambda^{2r-j+1} - (1+\lambda+a^{-r})(1+\lambda+a)^{2r-j}\lambda^{j-r}\} & \hbox{for
}r<j\le 2r.\end{array}\right.$$ Since the expressions within the parentheses in the first and third cases are increasing in $j$, there can be at most one change of sign among the first $r$ coefficients and at most one among the last $r$. Thus, regardless of the sign of the middle coefficient, $q_r$, there are at most three changes of signs in the coefficients of $Q$ with the exact number depending upon the signs of $q_0,q_{r-1},q_r.q_{r+1},q_{2r}$. (One may check that $q_{2r}$ is always positive for $a>1$, while $q_0$ is negative when $\lambda\le1$ (i.e $\pi\ge1/2i$) or when $a\le1$.) By Descartes’s rule of signs, the number of positive roots of $Q(x)=0$ does not, therefore, exceed 3.
It follows directly from the definition of $Q$ that $Q(a)=0$. However, one may check that $x=a$ is in fact a double root for all positive $a$. To see this, compute from (\[eqn3.7\]), $$\label{eqn3.9}
\begin{array}{rcl}
Q'(x)&=&rx^{r-1}\lambda((1+\lambda)ax+x+\lambda a)^r\\
&&\quad + r((1+\lambda)a+1)(1+\lambda+a^r+\lambda x^r)((1+\lambda)ax+x+\lambda a)^{r-1} \\
&&\quad -((1+\lambda)a^r+1)rx^{r-1}(1+\lambda+a+\lambda x)^r\\
&&\quad -r\lambda((1+\lambda)a^rx^r+x^r+\lambda a^r)(1+\lambda+a+\lambda x)^{r-1}\\
\end{array}$$ so that after simplification $$\begin{aligned}
Q'(a) &=& r(1+\lambda)^ra^{r-1}(a+1)^{r-1}\{\lambda a^r(a+1)+((1+\lambda)a+1)(1+a^r) \\
&&\quad -((1+\lambda)a^r+1)(1+a)-\lambda a(a^r+1)\}=0\end{aligned}$$ for any $a$. Since this implies that $x-a$ is a double root of $Q$, it follows from Descartes’s rule of signs that $Q$ has either two or three positive roots. In either case, we need to know that the root at $x=a$ is the largest positive root. To show this, differentiate (\[eqn3.9\]) to obtain $$\begin{aligned}
r^{-1}Q''(x) &=&\lambda(r-1)x^{r-2}((1+\lambda)ax+\lambda a+x)^r + 2\lambda rx^{r-1}((1+\lambda)a+1)((1+\lambda)ax+\lambda a+x)^{r-1} \\
&&\quad +(r-1)((1+\lambda)a+1)^2(1+\lambda+a^r+\lambda x^r)((1+\lambda)ax+\lambda a+x)^{r-2} \\
&&\quad -((1+\lambda)a^r+1)\{(r-1)x^{r-2}(1+\lambda+a+\lambda x)^r+2\lambda rx^{r-1}(1+\lambda+a+\lambda x)^{r-1}\} \\
&&\quad -\lambda^2(r-1)((1+\lambda)a^rx^r+x^r+\lambda a^r)(1+\lambda+a+x)^{r-2}\end{aligned}$$ from which $$\begin{aligned}
Q''(a) &=&r(1+\lambda)^{r-1}a^{r-2}(a+1)^{r-2}\{\lambda(1+\lambda)(r-1)a^r(a+1)^2+2\lambda ra^r((1+\lambda)a+1)(a+1)\\
&&\quad +(r-1)((1+\lambda)a+1)^2(1+a^r)-(1+\lambda)(r-1)(a+1)^2((1+\lambda)a^r+1)\\
&&\quad -2\lambda ra(a+1)((1+\lambda)a^r+1)-\lambda^2(r-1)a^2(a^r+1)\}.\\\end{aligned}$$ By grouping the terms within the parentheses here according to powers of $a$, this becomes $$Q''(a)=(1+\lambda)^{r-1}ra^{r-2}(a+1)^{r-2}\{\lambda(r-1)(a^{r+2}-
1)+2\lambda r(a^{r+1}-a)+\lambda(r+1)(a^r-a^2)\}.$$ Thus, for $r\ge2$ ($m\ge3$), $Q''(a)$ is positive, negative or zero according as $a>1$, $a<1$ or $a=1$. This implies in particular that when $a=1$, $x=1$ is a triple root, and hence the only root by Descartes’s rule of signs. Thus, when $a=1$, $x=1$ is the only positive root, insuring that $Q(x)>0$ for all $x>1$. For $a>1$, the fact that $Q''(a)>0$ shows that this double root at $x=a$ is a local minimum. Since by (\[eqn3.8\]) the leading coefficient, $q_{2r}$, is positive for all $\lambda$ and all $a>1$, this insures again that $x=a$ is the largest real root of $Q(x)=0$, thereby establishing that $Q(x)>0$ for all $x>a$ whenever $a\ge1$. This completes the proof of
\[thm3.1\] The random mixture, $G(m, \pi p+(1-\pi)\beta, \pi p'+(1-\pi)\beta')$, of two fair games, $G(m,\beta,\beta')$ and $G(m, p, p')$ is a winning game whenever $m\ge 3$ and $\frac{1}{2}\le\beta<p\le 1$.
\[cor3.2\] There exist losing games, the random mixture of which is a winning game.
By Corollary \[cor2.2\], the expression whose sign determines whether a game is winning, losing or fair, is a continuous function of its variables. It is therefore clear that for the games appearing in the statement of Theorem \[thm3.1\], one may make a sufficiently small change in the parameters $(\beta,\beta')$ and $(p,p')$ to make the associated fair games become losing ones, while preserving the inequality that ensures that the random mixture of the two remains a winning game.
The example presented in Harmer and Abbott (1999a) may now be described as follows. Take $m=3$, $\beta=\frac{1}{2}=\beta'$, $p=\frac{3}{4}$ and $p'=\frac{1}{10}$. The games $G(3, \frac{1}{2}, \frac{1}{2})$ and $G(3,
\frac{3}{4}, \frac{1}{10})$ are fair by Corollary \[cor2.2\], so that by Theorem \[thm3.1\], the mixture $G(3, \frac{5}{8}, \frac{3}{10})$ is a winning game. Consider now the games used by these authors, $G(3, \frac{3}{4}-{\varepsilon},
\frac{1}{10}-{\varepsilon})$ and $G(3, \frac{1}{2}-{\varepsilon}, \frac{1}{2}-{\varepsilon})$, and their random mixture $G(3, \frac{5}{8}-{\varepsilon}, \frac{3}{10}-{\varepsilon})$. It is clear that the first two are losing games for each positive ${\varepsilon}<1/10$ and that there would be some positive value ${\varepsilon}_0\le1/10$ for which the mixture remains a winning game whenever $0<{\varepsilon}<{\varepsilon}_0$, as postulated in Harmer and Abbott (1999a).
In this section we have considered the random mixing of two ${G(m, \bp)}$ walks. One is also interested in deterministic mixtures. Simulations in Harmer and Abbott(1999a) indicate that deterministic mixtures of the two games proposed by Parrondo turn their separate losing nature into a winning combination. It is difficult in gneral to analyze such deterministic mixtures since it requires computing the stationary probabilities of the product of the associated stochastic matrices. To expand upon this, suppose one has two distinct ${G(m, \bp)}$ games called $A$ and $B$ with parameters $a_j,0,1-a_j$ and $b_j,0,1-b_j$, respectively. By Lemma 3.2, the probabilities $p_m^\ast(A)$ and $p_m^\ast(B)$ for the two games would equal $1/2$ (i.e., the games would be fair) if and only if $$\label{eqn3.10}
\prod^{m-1}_{j=0}\frac{a_j}{1-a_j}=1=\prod^{m-1}_{j=0}\frac{b_j}{1-b_j}.$$ Consider now the random walk formed by alternating the transition probabilities of these two. Then the two-step process is also a random walk, though one with jumps of two units and with non-zero probabilities $r_i$ of zero jumps. That is, the alternation of two ${G(m, \bp)}$ games is a ${G(m, \bp,\bq)}$ game. This $2$-step process is then reducible with two classes, the odd and the even integers. If the walk starts in state $'0'$, for example, the corresponding quotient of relevant parameters is $$\label{eqn3.11}
\frac{(a_0b_1)(a_2b_3)\cdots (a_{m-2}b_{m-1})}{(1-a_0)(1-b_1)\cdots (1-a_{m-2})(1-b_{m-1})}.$$ Since only half of the parameters enter here, it is clear that this ratio may be greater or less than or equal to $1$ even when the separate games are fair. This implies that when $m$ is even, the alternation of two fair games may be either fair, winning or losing. Notice that even if one imposes the natural restriction that a fair game must be fair for all sarting states one gains nothing more since, for example, the condition for fairness starting in state $'1'$ , namely, $$\frac{(a_1b_2)(a_3b_4)\cdots (a_{m-1}b_0)}{(1-a_1)(1-b_2)\cdots (1-a_{m-1})(
1-b_0)}=1,$$ is equivalent under (\[eqn3.10\]) to the expression in (\[eqn3.11\]) being set equal to $1$.
When $m$ is odd, the alternation of fair games is fair as can be seen by considering the two-step game as a mod $2m$ game for which fairness requires by Lemma 3.1 that the product of (\[eqn3.11\]) and the following displayed quotient be equal to $1$, which follows from (\[eqn3.10\]). Thus the alternation of these fair games cannot result in winning ones when $m$ is odd.
The story is different, however, for $[AABB]$, the mixture in which two plays of game $A$ are alternated with two plays of $B$. In view of the previous paragraphs, this game is equivalent when $m$ is odd to an alternating $[AB]$ game [*but*]{} one in which both $A$ and $B$ are ${G(m, \bp,\bq)}$ games. For $m=3$ this is reasonably tractable. In particular, if one of the games is the classical simple random walk one can show that the mixture is indeed a winning game under a natural restriction on the second game. For the special case of $AABB$ in which $A$ and $B$ are the fair games $G(3, \frac{1}{2}, \frac{1}{2})$ and $G(3,
\frac{3}{4}, \frac{1}{10})$ corresponding to Parrondo’s example, one can show that the asymptotic average gain is $0.0218363>0$
Direct Calculation of the Asymptotic Expected Average Gain for a ${G(m, \bp)}$ Game
===================================================================================
By (\[eqn2.8\]), since $p^\ast_m$ has been evaluated, the asymptotic average gain (or loss) would be known once $E(T_2-T_1)$ is computed. A closed form for this expected inter-occurrence time is discussed below since it is of interest in its own right for these processes. However, the asymptotic average gain, $\lim_{n\to\infty} S_n/n$, being a limit of bounded r.v.’s, may also be derived directly by obtaining the limit of the corresponding expectations. We do this as follows.
Consider the game ${G(m, \bp)}$. Define $$\begin{aligned}
\label{eqn4.1}
\mu^{(j)}_k:&=&E(S_{n+k}-S_n|S_n\equiv j \mod{m}) \\
&=&E(\sum^k_{i=1}E(X_i|S_0\equiv j \mod{m}),\nonumber\end{aligned}$$ emphasizing by the notation the fact that the expectation depends only upon the congruence class of $S_n$ modulo $m$ and not upon the actual value of $S_n$ nor of $n$. In fact, the random walk $S_n$ is equivalent to the random walk on the circular group of integers $\mod{m}$ where a positive move is taken to be in the clockwise direction. Clearly, $$\begin{aligned}
\mu^{(0)}_{k+1} & = & p_0(1+\mu^{(1)}_k) + q_0(-1+\mu_k^{(m-1)}) \\
& = & p_0-q_0 + p_0\mu_k^{(1)} + q_0\mu_k^{(m-1)}.\end{aligned}$$ Similarly, for $j=1,2,\ldots, m-1$, $$\label{eqn4.15}
\mu^{(j)}_{k+1} = p_j-q_j + p_j\mu^{(j+1)}_k + q_j\mu^{(j-1)}_k$$ where we equate $\mu^{(m)}_k = \mu^{(0)}_k$ and $\mu^{(-1)}_k =
\mu^{(m-1)}_k$. To express this conveniently in matrix form, write ${\bmu}_k =
(\mu^{(0)}_k, \ldots, \mu^{r}_k)'$ and $\boldb=
(p_0-q_0, p_1-q_1, \ldots, p_r-q_r)'$ as $m\times 1$ column vectors and set $$\label{eqn4.2}
\bC = \left[ \begin{array}{ccccccc}
0 & p_0 & 0 & 0 & \cdots & \cdots & q_0 \\
q_1 & 0 & p_1 & 0 & \cdots & \cdots & 0 \\
0 & q_2 & 0 & p_2 & \cdots & \cdots & 0 \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \\
0 & 0 & 0 & 0 & q_{r-1} & 0 & p_{r-1} \\
p_r & 0 & 0 & 0 & 0 & q_r & 0 .
\end{array}\right]$$ where again $r=m-1$. Since $\bmu_1=\boldb$, it is clear from (\[eqn4.15\]) that $$\bmu_k=\boldb+\bC\boldb + \bC^2\boldb + \cdots + \bC^{k-1}\boldb.$$ which implies that $$\label{eqn4.3}
\lim_{n\to\infty}\bmu_n/n = \left(\lim_{n\to\infty} \frac{1}{n}
\sum^{n-1}_{i=0} \bC^i\right)\boldb.$$
The reader should note that if $\{S_n: n \ge 0\}$ were a more general $\mod{m}$ Markov chain, the vector $\boldb$ would be given by $$b_i \equiv E(X_{n+1} | S_n = i \mod{m}) = \sum_j (j-i)p_{ij}$$ and $\bC$ would be determined by $$C_{ij}=P(S_{n+1}=j \mod{m} |S_n=i \mod{m}) = \sum_{k\in {\mathbb{Z}}}p_{i,j-i+km}.$$ That is, the transition matrix $\bC$ for the Markov chain of congruence classes of $\{S_n\}$ is formed from the original chain’s transition matrix $\bP$ by summing over all states in the appropriate congruence class. With these defintions, the limit of (\[eqn4.3\]) applies to a general $\mod{m}$ Markov chain. We shall continue, however, with the ${G(m, \bp)}$ case in order to obtain explicit values.
The value of this limit depends upon the periodicity of $\bC$. Suppose first that $m$ is [*o*dd]{}. In this case, $\bC$ is an irreducible aperiodic stochastic matrix provided only that $p_jq_j\ne 0$ for each $j$. Thus the limit exists and is a stochastic matrix, each of whose rows is the row vector of stationary probabilities associated with $\bC$, $\bpi=(\pi_0, \pi_1, \ldots, \pi_{m-1})$, say. It is a known result of G. Mihoc (cf. Fréchet (1952), pp. 114-116) that the entries in $\bpi$ are proportional to the diagonal cofactors of $\bI-\bC$. (See Appendix A below for this and other results to be used below.)
Let $\gamma_{im}$ denote the $(i,i)$-th cofactor of $\bI-\bC$. These are tractable for reasonable values of $m$. Due to the cyclic structure underlying the matrix $\bC$ it is necessary only to obtain the first cofactor for each $m$. The first few values are: $$\begin{aligned}
\label{eqn6.12}
\gamma_{13}&=&1-p_1q_2,\qquad\gamma_{14}=1-p_1q_2-p_2q_3, \\
\gamma_{15}&=&1-p_1q_2-p_2q_3-p_3q_4+p_1q_2p_3q_4 \nonumber\end{aligned}$$ and $$\begin{aligned}
\gamma_{16}&=&(1-p_1q_2-p_2q_3)(1-p_3q_4-p_4q_5)-p_2p_3q_3q_4\\
&=&1-p_1q_2-p_2q_3-p_3q_4-p_4q_5+p_1q_2p_3q_4+p_1q_2p_4q_5+p_2q_3p_4q_5\end{aligned}$$ The remaining diagonal cofactors are then obtained for each $m$ by successively applying the cyclic permutation of $(p_0, p_1, \ldots,
p_{m-1})$ into $(p_1, p_2, \ldots, p_{m-1}, p_0)$. For the case of a Parrondo ${G(m, p, p')}$ game with $m=3$, the situation studied in Harmer and Abbott (1999a), (\[eqn6.12\]) implies that $$\gamma_{13}=1-pq, \ \gamma_{23}=1-pq',\ \gamma_{33}=1-p'q.$$
A general formula, presumably known, is possible for these cofactors, namely, $$\begin{aligned}
\label{eqn6.13}
\gamma_{1m}=1-\sum^{m-2}_{i=1}p_iq_{i+1} & + & \sum_{1\le i<j-1\le
m-3}p_iq_{i+1}p_jq_{j+1} \nonumber \\
& - & \sum_{1\le i<j-1<k-2\le m-4}p_iq_{i+1}p_jq_{j+1}p_kq_{k+1}+\cdots\end{aligned}$$ with the series continuing as long as the largest subscript does not exceed $m-1$. Thus for $l=[(m-1)/2]$, the last term has sign $(-1)^l$ and involves $l$ subscripts $i_1, \ldots, i_l$ satisfying $$1\le i_1<i_2-1<i_3-2<\cdots i_l-l+1\le m-l.$$ As indicated by its appearance, (\[eqn6.13\]) follows from an inclusion-exclusion argument based on the number of pairs of adjacent diagonal $1$’s used in the evaluation of the cofactor’s determinant. (All diagonal cofactors are of course equal for each value of $m\ge 3$ whenever the parameters $p_j$ and $q_j$ do not depend on $j$.)
As mentioned earlier, the stationary probabilities associated with $\bC$ are proportional to these diagonal cofactors so that in our previous notation $\pi_i=\gamma_{i+1,m}/\gamma_{\cdot m}$ where $\gamma_{\cdot m} =
\gamma_{1m} + \cdots + \gamma_{mm}$.
An early reference for the study of the general cyclical random walk on the integers modulo $m$, the one whose transition matrix is $\bC$, is Fréchet ((1952), pp. 122–125. This is in effect a 1938 reference for this random walk, called by Fréchet, “mouvement circulaire”, since the material is present in the 1938 first edition of his book. He works out as an example the stationary probabilities for the case of $m=4$. He obtains $\gamma_{14}$ as $p_2p_3+q_1q_2$ which is easily seen to agree with the expression given above in (\[eqn6.12\]).
The asymptotic average gain given by (\[eqn4.3\]) now follows directly from the above for the case when $m$ is odd. It is of the form $\lambda_m(1, 1, \ldots,
1)'$ with $$\label{eqn4.7}
\lambda_m=\bpi_m\boldb\equiv \frac{1}{\gamma_{\cdot m}} \sum^m_{i=1}\gamma_{im}
(p_{i-1}-q_{i-1}).$$
Consider now the case of $m$ *even*, say $m=2k$ for $k\ge 2$. Then $\bC$ is the stochastic matrix of a periodic Markov chain of period 2. By clustering the even and odd rows and columns, it may be written in the form $$\label{eqn4.8}
\bC=\left[\begin{array}{cc}
0 & A \\
B & 0
\end{array}\right]$$ in which $A$ and $B$ are $k\times k$ stochastic matrices. Consequently, $$\bC^2=\left(\begin{array}{cc}
AB & 0 \\
0 & BA \end{array}\right), \bC^{2s} = \left(\begin{array}{cc}
(AB)^s & 0 \\
0 & (BA)^s \end{array}\right), \bC^{2s+1} = \left(\begin{array}{cc}
0 & A(BA)^s \\
B(AB)^s & 0 \end{array}\right)$$ in which both $AB$ and $BA$ are irreducible aperiodic recurrent stochastic matrices. If $\bdelta$, $\brho$ represent the vectors of limiting stationary probabilities for $AB$ and $BA$, respectively, and if $D$ and $R$ are the matrices all of whose rows are $\bdelta$ and $(\brho)$, respectively, then $$\lim_{s\to\infty}\bC^{2s} = \left(\begin{array}{cc}
D & 0 \\
0 & R \end{array}\right),\qquad \lim_{s\to\infty}\bC^{2s+1} =
\left(\begin{array}{cc}
0 & R \\
D & 0 \end{array}\right)$$ and so (\[eqn4.3\]) becomes in the case of $m$ even, $$\label{eqn4.9}
\lim_{n\to\infty}\bmu_{n}/n=\frac{1}{2}\left(\begin{array}{cc}
0 & R \\
D & 0 \end{array}\right) \boldb.$$ By the result of Mihoc, the elements of the common rows $\bdelta$ and $\brho$ of $D$ and $R$ are proportional to the diagonal cofactors of $AB$ and $BA$, respectively. However, as shown in the Appendix below, the diagaonl cofactors of $\bI-\bC$ are made up of those of $\bI_{m/2}-AB$ and $\bI_{m/2}-BA$ and that the column sums of the latter are equal and equal to $1/2$ of the sum of the diagonal cofactors of $\bI-\bC$; cf (\[eqna6\]) below. In view of (\[eqn4.9\]) it follows that (\[eqn4.7\]) holds true as well when $m$ is even. We summarize this as
\[thm4.1\] For the general ${G(m, \bp)}$ game, with probability one, $$\label{eqn4.10}
\lim_{n\to\infty} \frac{S_n}{n} \equiv\lambda_m=\bpi_m\boldb\equiv \frac{1}{\gamma_{\cdot m}} \sum^m_{i=1}\gamma_{im}
(p_{i-1}-q_{i-1}).$$ in which the $\gamma_i$ are the diagonal cofactors of $\bI-\bC$ and $\gamma_{\cdot m}$ is their sum.
For the special case of a ${G(m, p, p')}$ walk, the limit of interest in (\[eqn4.10\]) becomes $$\begin{aligned}
\label{eqn4.4}
\lambda_m & = & \{(p'-q')\gamma_{1m} + (p-q)(\gamma_{\cdot
m}-\gamma_{1m})\}/\gamma_{\cdot m} \nonumber \\
& = & 2p-1 + 2(p'-p)\gamma_{1m}/\gamma_{\cdot m}.\end{aligned}$$ From (\[eqn6.12\]), the first few values of $\gamma_{\cdot m}$ for a ${G(m, p, p')}$ walk are $$\begin{aligned}
\gamma_{\cdot 3} & = & 3-pq-pq'-p'q = 2 + p'p^2+q'q^2 \\
\gamma_{\cdot 4} & = & 4-4pq-2pq'-2p'q = 2(1-pq)+2(p'p^2 + q'q^2) \\
\gamma_{\cdot 5} & = & 5-9pq-3pq' - 3p'q + pq(pq+2pq'+2p'q).\end{aligned}$$ For Game B of Harmer and Abbott (1999a), in which $m=3$, $p=3/4-{\varepsilon}$ and $p'=
1/10-{\varepsilon}$, one obtains $$\gamma_{13}=13/16-{\varepsilon}/2+{\varepsilon}^2,\qquad \gamma_{\cdot 3}=
\frac{169}{80}-\frac{{\varepsilon}}{5}+3{\varepsilon}^2,$$ from which the limit in (\[eqn4.4\]) becomes $$\label{eqn4.5}
\lambda_3 = -2{\varepsilon}\frac{147-24{\varepsilon}+240{\varepsilon}^2}{169-16{\varepsilon}+240{\varepsilon}^2} \cong
-1.74{\varepsilon}-.16 {\varepsilon}^2+O({\varepsilon}^2)$$
This value appears to differ from the one implied by the simulated curve for Game B shown in Fig. 3 of Harmer and Abbott (1999a). The value for the curve given there for $n=100$ is approximately $-1.35$, whereas for ${\varepsilon}=.005$ and $n=100$, the value from (\[eqn4.5\]) is approximately $n\lambda_3\cong-1.74/2=-.87$. The difference is that the slope of the simulated curve is affected by the early transient behavior; in a private communication, Harmer and Abbott confirm the agreement with this theoretical limit of their simulated slope when the first 100 plays are excluded. The analogous value for their Game A (where $p=p'=
\frac{1}{2}-{\varepsilon}$) is $n\lambda_3=(-2{\varepsilon})n=-1$ which agrees with the curve for Game A given in their Fig. 3.
For the randomized game that chooses between Games A and B with probability $1/2$, one obtains $p=\frac{5}{8}-{\varepsilon}$ and $p'
= \frac{3}{10}-{\varepsilon}$ for which $$\gamma_{13} = \frac{49}{64} - \frac{{\varepsilon}}{4} + {\varepsilon}^2, \qquad \gamma_{\cdot 3} =
\frac{709}{320}-\frac{{\varepsilon}}{10}+3{\varepsilon}^2.$$ Thus in this randomized case the asymptotic slope of $S_n/n$ is by (\[eqn4.4\]) $$\label{eqn4.6}
\lambda_3 = \frac{1}{4} - \frac{13\times 49}{4\times 709} - {\varepsilon}\left\{2 -
\frac{52\times 611}{(709)^2}\right\}+ O({\varepsilon}^2) \cong .0254 - 1.9368{\varepsilon}+ O({\varepsilon}^2);$$ the expansion used in the first step requires only that ${\varepsilon}< .876$. For the parameters $n=100$ and ${\varepsilon}=.005$ of Fig. 3 of Harmer and Abbott (1999a) the asymptotic approximation becomes $n\lambda_3\cong 2.54-.98=1.57$. This differs from their simulated value of about 1.26, again due to early outcome effects. The reader might note that the graphs in the insert of Fig. 3 seem to be closer to those of (\[eqn4.5\]) and (\[eqn4.6\]).
As an illustration for even $m$, consider $m=4$ for which the matrices become $$A=\left(\begin{array}{cc}
p' & q' \\
q & p \end{array}\right),\qquad B=\left(\begin{array}{cc}
q & p \\
p & q \end{array}\right),$$ and $$AB = \left(\begin{array}{cc}
p'p+q'q & -p'p-q'q \\
2pq-1 & 1-2pq \end{array}\right).$$ Hence $\delta_{14} = (1-2pq)(1-2pq+p'p+q'q)^{-1}$ and thus $$\label{eqn4.13}
\lambda_4 = \frac{(p'-p)(1-2pq)}{p'p^2+q'q^2+1-pq}+p-q =
\frac{2(p'p^3-q'q^3)}{p'p^2+q'q^2+1-pq};$$ see also (\[eqn5.6\]) below.
In this section, we restricted consideration to ${G(m, \bp)}$ games. The approach applies as well to ${G(m, \bp,\bq)}$ games but with the simplifying zero diagonal of $\bC$ being replaced with the $r_j$’s.
Expected Interoccurrence Times of Visits to ${\mathbb{Z}}_m$
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Set $\tau_j=E(T_1|S_0=j)$ for $j=0,\pm 1,\ldots, \pm (m-1)$ to denote the expected time of the first visit to ${{\mathbb{Z}}_m}$ of a ${G(m, \bp)}$ walk $\{S_n\}$ starting at $j$. In the expression (\[eqn2.8\]) for the asymptotic average gain, the denominator $E(T_2-T_1)$ is equal to $\tau_0$. Hence, an alternate derivation of the asymptotic average gain would be, in view of Lemma (\[lem6.2\]), to derive $\tau_0$. This may be done by solving the recursion relations satisfied by the $\tau_j$’s, namely, $$\label{eqn5.1}
\tau_j = p_j\tau_{j+1}+q_j\tau_{j-1}+1, \qquad \hbox{for } j=0,\pm 1,\ldots, \pm (m-1),$$ with boundary conditions $\tau_{-m} = \tau_m = 0$, where for negative $j$ we have $p_j=p_{j+m}$ and $q_j=q_{j+m}$ for a $\mod{m}$ walk. The solution of (\[eqn5.1\]) is given for example in Chung(1967, I.12.(8)) in which the reader should note that the $\rho_i$’s in this reference are related to the *reciprocals* of those used here.
The expression that one obtains in this way is quite complicated even in the case of $m=3$ and difficult to simplify into the more tractable expressions that can be obtained by direct solution of (\[eqn5.1\]) by matrix inversion. For if $\btau:=(\tau_{m-1}, \ldots, \tau_{1}, \tau_0, \tau_{-1}, \ldots,
\tau_{-m+1})'$ is the $(2m-1)$-dimensional column vector of expected occurrence times, ${\mathbf{1}}$ is the $(2m-1)$-dimensional column vector of ones and ${\mathbb{G}}$ denotes the $(2m-1)\times(2m-1)$ matrix of coefficients in (\[eqn5.1\]) then the system (\[eqn5.1\]) may be expressed as $\btau={\mathbb{G}}\btau+{\mathbf{1}}$ whose solution, with $\bH \equiv \bI-{\mathbb{G}}$ is expressible by $$\label{eqn5.3}
\btau=(\bI-{\mathbb{G}})^{-1}{\mathbf{1}}= \bH^{-1}{\mathbf{1}}.$$ Thus the expected interoccurrence times of ${\mathbb{Z}}_m$ are given as the row sums of the matrix $\bH^{-1}$. The matrix $\bH$ whose inverse is needed is a Jacobi matrix with $-p_i$’s below a diagonal of $1$’s and $-q_i$’s above it, namely, $$\bH = \left[\begin{array}{ccccccccc}
1 & -q_{m-1} & 0 & && & 0 & 0 & 0 \\
-p_{m-2} & 1 & -q_{m-2} & \\
& \cdot & \cdot & \cdot & \cdot & \cdot &\\
0 & & & 1 & -q_1 & 0 & & & 0 \\
0 & & & -p_0 & 1 & -q_0 && & 0 \\
0 & & & 0 & -p_{m-1} & 1 & & & 0 \\
&&& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
0 &&&&&&-p_2 & 1 & -q_2 \\
0 &&&&&&0 & -p_1 & 1 \end{array}\right].$$ In particular, by (\[eqn2.8\]) the required quantity, $E(T_2-T_1)=\tau_0$, in the computation of $p_m^\ast$, is the sum of the middle row of $\bH^{-1}$. Thus, if $H_{i,j}$ denotes the $\{i,j\}$-cofactor of $\bH=\bI-{\mathbb{G}}$ and $|\bH|$ denotes the determinant of $\bH$, then $\tau_0=H_{\cdot m}/
|\bH|$ where $H_{\cdot m}=H_{1m}+\cdot \cdot \cdot+H_{2m-1,m}$.
When $m=3$, $\bH$ is a $5\times 5$ matrix whose middle cofactors are straightforwardly shown to be $$\begin{aligned}
H_{13} & = & p_1p_0(1-p_1q_2),\qquad H_{23} = p_0(1-p_1q_2),\qquad H_{33}=(1-p_1q_2)^2, \\
H_{43} & = & q_0(1-p_1q_2)\quad \hbox{ and } \quad H_{53}=q_0q_2(1-p_1q_2).\end{aligned}$$ Hence $$H_{\cdot 3}=(1-p_1q_2)(3-p_1q_2-p_2q_0-p_0q_1)$$ and $$|\bH|=(1-p_1q_2)(1-p_1q_2-p_2q_0-p_0q_1)=(1-p_1q_2)(p_0p_1p_2+q_0q_1q_2).$$ Therefore, for $m=3$, $$\tau_0=E(T_2-T_1)=1+2/(p_0p_1p_2+q_0q_1q_2).$$ By (\[eqn2.8\]) and (\[eqn2.2\]), this implies that with probability one, $$\lambda_3 = \lim_{n\to\infty}\frac{S_n}{n}
= \frac{3(p_0p_1p_2-q_0q_1q_2)}{2+p_0p_1p_2+q_0q_1q_2}.$$ The reader may check that this agrees with the expression given for $\lambda_3$ in (\[eqn4.10\]).
For $m=4$, ${\mathbb{G}}$ is a $7\times 7$ matrix, and the middle column’s cofactors for the corresponding $\bH\equiv(\bI-{\mathbb{G}})^{-1}$ are easily computed to be $$\begin{array}{lll}
H_{14}=p_0p_1p_2|\bK|, & H_{24}=p_0p_1|\bK|,
& H_{34}=p_0(1-p_2q_3)|\bK|,\\[2mm]
H_{44}=|\bK|^2 , &
H_{54}=q_0(1-p_1q_2)|\bK|, & H_{64}=q_0q_3|\bK| \\[2mm]
H_{74}=q_0q_3q_2|\bK|.
\end{array}$$ where $\bK$ is the upper left ( *and* lower right) $(m-1)\times(m-1)$ corner matrix of $\bH$. This gives $$H_{\cdot 4} = |\bK|[3-p_0q_1-p_1q_2-p_2q_3-p_3q_0+(p_1-q_3)(p_2-q_0)]$$ and, by expansion along the middle column, the determinant of $\bH$ is $$|\bH|=|\bK|[p_0p_1p_2p_3+q_0q_1q_2q_3].$$ Therefore, after simplification, $$\label{eqn5.5}
\tau_0 = \frac{\bH_{\cdot 4}}{|\bH|} =
\frac{2(p_0p_1+p_2p_3+q_0q_3+q_2q_1)}{p_0p_1p_2p_3+q_0q_1q_2q_3}.$$ So that by (\[eqn2.8\]) and (\[eqn2.2\]) the asymptotic slope of the random walk for $m=4$ is $$\label{eqn5.6}
\lambda_4=\lim_{n\to\infty}\frac{S_n}{n} = \frac{2(p_0p_1p_2p_3-q_0q_1q_2q_3)}
{p_0p_1+p_2p_3+q_0q_3+q_2q_1}$$ with probability one. This is consistent with the result obtained by the methods of Section 4; see (4.10).
The above discussion focuses on ${G(m, \bp)}$ games rather than the more general ${G(m, \bp,\bq)}$ games. Only minor modifications for the latter are needed. The term $r_j\tau_j$ is added to the right hand side of the equations (\[eqn5.1\]). This results in a substitution of $p_j/(p_j+q_j)$ and $q_j/(p_j+q_j)$ for the parameters of the walk, and, more significantly, a replacement of the vector $\bI$ in the solution (\[eqn5.3\]) by the vector of the reciprocals, $p_j+q_j$. A benefit of working out the more general case would be that whenever $m$ is even, one could reduce the problem to one of order $m/2$ by observing that the embedded walk on ${{\mathbb{Z}}_m}$ is equivalent in its asymptotic behavior to that of the $2$-step random walk in which the parameters would become the products, $p_0p_1,
q_0q_{m-1}$, etc. One can see this already in the example of $m=4$ above, which the reader may compare to the case of $m=2$ for the associated $2$-step case.
A diffusion analogue of a general random walk
=============================================
Partition the real line into intervals $J_j=(j, j+1]=j+(0,1]$, for $j=0, \pm 1, \pm 2, \cdots$. Let $\bmu=\{\mu_j:j=0, \pm 1, \cdots\}$ be given constants. For real $x$ set $$\label{eqn7.1}
\mu(x)= \sum_j\mu_j1_{ J_j}(x).$$ Now define a diffusion $\{W_t:t\ge 0\}$ in terms of a standard Brownian motion $\{B_t:t\ge 0\}$ by $$\label{eqn7.2}
dW_t=dB_t+\mu(W_t)dt,$$ for $t>0$. For this process, introduce the probabilities of transition between consecutive integers, namely, $$\label{eqn7.3}
p_j=p_j(\mu_j,\mu_{j-1})=P[W_\cdot\hbox{ hits }j+1\hbox{ before hitting
}j-1|W_0=j]$$ and let $q_j=1-p_j$. Observe that $q_j(\mu_j, \mu_{j-1})=p_j(-\mu_{j-1},
-\mu_j)$ by reflection.
To obtain expressions for the $p_j$ in terms of the pertinent drift rates, $\mu_j$ and $\mu_{j-1}$, we will use the scale function of the diffusion. For this, fix constants $a<b$ and define for $x\in[a,b]$ the first passage probabilities $$\label{eqn7.4}
u(x)=P[W_\cdot\hbox{ hits }b\hbox{ before }a|W_0=x].$$ The backward equations for the Markov process $W_{\cdot}$ imply that $u$ satisfies the second order differential equation $u''+2{\mu}u'=0$, the solution of which is of the form $$u(x)=c\int^x_a \exp{\{-2\int^y_a\mu(z)dz\}}dy +b.$$ The boundary conditions, $u(a)=0, u(b)=1$ then give $$\label{eqn7.5}
u(x) = \frac{\int^x_a \exp{\{-2\int^y_a\mu(z)dz\}}dy}{\int^b_a \exp{\{-2\int^y_a\mu(z)dz\}}dy}.$$ Note that in the case of $\mu_j=\mu$ for every $j$, this becomes the formula of Anderson(1960, Theorem 4.1); for $a<0<b$ $$\label{eqn7.22}
P[B_t+\mu t\hbox{ hits }b\hbox{ before }a|B_0=0]
=\D{\frac{1-e^{2a\mu}}{1-e^{-2(b-a)\mu}}}$$ when $\mu\ne 0$, and equals $\frac{1}{2}$ when $\mu=0$.
A scale function for the diffusion, a function, $S$ say, which satisfies $u(x)=\{S(x)-S(a)\}
/\{S(b)-S(a)\}$, may be deduced from (\[eqn7.5\]) to be $$\label{eqn7.6}
S(x)=2\int^x_0\exp{\{-2\int^y_0\mu(z)\}}dy,$$ the scalar $2$ being inserted for later simplicity.
For the step function $\mu$ considered here, the above may be integrated out for all $x$. However, our interests here require $S$ only for integer values of $x=n$, and in this case, $S(0) = 0$ and $$\label{eqn7.7}
u(x) = \left\{
\begin{array}{ll}
\D{\sum^{n-1}_{k=0} r(\mu_k)\exp{\{-2\sum^k_{j=0}\mu_j\}}} & \hbox{if }n>0, \\
\D{}\\
\D{-\sum^{-1}_{k=n} r(\mu_k)\exp{\{2\sum^{-1}_{j=k+1}\mu_j\}}} & \hbox{if }n<0.
\end{array}\right.$$
The desired transition probabilities $p_j$ follow directly now from (\[eqn7.7\]). It suffices to consider $j=0$. Since $p_0 =u(0)$ when $b=1=-a$, (\[eqn7.7\]) implies that $$\label{eqn7.8}
p_0\equiv p(\mu_0, \mu_{-1})\equiv \frac{S(0)-S(-1)}{S(1)-S(-1)}=\frac{r(\mu_{-1})}{r(\mu_0)e^{-2\mu_0}+ r(\mu_{-1})}$$ where $r(u)=(e^{2u}-1)/u$ for $u\ne 0$ and $r(0)=2$. Note that $p(0,
0)=\frac{1}{2}$ as required for standard Brownian motion. Using the fact that $r(u)\exp{(-2u)}=r(-u)$, we summarize this as follows:
\[lem7.1\] For the diffusion defined by (\[eqn7.2\]), the transition probabilities of the embedded random walk on the integers that are defined by (\[eqn7.3\]) are given by $$\label{eqn7.10}
p_j = \frac{\mu_j(e^{2\mu_{j-1}}-1)}{\mu_j(e^{2\mu_{j-1}}-1)+ \mu_{j-1}(1-
e^{-2\mu_j})} = \frac{r(\mu_{j-1})}{r(\mu_{j-1}) + r(-\mu_j)}$$ for $j=0, \pm 1, \pm 2, \cdots$.
It is clear that the recurrence or transience of this diffusion agrees with that of the embedded random walk. By Section 6, this in turn depends upon the quotients, $p_1p_2\cdots p_k/q_1q_2\cdots q_k$. From (\[eqn7.10\]), $$\label{eqn7.11}
\frac{p_j}{q_j} = \frac{r(\mu_{j-1})}{r(\mu_j)} e^{2\mu_j}.$$ Then for any $k\ge 1$, $$\label{eqn7.12}
\prod^k_{j=1}
\frac{p_j}{q_j}=\frac{r(\mu_0)}{r(\mu_k)}\exp\left\{2\sum^k_{j=1}\mu_j\right\}$$ with a similar expression for negative indices. Substitution of these into (\[eqn6.1\]) and (\[eqn6.2\]) would then determine recurrence or not.
It is of interest to point out that the $p_j$’s may be evaluated directly from (\[eqn7.6\]) without finding the scale function. To see this, set $b=1=-a$ and let $x\in[-1,1]$. By partitioning the event $[W_\cdot\hbox{ hits }1\hbox{ before }-1]$ according to hitting $0$ or not before 1 and $-1$, the Markov property and Anderson’s result (\[eqn7.22\]) yield $$\label{eqn7.13}
u(x) = \left\{
\begin{array}{ll}
\D{\frac{1-e^{-2x\mu_1}}{1-e^{-2\mu_1}}} +
\D{\frac{e^{-2x\mu_1}-e^{-2\mu_1}}{1-e^{-2\mu_1}}u(0)} & \hbox{if }x>0, \\
\D{}\\
\D{\frac{1-e^{-2(1-x)\mu_2}}{1-e^{-2\mu_2}}u(0)} & \hbox{if }x<0.
\end{array}\right.$$ It therefore remains to derive $u(0)$.
For $\alpha\in(0,1]$ let $v(\alpha)$ denote the value of $u(0)$ when the barriers at $\pm 1$ are replaced by $\pm\alpha$. That is, $v(\alpha)$ is the probability of hitting $\alpha$ before $-\alpha$ given the process starts at zero. By partitioning the event of hitting 1 before $-1$ according to which of $\alpha$ or $-\alpha$ is hit first, one obtains $$\label{eqn7.14}
u(0) \equiv v(1) = v(\alpha)u(\alpha) + [1-v(\alpha)]u(-\alpha) .$$ Upon substitution of (\[eqn7.13\]) and then solving for $v(1)$ one obtains $$\label{eqn7.15}
v(1)=\frac{f(\alpha)}{1/v(\alpha)+f(\alpha)-1}$$ with $$f(\alpha) =
\frac{(1-e^{-2\alpha\mu_0})(1-e^{-2\mu_{-1}})}{(e^{-2(1-\alpha)\mu_{-1}}-e^{-2\mu_{-1}})(1-e^{-2\mu_0})}.$$ Observe that the limit of $f(\alpha)$ as $\alpha\searrow 0$ is $$f(0+) = \left(\frac{\mu_0}{\mu_{-1}}\right) \frac{e^{2\mu_{-1}}-1}{1-e^{-2\mu_0}}
= \frac{r(\mu_{-1})}{r(-\mu_0)}.$$ By scaling, $p(\alpha)$ is the same as $v(1)$, *but* with $\mu_1,\mu_2$ replaced by $\alpha\mu_1, \alpha\mu_2$. Thus one concludes that $p(0+)=\frac{1}{2}$. Substitution of these limits into the right hand side of (\[eqn7.13\]) leads to $$\label{eqn7.16}
v(1)\equiv p(\mu_0, \mu_{-1})= p_0 = \frac{r(\mu_{-1})}{r(\mu_2)+r(-\mu_0)}$$ as desired.
For our interests here, consider the $\mod{m}$ *shift diffusions* in which $\mu_j=\mu_l$ whenever $j\equiv l\bmod{m}$. In this case, Lemma \[lem7.1\] implies that $$\label{eqn7.16b}
\rho_m = \prod^{m-1}_{j=0} \frac{p_j}{q_j} = \exp\left\{2\sum^{m-1}_{j=0}
\mu_j\right\}$$ so that by Lemma \[lem6.1\], the embedded $\mod{m}$ random walk, and hence the $\mod{m}$ shift diffusion, is recurrent, transient toward $+\infty$ or transient toward $-\infty$ according as $$\label{eqn7.17}
\sum^{m-1}_{j=0}\mu_j=0,\qquad >0\qquad \hbox{ or }\qquad <0.$$ Then, by Lemma \[lem6.2\], the constant probability of “success” on ${\mathbb{Z}}_m$ is $$\label{eqn7.18}
p^\ast_m=1/\left(1+\exp\left\{-2\sum^{m-1}_{j=0}\mu_j\right\}\right).$$ Observe that the walk is fair $(p^\ast_m=\frac{1}{2})$ if and only if $\mu_0+\mu_1+\cdots+\mu_{m-1}=0$.
If one is given the $p_j$’s, one may solve the system of equations given by (\[eqn7.10\]) for $j=0, 1, \ldots, m-1$ to find the shift rates $\mu_0,
\ldots, \mu_{m-1}$ for the associated $\mod{m}$ shift diffusion. For example, for the random walk related to Game B of Harmer and Abbott (1999a) in which $m=3$, $p_0=1/10$ and $p_1=p_2=3/4$, the drift rates are $$\mu_0=-.687032,\qquad\mu_1=2.748128, \qquad\mu_2=-2.06109.$$ Note that these are proportional to $(-1, 4, -3)$. In fact, for any fair game $G(3,p,p')$, the associated drift rates $(\mu_0,\mu_1,\mu_2)$ are equal to $\mu_1(-q,1,-p)$ as given by
\[thm7.2\] If the transition probabilities, $p_0,p_1,p_2$, of a recurrent $\bmod{(m)}$ shift diffusion are known, the associated drift rates may be determined uniquely as follows:
[i)]{}. If each $p_i$ equals 1/2, then each $\mu =0$;
[ii)]{}. If exactly one of the $p_i$’s, say $p_2$, is equal to $1/2$ then $(\mu_0,\mu_1,\mu_2)=(0,x,-x)$ with $x$ being the solution of $p_0/q_0=(1-e^{-2x})/2x$;
[iii)]{}. If none of the $p_i$’s are equal to $1/2$, then $$(\mu_0,\mu_1,\mu_2)=(\frac{1}{2}\ln{w})(-(1-\theta),1,-\theta))$$ in which $\theta =(1-q_1/p_1)/(1-p_0/q_0)$ and $w\equiv e^{2\mu_1}$ is the positive solution other than 1 of the equation $$\label{eqn7.20}
\alpha w-w^{\theta}+(1-\alpha)=0$$ where $\alpha=(q_2/p_2)\theta$.
We prove case iii) first. Write $x=\mu_1$ and $y=\mu_2$. Set $a=p_1p_2/q_1q_2$ and $b=p_2/q_2$, neither of which equals $1$. By (\[eqn7.11\]) the equations to be solved are $$\label{eqn7.21}
a = r(x+y)/r(y) \quad \hbox{ and }\quad b =r(x)e^{2y}/r(y)=r(x)/r(-y).$$ Observe that $$\frac{x+y}{y}a=\frac{e^{2(x+y)}-1}{e^{2y}-1}=1+
\frac{e^{2x}-1}{e^{2y}-1}e^{2y}=1+\frac{xb}{y},$$ noting that the arguments of $r$ are not zero in this case. Hence, if we set $u=x+y$, we must have $x=cu$ and $y=(1-c)u$ with $c=(a-1)/(b-1)$. Substituion into the second equation of (\[eqn7.21\]) gives $$b=(\frac{1-c}{c})\frac{e^{2cu}-1}{1-e^{2(c-1)u}}.$$ By setting $w=e^{2x}=e^{2cu}$ this equation becomes $$\frac{\theta}{b}w-w^\theta +(1-\frac{\theta}{b})=0.$$ which completes the proof of iii).
Case i) is clear. For ii), the constant $b$ above is equal to $1$. Since $r$ is an increasing function, this means $x=-y$. The first equation then becomes $a=r(0)/r(y)=2/r(y)$ which is equivalent to the equation given in the statement of case ii).
When $p$ is rational, the equation (\[eqn7.20\]) of Theorem \[thm7.2\] becomes a polynomial. Here are two other examples: For the fair game $G(3,2/3,1/5)$, $\theta =2/3$ and $\alpha=\theta/2$. The equation that determines $\mu_1=\frac{1}{2}\log{w}$ is by (\[eqn7.20\]), $w-3w^{2/3}+2=0$. Upon setting $z=w^{1/3}$, the equation becomes $z^3-3z^2+2=0$, or, equivalently, after factoring out $z=1$, $z^2-2z-2=0$. Its desired positive solution is $z=1+\sqrt 3$ so that $\mu_1=(3/2)\log{(1+\sqrt 3)}$. This implies by Theorem \[thm7.2\] that the drift rates are $$\mu_0=-.502526,\qquad\mu_1=1.507579, \qquad\mu_2=-1.005053.$$ For the fair Parrondo game $G(3,4/5,1/17)$, $\theta=4/5$ and $\alpha=1/5$ so that (\[eqn7.20\]) becomes $w-5w^{4/5}+4=0$. With $z=w^{1/5}$, this becomes, after factoring out $z=1$, $z^4-4z^3-4z^2-4z-4=0$. The unique positive root (by Mathematica) is $z=4.99357$ so that $\mu_1 = (5/2)\log{z}=4.020378$ so that by Theorem \[thm7.2\] the drift rates are $$\mu_0=-.804076,\qquad\mu_1=4.020378, \qquad\mu_2=-3.216302 .$$
Results about stationary probabilities of Markov Chains
=======================================================
We begin with a 1934 result of G. Mihoc that expresses stationary probabilities of a finite state Markov chain in terms of cofactors: See Fréchet (1952), pp. 114–6. (Mihoc’s original paper was in Romanian, and Fréchet elaborated upon it in his 1938 first edition of the cited reference.) Let $\PP$ be any $k\times k$ stochastic matrix. For any $j\in \{1, 2, \ldots,
k\}$, the following two determinants are equal, since the matrix in the second is obtained from that in the first by replacing the $j$th column with the sum of all columns: For $0\le s<1$, $$\Delta(s):=|s\bI-\PP|=\left|\begin{array}{ccccccc}
s-p_{11} & -p_{12} & -p_{1j-1} & s-1 & p_{1j+1}, & \cdots, & -p_{1k} \\
-p_{21} & s-p_{22} & & s-1 & \\
\vdots & & & \vdots & \\
-p_{k1} & & & s-1 & & & s-p_{kk}
\end{array}\right|$$ and so $$\label{eqna1}
\lim_{s\nearrow 1} \frac{\Delta(s)}{s-1} = \left|\begin{array}{cccc}
1-p_{11} & -p_{12} & 1 & -p_{1k} \\
& & 1 \\
& & \vdots \\
-p_{k1} & & 1 & 1-p_{kk} \end{array}\right|.$$ Observe that the left hand side does not depend upon $j$. Hence the right hand side evaluated by expanding along the $j$-th column does not depend upon $j$. That is, if $\Delta_{ij}$ denotes the $(i, j)$-th cofactor of $\Delta(1)$, $$\label{eqna2}
\Delta_{\cdot j}:=\Delta_{1j}+\cdots+\Delta_{kj} = \Delta_{\cdot 1},\qquad
j=1,2,\ldots, k.$$ On the other hand, direct evaluation of $\Delta(1)=|\bI-\PP|$ by expansion along the $j$-th column gives $$\Delta(1) = \sum^k_{i=1}\Delta_{ij}(\delta_{ij}-p_{ij}).$$ Since $\Delta(1)=0$ this shows that $$\label{eqna3}
\Delta_{jj}=\sum^k_{i=1}\Delta_{ij}p_{ij}.$$ If $1$ is a *simple* root of $\Delta(s)=0$, (when the corresponding Markov chain has a single recurrent class) the derivative in (\[eqna1\]) is non-zero so that the common sums $\Delta_{\cdot j}$ are non-zero. In this case, (\[eqna3\]) implies that for each $j$, $(\Delta_{1j}, \Delta_{2j},
\ldots, \Delta_{kj})/\Delta_{\cdot 1}$ is a solution in $\bfx$ of $$\label{eqna4}
\bfx = \bfx\PP,\qquad \sum^k_{i=1}x_i=1.$$ Thus if $\PP$ is also such that (\[eqna4\]) has a unique solution, which is the case of $\PP^n$ converging as $n\to\infty$, these solutions must all agree (with the common row elements of that limit) so that the numbers $\Delta_{ij}/\Delta_{\cdot j}\equiv\Pi_j$ say, do not depend upon $j$. Equivalently, the cofactors of $\bI-\PP$ form a matrix all of which columns are equal whenever (\[eqna4\]) has a unique solution. (The reader will note the relationship to Cramer’s rule for solving simultaneous linear equtions.)
Even when $\PP^n$ does not converge the columns of cofactors are still all the same as long as the corresponding Markov chain has only one recurrent class. Here is the case of a periodic chain of period 2 which is neede for this paper.
Suppose the stochastic matrix $\PP$ in the above discussion is a periodic matrix $\bC$ of period 2 of the form given in (\[eqn4.8\]), namely, $$\bC=\left(\begin{array}{cc}
0 & A \\
B & 0 \end{array}\right),$$ in which $A$ is $r\times t$ and $B$ is $t\times r$ with $k=r+t$. (The non-square nature of $A$ and $B$ makes this slightly different from the $\bC$ of (\[eqn4.7\]).) Then, $$\label{eqna5}
\Delta(s)=|s\bI_k-\bC|=\left|\begin{array}{cc}
s\bI_r & -A \\
-B & s\bI_t\end{array}\right|.$$
It is known (e.g., Rao (1973), p. 32) that determinants of this form can be evaluated in two ways giving $$\Delta(s)=s|s\bI_t-s^{-1}BA|=s|s\bI_r-s^{-1}AB|.$$ Therefore, for $u=s^2$, $$\Delta(\sqrt{u})=|u\bI_t-BA|=|u\bI_r-AB|$$ and so $$\lim_{u\nearrow 1} \frac{|u\bI_t-BA|}{u-1} = \lim_{u\nearrow 1}\frac{|u\bI_r
- AB|}{u-1}.$$ By (\[eqna1\]) and (\[eqna2\]) above, this means that the common column sum of cofactors of $\bI_t-BA$ is equal to that of the column sums of cofactors of $\bI_r-AB$. Moreover, since $$\lim_{s\nearrow 1}\frac{\Delta(s)}{s-1} = 2\lim_{u\nearrow
1}\frac{\Delta(\sqrt{u})}{u-1},$$ each of these column sums is exactly half of the equal column sums of cofactors of $\bI_k-\bC$.
It is possible also to show that the set of diagonal cofactors of $\bI-\bC$ is made up of the diagonal cofactors of $\bI_t-BA$ and $\bI_r-AB$. Write $\bfalpha$ and $\bfbeta$ for the first row and column of $A$ and $B$, respectively, so that $$A = \left(\begin{array}{c}
\bfalpha \\
A^\ast\end{array}\right)\hbox{ and }b=(\bfbeta, B^\ast).$$ Then the cofactor $\Delta_{11}$ of $\bI_k-\bC$ is $$\Delta_{11}=\left|\begin{array}{cc}
\bI_{r-1} & A^\ast \\
B^\ast & \bI_t\end{array}\right| = |\bI_{r-1}-A^\ast B^\ast|.$$ But since $$\bI_r-AB = \left(\begin{array}{cc}
1-\bfalpha\bfbeta & -\alpha B^\ast \\
-A^\ast\beta & \bI_{r-1}-A^\ast B^\ast\end{array}\right)$$ it is clear that its first diagonal cofactor is also $|\bI_{r-1}-A^\ast
B^\ast|$. On the other hand if we use instead the partitioning $$A=(\bfalpha^\ast, A^{\ast\ast}), B=\left(\begin{array}{cc}
\bfbeta^\ast\\
B^{\ast\ast}\end{array}\right)$$ in which $\bfalpha^\ast$ and $\bfbeta^\ast$ are the first column and first row of $A$ and $B$, respectively, then $$\bI_k-\bC=\left(\begin{array}{cc}
\bI_r & (\bfalpha^\ast A^{\ast\ast}) \\
\left(\begin{array}{cc}
\bfbeta^\ast \\
B^{\ast\ast} \end{array}\right) & \bI_t\end{array}\right).$$ Therefore, the $(r+1)$-th diagonal cofactor of $\bI-\bC$ is $$\left|\begin{array}{cc}
\bI_r & A^{\ast\ast} \\
B^{\ast\ast} & \bI_{t-1}\end{array}\right| =
|\bI_{t-1}-B^{\ast\ast}A^{\ast\ast}|.$$ But since $$\bI_t-BA = \left( \begin{array}{cc}
1-\bfbeta^\ast\bfalpha^\ast & -\bfbeta^\ast A^{\ast\ast} \\
-B^{\ast\ast}\bfalpha^\ast & \bI_{t-1}-B^\ast A^\ast\end{array}\right)$$ its first diagonal cofactor is $|\bI_{t-1}-B^\ast A^\ast|$ as well.
By cyclically permuting the first $j$ columns and rows when $1\le j\le r$, or the $(r+1)$-th through $j$-th columns and rows when $r<j\le k$, the above arguments prove that the first $r$ diagonal cofactors of $\bI_k-\bC$ are those of $\bI_r-AB$ and the last $t$ of them are the diagonal cofactors of $\bI_t-BA$.
In view of the above results, the stationary probability vectors $\bdelta$ and $\brho$ for $AB$ and $BA$, respectively, that were introduced for (\[eqn4.8\]) may be expressed in the notation of Section 4 as $$\label{eqna6}
\bdelta=\frac{2}{\gamma_{\cdot m}}(\gamma_{1m}, \ldots, \gamma_{km})\hbox{
and }\brho=\frac{2}{\gamma_{\cdot m}}(\gamma_{k+1,m}, \ldots, \gamma_{mm}).$$ In particular, this verifies the equivalence of (\[eqn4.7\]) and (\[eqn4.9\]), showing that (\[eqn4.9\]) applies for all $m$, whether even or odd.
: The author is grateful to Derek Abbott and Greg Harmer for their encouragement hnd for their helpful comments on early drafts of this paper.
|
---
abstract: 'Using angle-resolved photoemission, we have investigated the development of the electronic structure and the Fermi level pinnning in Ga$_{1-x}$Mn$_{x}$As with Mn concentrations in the range 1–6%. We find that the Mn-induced changes in the valence-band spectra depend strongly on the Mn concentration, suggesting that the interaction between the Mn ions is more complex than assumed in earlier studies. The relative position of the Fermi level is also found to be concentration-dependent. In particular we find that for concentrations around 3.5–5% it is located very close to the valence-band maximum, which is in the range where metallic conductivity has been reported in earlier studies. For concentration outside this range, larger as well as smaller, the Fermi level is found to be pinned at about 0.15 eV higher energy.'
address:
- 'Department of Experimental Physics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden'
- |
Department of Experimental Physics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden\
and Institute of Physics, Polish Academy of Sciences, PL-02-668 Warszawa, Poland
- 'Department of Materials Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden'
author:
- 'H. Åsklund, L. Ilver, and J. Kanski'
- 'J. Sadowski'
- 'R. Mathieu'
date: 'Submitted: May 4, 2001'
title: |
Photoemission studies of Ga$_{1-x}$Mn$_{x}$As:\
Mn-concentration dependent properties
---
[2]{}
Introduction
============
The possibility to include magnetic impurities at relatively high concentrations in GaAs by means of low-temperature molecular beam epitaxy (LT-MBE) has opened new exciting prospects of combining magnetic phenomena with high-speed electronics and optoelectronics. The numerous investigations of Ga$_{1-x}$Mn$_{x}$As alloys that have been carried out so far have revealed interesting material properties, the most notable being carrier-induced ferromagnetism, with reported Curie temperatures as high as 110 K.[@Ohno01] Other interesting properties are anomalous Hall effect, negative magnetoresistance and photoinduced ferromagnetism. Although there is general consensus concerning the importance of Mn-induced holes, the detailed mechanisms behind the ferromagnetic ordering of the Mn spins remain a subject of debate.[@Dietl01; @Inoue00; @Konig00; @Akai98; @Litvinov01; @Schneider87] The electronic state of the Mn ions in samples with high Mn content is also discussed, though at low concentrations (below 1%) the $d^{5}$+hole configuration is established.[@Sanvito00; @Szczytko99] It is clear that further spectroscopic studies related to these problems are strongly motivated. In the present work we have used photoemission to study two key features in the electronic structure of Ga$_{1-x}$Mn$_{x}$As alloys for a range of Mn concentrations: the Mn-related modifications of the electronic structure and the Fermi level position relative to the VBM.
Experiment
==========
The experiments were performed on the toroidal grating monochromator beamline (BL 41) at the MAX I storage ring of the Swedish National Synchrotron Radiation Center MAX-lab, where a dedicated system for molecular beam epitaxy (MBE) is attached to the photoelectron spectrometer. This configuration allows samples to be transferred between the growth and analytical chambers under UHV conditions. In the transfer system the vacuum was in the low 10$^{-9}$ torr range, and in the electron spectrometer in the low 10$^{-10}$ torr range. The ability to transfer samples means that no post-growth treatment was needed to prepare the surfaces for the photoemission measurements. This is a point worth stressing in the present context, as the samples are prepared under rather extreme conditions and change their properties with annealing even at temperatures well below that at which MnAs segregates.[@Hayashi01] Indeed, the spectra presented here are somewhat different from those obtained on sputtered and annealed surfaces.[@Okabayashi99]
The MBE system contains six sources, including an As$_{2}$ valved cracker. It is also equipped with a 10 keV electron gun for reflection high-energy electron diffraction (RHEED). The samples were approximately 10$\times$10 mm$^{2}$ pieces of epi-ready $n$-type GaAs(100) wafers, which were In-glued on tranferrable Mo holders. Each sample preparation started with a 1000 [Å]{} buffer, grown at a substrate temperature ($T_{s}$) of 590 $^{\circ}$C, $T_{s}$ was then lowered to the growth temperature of LT-GaAs and GaMnAs, which was typically 220 $^{\circ}$C. At the low temperature the growth started always with a 200–300 [Å]{} LT-GaAs buffer layer. The As$_{2}$/Ga flux ratio was maintained at values around 10. During deposition of this layer the LT-GaAs growth rate was measured by recording RHEED intensity oscillations. After opening the Mn shutter the RHEED oscillations were observed again during the GaMnAs growth.[@Sadowski00] At this low growth temperature the reevaporation of Mn and Ga from the surface is negligible, so the growth rate increase is proportional to the Mn content. The Mn-concentrations quoted below are estimated to be accurate within 0.5%.
Immediately after transfer the surfaces were checked with low energy electron diffraction (LEED). All Mn-containing samples exhibited 1$\times$2 reconstructed surfaces in RHEED as well in LEED, while the clean reference GaAs sample displayed a $c$(4$\times$4) LEED pattern with sharp integer order and less distinct fractional order spots. Photoemission was excited with mainly $p$-polarized light incident at 45$^{\circ}$ relative to the surface normal, the samples being oriented with the \[110\] azimuth (i.e. the 1-fold periodicity) in the plane of incidence. The electron energy distribution curves were obtained using a hemispherical electron energy analyzer with an angular resolution of 2$^{\circ}$, and the overall energy resolution was around 0.3 eV. A clean Ta foil, in contact with the sample holder, was used to determine the Fermi level position in each case. The counting rates were normalized to the incident beam intensity by means of photocurrent from a gold mesh in the beam path.
Results and discussion
======================
Considering the intrinsic surface sensitivity of photoemission, and that surface compositions in alloy systems often deviate from those in the bulk, it is well motivated to start with a brief comment on this point. The fact that clear and actually unusually persistent RHEED oscillations are observed during growth of GaMnAs, shows that the atoms are still mobile in the surface layer despite the low-temperature conditions. However, once accommodated in lattice sites, further mobility that would lead to phase separation is efficiently inhibited under the low-temperature growth conditions. Thus, it is well motivated to expect that the sample compositions are uniform, including the first atomic layers. We should mention, however, that applying secondary ion mass spectroscopy (SIMS) as well as Auger microprobe analysis on samples exposed to atmospheric pressure, we have observed pronounced enrichment (by a factor of 2) of Mn in the surface layer (and a corresponding depletion in the underlying region), which is clearly associated with oxidation. Typically these redistributions range over a thickness of around 150 [Å]{}. This clearly emphasises the significance of carrying out surface sensitive experiments on [*in situ*]{} prepared samples. Photoemission from Mn 3$d$ states in dilute systems like Ga$_{1-x}$Mn$_{x}$As is easily identified via resonant enhancement of the 3$d$ cross section at the 4$p$ excitation threshold, which occurs at 50 eV photon energy. Since the spectral shape changes quite much in this energy range, and since our aim is to compare spectra recorded from a series of different samples, we have chosen to use a photon energy well above this resonant range (81 eV). Although the absolute cross section of Mn 3$d$ is smaller at 81 eV than that just above 50 eV photon energy, the cross sections of the GaAs valence states are also reduced in a similar way and therefore the Mn 3$d$-induced spectral features are still readily detected. Fig. \[Mn\]a shows a set of such valence band spectra from samples with different Mn contents, together with a reference spectrum from clean LT-GaAs. It is worth pointing out that just like the GaMnAs, such LT-GaAs contains large concentrations of point defects (mostly As antisites), and that spectra from such layers are found to be somewhat different relative those obtained from MBE layers grown at high temperature.[@Aasklund01] Considering that until now only one independent valence-band photoemission study of such materials has been published,[@Okabayashi99] and that the samples are produced under rather extreme growth conditions, it is well motivated to start the discussion with a direct comparison between the present data and the published ones. We note then that the data contain some similarities, but also some significant differences. The main Mn-induced feature is the peak at 3.4 eV below the VBM for the most Mn-rich samples. Its Mn origin is clearly revealed by the resonant enhancement mentioned above. A similar resonant structure was found in Ref. \[RefOkabayashi99\], but at a binding energy of 4.5 eV relative to the Fermi level. Since the Fermi level is located about 0.13 eV above the VBM (see below), there seems to be a discrepancy of almost 1 eV between the two results. Furthermore, the spectra in Ref. \[RefOkabayashi99\] contain a second pronounced peak at about 2.5 eV larger binding energy. This structure is completely missing in our data. The weak asymmetric peak seen in all spectra around 6.5–7 eV in Fig. \[Mn\]a reflects the $X_{3}$ critical point emission. Such density of states (DOS) structures are seen at all photon energies and all emission angles due to diffuse elastic scattering of the direct interband excitation of this state. Altogether we thus find that the present spectra are significantly different from those found in literature, and although the reason for these deviations is not clear, it is natural at this point to suspect that the different surface preparations could be the cause. This would then underline the importance of carrying out these experiments on [*in situ*]{} grown samples.
It is immediately clear in Fig. \[Mn\] that the Mn-induced spectral changes vary with the Mn content. To examine this variation in some more detail, we have generated consecutive difference spectra, as displayed in Fig. \[Mn\]b. The first spectrum in this sequence shows that with 1% Mn the spectral intensity is increased over a range 1–4 eV below the VBM, with a peak centered around 3 eV and a shoulder at 1 eV below the VBM. The main increase coincides with weak structures in the clean GaAs spectrum (around 3 eV and 4 eV below the VBM). As these structures are due to excitations at the high DOS regions at the $X_{5}$ and $\Sigma^{min}_{1}$ points, one could suspect that the Mn-induced changes are in this case caused by disorder-related increase of diffuse scattering. However, from the fact that no corresponding increase is seen for the $X_{3}$ critical point emission, and from the following development of the 3-eV peak, we can safely conclude that these spectral changes do indeed reflect Mn-derived states. With the Mn content raised to 3% we see that the incremental change is somewhat different than the initial one. The peak at 3 eV is increased further, but the range around 1 eV remains essentially unchanged. Increasing the Mn content further results in another change: the main additional spectral contribution appears as an asymmetric peak around 3.8 eV below the VBM, i.e. clearly shifted relative to that found at lower concentrations. Thus, the peak observed at 3.4 eV in the corresponding full spectrum (Fig. \[Mn\]) can be concluded to represent an average of several contributions. Finally, the additional spectral changes with further increase of Mn content are found to be less distinct, the intensity is increased rather uniformly over a range 2–6 eV below the VBM. The next difference spectrum is also essentially a peak centered at 3 eV below the VBM, though it is clearly narrower on the high-energy side.
The important conclusion from the data in Fig. \[Mn\] is that the character of the Mn states in Ga$_{1-x}$Mn$_{x}$As depends on the Mn concentration. Since supplementary X-ray diffraction analysis of our samples shows high degree of perfection in the layers, we have no reason to suspect that the variations seen here are due to varying sample structure quality, but ascribe them to the different Mn contents. No such dependence has been reported in any of the earlier studies. Previous analysis of Mn 3$d$ partial DOS in GaMnAs with 6.9% Mn was successful in modelling the observed spectrum using a configuration interaction model involving Mn 3$d$ and ligand states in a MnAs4 cluster.[@Okabayashi99] Obviously, this kind of model can not account for concentration-dependent properties like those reported here. With an average distance between two impurities of around 25 [Å]{}, it is clear that the explanation must be based on a model in which long range interactions are taken into account.
GaMnAs is also known to exhibit unusual conductivity characteristics:[@Oiwa97] at low Mn concentrations the system is semiconducting, around 4–5% Mn metallic conductivity is reported and with further increase of the Mn content the material becomes again insulating. Interestingly enough, the Curie temperature also exhibits a maximum around the same Mn concentration. These two observations suggest that the density of holes is actually decreasing with Mn concentrations above 5%, and this might be directly reflected by the Fermi level position relative the VBM. In Fig. \[VB\] we show a set of valence-band spectra from samples with varying Mn contents, aligned at the Fermi level. The photon energy used in this case was 38.5 eV, chosen to probe the phase space region around the $X_{3}$ point. This emission is reflected by the prominent peak around 7.5 eV. Considering the high density of defects in LT-GaAs and in GaMnAs, (in the range of 10$^{20}$/cm$^{2}$), it is reasonable to assume that the surface Fermi level does not deviate from that in the bulk. This assumption is supported by the fact that no additional spectral broadening that could be expected due to a emission from a very narrow band bending region was detected in any spectra. Focusing on the $X_{3}$ emission we see that its position is changing with Mn content. This variation is shown more clearly in Fig. \[EF\], where we have plotted the energy separation $E_{\text{F}}-X_{3}$ for a larger set of samples. Starting at a value of 7.35 eV for clean LT-GaAs it is reduced, and settles at a value of 7.1 eV around 1.5% Mn concentration. This pinning position remains stable for Mn concentrations up to around 3.5%, and is likely due to the Mn acceptor level known to be located 113 meV above the VBM.[@Schneider87] With this interpretation we deduce the VBM to be located around 6.95 eV above the $X_{3}$ point, a value well in the range of literature data[@Chiang80] (6.70–7.1 eV) based on angle resolved photoemission and X-ray photoemission. A very interesting feature is observed around 4–5% Mn concentrations, where $E_{\text{F}}$ appears to drop to a position close to the VBM. As already mentioned, samples in this concentration range are reported to exhibit metallic conductivity. The present observations are fully consistent with such behaviour. We also note that the low position of $E_{\text{F}}$ implies an increased density of holes, which in turn may be the explanation for the relatively high Curie temperatures found in this range of Mn concentrations. The most intriguing observation is the shift of $E_{\text{F}}$ back into the band gap region with further increased Mn content. This is consistent with the reported metal-insulator transition,[@Oiwa97] though the present results suggest that the reason for the insulating properties is not impurity scattering, but rather a true reduction of charge carriers.
Conclusions
===========
The present investigations of valence band photoemission from Ga$_{1-x}$Mn$_{x}$As compounds show two new effects. Firstly we find that the spectral changes induced by the Mn atoms depend on the Mn concentration, and secondly we observe that the position of the Fermi level also changes with Mn content. None of these features has been reported previously. The varying shape of the Mn-induced valence-band structures directly shows that the Mn-host interaction cannot be treated with a local model. As to the Fermi level variations, we note that the minimum observed around 3.5–5% Mn content coincides with previously reported metallic conductivity and also with the range of maximum paramagnetic-ferromagnetic transition temperatures.
Acknowledgements {#acknowledgements .unnumbered}
================
We are pleased to acknowledge the technical support of the MAX-lab staff. This work was supportet by grants from the Swedish Natural Science Research Council (NFR), the Swedish Research Council for Engineering Sciences (TFR), and, via co-operation with the Nanometer Structure Consortium in Lund, the Swedish Foundation for Strategic Research (SSF).
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‘=11 =manfnt @tchout\[\#1\][[tempcnta\#1 whilenumtempcnta>@]{}]{} @tchout dubious\[\#1\]
tempboxa tempdimatempboxa
W@tchout\#1[@tchout\[\#1\]]{} ‘=12
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SLAC-PUB-16169, MITP/14-095\
<span style="font-variant:small-caps;">Ye Li$^{(1)}$, Andreas von Manteuffel$^{(2)}$,\
Robert M. Schabinger$^{(2)}$, and Hua Xing Zhu$^{(1)}$\
</span>
**Abstract**
In this paper, we compute the soft-virtual corrections to Higgs boson production in gluon fusion for infinite top quark mass at next-to-next-to-next-to-leading order in QCD. In addition, we present analogous soft-virtual terms for both Drell-Yan lepton production in QCD and scalar pair production in $\Neqfour$ super Yang-Mills theory. The result for Drell-Yan lepton production is derived from the result for Higgs boson production using Casimir scaling arguments together with well-known results available in the literature. For scalar pair production in the $\Neqfour$ model, we show by explicit calculation that the result is equal to the part of the Higgs boson soft-virtual term which is of maximal transcendentality weight.
Introduction {#sec:intro}
============
The discovery of a scalar particle with properties compatible with the Standard Model Higgs boson was announced by the ATLAS and CMS collaborations on July 4, 2012 [@1207.7214; @1207.7235]. After the discovery of this Higgs boson-like particle, detailed measurements of its properties became one of the top priorities of the ongoing experimental physics program at the Large Hadron Collider (LHC). To fully realize the physics potential of the machine in this regard, accurate and precise theoretical predictions are required for parton-initiated scattering processes involving the Standard Model Higgs boson.
As has long been known, the primary mechanism for Standard Model Higgs boson production at the LHC is gluon-gluon fusion [@PHLTA.40.692]. The leading order (LO) cross section is of order $\alpha_s^2$ in Quantum Chromodynamics (QCD) and is, unfortunately, neither accurate nor precise; the next-to-leading order (NLO) QCD corrections to this process in the limit of large top-quark mass were found to be about 100% of the LO result [@NUPHA.B359.283; @PHLTA.B264.440; @hep-ph/9504378] and not covered by the conventional LO scale variation. The importance of the gluon fusion process and the size of the NLO QCD corrections to it have motivated studies of the next-to-next-to-leading order (NNLO) QCD corrections, as well as top quark mass effects and electroweak corrections. The next-to-next-to-leading order (NNLO) QCD corrections to the total cross section in the large top-quark mass limit were computed in references [@hep-ph/0201206; @hep-ph/0207004; @hep-ph/0302135] and, in the same approximation scheme, the phenomenologically relevant, fully differential NNLO parton-level calculations were subsequently carried out in references [@hep-ph/0501130; @hep-ph/0703012; @0707.2373; @0801.3232]. The effects stemming from the finite mass of the top quark have been studied as well, both at NLO [@Graudenz:1992pv] and at NNLO [@0801.2544; @0909.3420] in QCD. In addition, both two-loop electroweak corrections [@hep-ph/0404071; @hep-ph/0407249; @0809.3667] and three-loop mixed electroweak-QCD corrections [@0811.3458] have been considered.
Over the years, a number of soft-gluon or high-energy resummation techniques have been used to complement and consistently improve the available fixed-order QCD results [@hep-ph/9611272; @hep-ph/0306211; @hep-ph/0508265; @hep-ph/0508284; @hep-ph/0512249; @hep-ph/0603041; @hep-ph/0605068; @0809.4283; @1303.3590]. Currently, the state-of-art is a NNLO+NNLL QCD prediction which, however, still suffers from a nearly $10\%$ scale uncertainty [@1101.0593]. This is actually larger than the experimental error which can be achieved in the long run at LHC [@1307.7135] and it is therefore essential to go beyond the NNLO QCD approximation to further reduce the theoretical uncertainty. Important steps in this direction have already been taken by a number of groups. For example, both the effective Higgs-gluon coupling [@hep-ph/9708255; @hep-ph/0512058; @hep-ph/0512060] and the QCD beta function [@PHLTA.B93.429; @hep-ph/9302208; @hep-ph/9701390; @hep-ph/0411261] have been known to four-loop accuracy for quite some time. Somewhat more recently, three-loop virtual amplitudes for $gg \to H$ were calculated by two different groups [@0902.3519; @1001.2887; @1004.3653; @1010.4478]. In particular, all purely virtual ingredients for the calculation of $gg \to H$ at next-to-next-to-next-to-leading order (3lo) in QCD are well-known and available in the literature.
Of course, in order to obtain the physical cross section at 3lo, one still needs to deal with real radiation and convolve parton-level results with the appropriate splitting amplitudes [@1211.6559]. In contrast to the purely virtual corrections discussed above, a complete calculation of the real radiative corrections is still missing. Although the complete set of single-emission real-virtual corrections have now been calculated [@1311.1425; @1312.1296; @1411.3586; @1411.3587], even more substantial progress has been made by working in the threshold approximation where all secondary real radiation is soft. In fact, very recently, the next-to-threshold contribution to the cross section became available as well [@1411.3584]. Before the exact results became available, the single-emission, two-loop real-virtual corrections in the soft limit were computed by two different groups in references [@1309.4391; @1309.4393]. The triple-emission real corrections were calculated in the soft limit in reference [@1302.4379], and a subset of the exact master integrals required for these contributions were obtained in [@1407.4049]. The soft double-emission, one-loop real-virtual corrections were given implicitly in reference [@1403.4616] and more explicitly so in reference [@1404.5839]. Most importantly, the full set of 3lo soft plus virtual corrections (or, as we shall hereafter call them, soft-virtual corrections) were obtained in reference [@1403.4616].
In this paper, we continue our study of the 3lo soft-virtual corrections to the gluon-fusion Higgs production process. To that end, we calculate squares of time-ordered matrix elements of pairs of semi-infinite Wilson line operators with up to three massless partons in the final state. The object that we calculate to three-loop order is commonly referred to as a soft function for a pair of Wilson line operators and frequently appears in the study of observables in soft-collinear effective theory (SCET) [@hep-ph/0005275; @hep-ph/0011336; @hep-ph/0109045; @hep-ph/0206152] (see also reference [@1410.1892] for a review). In a precise sense, the soft function encodes the non-trivial part of the threshold limit of the QCD cross section, coming from correlations between soft partons radiated off of the incoming hard gluons responsible for the primary production process. Given that all purely virtual corrections are known, the calculation of the 3lo soft-virtual corrections of interest amounts to the calculation of the 3lo, three-loop soft function. In earlier work, we calculated both the single-emission, two-loop real-virtual [@1309.4391] and the double-emission, one-loop real-virtual [@1404.5839] contributions to the three-loop soft function. In this article, we complete the calculation by treating both the square of the single-emission, one-loop real-virtual corrections and the triple-emission real corrections. These single-emission contributions to the three-loop soft function can be straightforwardly obtained by using the soft-gluon current at one loop [@hep-ph/9903516; @hep-ph/0007142]. The triple-emission contributions, on the other hand, are highly non-trivial and their calculation requires the development of specialized computational technology.
Ultimately, we managed to overcome all technical hurdles and complete the calculation of the three-loop soft function initiated by two of us some time ago in reference [@1309.4391]. After briefly reviewing our SCET-based formalism in Section \[sec:formalism\], we present the complete three-loop soft function for Higgs boson production at threshold in Section \[sec:softHiggs\]. Then, by appropriately combining our results with the well-known purely virtual contributions to the cross section, we obtain the soft-virtual term for Higgs boson production at 3lo in Section \[sec:softvirt\]. We compare our results to a recent calculation [@1403.4616] and find full agreement. While our soft function calculation is performed with adjoint Wilson lines, it is straightforward to convert the results obtained using Casimir scaling arguments to the case of fundamental Wilson lines appropriate for Drell-Yan lepton production (see [*e.g.*]{} [@0901.1091; @0903.1126] for a discussion of the Casimir scaling principle). By combining the resulting three-loop Drell-Yan soft function and the known three-loop virtual amplitudes for $\gamma^* \to q \bar{q}$ [@0902.3519; @1001.2887; @1004.3653; @1010.4478], we derive the soft-virtual term for Drell-Yan lepton pair production at 3lo and present the result in Section \[sec:DY\]. Once again, we find full agreement with recent independent predictions of the previously unknown part of the result [@1404.0366; @1405.4827].
In fact, our highly-automated setup for the QCD squared eikonal matrix elements allows for an essentially trivial extension to completely general massless final-state partons and, in particular, we can easily calculate the analogous three-loop soft function for a $SU(\Nc)$ $\Neqfour$ super Yang-Mills theory [@Brink:1976bc]. In Section \[sec:SYM\], by making use of the $\Neqfour$ form factor computed in reference [@1112.4524], we give the 3lo soft-virtual term for color-singlet scalar pair production in $\Neqfour$ super Yang-Mills theory. Curiously, we find that this $\Neqfour$ soft-virtual term obeys a version of the principle of maximal transcendentality weight first proposed long ago for the anomalous dimensions of twist-two operators in $\Neqfour$ super Yang-Mills theory [@hep-th/0404092]. In this case, we find that our result for the 3lo soft-virtual term for scalar pair production in $\Neqfour$ super Yang-Mills theory coincides with the part of the 3lo soft-virtual term for Higgs boson production which is of maximal transcendentality weight. Finally, in Section \[sec:conclude\], we summarize our results and conclude the paper.
Formalism {#sec:formalism}
=========
In this section, we briefly review the SCET formalism that we use in what follows.[^1] Although we will ultimately be interested in other processes as well, our development in this section is specialized to the case of gluon-fusion Higgs boson production in order to keep the notation managable; our discussion of the other processes treated in this paper will be completely analogous. We are interested in the so-called threshold expansion of the partonic cross section, $\hat{\sigma}^{\mathrm{H}}_{gg} (\hat s, z, \alpha_s(\mu))$, in the limit $z\to 1$, where $z = M^2_{\rm H}/{\hat s}$, $M_{\rm H}$ is the Higgs mass, and $\hat s$ is the square of the partonic center of mass energy. In this regime, it can be written as $$\label{eq:partxsect}
\lim_{z\to 1}\left\{\hat{\sigma}^{\mathrm{H}}_{gg} (\hat s, z, \alpha_s(\mu))\right\} = \frac{\pi \lambda^2\left(\alpha_s(\mu),L_{\rm H}\right)}{ 8 (\Nc^2 - 1)} \, G^{\mathrm{H}}(z, L_{\rm H})\,,$$ where $L_{\rm H} = \ln(\mu^2/M_{\rm H}^2)$ and $\lambda\left(\alpha_s(\mu),L_{\rm H}\right)$ is the effective coupling of the Higgs boson to gluons in the limit of infinite top quark mass, $\mathcal{L}_{\mathrm{eff}} = - \frac{1}{4}\lambda H G^{\mu\nu,\,a}G^a_{\mu\nu}$. The factor out front of $G^{\mathrm{H}}(z, L_{\rm H})$ in Eq. (\[eq:partxsect\]) is the LO cross section in this limit. For the sake of convenience, we use a common factorization and renormalization scale $\mu_F = \mu_R = \mu$.
The coefficient function $G^{\rm H}(z, L_{\rm H})$ defined in this way can be shown to factorize in the framework of SCET as $z\to 1$: $$\begin{aligned}
G^{\rm H}(z, L_{\rm H}) = H^{\rm H} (L_{\rm H}) \bar{S}^{\rm \,H}(z, L_{\rm H})\,,
\label{eq:coef}\end{aligned}$$ where the renormalized hard function, $H^{\rm H} (L_{\rm H})$, captures all purely virtual effects and the renormalized soft function, $\bar{S}^{\rm \,H}(z, L_{\rm H})$, captures all effects coming from soft final-state gluon radiation very close to threshold. Note that Eq. (\[eq:coef\]) depends implicitly on the renormalized strong coupling constant, $\alpha_s(\mu)$.
As explained in reference [@1404.5839], at order $n$ in QCD perturbation theory, the $n$-th perturbative coefficient of the hard function in SCET for Higgs production can be extracted from the $n$-loop gluon form factor [@0902.3519; @1001.2887; @1004.3653; @1010.4478]. In fact, the relevant formulae are written down explicitly in reference [@1004.3653], Eqs. (7.6), (7.7), and (7.9). The hard coefficients that we need for our analysis can be derived by taking the complex square of Eq. (7.3) in that work, setting $\mu = M_{\rm H}$, and then expanding in the strong coupling to third order. The perturbative coefficients of the soft function, on the other hand, require a dedicated calculation and, in fact, we present the third-order coefficient explicitly for the first time in Section \[sec:softHiggs\].
The bare soft function for Higgs production can be written as the square of a time-ordered matrix element of a pair of semi-infinite Wilson line operators, $$\begin{aligned}
\label{eq:softdef}
S^{\rm H}\left(z, L_{\rm H}\right) = \frac{M_{\rm H}}{\Nc^2 - 1} \sum_{\mbox{\tiny $X_s$ }}
\langle 0 | T\Big\{Y_{n} Y_{\bar{n}}^\dagger\Big\} \delta\Big(\lambda - \hat{\bf P}^0\Big) | X_s\rangle\langle X_s | T\Big\{ Y_{\bar{n}} Y_{n}^\dagger \Big\} | 0 \rangle\,.\end{aligned}$$ As discussed in reference [@1404.5839], the soft function defined in Eq. (\[eq:softdef\]) depends on $z$ and the bare strong coupling constant $\alpha_s$. In this notation, the ratio of twice the energy of the soft QCD radiation to $M_{\rm H}$ is approximately $1-z$. The summation in Eq. (\[eq:softdef\]) is over all possible soft parton final states, $|X_s\rangle$. The operator $\hat{\bf P}^0$ acts on the final state $|X_s\rangle$ according to $$\hat{\bf P}^0|X_s\rangle = 2 E_{X_s} |X_s\rangle\,,$$ where $E_{X_s}$ is the energy of the soft radiation in final state $|X_s\rangle$. The Wilson line operators, $Y_{n}$ and $Y_{\bar{n}}^\dagger$, are respectively defined as in-coming, path-ordered $\left(\mathbf{P}\right)$ and anti-path-ordered $\left(\overline{\mathbf{P}}\right)$ exponentials [@hep-ph/0412110], $$\begin{aligned}
Y_{n}(x) &=& \mathbf{P} \exp \left( i g \int^0_{-\infty} {\mathrm{d}}s\, n \cdot A(n s+x) \right)
{\nonumber}\\
Y_{\bar{n}}^\dagger(x) &=& \overline{\mathbf{P}} \exp \left( -i g \int_{-\infty}^0 {\mathrm{d}}s\, \bar{n} \cdot A
(\bar{n}s+x) \right).\end{aligned}$$ In the above, $A_\mu = A_{\mu}^a T^a$, where the $T^a$ are adjoint $\mathfrak{su}(N_c)$ matrices. As usual, $n$ and $\bar{n}$ are light-like vectors whose space-like components are back-to-back and determine the beam axis. For a generic four-vector, $k^\mu$, we have $$n \cdot k = k^+ \qquad {\rm and} \qquad \bar{n} \cdot k = k^-$$ given the usual definitions $k^+ = k^0 + k^3$ and $k^- = k^0 - k^3$.
The soft function obeys a Renormalization Group (RG) equation which involves a convolution in momentum space. The convolution makes it somewhat inconvenient to work in momentum space directly and it is customary to take a Laplace transform to turn the convolution into a product, $$\begin{aligned}
\label{eq:Laplace}
{\widetilde{S}}^{\rm \,H}( L_\omega ) = \int^1_{-\infty} \! {\mathrm{d}}z \, \exp
\bigg( - \frac{ M_{\rm H} ( 1-z )}{e^{\gamma_E} \omega }\bigg) {\bar{S}^{\rm \,H}}(z, L_{\rm H})\,.\end{aligned}$$ In Eq. (\[eq:Laplace\]), $\omega$ is the variable Laplace-conjugate to $z$, $L_\omega =\ln \left(\mu^2/\omega^2\right)$, and $\gamma_E$ is the Euler-Mascheroni constant. The perturbative coefficients of the Laplace-transformed soft function are simple polynomial functions of $L_\omega$ of degree $2 n$ at $n$-loop order. In this case, the inverse Laplace transform can be evaluated in closed form: $$\begin{aligned}
\label{eq:inverseLaplace}
{\bar{S}^{\rm \,H}}(z, L_{\rm H}) = \lim_{\eta \to 0} \bigg\{{\widetilde{S}}^{\rm \,H}\left(-\partial_\eta\right) \frac{1}{(1-z)^{1 - 2 \eta} }
\left( \frac{M^2_{\rm H}}{\mu^2} \right)^\eta \frac{ \exp ( -2 \eta \gamma_E) }{\Gamma ( 2 \eta ) }\bigg\}\,.\end{aligned}$$ It is therefore possible to perform the entire analysis in Laplace space and then, at the very end, appropriately apply Eq. (\[eq:inverseLaplace\]) to recover the momentum space formulas of interest.
In Laplace space, the RG equation for the soft function is [@hep-ph/9808389] $$\begin{aligned}
\frac{{\mathrm{d}}{\widetilde{S}}^{\rm \,H}(L_\omega)}{{\mathrm{d}}\ln (\mu)} = \bigg( 2 {\Gamma_{\rm cusp}}^{\rm H}(a) L_\omega - 2 {\gamma_{s}}^{\rm H}(a) \bigg) {\widetilde{S}}^{\rm \,H}(L_\omega)\,,
\label{eq:RG}\end{aligned}$$ where ${\Gamma_{\rm cusp}}(a)$ and ${\gamma_{s}}(a)$ are, respectively, the cusp anomalous dimension and the soft anomalous dimension. They admit perturbative expansions of the form $$\begin{aligned}
{\Gamma_{\rm cusp}}^{\rm H} (a) = \sum_{n=1}^\infty a^{n} \Gamma^{\rm H}_{n - 1} \qquad {\rm and} \qquad {\gamma_{s}}^{\rm H} (a) = \sum_{n=1}^\infty a^{n} \gamma^{\rm H}_{n - 1} \,,
\label{eq:gdef}\end{aligned}$$ where we have defined $$\begin{aligned}
a = \frac{\alpha_s(\mu)}{4 \pi}\,. \end{aligned}$$ Their perturbative coefficients up to and including terms of $\Ord\left(a^3\right)$ are collected in the appendix. Eq. (\[eq:RG\]) can be solved straightforwardly order-by-order in $a$. Through to $\Ord(a^3)$, one can write down the results immediately by simultaneously making the replacements $L \to -L_\omega$, $\Gamma_i \to - \Gamma_i^{\rm H}$, and $\gamma_i^{\rm H} \to - \gamma_i^{\rm H}$ in the right-hand side of Eq. (55) of reference [@0803.0342], $$\begin{aligned}
\label{eq:expand}
{\widetilde{S}}^{\rm \,H}\left(L_\omega\right) = & \,1 + a \bigg[ \frac{1}{2} \Gamma_0^{\rm H} L^2_\omega - \gamma_0^{\rm H} L_\omega + c^{\rm H}_1 \bigg]
+ a^2 \bigg[ \frac{1}{8} \Big(\Gamma_0^{\rm H}\Big)^2 L^4_\omega +
\left( \frac{1}{6} \beta_0 \Gamma_0^{\rm H} - \frac{1}{2} \gamma_0^{\rm H} \Gamma_0^{\rm H} \right) L^3_\omega
\\
& +
\left( \frac{1}{2}\Gamma_1^{\rm H} + \frac{1}{2}c^{\rm H}_1\Gamma_0^{\rm H} - \frac{1}{2} \beta_0 \gamma_0^{\rm H} + \frac{1}{2} \Big( \gamma_0^{\rm H} \Big)^2\right) L^2_\omega
+ \Big( \beta_0 c^{\rm H}_1 - c^{\rm H}_1 \gamma_0^{\rm H} - \gamma_1^{\rm H} \Big) L_\omega + c^{\rm H}_2\bigg]
{\nonumber}\\
&
+ a^3 \bigg[ \frac{1}{48} \Big(\Gamma_0^{\rm H}\Big)^3 L_\omega^6 +
\left(\frac{1}{12} \beta_0 \Big( \Gamma_0^{\rm H}\Big)^2 - \frac{1}{8} \gamma_0^{\rm H} \Big( \Gamma_0^{\rm H}\Big)^2 \right) L_\omega^5
+ \left( \frac{1}{4} \Gamma_0^{\rm H} \Gamma_1^{\rm H} + \frac{1}{8} c^{\rm H}_1 \Big( \Gamma_0^{\rm H} \Big)^2 \right.
{\nonumber}\\
&
\left.+ \frac{1}{12} \Big(\beta_0\Big)^2 \Gamma_0^{\rm H} - \frac{5}{12} \beta_0 \gamma_0^{\rm H} \Gamma_0^{\rm H} + \frac{1}{4} \Big(\gamma_0^{\rm H}\Big)^2 \Gamma_0^{\rm H}\right) L_\omega^4
+ \left( \frac{1}{3} \beta_0 \Gamma_1^{\rm H} - \frac{1}{2}\gamma_0^{\rm H} \Gamma_1^{\rm H} + \frac{1}{6}\beta_1\Gamma_0^{\rm H} \right.
{\nonumber}\\
&
\left. + \frac{2}{3}\beta_0 c^{\rm H}_1\Gamma_0^{\rm H} - \frac{1}{2}c^{\rm H}_1\gamma_0^{\rm H} \Gamma_0^{\rm H} - \frac{1}{2}\gamma_1^{\rm H} \Gamma_0^{\rm H} - \frac{1}{3}\Big(\beta_0\Big)^2 \gamma_0^{\rm H}
+ \frac{1}{2} \beta_0 \Big(\gamma_0^{\rm H}\Big)^2 - \frac{1}{6} \Big(\gamma_0^{\rm H}\Big)^3 \right) L_\omega^3
{\nonumber}\\
&
+
\left( \frac{1}{2} \Gamma_2^{\rm H} + \frac{1}{2} c^{\rm H}_1 \Gamma_1^{\rm H} + \frac{1}{2} c^{\rm H}_2 \Gamma_0^{\rm H} + \Big(\beta_0\Big)^2 c^{\rm H}_1
- \frac{1}{2} \beta_1 \gamma_0^{\rm H} - \frac{3}{2} \beta_0 c^{\rm H}_1 \gamma_0^{\rm H}+ \frac{1}{2} c^{\rm H}_1 \Big( \gamma_0^{\rm H}\Big)^2\right.
{\nonumber}\\
&
\left. - \beta_0 \gamma_1^{\rm H} + \gamma_0^{\rm H} \gamma_1^{\rm H} \vphantom{\frac{1}{2} \Gamma_2^{\rm H}}\right) L_\omega^2
+ \Big( \beta_1 c^{\rm H}_1 + 2 \beta_0 c^{\rm H}_2 - c^{\rm H}_2 \gamma_0^{\rm H} - c^{\rm H}_1 \gamma_1^{\rm H} - \gamma_2^{\rm H} \Big) L_\omega + c^{\rm H}_3 \bigg]+\Ord\left(a^4\right)\,.
{\nonumber}\end{aligned}$$
While $\beta_0$ and $\beta_1$ are nothing but the one- and two-loop coefficients of the QCD beta function (see Appendix \[sec:append\]), the $c^{\rm H}_n$ in Eq. (\[eq:expand\]) are [*a priori*]{} unknown matching coefficients which must be determined by an explicit computation at each order in QCD perturbation theory. With the understanding that $c^{\rm H}_0 = 1$, we can write $$c_s^{\rm H}(a) = \sum_{n=0}^\infty a^n c^{\rm H}_n$$ in analogy with Eqs. (\[eq:gdef\]) above. The one- and two-loop matching coefficients, $c^{\rm H}_1$ and $c^{\rm H}_2$, were calculated for a soft function built out of fundamental representation Wilson line operators long ago [@hep-ph/9808389] and, as was pointed out in reference [@0809.4283], the results can be converted to the adjoint representation case relevant to Higgs production using a simple Casimir scaling argument. For a theory with $N_f$ light quark flavors, they read $$\begin{aligned}
c^{\rm H}_1 = & \,2 \zeta_2 C_A
{\nonumber}\\
c^{\rm H}_2 = & \,\frac{1}{2!} \Big(c^{\rm H}_1\Big)^2 + \Delta c^{\rm H}_2 \,,\end{aligned}$$ where $$\begin{aligned}
\Delta c^{\rm H}_2 = \left( \frac{2428}{81} + \frac{67\zeta_2}{9} - \frac{22\zeta_3 }{9} - 30 \zeta_4 \right)C^2_A + \left( -\frac{328}{81} - \frac{10\zeta_2}{9} + \frac{4\zeta_3}{9} \right) C_A N_f\end{aligned}$$ with $C_A = \Nc$ and $C_F = (\Nc^2 - 1)/(2\Nc)$.
Note that, in the above, we have explicitly factored out the terms that are fixed by the non-Abelian exponentiation theorem [@PHLTA.B133.90; @NUPHA.B246.231]. This formulation works well for our purposes since, at least through three-loop order, $c_s^{\rm H}(a)$ is naturally written as an exponential (see Section \[sec:DY\]): $$\begin{aligned}
c_s^{\rm H}(a) = \exp \Big( a c^{\rm H}_1 + a^2 \Delta c^{\rm H}_2 + a^3 \Delta c^{\rm H}_3 + \cdots \Big) \, .
\label{eq:csexp}\end{aligned}$$ One of the main goals of the present paper is to calculate the new contribution, $\Delta c^{\rm H}_3$, to the three-loop matching coefficient and this is the subject of the next section.
The Higgs soft function at 3lo {#sec:softHiggs}
==============================
![ Cut eikonal Feynman diagrams for final states with $(a)-(c)$ three gluons, $(d)$ one gluon and two fermions, $(e)$ one gluon and two scalars, and $(f)$ one scalar and two fermions. Diagrams such as the one shown in panel $(f)$ are only relevant for the calculation of the three-loop soft function in $\Neqfour$ super Yang-Mills theory. Note that diagram $(b)$ is planar since it can be drawn in the plane without any lines crossing. Soft contributions from non-planar diagrams such as $(c)$ vanish for all of the processes considered in this paper because of the color Jacobi identity.[]{data-label="diag"}](threeloopsoftfunc){width=".7\textwidth"}
In order to compute the threshold soft function for gluon-fusion Higgs boson production at 3lo, one must consider the following:
1. Interference between tree-level and two-loop single soft emission diagrams.
2. Interference between tree-level and one-loop double soft emission diagrams.
3. The square of the one-loop single soft emission diagrams.
4. The square of the tree-level triple soft emission diagrams.
As mentioned in the introduction, we computed the first two contributions in earlier work [@1309.4391; @1404.5839] and the third contribution is straightforward to derive using the one-loop soft gluon current [@hep-ph/9903516; @hep-ph/0007142]. The fourth and final contribution is more challenging and we therefore briefly describe its calculation below.
First, we generate all relevant tree-level triple soft emission diagrams using [QGRAF]{} [@Nogueira:1991ex]. All possible partonic final states are considered, including $ggg$, $gq\bar{q}$, $g \phi\phi^\dagger$, and $q\bar{q}\phi$, where $\phi$ is an adjoint scalar. Representative cut eikonal diagrams are depicted in Figure \[diag\]. To process the squared matrix element, we employ in-house [FORM]{} [@math-ph/0010025] and [Maple]{} routines to carry out all of the necessary numerator algebra. The spacetime dimension is set to $D = 4 - 2 \e$ and the polarization dimension is set to $D_s$; in conventional dimensional regularization (CDR), $D_s = D$, and in the four-dimensional helicity scheme (FDH), $D_s=4$ [@hep-ph/0202271].[^2] As observed in our calculation of the interference between the tree-level and one-loop double soft emission diagrams [@1404.5839], it is essential to partial fraction the raw integrand. Once all independent topologies are identified, we apply the integration by parts reduction [@PHLTA.B100.65; @NUPHA.B192.159; @hep-ph/0102033] algorithms for phase space integrals implemented in [Reduze2]{} [@1201.4330; @cs.sc/0004015; @fermat] and [LiteRed]{} [@1212.2685]. After reduction, a compact linear combination of seven master integrals is obtained. Our independent calculation of these integrals benefited from the development of a new computational technique for phase space integrals which will be described in a forthcoming paper by one of us [@zhu:2014].
Combining all of the contributions enumerated above, together with appropriate coupling constant and operator renormalization contributions[^3], we obtain the renormalized soft function for gluon-fusion Higgs boson production at 3lo. An important and immediate cross-check is that our explicit calculation is completely consistent with the prediction of renormalization group invariance, Eq. (\[eq:expand\]). The three-loop matching coefficient, $c^{\rm H}_3$, is most naturally written in the exponentiated form (see Eq. (\[eq:csexp\])) $$\begin{aligned}
c^{\rm H}_3 = & \frac{1}{3!}\Big(c^{\rm H}_1\Big)^3 + c^{\rm H}_1 \Delta c^{\rm H}_2 + \Delta c^{\rm H}_3\,,
\label{eq:expc3}\end{aligned}$$ where, in the CDR scheme defined above, the last term in Eq. (\[eq:expc3\]), $\Delta c^{\rm H}_3$ is given by $$\begin{aligned}
\Delta c^{\rm H}_3 = & \left(\frac{5211949}{13122} - \frac{20371\zeta_2}{729} - \frac{87052\zeta_3}{243} - \frac{9527\zeta_4}{27} - \frac{220\zeta_2\zeta_3}{9} - \frac{968\zeta_5}{9} \right.
{\nonumber}\\
&
\left. + \frac{1072\zeta_3^2}{9} + \frac{8506\zeta_6}{27} \right)C^3_A + \left( -\frac{412765}{6561} +\frac{2638\zeta_2}{729} +\frac{1216\zeta_3}{81} +\frac{928\zeta_4}{27} \right.
{\nonumber}\\
&
\left.- \frac{8\zeta_2\zeta_3}{9} -\frac{16\zeta_5}{3} \right) C_A^2 N_f + \left( -\frac{42727}{486} - \frac{55\zeta_2}{9} + \frac{2840\zeta_3}{81} + \frac{152\zeta_4}{9}\right.
{\nonumber}\\
&
\left. + \frac{16\zeta_2\zeta_3}{3} + \frac{224\zeta_5}{9} \right) C_A C_F N_f + \left(-\frac{256}{6561} - \frac{8\zeta_2}{81} + \frac{880\zeta_3}{243} + \frac{52\zeta_4}{27}\right)C_A N_f^2\,.
\label{eq:delc3}\end{aligned}$$ Eq. (\[eq:delc3\]) is one of the main results of this paper.
The soft-virtual term for Higgs production at 3lo {#sec:softvirt}
=================================================
We now have all the ingredients required to write down the 3lo soft-virtual corrections to gluon-fusion Higgs boson production. Combining Eqs. (\[eq:coef\]) and (\[eq:inverseLaplace\]), the coefficient function reads $$\begin{aligned}
\label{eq:GHiggs}
G^{\rm H}(z, L_{\rm H}) = H^{\rm H} (L_{\rm H}) ~\lim_{\eta \to 0} \bigg\{{\widetilde{S}}^{\rm \,H}\left(-\partial_\eta\right) \frac{1}{(1-z)^{1 - 2 \eta} }
\left( \frac{M^2_{\rm H}}{\mu^2} \right)^\eta \frac{ \exp (-2 \eta \gamma_E) }{\Gamma ( 2 \eta ) }\bigg\}\,.\end{aligned}$$ For simplicity, we present the result for $\mu = M_{\rm H}$, such that $L_{\rm H} = 0$. Introducing the convenient short-hand notation for the distributions which appear in the soft-virtual term, $$\begin{aligned}
\label{eq:dists}
\mathcal{D}_0 = \delta( 1 - z) \qquad {\rm and} \qquad \mathcal{D}_i = \left[ \frac{\ln^{i-1}( 1 - z)}{1 - z}\right]_+ \qquad \forall \, i > 0 \, ,\end{aligned}$$ we have $$\begin{aligned}
G^{\rm H}(z, 0) = & \dda + a \bigg\{ \bigg[ 16 \ddc + 8 \zeta_2 \dda\bigg] C_A \bigg\}
+ a^2 \bigg\{ \bigg[ 128 \dde - \frac{176}{3} \ddd + \left( \frac{1072}{9} - 160 \zeta_2 \right) \ddc
{\nonumber}\\
&
+ \left( -\frac{1616}{27} + \frac{176\zeta_2}{3} + 312 \zeta_3 \right) \ddb + \left( 93 + \frac{536\zeta_2}{9} - \frac{220\zeta_3}{3} - 2 \zeta_4 \right) \dda \bigg] C_A^2
{\nonumber}\\
&
+ \bigg[ \frac{32}{3} \ddd - \frac{160}{9} \ddc + \left( \frac{224}{27} - \frac{32\zeta_2}{3} \right) \ddb + \left( - \frac{80}{3} - \frac{80\zeta_2}{9} - \frac{8\zeta_3}{3} \right) \dda \bigg] C_A N_f
{\nonumber}\\
&
+ \bigg[ \left( -\frac{67}{3} + 16 \zeta_3 \right) \dda \bigg] C_F N_f \bigg\} + a^3 \bigg\{ \bigg[ 512 \ddg - \frac{7040}{9} \ddf + \left( \frac{59200}{27} - 3584 \zeta_2 \right) \dde
{\nonumber}\\
&
+ \left( -\frac{67264}{27} + \frac{11968}{3} \zeta_2 + 11584 \zeta_3 \right) \ddd + \left(\frac{244552}{81} - \frac{9728\zeta_2}{3} - \frac{22528\zeta_3}{3} \right.
{\nonumber}\\
&
\left. - 4928\zeta_4 \vphantom{-\frac{67264}{27}}\right) \ddc + \left( -\frac{594058}{729}+ \frac{137008\zeta_2}{81} + \frac{143056\zeta_3}{27} + \frac{2024\zeta_4}{3} - \frac{23200\zeta_2 \zeta_3}{3} \right.
{\nonumber}\\
&
\left. + 11904 \zeta_5 \vphantom{-\frac{594058}{729}}\right) \ddb + \left(\frac{215131}{81} + \frac{64604\zeta_2}{81} - 3276\zeta_3 - \frac{30514\zeta_4}{27} + \frac{7832\zeta_2 \zeta_3 }{3} - \frac{30316\zeta_5}{9} \right.
{\nonumber}\\
&
\left. + \frac{13216\zeta_3^2 }{3} - \frac{8012\zeta_6}{3} \vphantom{-\frac{594058}{729}}\right) \dda \bigg]C_A^3 + \bigg[ \frac{1280}{9} \ddf - \frac{10496}{27} \dde + \left(\frac{14624}{27} - \frac{2176\zeta_2}{3} \right) \ddd
{\nonumber}\\
&
+ \left( -\frac{67376}{81} +\frac{6016\zeta_2}{9} + \frac{2944\zeta_3}{3}\right) \ddc + \left( \frac{125252}{729} - \frac{34768\zeta_2}{81} - \frac{7600\zeta_3}{9} - \frac{272\zeta_4}{3}\right) \ddb
{\nonumber}\\
&
+ \left(-\frac{98059}{81} -\frac{4480\zeta_2}{81} + \frac{9848\zeta_3}{27} + \frac{10516\zeta_4}{27}- \frac{2000\zeta_2\zeta_3}{3} + \frac{6952\zeta_5}{9}\right) \dda \bigg] C_A^2 N_f
{\nonumber}\\
&
+ \bigg[ 32 \ddd + \Big( -504 + 384 \zeta_3 \Big) \ddc + \left(\frac{3422}{27} - 32\zeta_2 - \frac{608\zeta_3}{9} - 32 \zeta_4 \right) \ddb
{\nonumber}\\
&
+ \left( -\frac{63991}{81} - \frac{1136\zeta_2}{9} + 416 \zeta_3 + \frac{88\zeta_4}{9} + 192\zeta_2 \zeta_3 + 160\zeta_5 \right) \dda \bigg] C_A C_F N_f
{\nonumber}\\
&
+ \bigg[\left( \frac{608}{9} + \frac{592\zeta_3}{3} - 320 \zeta_5 \right) \dda\bigg] C_F^2 N_f + \bigg[ \frac{256}{27} \dde - \frac{640}{27} \ddd + \left( \frac{1600}{81} - \frac{256\zeta_2}{9} \right) \ddc
{\nonumber}\\
&
+ \left( -\frac{3712}{729}+ \frac{640\zeta_2}{27} + \frac{320\zeta_3}{27} \right)\ddb + \left( \frac{2515}{27} - \frac{2128\zeta_2}{81} + \frac{688\zeta_3 }{27} -\frac{304\zeta_4}{9} \right) \dda \bigg]C_A N_f^2
{\nonumber}\\
&
+ \bigg[\left( \frac{8962}{81} -\frac{184\zeta_2}{9} - \frac{224\zeta_3 }{3} - \frac{16\zeta_4}{9} \right) \dda\bigg] C_F N_f^2 \bigg\} \, .
\label{eq:Hsv}\end{aligned}$$ Remarkably, our calculation agrees completely both with the tower of plus distributions [@hep-ph/0508265; @hep-ph/0508284; @hep-ph/0512249; @hep-ph/0603041; @hep-ph/0605068; @0809.4283] and with the delta function terms computed for the first time in reference [@1403.4616]. It should be stressed that, at every stage of the calculation, different computer codes and computational techniques were used by the authors of [@1403.4616]. Therefore, our analysis constitutes a highly non-trivial and important confirmation of their result.
The soft-virtual term for Drell-Yan production at 3lo {#sec:DY}
=====================================================
It turns out that, once the three-loop soft function for gluon-fusion Higgs boson production is known, it is not difficult to obtain the 3lo soft-virtual corrections for Drell-Yan lepton production. First of all, the Drell-Yan hard function is known to $\Ord\left(a^3\right)$ and one can read off its one-, two-, and three-loop perturbative coefficients from Eqs. (7.3), (7.4), (7.5), and (7.8) of reference [@1004.3653]. The one- and two-loop perturbative coefficients of the Drell-Yan soft function were computed long ago in reference [@hep-ph/9808389] and, therefore, the three-loop soft function for the Drell-Yan process is the only non-trivial ingredient that remains. It was observed in the QCD literature some time ago that the soft limit of the Drell-Yan production process can be determined from the analogous result for Higgs boson production [@hep-ph/0512249; @hep-ph/0603041]. Here, we present the analysis in the framework of SCET.
The first three perturbative coefficients of the Drell-Yan soft function can be obtained from the analogous results for Higgs boson production by making the replacements $$\begin{gathered}
M_{\rm H} \to M_{\gamma^*}
\qquad\qquad
{\Gamma_{\rm cusp}}^{\rm H} (a) \to \frac{C_F}{C_A} {\Gamma_{\rm cusp}}^{\rm H} (a)
{\nonumber}\\
{\gamma_{s}}^{\rm H} (a) \to \frac{C_F}{C_A} {\gamma_{s}}^{\rm H} (a)
\qquad\qquad
c_s^{\rm H}(a) \to \left(c_s^{\rm H}(a)\right)^{\frac{C_F}{C_A}}
\label{eq:replacement}\end{gathered}$$ everywhere in the results for Higgs production presented in Eq. (\[eq:expand\]).[^4] In the above, $M_{\gamma^*}$ is the invariant mass of the off-shell photon intermediate state in the classical Drell-Yan process. It becomes clear now why it is useful to define $c_s^{\rm H}(a)$ as in Eq. (\[eq:csexp\]): although the soft matching coefficients $c_n^{\rm H}$ and $c_n^{\rm DY}$ need not be directly related, subtracting exponentiated lower-order contributions reveals a simple Casimir scaling relation between the remaining pieces, $\Delta c_n^{\rm H}$ and $\Delta c_n^{\rm DY}$, at least for $n \leq 3$.
In analogy to Eq. (\[eq:GHiggs\]), we have $$\begin{aligned}
G^{\rm DY}(z, L_{\rm DY}) = H^{\rm DY} (L_{\rm DY}) ~\lim_{\eta \to 0} \bigg\{{\widetilde{S}}^{\rm \,DY}\left(-\partial_\eta\right) \frac{1}{(1-z)^{1 - 2 \eta} }
\left( \frac{M_{\gamma^*}^2}{\mu^2} \right)^\eta \frac{ \exp ( -2 \eta \gamma_E) }{\Gamma ( 2 \eta ) }\bigg\}\,,\end{aligned}$$ where $L_{\rm DY} = \ln(\mu^2/M_{\gamma^*}^2)$. Going through the steps described above, we obtain the 3lo soft-virtual corrections for Drell-Yan lepton production. For simplicity, we present the result for $\mu = M_{\gamma^*}$, such that $L_{\rm DY} = 0$. Explicitly, we have $$\begin{aligned}
G^{\rm DY}(z, 0) = & \dda + a \bigg\{ \bigg[ 16 \ddc + \Big( -16 + 8 \zeta_2 \Big) \dda\bigg] C_F \bigg\} + a^2 \bigg\{ \bigg[ 128 \dde + \Big( -256 - 128\zeta_2\Big) \ddc
{\nonumber}\\ &
+ 256 \zeta_3 \ddb + \left( \frac{511}{4} - 70 \zeta_2 - 60 \zeta_3 + 4\zeta_4 \right) \dda \bigg] C_F^2 + \bigg[ - \frac{176}{3} \ddd + \left( \frac{1072}{9} \right.
{\nonumber}\\ &
\left. - 32 \zeta_2 \vphantom{\frac{1072}{9}}\right) \ddc + \left(-\frac{1616}{27} + \frac{176\zeta_2}{3} + 56 \zeta_3 \right) \ddb + \left( -\frac{1535}{12} + \frac{592\zeta_2}{9} + 28 \zeta_3 \right.
{\nonumber}\\ &
\left. - 6 \zeta_4 \vphantom{\frac{1535}{12}}\right) \dda \bigg] C_A C_F + \bigg[ \frac{32}{3} \ddd - \frac{160}{9} \ddc + \left( \frac{224}{27} - \frac{32\zeta_2}{3} \right) \ddb + \left(\frac{127}{6} - \frac{112\zeta_2}{9} \right.
{\nonumber}\\ &
\left. + 8\zeta_3 \vphantom{\frac{127}{6}}\right) \dda \bigg] C_F N_f \bigg\}+ a^3 \bigg\{ \bigg[ 512 \ddg + \Big( -2048 - 3072\zeta_2 \Big)\dde + 10240 \zeta_3 \ddd + \Big( 2044
{\nonumber}\\ &
+ 2976\zeta_2 - 960\zeta_3 - 7104 \zeta_4 \Big) \ddc+ \Big( -4096 \zeta_3 - 6144 \zeta_2 \zeta_3 +12288 \zeta_5 \Big) \ddb + \left( -\frac{5599}{6} \right.
{\nonumber}\\ &
\left. - \frac{130}{3}\zeta_2 - 460\zeta_3 + 206 \zeta_4 + 80 \zeta_2 \zeta_3 + 1328 \zeta_5 + \frac{10336}{3} \zeta_3^2 - \frac{23092}{9}\zeta_6 \vphantom{\frac{5599}{6}}\right) \dda \bigg] C_F^3
{\nonumber}\\ &
+ \bigg[ - \frac{7040}{9} \ddf + \left( \frac{17152}{9} - 512 \zeta_2 \vphantom{\frac{17152}{9}}\right) \dde + \left( -\frac{4480}{9} + \frac{11264\zeta_2}{3} + 1344\zeta_3 \right) \ddd
{\nonumber}\\ &
+ \left(-\frac{35572}{9} - \frac{11648}{9} \zeta_2 - 5184 \zeta_3 + 1824 \zeta_4 \right) \ddc + \left( \frac{25856}{27} - \frac{12416\zeta_2}{27} + \frac{26240\zeta_3}{9}\right.
{\nonumber}\\ &
\left. + \frac{3520\zeta_4}{3} - 1472 \zeta_2 \zeta_3 \right) \ddb + \left( \frac{74321}{36} - \frac{13186\zeta_2}{27} - \frac{20156\zeta_3}{9} - \frac{832\zeta_4}{27} + \frac{28736\zeta_2 \zeta_3}{9} \right.
{\nonumber}\\ &
\left. - \frac{39304\zeta_5}{9} + \frac{3280\zeta_3^2}{3} - \frac{2602\zeta_6}{9} \right) \dda \bigg] C_A C_F^2 + \bigg[ \frac{7744}{27} \dde + \left( -\frac{28480}{27} + \frac{704\zeta_2}{3} \right) \ddd
{\nonumber}\\ &
+ \left( \frac{124024}{81} - \frac{12032\zeta_2}{9} - 704 \zeta_3 + 352 \zeta_4 \vphantom{\frac{124024}{81}}\right) \ddc + \left( -\frac{594058}{729} + \frac{98224\zeta_2}{81} + \frac{40144\zeta_3}{27} \right.
{\nonumber}\\ &
\left. - \frac{1496\zeta_4}{3} - \frac{352\zeta_2 \zeta_3}{3} - 384\zeta_5 \vphantom{\frac{594058}{729}}\right) \ddb + \left( -\frac{1505881}{972} + 843 \zeta_2 + \frac{82385\zeta_3}{81} + \frac{14611\zeta_4}{54} \right.
{\nonumber}\\ &
\left. - \frac{884\zeta_2 \zeta_3}{3} - 204 \zeta_5 - \frac{400\zeta_3^2}{3} + \frac{1658\zeta_6}{9} \right)\dda \bigg] C_A^2 C_F + \bigg[\frac{1280}{9} \ddf - \frac{2560}{9} \dde
{\nonumber}\\ &
+ \left( \frac{544}{9} - \frac{2048\zeta_2}{3} \right) \ddd + \left( \frac{4288}{9} + \frac{2048\zeta_2}{9} + 1280\zeta_3 \right) \ddc + \left(\vphantom{\frac{736\zeta_4}{3}} -6 + \frac{1952\zeta_2}{27} \right.
{\nonumber}\\ &
\left.- \frac{5728\zeta_3}{9} - \frac{736\zeta_4}{3} \right) \ddb+ \left( -\frac{421}{3} + \frac{2632\zeta_2}{27} + \frac{3512\zeta_3}{9} + \frac{136\zeta_4}{27} - \frac{5504\zeta_2 \zeta_3}{9} \right.
{\nonumber}\\ &
\left. + \frac{5536\zeta_5}{9} \right) \dda \bigg]C_F^2 N_f + \bigg[ - \frac{2816}{27} \dde + \left( \frac{9248}{27} - \frac{128\zeta_2}{3} \right) \ddd + \left( -\frac{32816}{81} \right.
{\nonumber}\\ &
\left. + 384\zeta_2 \vphantom{\frac{32816}{81}}\right) \ddc + \left( \frac{125252}{729}- \frac{29392\zeta_2}{81} -\frac{2480\zeta_3}{9} + \frac{368\zeta_4}{3} \right) \ddb + \left( \frac{110651}{243} \right.
{\nonumber}\\ &
\left. - \frac{28132\zeta_2}{81} - \frac{6016\zeta_3}{81} - \frac{2878\zeta_4}{27} + \frac{208\zeta_2 \zeta_3}{3} - 8 \zeta_5 \right) \dda \bigg] C_A C_F N_f
{\nonumber}\\ &
+ \bigg[ \frac{256}{27} \dde - \frac{640}{27} \ddd + \left( \frac{1600}{81} - \frac{256\zeta_2}{9} \right) \ddc + \left( -\frac{3712}{729} + \frac{640\zeta_2}{27} + \frac{320\zeta_3}{27} \right) \ddb
{\nonumber}\\ &
+ \left( -\frac{7081}{243} + \frac{2416\zeta_2}{81} - \frac{1264\zeta_3}{81} + \frac{320\zeta_4}{27} \right) \dda \bigg] C_F N_f^2
{\nonumber}\\ &
+ \bigg[ \left( 4 + 10 \zeta _2+\frac{14 \zeta _3}{3} - \zeta _4 - \frac{80 \zeta _5}{3} \right) \dda \bigg] \frac{d_{abc}d_{abc}}{\Nc} N_{q\gamma} \bigg\}\,,
\label{eq:DYsv}\end{aligned}$$ where $N_{q\gamma} = (1/e_q){\sum_{q^\prime} e_{q^\prime}}$ is the charge-weighted sum of the $N_f$ quark flavors normalized to the charge of the primary quark $q$ and $d_{abc}d_{abc} = (\Nc^2 - 1)(\Nc^2 - 4)/\Nc$. Our calculation agrees completely both with the well-known tower of plus distributions [@hep-ph/0508265; @hep-ph/0508284; @hep-ph/0603041; @hep-ph/0605068] and with the delta function terms predicted by two groups independently using an approach different than the one described above [@1404.0366; @1405.4827].
A 3lo soft-virtual term for the $\Neqfour$ model {#sec:SYM}
================================================
Loop corrections in $\Neqfour$ super Yang-Mills theory often have a simple structure and it is therefore interesting to consider an analog of the soft-virtual terms treated above in such a model. In fact, it is natural to consider the soft-virtual corrections in a $SU(\Nc)$ $\Neqfour$ super Yang-Mills theory for the simple case of scalar pair production because the relevant hard function has already been computed through to three-loop order [@1112.4524; @vanNeerven:1985ja]. As usual, $\Neqfour$ super Yang-Mills theory refers to a theory with a single massless $\Neqfour$ supermultiplet in the adjoint representation of $SU(N_c)$. The massless $\Neqfour$ supermultiplet contains a massless gauge field, four massless Majorana fermions, and three massless complex scalars. It therefore follows that, to obtain the desired result in the $\Neqfour$ model, we need to calculate diagrams with scalars in the loops or in the final state and appropriately adjust the color coefficients for the fermionic contributions. No additional master integrals are required beyond those which we have already calculated. It is worth emphasizing that it is inconvenient to use the CDR scheme for this calculation because it does not preserve supersymmetry. Instead, we employ the FDH scheme [@hep-ph/0202271] in which the diagrammatic numerator algebra is performed in $D_s = 4$ dimensions, while the required integral reductions are carried out in $D = 4 - 2 \e$ dimensions. As is well-known, the beta function vanishes in all $\Neqfour$ models to all orders in perturbation theory [@Sohnius:1981sn; @Howe:1983wj; @Howe:1983sr; @Brink:1982wv; @Brink:1982pd; @Lemes:2001vf]. Among other things, this implies that no additional renormalization related to so-called $\e$-scalar contributions is necessary [@1404.2171].
Following the formalism first outlined in reference [@vanNeerven:1985ja], we consider the annihilation of two scalars $\phi^{a\dagger}_r$ and $\phi^a_t$ into a color-singlet scalar current $J_{rt}$ plus anything, $$\begin{aligned}
\phi^{a\dagger}_r + \phi^a_t \to J_{rt} + X \,,\end{aligned}$$ where the color-singlet scalar current reads $$\begin{aligned}
J_{rt} = \frac{1}{2} ( \phi^{a\dagger}_r \phi^a_t + \phi^{a\dagger}_t \phi^a_r ) - \frac{1}{3} \delta_{rt} \phi^{a\dagger}_s \phi^a_s \,.
\label{eq:scalarcurrent}\end{aligned}$$ Along the lines described above, we compute the relevant three-loop soft function and see that, given an appropriate definition of transcendentality weight for the distributions that appear in the result, the soft function obeys a so-called maximal transcendentality weight principle.
In order to discuss the observed correspondence, let us first define the notion of transcendentality weight for the relevant terms. As is well-known, the transcendentality weight is $0$ for rational numbers, $1$ for a logarithm, and $i$ for $\zeta_i$. The weight of a product of these quantities is the sum of the weights of the individual factors in the product. Taking into account the integration to be performed over $z$, we associate to the distributions introduced in Eqs. (\[eq:dists\]), $\mathcal{D}_i$, the transcendentality weight $i$. Given this definition, the three-loop soft function for scalar pair production in $SU(\Nc)$ $\Neqfour$ super Yang-Mills coincides with the part of the three-loop Higgs boson production soft function which is of maximal transcendentality weight (six in this case). In fact, each of the contributions enumerated at the beginning of Section \[sec:softHiggs\] separately satisfies this maximal transcendentality weight principle, similar and yet distinct from the original principle observed long ago in the context of the anomalous dimensions of $\Neqfour$ twist-two operators [@hep-th/0404092]. In the $\Neqfour$ theory, similar observations have subsequently been made for form factors [@1112.4524; @1201.4170; @1203.0454] and for the tower of plus distributions in the soft-virtual term [@hep-ph/0508265]. Our analysis extends these observations to the complete set of soft-virtual corrections at 3lo. Combining all of the relevant ingredients together, we find $$\begin{aligned}
G^{\Neqfour}(z, 0) = &\dda + a \bigg\{ \bigg[ 16 \ddc + 8 \zeta_2 \dda\bigg] C_A \bigg\} + a^2 \bigg\{ \bigg[ 128 \dde - 160 \zeta_2 \ddc + 312 \zeta_3 \ddb - 2 \zeta_4 \dda \bigg] C_A^2\bigg\}
{\nonumber}\\ &
+ a^3 \bigg\{ \bigg[ 512 \ddg - 3584 \zeta_2 \dde + 11584 \zeta_3 \ddd - 4928 \zeta_4 \ddc
{\nonumber}\\ &
+ \left(- \frac{23200 \zeta_2 \zeta_3}{3} + 11904 \zeta_5 \right) \ddb + \left( \frac{13216 \zeta_3^2}{3} - \frac{8012 \zeta_6}{3} \right) \dda \bigg] C_A^3\bigg\}
\label{eq:Neqfour}\end{aligned}$$ for the soft-virtual corrections to scalar pair production in $SU(\Nc)$ $\Neqfour$ super Yang-Mills theory. One can explicitly check by comparing Eqs. (\[eq:Hsv\]) and (\[eq:Neqfour\]) that the principle of maximal transcendentality weight is satisfied.
Summary and Outlook {#sec:conclude}
===================
In this work, we presented the 3lo soft-virtual corrections to both gluon-fusion Higgs boson production and Drell-Yan lepton production in QCD and to color-singlet scalar pair production in $SU(\Nc)$ $\Neqfour$ super Yang-Mills theory. Our main results are given in Eqs. (\[eq:delc3\]), (\[eq:Hsv\]), (\[eq:DYsv\]), and (\[eq:Neqfour\]). In particular, we found results in full agreement with a calculation of the Higgs soft-virtual term performed recently by a different group [@1403.4616]. By performing a Casimir scaling of the Higgs result, we reproduced the 3lo Drell-Yan soft-virtual term derived in earlier work [@1404.0366; @1405.4827]. Finally, we observed a maximal transcendentality weight principle whereby one can predict the complete result for color-singlet scalar pair production in the $\Neqfour$ theory from the part of the soft-virtual term for Higgs production which is of maximal transcendentality weight.
The 3lo Higgs soft-virtual term presented in this paper provides an important reference point for the full fixed-order calculation. In light of recent work on the subject [@1411.3584; @1405.4827; @1405.5685; @1408.6277; @1405.3654], it is not clear that N$^3$LL soft-gluon resummation alone will improve the prediction of QCD perturbation theory to the required extent. Therefore, a full three-loop, fixed-order calculation of the partonic cross section is highly desirable and should be carried out in the near future in order to obtain a full N$^3$LO+N$^3$LL QCD prediction for the gluon-fusion Higgs boson production cross section.
0.5cm [**Acknowledgments**]{} 0.3cm YL and HXZ thank Lance Dixon, Michael Peskin, and Stefan Höche for useful discussions. HXZ thanks Adrian Signer for insightful comments on the use of the FDH scheme. YL and HXZ would also like to thank the Institute of High Energy Physics in China for hospitality while part of this work was carried out. Our figure was generated using [Jaxodraw]{} [@hep-ph/0309015], based on [AxoDraw]{} [@CPHCB.83.45]. The research of YL and HXZ is supported by the US Department of Energy under contract DE-AC02-76SF00515. The research of RMS is supported in part by the ERC Advanced Grant EFT4LHC of the European Research Council, the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA-EXC 1098).
Anomalous dimensions and beta function coefficients {#sec:append}
===================================================
In this appendix, we present the anomalous dimensions and beta function coefficients relevant to the calculation of the terms in the Laplace-transformed soft function for gluon-fusion Higgs boson production which are fixed by RG invariance. Through to three loops, the perturbative coefficients of the cusp anomalous dimension, ${\Gamma_{\rm cusp}}^{\rm H}(a)$, are given by [@hep-ph/0403192] $$\begin{aligned}
\Gamma^{\rm H}_0 = & \,4 C_A
\\
\Gamma^{\rm H}_1 = & \left( \frac{268}{9} - 8 \zeta_2 \right) C_A^2 - \frac{40}{9} C_A N_f
\\
\Gamma^{\rm H}_2 = & \left( \frac{490}{3} - \frac{1072\zeta_2}{9} + \frac{88\zeta_3}{3} + 88 \zeta_4 \right) C^3_A + \left( -\frac{836}{27} + \frac{160\zeta_2 }{9}- \frac{112\zeta_3}{3} \right)C_A^2 N_f
{\nonumber}\\ &
+ \left( - \frac{110}{3} + 32 \zeta_3 \right)C_A C_F N_f - \frac{16}{27} C_A N_f^2\,.\end{aligned}$$
The perturbative coefficients of the soft anomalous dimension, ${\gamma_{s}}^{\rm H}(a)$, can be extracted from the pole terms of the perturbative coefficients of the gluon form factor [@hep-ph/0508055]. Through to three loops, they read $$\begin{aligned}
\gamma_0^{\rm H} = & \,0
\\
\gamma_1^{\rm H} = & \left( -\frac{808}{27} + \frac{22\zeta_2}{3} + 28 \zeta_3 \right)C_A^2 + \left( \frac{112}{27} - \frac{4 \zeta_2}{3} \right) C_A N_f
\\
\gamma_2^{\rm H} = & \left( - \frac{136781}{729} + \frac{12650\zeta_2}{81} + \frac{1316\zeta_3}{3} - 176 \zeta_4 - \frac{176\zeta_2 \zeta_3}{3} - 192 \zeta_5 \right) C_A^3
{\nonumber}\\ &
+\left( \frac{11842}{729} - \frac{2828\zeta_2}{81} - \frac{728\zeta_3}{27} + 48\zeta_4 \right) C_A^2 N_f + \left( \frac{1711}{27} - 4 \zeta_2 - \frac{304\zeta_3}{9} - 16\zeta_4 \right) C_A C_F N_f
{\nonumber}\\ &
+ \left( \frac{2080}{729} + \frac{40\zeta_2}{27} - \frac{112\zeta_3}{27} \right) C_A N_f^2\, .\end{aligned}$$
Finally, the first two coefficients of the QCD beta function are required. They are given by (see [*e.g.*]{} [@hep-ph/9701390]) $$\begin{aligned}
\beta_0 = &\frac{11}{3} C_A - \frac{2}{3} N_f
\\
\beta_1 = & \frac{34}{3} C_A^2 - \frac{10}{3} C_A N_f - 2 C_F N_f\, .\end{aligned}$$
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[^1]: We invite readers unfamiliar with SCET to consult reference [@1410.1892] for a pedagogical introduction.
[^2]: The number of gluonic polarization states is $2 - 2 \e$ in the CDR scheme, and $2$ in the FDH scheme.
[^3]: See [*e.g.*]{} Section II of reference [@1408.5134] for a discussion of the appropriate multiplicative renormalization.
[^4]: It is not known whether these replacement rules continue to hold at higher orders in QCD perturbation theory.
|
---
abstract: 'The aim of this paper is to study linear preservers of the trace of Kronecker sums $A\oplus B$ and their connection with preservers of determinants of Kronecker products. The partial trace and partial determinant play a fundamental role in characterizing the preservers of the trace of Kronecker sums and preservers of the determinant of Kronecker products respectively.'
address:
- |
School of Mathematics,\
University of the Witwatersrand,\
Johannesburg,\
Private Bag 3,\
Wits 2050,\
South Africa
- |
Faculty of Management,\
University of Primorska,\
Cankarjeva 5,\
SI-6000 Koper,\
Slovenia
author:
- Yorick Hardy
- Ajda Fošner
bibliography:
- 'ksumlinpreserve.bib'
title: Preserving the trace of the Kronecker sum
---
Introduction
============
For positive integers $m,n>2$, let $M_n$ be the algebra of all $n\times n$ matrices over some field ${\mathbb{F}}$ and let $M_{mn} = M_m\otimes M_n = M_m(M_n)$. Here we consider $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$ and define $H_n\subset M_n$ to be the Hermitian matrices in $M_n$. Linear maps preserving properties of Kronecker products of matrices have received considerable attention in recent years. Such maps are closely connected to quantum information science (see, e.g., [@AF]). More recently, Ding et. al. considered linear preservers of determinants of Kronecker products of Hermitian matrices [@ding17], i.e., linear maps $\phi:H_{mn}\to H_{mn}$ satisfying $$\det(\phi(A\otimes B)) = \det(A\otimes B)$$ where $A$ and $B$ are Hermitian. A few of the results in [@ding17] are restricted to the case when $A$ and $B$ are positive or negative semidefinite matrices. In order to study this problem more generally, we make use of the identity $$\det\left(e^A\otimes e^B\right) = e^{\operatorname{tr}(A\oplus B)}$$ where $\oplus$ is the Kronecker sum, i.e., $$A\oplus B := A\otimes I_n + I_m\otimes B,$$ where $I_k$, $k=m,n$ denotes the $k\times k$ identity matrix. Under exponentiation of Hermitian matrices, the Kronecker sum arises naturally as the unique map $\oplus: H_m\times H_n\to H_m\otimes H_n$ satisfying $$e^{A\oplus B} = e^A\otimes e^B, \qquad A\in H_m,\,B\in H_n.$$ Moreover, $A\in M_n$ over $\mathbb{C}$ is non-singular if and only if $A=e^B$ for some $B\in M_n$ [@horn91 Example 6.2.15]. Thus, in studying linear maps $\psi:M_{mn}\to M_{mn}$ preserving determinants of (non-singular) Kronecker products $$\det(\psi(A\otimes B)) = \det(A\otimes B)$$ of non-singular matrices $A$ and $B$, it suffices to study maps $\phi:M_{mn}\to M_{mn}$ preserving the trace of Kronecker sums $$\label{eq:pr} \operatorname{tr}(\phi(A\oplus B)) = \operatorname{tr}(A\oplus B).$$ In this article we will confine our attention to linear maps $\phi$ satisfying . We denote by $GL_n({\mathbb{F}})$ and $SL_n({\mathbb{F}})$ the general and special linear groups in $M_n$ respectively. Thus, we are also interested in preservers of the determinant of Kronecker product in $GL_{mn}({\mathbb{F}})$ in terms of linear preservers of the Kronecker sum.
In what follows, $m,n$ are positive integers. For a positive integer $k$, $I_k$ denotes the $k\times k$ identity matrix, $0_k$ the $k\times k$ zero matrix, and $E_{ij}^{(k)}$, $1\le i,j\le k$, the $k\times k$ matrix whose entries are all equal to zero except for the $(i,j)$-th entry which is equal to one. As usual, the symbol $\delta _{ij}$ denotes the Kronecker delta, i.e., $$\delta_{ij} =
\begin{cases}
1, & \textrm{if $i=j$}~;\\
0, & \textrm{if $i\ne j$}~.
\end{cases}$$
The partial trace plays a central role in our investigation. Let $A\in M_{mn} = M_m(M_n) = M_m\otimes M_n$. Then $A$ can be written as a block matrix $A = (A_{ij})_{ij}$ where $A_{ij}\in M_n$ and $i,j=1,\ldots, m$. In terms of the Kronecker product we write $$A = \sum_{i,j=1}^m E_{i,j}^{(m)}\otimes A_{ij}.$$ The second partial trace ($\operatorname{tr}_2$) maps each $n\times n$ block of $A$ to its trace, i.e., $\operatorname{tr}_2:(A_{ij})_{ij}\mapsto (\operatorname{tr}(A_{ij}))_{ij}$, or equivalently $$\operatorname{tr}_2(A) = \sum_{i,j=1}^m E_{i,j}^{(m)}\otimes \operatorname{tr}(A_{ij}) = \sum_{i,j=1}^m \operatorname{tr}(A_{ij})E_{i,j}^{(m)}.$$ Like the trace, the partial trace is a linear operation. Furthermore, the partial trace preserves the trace $$\operatorname{tr}(A) = \operatorname{tr}(\operatorname{tr}_2(A)).$$ Similarly, we can define the first partial trace ($\operatorname{tr}_1$). This definition provides $$\operatorname{tr}_1(A) = \sum_{j=1}^m A_{jj}.$$ Finally, we note that for any $B\in M_n$ $$\operatorname{tr}_1(A(I_m\otimes B))
= \sum_{j=1}^m A_{jj}B = \operatorname{tr}_1(A)B$$ and similarly for any $C\in M_m$ $$\operatorname{tr}_2(A(C\otimes I_n))
= \operatorname{tr}_2(A)C.$$ The transpose of the matrix $A\in M_n$ will be denoted by $A^T$. First, we define RT-symmetry, which plays a similar role in our analysis similar to that of symmetry in matrix analysis.
RT-Symmetry
===========
Noting that $M_n$ is an $n^2$-dimensional space, in general we have $$\phi(A) = \sum_{j,k,u,v=1}^n \alpha_{jk;uv}E_{jk}^{(n)}AE_{uv}^{(n)}$$ for some $\alpha_{jk;uv}$ in the underlying field. We define the linear transform $\phi'$ of $\phi$ by $$\phi'(A) := \sum_{j,k,u,v=1}^n \alpha_{uv;jk}E_{jk}^{(n)}AE_{uv}^{(n)}
\equiv \sum_{j,k,u,v=1}^n \alpha_{jk;uv}E_{uv}^{(n)}AE_{jk}^{(n)}.$$ Clearly, $(\phi')'=\phi$. Let $\Phi$ be the matrix representing the linear map $\phi$ in the standard basis, and $\Phi'$ be the matrix representing $\phi'$. Since $$\phi(E_{pq}^{(n)}) = \sum_{j,v=1}^n \alpha_{jp;qv}E_{jv}^{(n)}$$ it follows that $$\Phi = \sum_{j,p,q,v=1}^n\alpha_{jp;qv}E_{jp}^{(n)}\otimes E_{vq}^{(n)}, \quad
\Phi' = \sum_{q,k,u,p=1}^n\alpha_{qk;up}E_{up}^{(n)}\otimes E_{kq}^{(n)}.$$ Let $P$ denote the perfect shuffle (also known as the vec-permutation matrix) on $\mathbb{F}^n\otimes\mathbb{F}^n$ [@henderson81a; @vanloan00a], i.e. $P^T(A\otimes B)P=B\otimes A$. Then $$\Phi^T = P^T\Phi' P.$$ Consequently, $\phi=\phi'$ if and only if $\Phi^T=P^T\Phi P$. Equivalently, $\phi=\phi'$ if and only if $R(\Phi)^T = R(\Phi^T)$ where $R$ is the rearrangement operator [@vLP]. The rearrangement operator $R$ is linear, and defined by $R(A\otimes B)=(\operatorname{vec}A)(\operatorname{vec}B)^T$ where $\operatorname{vec}$ is the vec operator [@henderson81a].
A linear map $\phi:M_n\to M_n$ satisfying $\phi=\phi'$ is said to be *RT-symmetric*. If $\phi=-\phi'$ then $\phi$ is said to be *skew RT-symmetric*.
The following lemma follows immediately from $\phi=\frac12(\phi+\phi')+\frac12(\phi-\phi')$.
If the underlying field has characteristic not equal to 2, then every linear map $\phi:M_n\to M_n$ is the sum of an RT-symmetric map and a skew RT-symmetric map.
A linear map $\phi:M_n\to M_n$, over $\mathbb{C}$, satisfying $\phi=\overline{\phi'}$ is said to be *RT-Hermitian*. If $\phi=-\overline{\phi'}$ then $\phi$ is said to be *skew RT-Hermitian*.
Linear trace preservers of Kronecker sums
=========================================
We may write a linear map $\phi:M_{mn}\to M_{mn}$ in the operator-sum form $$\phi(M) = \sum_{i=1}^r P_iMQ_i$$ for some matrices $P_i,Q_i$, $i=1,\ldots, r$, of the appropriate sizes. If $\phi$ preserves the trace of a Kronecker sum $M$, then the cyclic property of the trace yields $$\operatorname{tr}(M) = \operatorname{tr}(\phi(M)) = \sum_{i=1}^r \operatorname{tr}(Q_iP_iM)$$ and so we need only consider preservers of the form $$\phi(M) = \sum_{i=1}^r Q_iP_iM = PM,$$ where $$\label{eq:repr} P := \sum_{i=1}^r Q_iP_i$$ and the remaining preservers are all obtained by representations of $P$. First we consider maps of the form $\phi(M)=PM$, $M\in M_{mn}$.
\[thm:mainpart\] Let $\phi:M_{mn}\to M_{mn}$ be a map given by $\phi:M\mapsto PM$ for some $P\in M_{mn}$. Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$, if and only if $$\operatorname{tr}_1(P)=\operatorname{tr}_1(I_{mn}) \quad\text{and}\quad \operatorname{tr}_2(P)=\operatorname{tr}_2(I_{mn}).$$
First, let us write $P$ in block matrix form, $P=(P_{kl})$ where each $P_{kl}\in M_n$ for $k,l=1,\ldots,m$. In other words, $$P = \sum_{k,l=1}^m E_{kl}^{(m)}\otimes P_{kl}.$$ Since $\phi$ is linear, the map $\phi$ preserves the trace of Kronecker sums if and only if $\phi$ preserves traces of Kronecker products of the form $E_{ij}^{(m)}\otimes I_n$ and of the form $I_m\otimes E_{kl}^{(n)}$. Thus, we have $$n\delta_{ij} = \operatorname{tr}(E_{ij}^{(m)}\otimes I_n)
= \operatorname{tr}(\phi(E_{ij}^{(m)}\otimes I_n))$$ and $$\operatorname{tr}(\phi(E_{ij}^{(m)}\otimes I_n))
= \sum_{kl=1}^m\delta_{jk}\delta_{il}\operatorname{tr}(P_{kl})
= \operatorname{tr}(P_{ij}).$$ It follows that $$\operatorname{tr}_2(P) = \sum_{k,l=1}^m \operatorname{tr}(P_{kl})E_{kl}^{(m)} = nI_m.$$ For Kronecker products of the form $I_m\otimes E_{kl}^{(n)}$, we find $$m\delta_{kl} = \operatorname{tr}(I_m\otimes E_{kl}^{(n)})
= \operatorname{tr}(\phi(I_m\otimes E_{kl}^{(n)})),$$ where $$\operatorname{tr}(\phi(I_m\otimes E_{kl}^{(n)}))
= \sum_{i,j=1}^m \delta_{ij}\operatorname{tr}(P_{ij}E_{kl}^{(n)})
= \sum_{j=1}^m(P_{jj})_{lk}.$$ Consequently, $$\operatorname{tr}_1(P) = \sum_{i,j=1}^m P_{jj} = mI_n.$$ Conversely, suppose that $\operatorname{tr}_1(P) = mI_n$ and $\operatorname{tr}_2(P) = nI_m$. Then $$\begin{aligned}
\operatorname{tr}(\phi(A\oplus B))
&= \operatorname{tr}(\operatorname{tr}_2(P(A\otimes I_n))) + \operatorname{tr}(\operatorname{tr}_1(P(I_m\otimes B))) \\
&= \operatorname{tr}(\operatorname{tr}_2(P)A) + \operatorname{tr}(\operatorname{tr}_1(P)B) = n\operatorname{tr}(A) + m\operatorname{tr}(B) = \operatorname{tr}(A\oplus B).
\qedhere
\end{aligned}$$
Let $\phi:M_{mn}\to M_{mn}$ be a map given by $\phi:M\mapsto PM$ for some $P\in M_{mn}$, where $$P = I_{mn} + \sum_{j=1}^r A_j\otimes B_j.$$ Here, $r$ is the tensor rank of $P-I_{mn}$ over $M_m\otimes M_n$ and $A_j\in M_m$, $B_j\in M_n$ for $j=1,\ldots,r$. Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$, if and only if $\operatorname{tr}(A_j)=\operatorname{tr}(B_j)=0$ for $j=1,\ldots,r$.
By theorem \[thm:mainpart\] we need only show that $\operatorname{tr}_1(P)=mI_n$ and $\operatorname{tr}_2(P)=nI_m$ if and only if $\operatorname{tr}(A_j)=\operatorname{tr}(B_j)=0$ for $j=1,\ldots,r$. The proof of $(\Leftarrow)$ is immediate. For $(\Rightarrow)$, suppose $\operatorname{tr}_1(P)=mI_n$ and $\operatorname{tr}_2(P)=nI_m$. It follows that $$\sum_{j=1}^r \operatorname{tr}(A_j)B_j = 0_n, \qquad
\sum_{j=1}^r \operatorname{tr}(B_j)A_j = 0_m.$$ Since $r$ is the tensor rank of $P-I_{mn}$, the set $\{\,B_1,\ldots,\, B_r\}$ is a linearly independent set and $\operatorname{tr}(A_j)=0$ for $j=1,\ldots,r$. Similarly, $\operatorname{tr}(B_j)=0$ for $j=1,\ldots,r$.
As a consequence of Theorem \[thm:mainpart\], we have that $\phi:M\mapsto PM$ satisfies $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ if and only if $\operatorname{tr}_1(\phi(I_{mn})) = \operatorname{tr}_1(I_{mn})$ and $\operatorname{tr}_2(\phi(I_{mn})) = \operatorname{tr}_2(I_{mn})$. In general, this statement is true modulo a traceless matrix. We note that any linear map $\phi:M_{mn}\to M_{mn}$ can be written in the form $$\phi(M) = M + \sum_{j=1}^r (A_j\otimes C_j)M(B_j\otimes D_j),$$ where $A_j,B_j\in M_m$ and $C_j,D_j\in M_n$ for $j=1,\ldots,r$. In the following we will use the commutation operation $[A,B]=AB-BA$ corresponding to the Lie product of matrices $A$ and $B$ of the appropriate sizes.
\[lem:ident\] Let $\phi:M_{mn}\to M_{mn}$ be a linear map given by $$\phi(M) = M + \sum_{j=1}^r (A_j\otimes C_j)M(B_j\otimes D_j).$$ Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$, if and only if $$\operatorname{tr}_1(\phi(I_{mn})-I_{mn})
= \sum_{j=1}^r \operatorname{tr}(A_jB_j) [C_j,D_j]$$ and $$\operatorname{tr}_2(\phi(I_{mn})-I_{mn})
= \sum_{j=1}^r \operatorname{tr}(C_jD_j) [A_j,B_j].$$
The linear map $\phi:M_{mn}\to M_{mn}$ can be written in the form $$\phi(M) = M + \sum_{j=1}^r (A_j\otimes C_j)M(B_j\otimes D_j)$$ where $A_j,B_j\in M_m$ and $C_j,D_j\in M_n$ for $j=1,\ldots,r$. Since $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ if and only if $\operatorname{tr}\phi(A\otimes I_n) = \operatorname{tr}(A\otimes I_n)$ and $\operatorname{tr}\phi(I_m\otimes B) = \operatorname{tr}(I_m\otimes B)$ for all $A\in M_m$ and $B\in M_n$, we consider these two cases separately. In the first case we have $$\operatorname{tr}\phi(A\otimes I_n) = n\operatorname{tr}(A) + \operatorname{tr}\left(\left(\sum_{j=1}^r \operatorname{tr}(C_jD_j) B_jA_j\right)A\right)
= n\operatorname{tr}(A)$$ for all $A\in M_m$. This equation holds if and only if $$\begin{aligned}
0_m &= \sum_{j=1}^r \operatorname{tr}(C_jD_j) B_jA_j = \sum_{j=1}^r \operatorname{tr}(C_jD_j) ([B_j,A_j] + A_jB_j) \\
&= -Q + \sum_{j=1}^r \operatorname{tr}(C_jD_j)A_jB_j \\
&= -Q + \operatorname{tr}_2\phi(I_{mn}) - \operatorname{tr}_2 I_{mn}
\end{aligned}$$ where $[A,B]:=AB-BA$ is the commutator and $$Q := \sum_{j=1}^r \operatorname{tr}(C_jD_j) [A_j,B_j] = \operatorname{tr}_2\phi(I_{mn}) - \operatorname{tr}_2 I_{mn}$$ is traceless (i.e., $\operatorname{tr}(Q)=0$). Similarly, the second case yields that $\operatorname{tr}\phi(I_m\otimes B) = \operatorname{tr}(I_m\otimes B)$ if and only if $$\sum_{j=1}^r \operatorname{tr}(A_jB_j) [C_j,D_j] = \operatorname{tr}_1\phi(I_{mn}) - \operatorname{tr}_1 I_{mn}.
\qedhere$$
The commutators in this lemma highlight the traceless character. However, the anti-commutator plays a similar role. Here, the anti-commutator of matrices $A$ and $B$ is given by $[A,B]_+=AB+BA$. We state the following lemma without proof, which is almost identical to the previous.
\[lem:ac\] Let $\phi:M_{mn}\to M_{mn}$ be a linear map given by $$\phi(M) = M + \sum_{j=1}^r (A_j\otimes C_j)M(B_j\otimes D_j).$$ Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$, if and only if $$\operatorname{tr}_1(\phi(I_{mn})-I_{mn})
= \sum_{j=1}^r \operatorname{tr}(A_jB_j) [C_j,D_j]_+$$ and $$\operatorname{tr}_2(\phi(I_{mn})-I_{mn})
= \sum_{j=1}^r \operatorname{tr}(C_jD_j) [A_j,B_j]_+.$$
Lemma \[lem:ident\] shows that the partial traces of the identity matrix must be preserved modulo a traceless matrix. However, this traceless matrix is not arbitrary but precisely defined in terms of $\phi$. The following theorem shows that $\phi'$ plays a fundamental role in the characterization of $\phi$, and provides an succint characterization for RT-symmetric and skew RT-symmetric maps in the subsequent two corollaries.
\[thm:ident\] Let $\phi:M_{mn}\to M_{mn}$ be a linear map. Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$ if and only if $$\operatorname{tr}_1\phi'(I_{mn}) = \operatorname{tr}_1(I_{mn})
\quad\text{and}\quad
\operatorname{tr}_2\phi'(I_{mn}) = \operatorname{tr}_2(I_{mn}).$$
Using the representation of $\phi$ from Lemma \[lem:ident\] provides $$\begin{aligned}
\operatorname{tr}_1(\phi(I_{mn})) &= \operatorname{tr}_1 I_{mn} + \sum_{j=1}^r \operatorname{tr}(A_jB_j) C_jD_j, \\
\operatorname{tr}_1(\phi'(I_{mn})) &= \operatorname{tr}_1 I_{mn} + \sum_{j=1}^r \operatorname{tr}(B_jA_j) D_jC_j
\end{aligned}$$ and subtracting these two equations yields $$\operatorname{tr}_1((\phi-\phi')(I_{mn})) = \sum_{j=1}^r \operatorname{tr}(A_jB_j)[C_j,D_j].$$ Similarly, $$\operatorname{tr}_2((\phi-\phi')(I_{mn})) = \sum_{j=1}^r \operatorname{tr}(C_jD_j)[A_j,B_j].$$ From Lemma \[lem:ident\], $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$ if and only if $$\operatorname{tr}_1(\phi(I_{mn})-I_{mn}) = \operatorname{tr}_1((\phi-\phi')(I_{mn}))
\quad\text{and}\quad
\operatorname{tr}_2(\phi(I_{mn})-I_{mn}) = \operatorname{tr}_2((\phi-\phi')(I_{mn}))$$ if and only if $$\operatorname{tr}_1(\phi'(I_{mn})) = \operatorname{tr}_1(I_{mn})
\quad\text{and}\quad
\operatorname{tr}_2(\phi'(I_{mn})) = \operatorname{tr}_2(I_{mn}).
\qedhere$$
\[cor:ident\] Let $\phi:M_{mn}\to M_{mn}$ be an RT-symmetric map. Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$ if and only if $$\operatorname{tr}_1\phi(I_{mn}) = \operatorname{tr}_1(I_{mn})
\quad\text{and}\quad
\operatorname{tr}_2\phi(I_{mn}) = \operatorname{tr}_2(I_{mn}).$$
\[cor:skident\] Let $\phi:M_{mn}\to M_{mn}$ be a skew RT-symmetric map. Then $\operatorname{tr}\phi(A\oplus B) = \operatorname{tr}(A\oplus B)$ for all $A\in M_m$ and $B\in M_n$ if and only if $$\operatorname{tr}_1\phi(I_{mn}) = -\operatorname{tr}_1(I_{mn})
\quad\text{and}\quad
\operatorname{tr}_2\phi(I_{mn}) = -\operatorname{tr}_2(I_{mn}).$$
It is straightforward to extend Corollaries \[cor:ident\] and \[cor:skident\] to the RT-Hermitian and skew RT-Hermitian cases since $\operatorname{tr}_1(\phi'(I_{mn})) = \operatorname{tr}_1(I_{mn})$ if and only if $\operatorname{tr}_1(\overline{\phi'(I_{mn})}) = \operatorname{tr}_1(I_{mn})$.
Now we are ready to consider the connection with the work in [@ding17]. The connection is provided by the exponential map, i.e., $$\det(e^A\otimes e^B) = e^{\operatorname{tr}(A\oplus B)}.$$
Determinant preservers of Kronecker products
============================================
The condition given in Lemma \[lem:ident\] implies that we may characterize a class of determinant preservers of Kronecker products in terms of partial determinants. However, the relationship between the partial trace and the partial determinant is not straightforward. If we restrict our attention to matrices over the complex numbers, $M_{mn}(\mathbb{C})$, then we have [@hardy17] $$\operatorname{Det}(e^A\otimes e^B) = e^{\operatorname{Tr}(A\oplus B)} R_{mn}$$
where $\operatorname{Det}(A) := \sqrt[n]{\det(A)}R_n$ and $\operatorname{Tr}(A) := \operatorname{tr}(A)/n$ for $A\in M_n$, and $R_n$ is the multiplicative group of $n$-th roots of unity in $\mathbb{C}$. Furthermore, [@hardy17] showed that $$\operatorname{Det}_1(e^A\otimes e^B) = e^{\operatorname{Tr}_1(A\oplus B)} R_{m}
\quad\text{and}\quad
\operatorname{Det}_2(e^A\otimes e^B) = e^{\operatorname{Tr}_2(A\oplus B)} R_{n}.$$
Let $\Omega_{mn}\subset M_{mn}(\mathbb{C})$ denote the set of matrices in $M_{mn}(\mathbb{C})$ with each eigenvalue $\lambda$ satisfying $\operatorname{Im}(\lambda)\in(-\pi,\pi]$. Thus we associate with every non-singular matrix $A$ a unique matrix $M\in\Omega_{mn}$ such that $A=e^M$. Let $\phi:M_{mn}(\mathbb{C})\to M_{mn}(\mathbb{C})$ be a linear map and let $\psi:GL_{mn}(\mathbb{C})\to GL_{mn}(\mathbb{C})$ be the non-linear map $$\psi(e^M) = e^{\phi(M)}.$$ The map is well defined since $M\in\Omega_{mn}$ is uniquely determined for every matrix in $GL_{mn}(\mathbb{C})$. We have $$\det\psi(e^M) = \det e^{\phi(M)}
= e^{\operatorname{tr}\phi(M)}$$ so that $\det\psi(e^M)=\det(e^M)$ if and only if $e^{\operatorname{tr}\phi(M)}=e^{\operatorname{tr}M}$. By linearity of the trace and $\phi$, this holds if and only if $\operatorname{tr}\phi(M)=\operatorname{tr}M$. Clearly, linear preservers of the trace of Kronecker sums also preserve the $\operatorname{Tr}$ of Kronecker sums. Thus, Lemma \[lem:ident\] provides the following corollary. We use the same form for $\phi$ as in Lemma \[lem:ident\].
Let $\phi:\Omega_{mn}(\mathbb{C})\to M_{mn}(\mathbb{C})$ be a linear map. The map $\psi:GL_{mn}(\mathbb{C})\to GL_{mn}(\mathbb{C})$ given by $$\psi(e^M) = e^{\phi(M)}$$ satisfies $\operatorname{Det}(\psi(A\otimes B)) = \operatorname{Det}(A\otimes B)$ if and only if $\operatorname{Det}_1(\psi(I)) = e^{I_n}U R_m$ for $U\in SL_{n}(\mathbb{C})$ and $\operatorname{Det}_2(\psi(I)) = e^{I_m}V R_n$ for $V\in SL_{m}(\mathbb{C})$, where $$U = \exp\left(\frac1m\sum_{j=1}^r \operatorname{tr}(A_jB_j) [C_j,D_j]\right),\qquad
V = \exp\left(\frac1n\sum_{j=1}^r \operatorname{tr}(C_jD_j) [A_j,B_j]\right).$$
The matrices $U\in SL_{n}(\mathbb{C})$ and $V\in SL_{m}(\mathbb{C})$ are not arbitrary. Theorem \[thm:ident\] and Corollaries \[cor:ident\] and \[cor:skident\] provide a stronger condition, which we present as our final theorem.
\[thm:detthm\] Let $\phi:\Omega_{mn}(\mathbb{C})\to M_{mn}(\mathbb{C})$ be an RT-symmetric or RT-Hermitian map. The map $\psi:GL_{mn}(\mathbb{C})\to GL_{mn}(\mathbb{C})$ given by $$\psi(e^M) = e^{\phi(M)}$$ satisfies $\operatorname{Det}(\psi(A\otimes B)) = \operatorname{Det}(A\otimes B)$ if and only if $$\operatorname{Det}_1(\psi(e^{I_{mn}})) = e^{I_n} R_m \quad\text{and}\quad
\operatorname{Det}_2(\psi(e^{I_{mn}})) = e^{I_m} R_n.$$
\[thm:detthm2\] Let $\phi:\Omega_{mn}(\mathbb{C})\to M_{mn}(\mathbb{C})$ be a skew RT-symmetric or skew RT-Hermitian map. The map $\psi:GL_{mn}(\mathbb{C})\to GL_{mn}(\mathbb{C})$ given by $$\psi(e^M) = e^{\phi(M)}$$ satisfies $\operatorname{Det}(\psi(A\otimes B)) = \operatorname{Det}(A\otimes B)$ if and only if $$\operatorname{Det}_1(\psi(e^{I_{mn}})) = e^{-I_n} R_m \quad\text{and}\quad
\operatorname{Det}_2(\psi(e^{I_{mn}})) = e^{-I_m} R_n.$$
Funding {#funding .unnumbered}
=======
The first author is supported by the National Research Foundation (NRF), South Africa. This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers: 105968). Any opinions, findings and conclusions or recommendations expressed is that of the author(s), and the NRF accepts no liability whatsoever in this regard.
|
---
abstract: 'We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the “giant component” and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all. These results are based on some new ingredients and in particular on a new bound of the number of solutions of polynomial equations in cosets of multiplicative subgroups in finite fields, which generalises previous results of P. Corvaja and U. Zannier (2013).'
address:
- 'Steklov Mathematical Institute, 8, Gubkin Street, Moscow, 119991, Russia'
- 'Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow, 143026, Russia'
- |
Department of Pure Mathematics, University of New South Wales\
2052 NSW, Australia.
- |
Institute for Information Transmission Problems RAS\
19, Bolshoy Karetny per., Moscow, 127051, Russia\
and\
Department of Mathematics, Higher School of Economics\
6, Usacheva Street, Moscow, 119048, Russia.
author:
- 'Sergei V. Konyagin'
- 'Sergey V. Makarychev'
- 'Igor E. Shparlinski'
- 'Ilya V. Vyugin'
title: On the New Bound for the Number of Solutions of Polynomial Equations in Subgroups and the Structure of Graphs of Markoff Triples
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Introduction
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Background and motivation
-------------------------
We recall that the set $\cM$ of [*Markoff triples*]{} $(x, y, z) \in \N^3$ is the set of positive integer solutions to the Diophantine equation $$\label{eq:Markoff}
x^2 + y^2 + z^2 = 3 x yz, \qquad (x, y, z)\in \Z^3.$$ One easily verifies that the map $$\cR_1: (x, y, z) \mapsto (3 yz - x, y, z)$$ and similarly defined maps $\cR_2$, $\cR_3$ (which are all involutions), send one Markoff triple to another. It is also obvious that so do permutations $ \Pi \in \rS_3$ of the components of $(x, y, z)$.
By a classical result of Markoff [@Mar1; @Mar2] one can get all integer solutions to starting from the solution $(1,1,1)$ and then applying the above transformations. More formally, let $\Gamma$ be the group of transformations generated by $\cR_1, \cR_2, \cR_3$ and permutations $\Pi \in \rS_3$. Then the orbit of $(1,1,1)$ under $\Gamma$ contains $\cM$. Hence, if one defines a [*functional graph*]{} on Markoff triples, where, starting from the “root” $(1,1,1)$, the edges $(x_1, y_1,z_1) \to (x_2, y_2,z_2)$ are governed by $(x_2, y_2,z_2) = \cT(x_1, y_1,z_1)$, where $$\label{eq:Transf}
\cT=\{\cR_1,\cR_2,\cR_3\}\cup \rS_3,$$ then this graph is connected.
Baragar [@Bar Section V.3] and, more recently, Bourgain, Gamburd and Sarnak [@BGS1; @BGS2] conjecture that this property is preserved modulo all sufficiently large primes and the set of non-zero solutions $\cM_p$ to considered modulo $p$ can be obtained from the set of Markoff triples $\cM$ reduced modulo $p$. This conjecture means that the functional graph $\cX_p$ associated with the transformation remains connected.
Accordingly, if we define by $\cC_p\subseteq \cM_p$ the set of the triples in largest connected component of the above graph $\cX_p$ then we can state:
\[conj:BGS\] For every prime $p$ we have $\cC_p = \cM_p$.
Bourgain, Gamburd and Sarnak [@BGS1; @BGS2] have obtained several major results towards Conjecture \[conj:BGS\], see also [@CGMP; @GMR]. For example, by [@BGS1 Theorem 1] we have $$\label{eq:except set}
\#\(\cM_p \setminus \cC_p\) = p ^{o(1)}, \qquad \text{as} \ p \to \infty,$$ and also by [@BGS1 Theorem 2] we know that Conjecture \[conj:BGS\] holds for all but maybe at most $X^{o(1)}$ primes $p \le X$ as $X\to \infty$.
Here, in Theorem \[thm:except set\] below, we obtain a more precise form of the bound . This result is based on a new bound, given in Theorem \[thm:MV\] below, on the total number of zeros in cosets of multiplicative subgroup of $\ovFp$ for several polynomials, which generalises a series of previous estimates of similar type that refer to only one polynomial see [@CoZa; @GaVo; @HBK] ot to a system of linear equations [@ShkVyu]. We believe that Theorem \[thm:MV\] is of independent interest and may find several other applications.
Furthermore, Bourgain, Gamburd and Sarnak [@BGS1; @BGS2] have also proved that the size of any connected component of the graphs $\cX_p$ is at least $c (\log p)^{1/3}$ for some absolute constant $c > 0$. This bound is based on proving that any component contains a path of length at least $c (\log p)^{1/3}$. Here we use an additional argument and show that a positive proportion of nodes along this path have “secondary” paths attached to them which are not also sufficiently long. Finally, we show that “many” of the elements of these “secondary” paths, have “tertiary” paths that are long as well. This allows us to improve the exponent $1/3$ to $7/9$, see Theorem \[thm:lower\].
New results
-----------
For a bivariate irreducible polynomial $$\label{eq:P}
P(X,Y)=\sum_{i+j\le d} a_{ij}X^{i}Y^{j} \in \ovFp[X,Y]$$ of total degree $\deg P \le d$, we define $P^{\sharp}(X,Y)$ as the homogeneous polynomial of degree $d^{\sharp}=\min\{ i+j:~a_{ij}\ne 0\}$ given by $$\label{eq:Pmin}
P^{\sharp}(X,Y)=\sum_{i+j=d^{\sharp}} a_{ij}X^{i}Y^{j}.$$ We also consider the set of polynomials $\cP$: $$\label{eq: set P}
\cP=
\{ P(\lambda X,\mu Y) \mid \lambda,\mu
\in\overline{\mathbb{F}}_p^* \}.$$ Define $g$ as the greatest common divisor of the following set of differences $$\label{eq:gk}
g=\gcd\{ i_1+j_1-i_2 -j_2 ~:~ a_{i_1,j_1} a_{i_2,j_2} \ne 0\}.$$
Given a multiplicative subgroup $\cG \subseteq \ovFp$, we say that two polynomials $P, Q \in \ovFp[X,Y]$ are [*$\cG$-independent*]{} if there is no $(u,v)\in \cG^2$ and $\gamma\in \ovFp^*$ such that polynomials $P(X,Y)$ and $\gamma
Q(uX,vY)$ coincide.
We now fix $h$ polynomials $$\label{eq:Pk}
P_k(X,Y) = P(\lambda_k X,\mu_k Y) \in\cP,\qquad k=1,\ldots,h,$$ which are $\cG$-independent.
The following result generalises a series of previous estimates of a similar type, see [@CoZa; @GaVo; @HBK; @ShkVyu] and references therein.
\[thm:MV\] Suppose that $P$ is irreducible, $$\deg_X P = m \mand \deg_Y P = n$$ and also that $P^{\sharp}(X,Y)$ consists of at least two monomials. There exists a constant $c_0(m,n)$, depending only on $m$ and $n$, such that for any multiplicative subgroup $\cG \subseteq \ovFp$ of order $t = \# \cG$ satisfying $$\frac{1}{2}p^{3/4}h^{-1/4}\ge t\ge \max\{h^2, c_0(m,n)\},$$ and $\cG$-independent polynomials we have $$\sum_{i=1}^h\#\left\{ (u,v)\in \cG^2 ~:~P_i(u,v)=0\right\} <12mn\gmax h^{2/3}t^{2/3}.$$
Using Theorem \[thm:MV\] we then derive:
\[thm:except set\] We have, $$\#\(\cM_p \setminus \cC_p\) \le \exp\((\log p)^{1/2+o(1)}\) , \qquad \text{as} \ p \to \infty.$$
We also obtain the following improvement of a lower bound from [@BGS1; @BGS2] on the size of individual components of $\cX_p$,
\[thm:lower\] The size of any connected component of $\cX_p$ is at least $c(\log p)^{7/9}$, where $c>0$ is an absolute constant.
Solutions to polynomial equations in subgroups of finite fields
===============================================================
Stepanov’s method
-----------------
Consider a polynomial $\Phi\in \overline{\mathbb{F}}_p[X,Y,Z]$ such that $$\deg_X\Phi<A,\quad \deg_Y\Phi<B,\quad \deg_Z\Phi<C,$$ that is, $$\label{poly-step2}
\Phi(X,Y,Z)=\sum_{0 \le a <A} \sum_{0 \le b< B} \sum_{0 \le c<C}\omega_{a,b,c}X^aY^bZ^c.$$ We assume $$A<t$$ where $t= \# \cG$ is the order of the subgroup $\cG\subseteq \F_p^*$, and consider the polynomial $$\Psi(X,Y)=Y^t\Phi(X/Y,X^{t},Y^t).$$ Clearly $$\deg \Psi \le t + t(B-1) + t(C-1) = (B+C-1)t .$$ We now fix some $\cG$-independent polynomials and define the sets $$\label{eq: Fi E}
\cF_i = \(\lambda_i^{-1}\cG\times\mu_i^{-1} \cG\), \quad i =1, \ldots, h,
\mand \cE = \bigcup_{i=1}^h \cF_i.$$ We also consider the locus of singularity $$\cM_{sing} = \left\{(X,Y) \mid X Y=0 \ \text{or} \ \frac{\partial{}}{\partial{Y}}P(X, Y)= P(X,Y)=0\right\}.$$
\[lem:Msing\] Let $P(X,Y)$ be an irreducible polynomial of bi-degree $$\(\deg_X P, \deg_Y P\) = (m,n)$$ and let $n \ge 1$. Then for the cardinality of the set $\cM_{sing}$ the following holds: $$\# \cM_{sing} \le (m+n)^2.$$
If the polynomial $P(X,Y)$ is irreducible, then the polynomials $P(X,Y)$ and $\frac{\partial P}{\partial Y}(X,Y)$ are relatively prime. Thus the Bézout theorem yields the bound $L\le (m+n)(m+n-1)$, where $L$ is the number of roots of the system $$\frac{\partial{}}{\partial{Y}}P(X, Y)= P(X,Y)=0.$$ Actually, the number of $x$ with $P(X,0)=0$ is less than or equal to $\deg_X P(X,Y)=m$, the number of pairs $(0,Y)$ on the curve $$\label{eq:curve P}
P(X,Y)=0$$ where $P$ is given by , is less than or equal to $\deg_Y P(X,Y)=n$. The total numbers of such pairs is at most $L+ m+n \le (m+n)^2$.
Assume that the polynomial $\Psi$ and $\cG$-independent polynomials satisfy the following conditions:
- all pairs in the set $$\left\{ (X,Y) \in \cE\setminus \cM_{sing} \mid P(X,Y)=0 \right \}$$ are zeros of orders at least $D$ of the function $\Psi(X,Y)$ on the curve ;
- the polynomials $\Psi(X,Y)$ and $P(X,Y)$ are relatively prime.
If these conditions are satisfied then the [*Bézout theorem*]{} gives us the upper bound $D^{-1} \deg \Psi \deg P+\#\cM_{sing}$ for the number of roots $(x,y)$ of the system $$\Psi(X,Y) =P(X,Y)=0, \qquad (X,Y) \in \cG.$$ Since the polynomials $P_k$ are $\cG$-independent, the sets $\cF_k$ are disjoint and also there is a one-to-one correspondence between the zeros: $$\begin{aligned}
P_k(X,Y)=0, &\ (X,Y)\in\cG^2, \\
& \Longleftrightarrow P(u,v)=0, \ (u,v) =
(\lambda_k^{-1}X, \mu_k^{-1} Y)\in\cF_k.\end{aligned}$$ Therefore, we obtain the bound $$\begin{split}
\label{eq:Nh}
N_h& \le \frac{\deg \Psi \cdot \deg P}{D}+\# \cM_{sing}\\
& \le \frac{(m+n) (B+C-1)t}{D}+\# \cM_{sing}
\end{split}$$ on the total number of zeros of $P_k$ in $\cG^2$, $k=1,\ldots,h$: $$N_h=\sum_{k=1}^h\#\{ (u,v)\in \cG^2 ~:~P_k(u,v)=0\} .$$ For completeness, we present proofs of several results from [@MV] which we use here as well.
Some divisibilities and non-divisibilities
------------------------------------------
We begin with some simple preparatory results on the divisibility of polynomials.
\[lem-nonzero\] Suppose that $Q(X,Y)\in \F_p[X,Y]$ is irreducible polynomial $$Q(X,Y)\mid\Psi(X,Y)$$ and $Q^{\sharp}(X,Y)$ consists of at least two monomials. Then $$Q^{\sharp}(X,Y)^{\lfloor t/e \rfloor }\mid\Psi^{\sharp}(X,Y),$$ where $Q^{\sharp}(X,Y)$, $\Psi^{\sharp}(X,Y)$ are defined as in and $e$ is defined as $g$ in .
Consider $\rho \in \cG$ and substitute $X=\rho \widetilde{X}$ and $Y=\rho \widetilde{Y}$ in the polynomials $Q(X,Y)$ and $\Psi(X,Y)$. Then $$Q(X,Y)\longmapsto Q_\rho (\widetilde{X},\widetilde{Y})=Q(\rho \widetilde{X},\rho \widetilde{Y}),$$ and $$\begin{aligned}
\Psi(X,Y) &= \Psi(\rho \widetilde{X},\rho \widetilde{Y})\\
& =(\rho \widetilde{Y})^t\Phi((\rho \widetilde{X})/(\rho \widetilde{Y}),
(\rho \widetilde{X})^{t},(\rho \widetilde{Y})^{t})
=\Psi(\widetilde{X},\widetilde{Y}),\end{aligned}$$ because $\rho ^t=1$. Hence for any $\rho \in \cG$ we have $$Q_\rho (X,Y)\mid \Psi(X,Y),$$ and we also note that $Q_\rho (X,Y)$ is irreducible.
Clearly, there exist at least $s = \fl{t/e}$ elements $\rho _1,\ldots,\rho _{s}\in \cG$ such that $$\label{ratio}
Q_{\rho _i}(X,Y)/Q_{\rho _j}(X,Y)\notin \overline{\mathbb{F}}_p, \qquad 1\le i<j \le s .$$ Obviously the polynomials $Q_{\rho _1}(X,Y),\ldots, Q_{\rho _{s}}(X,Y)$ are pairwise relatively prime, because they are irreducible and satisfy . Polynomials $Q_{\rho _i}^{\sharp}(X,Y)$ are homogeneous of degree $d^{\sharp}$ and the following holds $$\rho _1^{-d^{\sharp}}Q_{\rho _1}^{\sharp}(X,Y)=\ldots = \rho _{s}^{-d^{\sharp}}Q_{\rho _{s}}^{\sharp}(X,Y).$$ So, we have $$Q_{\rho _1}(X,Y)\cdot\ldots\cdot Q_{\rho _s}(X,Y)\mid \Psi(X,Y),$$ consequently, $$Q_{\rho _1}^{\sharp}(X,Y)\cdot\ldots\cdot Q_{\rho _{s}}^{\sharp}(X,Y)\mid \Psi^{\sharp}(X,Y).$$ Since $$Q_{\rho _1}^{\sharp}(X,Y)\cdot\ldots\cdot Q_{\rho _{s}}^{\sharp}(X,Y)=(\rho _1\cdot\ldots\cdot \rho _{s})^{d^{\sharp}} Q^{\sharp}(X,Y)^{s}$$ we obtain the desired result.
\[lem-hH\] Let $G(X,Y), H(X,Y)\in \F_p[X,Y]$ be two homogeneous polynomials. Also suppose that $G(X,Y)$ consists of at least two nonzero monomials and the number of monomials of the polynomial $H(X,Y)$ does not exceed $s$ for some positive integer $s<p$. Then $$G(X,Y)^s\nmid H(X,Y).$$
Let us put $y=1$. If $G(X,Y)^s\mid H(X,Y)$ then $G(X,1)^s\mid H(X,1)$. The polynomial $G(X,1)$ has at least one nonzero root. It has been proved in [@HBK Lemma 6] that such a polynomial $H(X,1)$ cannot have a nonzero root of order $s$ and the result follows.
Since the number of monomials of $\Psi^{\sharp}(X,Y)$ does not exceed $AB$ and we can combine Lemmas \[lem-hH\] and \[lem-nonzero\] (applied to irreducible divisors of polynomials $P_k$).
\[lem:AB Pk\] If $AB<t/\gmax$ then for polynomial we have $$P(X,Y)\nmid \Psi(X,Y).$$
Derivatives on some curves
--------------------------
There we study derivatives on the algebraic curve and define some special differential operators. Thought this section we use $$\frac{\partial} {\partial X}, \quad \frac{\partial} {\partial Y}
\mand \frac{d }{dX}$$ for standard partial derivatives with respect to $X$ and $Y$ and for a derivative with with respect to $X$ along the curve . In particular $$\begin{aligned}
\label{eq:ddx}
\frac{d}{dX}=\frac{\partial }{\partial X}+\frac{dY}{dX}\frac{\partial}{\partial Y},\end{aligned}$$ where by the implicit function theorem from the equation we have $$\frac{dY}{dX}=-\frac{\frac{\partial P}{\partial X}(X,Y)}{\frac{\partial P}{\partial Y}(X,Y)}.$$
We also define inductively $$\frac{d^k}{dX^k}=\frac{d}{dX}\frac{d^{k-1}}{dX^{k-1}}$$ the $k$-th derivative on the curve .
Consider the polynomials $q_k(X,Y)$ and $r_{k}(X,Y)$, $k\in\mathbb{N}$, which are defined inductively as $$q_1(X,Y)=-\frac{\partial}{\partial X}P(X,Y),\qquad r_1(X,Y)=\frac{\partial}{\partial Y}P(X,Y),$$ and $$\begin{split}
\label{eq: qkrk}
q_{k+1}(X,Y)&=\frac{\partial q_k}{\partial X}\left(\frac{\partial P}{\partial Y}\right)^2\\
&\qquad -\frac{\partial q_k}{\partial Y}\frac{\partial P}{\partial X}\frac{\partial P}{\partial Y}
-(2k-1)q_k(X,Y)\frac{\partial^2 P}{\partial X\partial Y}\frac{\partial P}{\partial Y} \\
&\qquad \qquad \qquad \qquad \qquad +(2k-1)q_k(X,Y)\frac{\partial^2 P}{\partial Y^2}\frac{\partial P}{\partial X},\\
r_{k+1}(X,Y)&=r_k(X,Y)\left(\frac{\partial P}{\partial Y}\right)^2=\left(\frac{\partial P}{\partial Y}\right)^{2k+1}.
\end{split}$$ We now show by induction that $$\label{eq: dk}
\frac{d^k}{dX^k}Y=\frac{q_k(X,Y)}{r_k(X,Y)}, \qquad k\in\mathbb{N}.$$ The base of induction is $$\frac{d}{dX}Y=-\frac{\frac{\partial}{\partial X}P(X,Y)}{\frac{\partial}{\partial Y}P(X,Y)} =\frac{q_1(X,Y)}{r_1(X,Y)}.$$ One can now easily verifies that assuming and we have $$\begin{aligned}
\frac{d^{k+1}}{dX^{k+1}}Y = \frac{d}{dX}\frac{d^{k}}{dX^{k}}Y=
\frac{d}{dX} \frac{q_k(X,Y)}{r_k(X,Y)}
=\frac{q_{k+1}(X,Y)}{r_{k+1}(X,Y)}, \end{aligned}$$ where $q_{k+1}$ and $r_{k+1}$ are given by , which concludes the induction and proves the formula .
The implicit function theorem gives us the derivatives $\frac{d^{k+1}}{dX^{k+1}}Y$ at a point $(X,Y)$ on the algebraic curve , if the denominator $r_{k}(X,Y)$ is not equal to zero. Otherwise $r_k(X,Y)=0$ if and only if the following system holds $$\frac{\partial{}}{\partial{Y}}P(X, Y)= P(X,Y)=0.$$
Let us give the following estimates
\[d:deriv\] For all integers $k \ge 1$, the degrees of the polynomials $q_k(X,Y)$ and $r_k(X,Y)$ satisfy the bounds $$\begin{aligned}
& \deg_X q_k \le (2k-1)m-k,\qquad \deg_Y q_k \le (2k-1)n-2k+2,\\
&\deg_X r_k \le (2k-1)m, \qquad \deg_Y r_k \le (2k-1)(n-1).\end{aligned}$$
Direct calculations show that $$\deg_X q_1\le m-1 \mand \deg_Yq_1\le n,$$ and using (with $k-1$ instead of $k$) and examining the degree of each term, we obtain the inequalities $$\begin{aligned}
&\deg_X q_k \le\deg_Xq_{k-1} +2m-1\le (2k-1)m-k,\\
& \deg_Y q_k \le\deg_{y}q_{k-1} +2n-2\le (2k-1)n-2k+2.\end{aligned}$$ We now obtain the desire bounds on $ \deg_X q_k$ and $\deg_Y q_k$ by induction.
For the polynomials $r_{k}$ the statement is obvious.
\[P:div\] Let $Q(X,Y)\in \mathbb{F}_p[X,Y]$ be a polynomial such that $$\label{CondQ}
\deg_X Q(X,Y)\le \mu,\quad \deg_Y Q(X,Y)\le \nu,$$ and $P(X,Y)\in \mathbb{F}_p[X,Y]$ be a polynomial such that $$\deg_X P(X,Y)\le m,\quad \deg_Y P(X,Y)\le n.$$ Then the divisibility condition $$\label{DivPQ}
P(X,Y)\mid Q(X,Y)$$ on the coefficients of the polynomial $Q(X,Y)$ is equivalent to a certain system of $n((\nu-n+2)m+\mu)\le (\mu+\nu+1)mn$ homogeneous linear algebraic equations in coefficients of $Q(X,Y)$ as variables.
The dimension of the vector space $\cL$ of polynomials $Q(X,Y)$ that satisfy is equal to $(\mu+1)(\nu+1)$. Let us call the vector subspace of polynomials $Q(X,Y)$ that satisfy and by $\widetilde {\cL}$. Because $Q(X,Y)=P(X,Y)R(X,Y)$ where the polynomial $R(X,Y)$ is such that $$\label{eq: Poly R}
\deg_X R(X,Y)\le \mu-m \mand \deg_Y R(X,Y)\le \nu-n,$$ then the vector space $\widetilde {\cL}$ isomorphic to the vector space of the coefficients of the polynomials $R(x,y)$ satisfying . The dimension of the vector space $\widetilde {\cL}$ is equal to $$\dim \widetilde {\cL} = (\mu-m+1)(\nu-n+1).$$ It means that the subspace $\widetilde {\cL}$ of the space $\cL$ is given by a system of $$\begin{aligned}
(\mu+1)(\nu+1)&-(\mu-m+1)(\nu-n+1)\\
&=\mu n +\nu m-mn+m+n+1
\le(\mu+\nu+1)mn\end{aligned}$$ homogeneous linear algebraic equations.
As inin [@MV], we now consider the differential operators: $$\label{diff-oper}
D_k=\left(\frac{\partial P}{\partial Y}\right)^{2k-1}X^kY^k\frac{d^k}{d X^k},\qquad k \in\N,$$ where, as before, $\frac{d^k}{d X^k}$ denotes the $k$-th derivative on the algebraic curve with the local parameter $X$. We note now that the derivative of a polynomial in two variables along a curve is a rational function. As one can see from the inductive formula for $\frac{d^k}{d X^k}$, the result of applying any operator $D_k$ to a polynomial in two variables is again a polynomial in two variables.
Consider non-negative integers $a, b, c$ such that $ a<A,\, b<B,\, c<C.$ From the formulas for derivatives on the algebraic curve we obtain by induction the following relations $$\label{diff-oper-rel}
\begin{split}
& D_k \left(\frac{X}{Y}\right)^aX^{bt}Y^{(c+1)t}=R_{k,a,b,c}(X,Y)\left(\frac{X}{Y}\right)^{a}X^{bt}Y^{(c+1)t},\\
& D_k \Psi(X,Y)|_{x,y\in \cF}=R_{k,i}(X,Y)|_{x,y\in \cF_i},
\end{split}$$ where $\cF_i$ from formula , $$\label{Rki}
\begin{split}
R_{k,i}(X,Y) = \sum_{0 \le a < A} &
\sum_{0 \le b< B} \\
&\sum_{0 \le c<C} \omega_{a,b,c}R_{k,a,b,c}(X,Y)\left(\frac{X}{Y}\right)^{a}\lambda_i^{bt}\mu_i^{(c+1)t}
\end{split}$$ for some coefficients $\omega_{a,b,c}\in \F_p$, $a<A$, $b<B$, $c<C$, and $\lambda_i,\mu_i$ from .
We now define $$\label{eq: poly R-tilde}
\widetilde{R}_{k,i}(X,Y)=Y^{A-1}R_{k,i}(X,Y).$$
\[lem-R\] The rational functions $R_{k,a,b,c}(X,Y)$ and $\widetilde{R}_{k,i}(X,Y)$, given by and , are polynomials of degrees $$\deg_X R_{k,a,b,c}\le 4km, \qquad
\deg_Y R_{k,a,b,c}\le 4kn,$$ and $$\deg_X \widetilde{R}_{k,i} \le A+4km,\qquad \deg_Y \widetilde{R}_{k,i}\le A+4kn.$$
We have $$\label{deriv-sumform}
\begin{split}
\frac{d^k}{dX^k}X^{a+bt}Y^{(c+1)t-a}
&=\sum_{(\ell_1,\ldots,\ell_s)} C_{\ell_1,\ldots,\ell_s} X^{a+bt-k+\sum_{i=1}^s \ell_i}\\
& \qquad \quad Y^{(c+1)t-a-s}\left(\frac{d^{\ell_1}Y}{dX^{\ell_1}}\right)\ldots\left(\frac{d^{\ell_s}Y}{dX^{\ell_s}}\right),
\end{split}$$ where $(\ell_1,\ldots,\ell_s)$ runs through the all $s$-tuples of positive integers with $\ell_1+\ldots+\ell_s\le k$, $s=0,\ldots,k$ and $C_{\ell_1,\ldots,\ell_s}$ are some constant coefficients.
By the formula and the form of the operator we obtain that $R_{k,a,b,c}(x,y)$ are polynomials and $R_{k,i}(x,y)$ are rational functions. Actually, from the formulas and we easily obtain that the denominator of $$\frac{d^k}{dX^k} \left(\frac{X}{Y}\right)^aX^{bt}Y^{(c+1)t}$$ divides $\left(\frac{\partial P}{\partial Y}(X,Y) \right)^{2k-1}$. We obtain that $R_{k,a,b.c}(X,Y)$ are polynomials. From the formula we obtain that $R_{k,i}$ is a rational function with denominator divided by $Y^{A-1}$. Consequently, $\widetilde{R}_{k,i}$ are polynomials.
The result now follows from Lemma \[d:deriv\] and the formulas and .
Multiplicities points on some curves
------------------------------------
\[deriv:k-th\] If $P(X,Y)\mid\Psi(X,Y)$ and $P(X,Y)\mid D_j\Psi(X,Y)$, $j=1,\ldots,k-1$, then at least one of the following alternatives holds:
- either $(x,y)$ is a root of order at least $k$ of $\Psi(X,Y)$ on the algebraic curve ;
- or $(x,y) \in \cM_{sing}$.
If $D_j\Psi(X,Y)$ vanishes on the curve $P(X,Y)= 0$, then either $$\label{eq:altern1}
\frac{d^j}{dX^j}\Psi (x,y)=0,$$ where, as before, $\frac{d^j}{d X^j}$ is $j$-th derivative on the algebraic curve with the local parameter $X$, or $$\label{eq:altern2}
xy=0,$$ or $$\label{eq:altern3}
\frac{\partial P}{\partial Y}(x,y)=0,$$ on the curve .
If we have for $j=1,\ldots,k-1$ and also $\Psi(x,y)=0$ then the pair $(x,y)$ satisfies the first case of conditions of Lemma \[deriv:k-th\].
If we have or on the curve then the pair $(x,y)$ satisfies the second case of conditions of Lemma \[deriv:k-th\].
Multiplicative orders and divisors
==================================
Multiplicative orders and binary recurrences
--------------------------------------------
For $x \in \F_p^*$ we define $$\label{eq:tx}
t(x) = \ord \xi$$ as the order of $\xi \in \F_{p^2}^*$ which satisfies the equation $3x=\xi+\xi^{-1}$ (it is easy to see that this is correctly defined and does not depend on the particular choice of $\xi$).
Throughout the paper, as usual, we use the expressions $F \ll G$, $G \gg F $ and $F=O(G)$ to mean that $|F|\leq cG$ for some constant $c>0$.
\[lem:threeorders\] For any nonzero triple $(x,y,z)\in \cM_p$, we have $$t(x)t(y)t(z) \gg \log p.$$
As in [@BGS1; @BGS2] we note that the inequality between the arithmetic and geometric means implies that the equation , considered over $\C$ has no non-zero solution $(x,y, z)$ where $$3x = \xi +\xi^{-1}, \qquad 3y = \zeta+\zeta^{-1}, \qquad 3z =\eta +\eta^{-1}$$ with the roots of unity $\xi$, $\zeta$, $\eta$ (or more generally with any $|\xi|=|\zeta|=|\eta|=1$).
Thus if we denote by $\Phi_k$ the $k$th cyclotomic polynomial, and also define $$\begin{aligned}
F(U,V,W) = (U +U^{-1})^2& +(V+V^{-1})^2 +(W +W^{-1})^2 \\
& \quad -(U +U^{-1}) (V+V^{-1}) (W +W^{-1})\end{aligned}$$ then for any positive integers $r,s,t$, the system of polynomials equations $$U^2V^2W^2F(U,V,W) = \Phi_r(U) = \Phi_s(V) = \Phi_t(W) = 0$$ has no solutions (unless $r=s = t =4$). Using the effective Hilbert’s Nullstellensatz in the form given by D’Andrea, Krick and Sombra [@DKS Theorem 1] we see that for some polynomials $g_i(U,V,W) \in \Z[U,V,W]$, $i=1,\ldots, 4$ we have $$\begin{aligned}
U^2V^2W^2F(U,V,W)&g_1(U,V,W)+ \Phi_r(U)g_2(U,V,W) \\
& + \Phi_s(V)g_3(U,V,W)+ \Phi_t(W)g_4(U,V,W) = A\end{aligned}$$ with some positive integer $A$ with $\log A \ll rst$. This immediately implies the result.
We also use the following result which follows immediately from the explicit form of solutions to binary recurrence equations and a result [@CoZa Theorem 2].
\[lem:small\_intersect\] For two distinct elements $x_1,x_2\in \ovFp$ we consider the binary recurrence sequences $$u_{i,n+2} = 3x_iu_{i,n+1} - u_{i,n}, \qquad n = 1, 2, \ldots,$$ with nonzero initial values, $(u_{i,1}, u_{i,2}) \in \ovFp$, $i=1,2$. Then $$\begin{aligned}
\# \(\{u_{1,1},\dots,u_{1,t(x_1)}\}\cap\{u_{2,1},\dots,u_{2,t(x_2)}\} \)&\\
\ll \frac{t(x_1) t(x_2)}{p} &+ \(t(x_1) t(x_2)\)^{1/3}.\end{aligned}$$
Number of small divisors of integers
------------------------------------
For a real $z$ and an integer $n$ we use $\tau_z(n)$ to denote the number of integer positive divisors $d \mid n$ with $d \le z$. We present a bound on $\tau_z(n)$ for small values of $z$ (which we put in a slightly more general form than we need for our applications).
\[lem:tauz\] For any fixed real positive $\gamma< 1$, if $z \ge \exp\((\log n)^{\gamma + o(1)}\)$ then $$\tau_z(n) \le z^{1-\gamma + o(1)}$$ as $n \to\infty$.
As usual, we say that a positive integer is $y$-smooth if it is composed of prime numbers up to $y$. Then we denote by $\psi(x,y)$ the number of $y$-smooth positive integers that are up to $x$. Let $s$ be the number of all distinct prime divisors of $n$ and let $p_1, \ldots, p_s$ be the first $s$ primes. We note that $$\label{eq:tau psi}
\tau_z(n) \le \psi(z, p_s).$$
By the prime number theorem we have $n \ge p_1\ldots p_s = \exp(s + o(s))$ and thus $$\label{eq:ps z}
p_s \ll s \log s \le (\log n)^{1 + o(1)} \le (\log z)^{1/\gamma + o(1)}.$$ We now recall that for any fixed $\alpha>1$ we have $$\Psi(x,(\log x)^\alpha)=x^{1-1/\alpha+o(1)}$$ as $x\to \infty$, see, for example, [@HT Equation (1.14)]. Combining this with and we conclude the proof.
Proofs of main results
======================
Proof of Theorem \[thm:MV\]
---------------------------
We define the following parameters: $$A=\fl{\frac{t^{2/3}}{\gmax h^{1/3}}} ,\quad B=C=\fl{ h^{1/3}t^{1/3}},
\qquad
D=\fl{ \frac{t^{2/3}}{4 \gmax h^{1/3} mn}}.$$
If $P_i(x,y)=0$ for at least one $i=1,\ldots,h$, then $$\label{Deriv-3}
D_k\Psi(x,y)=0,\quad (x,y)\in \bigcup_{i=1}^h\cF_i,$$ with the operators , where the sets $\cF_i$ are as in , is given by the system of linear homogeneous algebraic equations in the variables $\omega_{a,b,c}$. The number of equations can be calculated by means of Lemmas \[P:div\] and \[lem-R\]. To satisfy the condition for some $k$ we have to make sure that the polynomials $\widetilde{R}_{k,i}(X,Y)$, $i=1,\ldots,h$, given by , vanish identically on the curve . The bi-degree of $\widetilde{R}_{k,i}(X,Y)$ is given by Lemma \[lem-R\]: $$\deg_X \widetilde{R}_{k,i} \le A+4km,\qquad \deg_Y \widetilde{R}_{k,i}\le A+4kn.$$ The number of equations on the coefficients that give us the vanishing of polynomial $\widetilde{R}_{k,i}(X,Y)$ on the curve is given by Lemma \[P:div\] and is equal to $(\mu+\nu+1)mn$, where $\mu,\nu$ are as Lemma \[P:div\] and $$\mu \le A+4km, \quad \nu \le A+4kn.$$ Finally, the condition for some $k$ is given by $h(\mu+\nu+1)mn \le mnh(2A+4k(m+n))$ linear algebraic homogeneous equations. Consequently, the condition for all $k=0,\ldots,D-1$ is given by the system of $$L = hmn\sum_{k=0}^{D-1}\(4k(m+n)+2A+1\)$$ linear algebraic homogeneous equations in variables $\kappa_{a,b,c}$. Now it is easy to see that $$\begin{aligned}
L&=h\left((2A+1)Dmn+2nm(m+n)D(D-1)\right)\\
& \le 2hADmn+2hmn(m+n)D^2 = 2hmn(AD+(m+n)D^2).\end{aligned}$$ The system has a nonzero solution if the number of equations is less than to the number of variables, in particular, if $$\label{number-of-equ-2}
2hmn(AD+(m+n)D^2)<ABC,$$ as we have $ABC$ variables. It is easy to get an upper bound for the left hand side of . For sufficiently large $t>c_0(m,n)$, where $c_0(m,n)$ is some constant depending only on $m$ and $n$, we have $$\label{system_has_a_solution}
\begin{split}
2hmn&(AD+(m+n)D^2)\\
& <2hmn\left(\frac{h^{-1/3}t^{2/3}}{\gmax}\frac{h^{-1/3}t^{2/3}}{4mn\gmax}+(m+n)\frac{h^{-2/3}t^{4/3}}{16m^2n^2\gmax^2}\right)\\
&<\frac{3}{4}\frac{h^{1/3}t^{4/3}}{\gmax^2}. \end{split}$$ On the other hand, assuming that $c_0(m,n)$ is large enough, we obtain $$ABC=\left\lfloor \frac{h^{-1/3}t^{2/3}}{\gmax}\right\rfloor \lfloor h^{1/3}t^{1/3}\rfloor^2>\frac{3}{4}\frac{h^{1/3}t^{4/3}}{\gmax^2},$$ which together with implies .
It is clearly that $$\gmax AB\le t.$$ We also require that the degree of the polynomial $\Psi(x,y)$ should be less than $p$, $$\deg\Psi(x,y)\le (B-1)t+Ct<p.$$ Actually, the inequality $(B-1)t+Ct<2h^{1/3}t^{4/3}<p$ is satisfied because $t<\frac{1}{2}p^{3/4}h^{-1/4}$.
Finally, recalling Lemmas \[lem:Msing\] and \[deriv:k-th\] and the inequality we obtain that $N_h$ satisfies the inequality $$\begin{aligned}
N_h & \le \# \cM_{sing} + (m+n)\frac{(B+C-1)t}{D}\\
&< (m+n)^{2} + \frac{2h^{1/3}t^{4/3}}{\fl{h^{-1/3}t^{2/3}/(4mn\gmax)}}
<12mn\gmax h^{2/3}t^{2/3}\end{aligned}$$ for sufficiently large $t>c_0(m,n)$, which concludes the proof.
Proof of Theorem \[thm:except set\]
-----------------------------------
Define the mapping $$\cT_0 \(x, y, z\) \mapsto \(x, z, 3xz-y\)$$ where $\cT_0 = \Pi_{1,3,2}\circ \cR_2$ is the composition of the permutations $$\Pi_{1,3,2} = (x, y,z) \mapsto (x, z,y)$$ and the involution $$\cR_2: (x, y,z) \mapsto (x, 3xz - y, z)$$ as in the above.
Therefore the orbit $\Gamma(x, y, z)$ of $(x, y,z)$ under the above group of transformations $\Gamma$ contains, in particular the triples $(x, u_n, u_{n+1})$, $n = 1, 2, \ldots$, where the sequence $u_n$ satisfies a binary linear recurrence relation $$\label{eq:bin rec}
u_{n+2} = 3xu_{n+1} - u_n, \qquad n = 1, 2, \ldots,$$ with the initial values, $u_1=y$, $u_2 = z$. This also means that $\Gamma(x, y, z)$ contains all triples obtained by the permutations of the elements in $(x, u_n, u_{n+1})$.
Let $\xi, \xi^{-1} \in \F_{p^2}^*$ be the roots of the characteristic polynomial $Z^2 -3xZ +1$ of the recurrence relation . In particular $3x=\xi+\xi^{-1}$. Then, it is easy to see tha unless $(x, y, z)=(0,0,0)$, which we eliminate from the consideration, the sequence $u_n$ is periodic with period $t(x)$ which is the order of $\xi$ in $\F_{p^2}^*$ as given by .
We now fix some $\ve>0$ and denote $$M_0=\exp((\log p)^{1/2+\ve}),\quad M_1 = M_0^{1/6}/2
> \exp((\log p)^{1/2+\ve/2}).$$
Assume that the remaining set of nodes $\cR=\cM_p \setminus \cC_p$ is of size $\# \cR>M_0$. Note that if $(x,y,z)\in \cR$ then aslo $\Pi(x,y,z)\in \cR$ for every $\Pi \in \rS_3$. Therefore, there are more that $M_0^{1/3}$ elements $x\in\F_p^*$ with $(x,y,z)\in \cR$ for some $y, z \in \F_p$.
Since there are obviously at most $T(T+1)/2$ elements $\xi \in \F_{p^2}^*$ of order at most $T$ we conclude that there is a triple $(x^*,y^*,z^*)\in \cR$ with $$\label{eq:large t}
t(x^*)>\sqrt{M_0^{1/3}} =2M_1.$$ Then the orbit $\Gamma(x^*,y^*,z^*)$ of this triple has at least $2M_1$ elements. Let $M$ be the cardinality of the set $\cM$ of projections along the first components of all triples $(x,y,z) \in \Gamma(x^*,y^*,z^*)$. Since the orbits are closed under the permutation of coordinates, and permutations of the triples $$(x^*, u_n, u_{n+1}), \qquad n = 1, \ldots, t(x^*),$$ where the sequence $u_n$ is defined as in with respect to $(x^*,y^*,z^*)$, produce the same projection no more than twice we obtain $$\label{eq:M and t}
M \ge \frac{1}{2} t(x^*).$$ Recalling , we obtain $$\label{eq:large M}
M \ge M_1 > \exp\((\log p)^{1/2+\ve/2}\).$$ We also notice, that by the bound we also have $$\label{eq:small M}
M = p^{o(1)}.$$
By Lemma \[lem:tauz\], applied with $\gamma = 1/2+\ve/2$ and the inequalities we have $$\begin{aligned}
\sum_{\substack{t\le M^{2/3+\ve/5}\\ t\mid p^2-1}} t
& \le M^{2/3+\ve/5} \tau_{M^{2/3+\ve/5}}(p^2-1)\\
& = M^{2/3+\ve/5} M^{(2/3+\ve/5)(1/2-\varepsilon/2) + o(1)} \\
& = M^{1 - \ve/30 -\ve^2/10 + o(1)} =o(M).\end{aligned}$$ For $t\mid p^2-1$ we denote $g(t)$ the number of $x\in \cM$ with $t(x)=t$. Since $$\sum_{t\mid p^2-1} g(t) =M$$ and $g(t)<t$ for any $t$, we conclude that $$\sum_{\substack{t > M^{2/3+\ve/3}\\ t\mid p^2-1}} g(t) = M + o(M)$$ Next, the same argument as used in the bound implies that $g(t)=0$ for $t>2M$. Applying Lemma \[lem:tauz\] and the inequalities again we see that for some integer $t_0\mid p^2-1 $ with $$\label{eq:t and M}
2M \ge t_0 > M^{2/3+\ve/3}$$ we have $$\label{estg}
\begin{split}
g(t_0)& \ge \frac{1}{\tau_{2M}(p^2-1)}\sum_{\substack{t > M^{2/3+\ve/3}\\ t\mid p^2-1}} g(t) \\
& = \frac{ M + o(M) }{\tau_{2M}(p^2-1)} \ge M^{1/2+\ve/2+o(1)}\ge M^{1/2+\ve/3},
\end{split}$$ provided that $p$ is large enough.
Let $\cL$ be the set of $x \in \cM$ with $t(x)=t_0$ thus $$\label{eq: set L}
\# \cL = g(t_0).$$ For each $x \in \cL$ we fix some $y,z \in \F_p$ such $(x,y,z)\in\Gamma (x^*,y^*,z^*)$ and again consider the sequence $u_n$, $n=1, 2, $, given by and of period $t(x)=t_0$, so we consider the set $$Z(x) = \{u_n~:~n =1, \ldots, t_0\}$$ Let $\cH$ be the subgroup of $\F_{p^2}^*$ of order $t_0$, and $\xi(x)$ satisfy the equation $3x=\xi(x)+\xi(x)^{-1}$. One can easily check, using an explicit expression for binary recurrence sequences via the roots of the characteristic polynomial, that $$\cZ(x) = \left\{\alpha(x) u+\frac{r (x)}{\alpha(x) u}~:~u\in \cH\right\},$$ where $$r(x)=\frac{(\xi(x)^2+1)^2}{9(\xi(x)^2-1)^2},$$ and $\alpha(x)\in\F_{p^2}^*$. If $\xi=\xi_0$ satisfies the equation $$r= \frac{(\xi^2+1)^2}{9(\xi^2-1)^2},$$ then other solutions are $-\xi_0, 1/\xi_0, -1/\xi_0$. Moreover, $3x=\xi+\xi^{-1}$ can take at most two values whose sum is $0$. Since every value is taken at most twice among the elements of the sequence $u_n$, $n =1, \ldots, t_0$, we have $$\# \cZ(x) \ge \frac{1}{2} t_0.$$
If we have $x_1, x_2\in \cL$ with $x_1\neq \pm x_2$ (the last condition guarantees that the orbits $Z(x_1)$ and $Z(x_2)$ do not coincide), then $\#(Z(x_1)\cap Z(x_2)$ is the number of solutions of the equation $$\alpha(x_1) u+\frac{r (x_1)}{\alpha(x_1) u}
=\alpha(x_2) v+\frac{r (x_2)}{\alpha(x_2) v}
\quad u,v\in \cH,$$ or, equivalently, $$P_{x_1,x_2}(u,v) = 0, \qquad u,v\in \cH,$$ where $$\begin{aligned}
P_{x_1,x_2}(X,Y) = \alpha(x_1) ^2\alpha(x_2) X^2 Y - \alpha(&x_1) \alpha(x_2)^2 X Y^2\\
& - \alpha(x_1) r (x_2) X + \alpha(x_2)r (x_1)Y.\end{aligned}$$
We now use Theorem \[thm:MV\] to estimate the size of these intersections, for different choices of pairs $(x_1,x_2), (x_1,x_3), \in \cL^2$ (sharing the first component). For this, we need to show that for $x_1, x_2, x_3\in \cL$ with $x_1\neq \pm x_2$, $x_1\neq \pm x_3$ and $x_2 \ne x_3$, the polynomials $P_{x_1,x_2}$ and $P_{x_1,x_3}$ are $\cH$-independent. Indeed, assume that $$P_{x_1,x_2}(X,Y) = \gamma P_{x_1,x_3}(uX,vY) .$$ We then derive $$\begin{aligned}
\alpha(x_1) X&+\frac{r (x_1)}{\alpha(x_1) X}
- \alpha(x_2) Y+\frac{r (x_2)}{\alpha(x_2) Y} \\
&= \gamma\(
\alpha(x_1) uX+\frac{r (x_1)}{\alpha(x_1) u X}
- \alpha(x_3) vY+\frac{r (x_3)}{\alpha(x_3) vY} \).\end{aligned}$$ Hence, we have $(\gamma,u) =(\pm 1, \pm 1)$, and in fact we can assume that $\gamma = u=1$. Then we obtain $\alpha(x_2)/\alpha(x_3) = v/u \in \cH$. However this means that $\cZ(x_2) = \cZ(x_3)$, which by Lemma \[lem:small\_intersect\] contradicts our choice of $x_2$ and $x_3$.
Now we consider $x_{1},\ldots,x_{h} \in \cL$ with $x_{i}\neq \pm x_{j}$ for $1\le i < j\le h$. We take $h=\fl{c_0t_0^{1/2}}$ for an appropriate small $c_0>0$. We can do it due to and . We now recall Theorem \[thm:MV\], which applies due to the upper and lower bounds , and . Hence, for for an appropriate choice of $c_0$, we conclude that for $i=1,\ldots,h$ we have $$\#\(\cZ\(x_{i}\)\setminus\bigcup_{j=1}^{i-1}\cZ\(x_{j}\) \) \ge \frac{1}{2} \#\cZ\(x_{i}\) \ge t_0/4.$$ Therefore, $$\# \bigcup_{i=1}^{h}\cZ\(x_{i}\) \ge t_0h/4\gg t_0^{3/2}$$ and thus, by , we have $$\# \bigcup_{i=1}^{h}\cZ\(x_{i}\) \ >M$$ provided that $t$ is large enough. This contradicts the choice of $M$.
Proof of Theorem \[thm:lower\]
------------------------------
We assume that consider that $p$ is large enough and fix a connected component $\cC$ of $\cM_p$.
Let $\cX$ be the set of $x\in \F_p$ such that $(x,y,z)\in\cC$ for some $y,z$. If $t(x)>(\log p)^{7/9}$ for some $x\in\cX$, then $\cC$ contains at least $t(x)$ triples $(x,y,z)$ and the desired result easily follows. Thus, we assume that $t(x)\le (\log p)^{7/9}$ for all $x\in \cX$. In particular, for $x_1, x_2 \in \cX$ the bound of Lemma \[lem:small\_intersect\] becomes $O\(\(t(x_1) t(x_2)\)^{1/3}\)$.
We consider first the case where there exists $x_0\in\cX$ such that $$\label{restrt()}
(\log p)^{0.15}\le t(x_0) \le (\log p)^{1/3}$$ (one can see from the argument below that the exponent $0.15$ can be replaced by any constant in the open interval $(1/7, 1/6)$).
With every $x_0$ satisfying , we associate the $t(x_0)$-periodic sequence $\{u_j\}$ as in . By Lemma \[lem:threeorders\] for any $j=1, 2, \ldots$ we have $$\max\{ t(u_j),t(u_{j+1})\} \ge \sqrt{t(u_j) t(u_{j+1})} \gg (\log p)^{1/2}t(x_0)^{-1/2}.$$ Hence, if we define $$\vartheta(x_0) = c (\log p)^{1/2}t(x_0)^{-1/2} \gg (\log p)^{1/3}$$ for an appropriate constant $c> 0$, then for any $j=1, 2, \ldots$ we have $$\max\{ t(u_j),t(u_{j+1})\} \ge \vartheta(x_0).$$ Therefore, there are at least $t(x_0)/2$ values $j$, $1\le j\le t(x_0)$, such that $t(u_j)\ge \vartheta(x_0)$. Since there are at most two $j$ with the same $t(u_j)$, there is a set $\cY(x_0)\subset \{u_1,\dots,u_{t(x_0)}\}$ with $\# \cY(x_0) \ge t(x_0)/4$ and $t(y)\ge \vartheta(x_0)$ for $y\in \cY(x_0)$.
We say that $y$ is associated with $x$ if $(x,y,z)\in\cC$ for some $z$. By our construction, all elements of $\cY(x_0)$ are associated with $x_0$.
Let $$s=\fl{c_0(\vartheta(x_0))^{1/3}}$$ where $c_0$ is a small positive constant. By the first inequality from we have $s\le t(x_0)/4$ (provided $c_0$ is small enough). Hence we can choose elements $y_1,\dots,y_s\in \cY(x_0)$. We order them so that $$t(y_1)\le\ldots\le t(y_s).$$
For $i=1,\ldots,s$, there is a set $\cZ(y_i)$ of elements associated with $y_i$ such that $\# \cZ(y_i)\ge t(y_i)/4$ and $$\label{eq:tz}
t(z)\gg (\log p)^{1/2}t(y_i)^{-1/2}$$ for $z\in \cZ(y_i)$.
Now we use that due to Lemma \[lem:small\_intersect\] for any $1\le j<i\le s$ we have $$\#\(\cZ(y_i)\cap \cZ(y_j)\) \ll (t(y_i)t(y_j))^{1/3} \ll t(x_i) \vartheta(x_0)^{-1/3}.$$ Taking into account the choice of $s$ we conclude that $$\sum_{j<i}\#\(\cZ(y_i)\cap \cZ(y_j)\) \le\frac12\cZ(y_i),$$ provided that $c_0$ is small enough. Hence, there are subsets $\cW(y_i) \subseteq \cZ(y_i)$ such that $$\# \cW(y_i) \ge \frac{1}{2} \# \cZ(y_i) \ge t(y_i)/8$$ which are pairwise disjoined, that is $$\cW(y_i)\cap \cW(y_j) = \emptyset, \quad 1 \le j < i \le s.$$
For any $i=1,\dots,s$ and $z\in \cW(y_i)$ we have $t(z)$ triples $(x,y,z)$ from $\cC$. Summing up the bound over $z\in\cW(y_i)$ we get $$\sum_{z\in\cW(y_i)} t(z) \gg (\log p)^{1/2}(t(y_i)^{1/2}) \ge (\log p)^{1/2}\vartheta(x_0)^{1/2}$$ triples from $\cC$. So, $$\# \cC \gg s (\log p)^{1/2}\vartheta(x_0)^{1/2} \gg (\log p)^{1/2}\vartheta(x_0)^{5/6}
\gg (\log p)^{7/9}$$ as required.
Now we consider the case where no element $x_0\in\cX$ satisfies . By Lemma \[lem:threeorders\] there exists $x_1\in\cX$ with $t(x_1)\gg(\log p)^{1/3}$. There are at least $ t(x_1)/2$ elements $y\in\cX$ associated with $x_1$. Among them there are at most $(\log p)^{0.3}$ elements $y$ with $t(y)<(\log p)^{0.15}$. Hence, there is a set $\cY(x_1)$ of elements associated with $x_1$ such that $\# \cY(x_1)\ge t(x_1)/3\gg(\log p)^{1/3}$ and $t(y)>(\log p)^{1/3}$ for any $y\in \cY(x_1)$. We now define $$s=\fl{c_1(\log p)^{1/9}},$$ where $c_1$ is a small positive constant, and take elements $y_1,\dots,y_s$ from $ \cY(x_1)$. The same argument as in the first case shows again that $$\# \cC\gg s (\log p)^{2/3} \gg (\log p)^{7/9}.$$ This completes the proof.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Peter Sarnak for the encouragement and useful comments.
This work was supported by the Program of the Presidium of the Russian Academy of Sciences no.01 “Fundamental Mathematics and its Applications”, Grant PRAS-18-01 (Konyagin), by the Russian Fund of Fundamental Research, Grants RFBR 17-01-00515 (Makarychev and Vyugin), and RFBR-CNRS 16-51-150005 (Vyugin), the Australian Research Council, Grant DP170100786 (Shparlinski) and by the Simons-IUM fellowship (Vyugin).
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---
abstract: 'We study the second-harmonic generation and localization of light in a reconfigurable waveguide induced by an optical vortex soliton in a defocusing Kerr medium. We show that the vortex-induced waveguide greatly improves conversion efficiency from the fundamental to the second harmonic field.'
author:
- 'José R. Salgueiro, Andreas H. Carlsson$^{\dagger}$, Elena Ostrovskaya, and Yuri Kivshar'
title: 'Second-harmonic generation in vortex-induced waveguides'
---
Spatial optical solitons have become a topic of active research promising many realistic applications and opening new directions in nonlinear physics [@book]. In its simplest form, a spatial soliton is a single self-guided beam of a specific polarization and frequency. Two (or more) mutually trapped components with different polarizations or frequencies can form[*a vector soliton*]{}.
One important application of spatial optical solitons is to induce stable nondiffractive steerable waveguides that can guide and direct another beam, thus creating a reconfigurable all-optical circuit. The soliton-induced optical waveguides have been studied theoretically and demonstrated experimentally in many settings [@wave1; @wave2; @wave3; @wave4]. It was also shown that soliton waveguides can be used for a number of important applications, including the second-harmonic generation [@OL_lan], directional couplers and beam splitters [@OL_guo], and optical parametric oscillators [@OL_lan2].
Dark solitons - localized dips on a background intensity - are more attractive for soliton waveguiding applications because of their greater stability and steerability [@book]. Their two-dimensional generalization, [*optical vortex solitons*]{} [@swartz], may have a number of potential advantages. Optical vortex solitons are light beams self-trapped in two spatial dimensions and carrying a phase dislocation. A systematic analysis of the waveguides created by vortex solitons in a Kerr medium [@vort2; @vort] demonstrates that an optical vortex can guide both weak and strong probe beams, and that in the latter case the vortex creates a stable vector soliton with its guided component [@book].
In this Letter, we study the second-harmonic generation in a reconfigurable vortex-induced waveguide and determine conditions for significant enhancement of the conversion efficiency. We also describe novel types of [*three-component vector solitons*]{} created by a vortex beam together with both fundamental and second-harmonic parametrically coupled localized modes guided by the vortex-induced waveguide.
We consider two incoherently coupled beams with frequencies $\omega_0$ and $\omega_1$ propagating in a bulk nonlinear Kerr medium. The $\omega_0$-beam propagates in a self-defocusing regime and carries a phase dislocation. We assume that the phase-matching conditions of the second-harmonic generation (SHG) are fulfilled for the fundamental wave of frequency $\omega_1$ guided by the vortex waveguide, so that it generates a second-harmonic (SH) wave with the frequency $2\omega_1$. The SH wave is parametrically coupled to the fundamental one and is also guided by the vortex waveguide. Evolution of the slowly varying beam envelopes of the vortex beam, the fundamental guided wave, and the SH wave can be described by the following system of three coupled dimensionless equations $$\label{eq1}
\begin{array}{l} {\displaystyle
i \frac{\partial u}{\partial z} +\Delta_{\perp} u - (|u|^2 +
\sigma |w|^2 + \rho |v|^2) u
= 0,
} \\*[9pt] {\displaystyle
i \frac{\partial w}{\partial z} +\Delta_{\perp} w + w^*v -
\sigma |u|^2 w
= 0,
} \\*[9pt] {\displaystyle
2i \frac{\partial v}{\partial z} +\Delta_{\perp} v -\beta v + \frac{1}{2}w^2 -\rho
|u|^2v
= 0.
} \end{array}$$ where $u$, $w$, and $v$ are the normalized slowly varying complex envelopes of the vortex beam, the fundamental field, and the SH field, respectively. Other notations are: the Laplacian $\Delta_{\perp}$ refers to the transverse coordinate ${\bf r}
=(x,y)$ measured in units of $r_0$, where $r_0^2 = 3\chi^{(3)}/16
\omega_1^2 [\chi^{(2)}]^2$ (see details in Ref. [@ole]), $z$ is the beam propagation coordinate measured in units of $z_0=2k_1r_0^2$. The parameter $\beta = 2z_0 \Delta k$ is proportional to the wavevector mismatch $\Delta k = 2k_1 -k_2$, whereas the nonlinear coupling coefficients $\sigma$ and $\rho$ are proportional to the corresponding third-order tensor components [@book], and the self-action effects for the fundamental and SH fields are neglected. Equations (\[eq1\]) are valid when spatial walk-off is negligible and the fundamental frequency $\omega_1$ and its second harmonic are far from resonance.
We emphasize that the model (\[eq1\]) is the simplest of its kind, which is most suitable for our feasibility study of SHG in vortex-induced waveguides. It is clear that modelling of particular experimental setups for realization of this concept would require modifications of Eqs. (\[eq1\]), according to the geometry of an experiment and properties of nonlinear materials. For example, in photorefractive crystals [@OL_lan] one should take into account the nonlinearity saturation effect.
First, we analyze the stationary solutions of the model (\[eq1\]) in the form of the (2+1)-dimensional radially symmetric nonlinear modes. We look for spatially localized solutions in the polar coordinates $(r,\phi)$ of the form $u= u(r)
e^{-iz}e^{i\phi}$, $w=w(r)e^{i\lambda z}$, and $v=v(r)e^{i2\lambda
z}$, with the following asymptotic: $u(r) \rightarrow 1$, and $(v(r), w(r)) \rightarrow 0$ for $r =\sqrt{x^2+y^2} \rightarrow
\infty$. Then, the mode amplitudes satisfy the system of $z$-independent equations $$\label{eq2}
\begin{array}{l} {\displaystyle
\Delta_{r} u - \frac{1}{r^2} u + u - (u^2 + 2w^2 + 8v^2)u
= 0,
} \\*[9pt] {\displaystyle
\Delta_{r} w - \lambda w + wv - 2u^2 w
= 0,
} \\*[9pt] {\displaystyle
\Delta_{r} v - (4\lambda + \beta)v +\frac{1}{2} w^2 - 8u^2 v
= 0.
} \end{array}$$ where $\Delta_r = (1/r)d/dr(r d/dr)$ is the radial part of the Laplacian, and for definiteness we have specified the parameters of the cross-phase modulation interaction, $\sigma =2$ and $\rho
=8$. In Eqs. (\[eq2\]), the real propagation constant $\lambda$ must be above cutoff, $\lambda > \lambda_c = {\rm max}(0,
-\beta/4)$, for $w$ and $v$ to be exponentially localized.
![Spatial profiles of the three-wave vector soliton components for the points A to F marked in Fig. \[fig1\]. Shown are: the vortex amplitude $u(r)$ (thin solid), the fundamental field $w(r)$ (thick solid), and the SH field $v(r)$ (dashed) at the indicated values of $\beta$ and $\lambda$. []{data-label="fig2"}](salgueiroF1.eps){width="3.0in"}
![Region of existence (shaded) of the three-component vector solitons of the model (\[eq1\]) in the plane $(\lambda,
\beta)$. Marked points correspond to the localized modes shown in Fig. \[fig2\]. []{data-label="fig1"}](salgueiroF2.eps){width="2.9in"}
Using the standard relaxation numerical technique, we find the families of localized solutions of the system (\[eq2\]) for allowed values of $\beta$ and $\lambda$. In Fig. \[fig2\] we show several examples of the profiles of the three-wave localized solutions for selected values of the parameters $\beta$ and $\lambda$. The numerical results are summarized in Fig. \[fig1\] which shows the existence domain as a shaded region of the plane $(\lambda, \beta)$ with the boundary $\lambda = \lambda_{\rm th}$ (solid curve) found numerically and the asymptotic lines $\lambda
=0$ and $\lambda = -\beta/4$ (dashed) found from a simple analysis of Eqs. (\[eq2\]).
All three-wave solutions of Eqs. (\[eq2\]) can formally be divided into [*two categories*]{} according to the dominant regime of their formation: (i) vortex-waveguiding regime, $\lambda <0$, and (ii) quadratic solitons regime, $\lambda >0$ . For $\lambda_{\rm th} < \lambda < 0$, the parametrically coupled modes $w$ and $v$ are localized only in the presence of the vortex, and can be regarded as two guided modes of the effective vortex-induced waveguide. Examples of such solutions are presented in Fig. \[fig2\] by the cases A, C, and D. For $\lambda >0$, the positive values of the propagation constant correspond to an effectively self-localizing parametric nonlinearity acting between the fields $w$ and $v$. These components can then become localized even without the vortex, in the form of a parametric quadratic soliton (see Fig. \[fig2\], the cases B, E, and F). Parametric coupling between the two fields is defined, as expected, by the value of the phase-matching parameter $\beta$.
In order to study the SHG process in the vortex-induced waveguide, we employ the stationary solutions obtained above and analyze numerically the evolution of the beams in the case when the SH component is absent at the input. We perform all our calculations for the case of [*a finite-extent*]{} input vortex beam, obtained by superimposing the stationary vortex profile $u(r)$ onto a broad super-Gaussian beam, $u_{\rm sG}= u(r) \exp [-(r^6/d]$, where $d=
10^8$. This form of initial conditions makes our predictions more suitable for experimental verifications.
The numerical results indicate that the generation of the SH field from such an input differs dramatically for the vortex-waveguiding and quadratic soliton regimes. Indeed, for $\lambda <0$ we observe a good correspondence with the SHG theory. For large $\beta$, the generated SH field is weak and the process corresponds to the so-called nondepleted pump approximation in the SHG theory. Almost perfect SHG is observed for $\beta$ close to zero, and in all such cases the distortion of the vortex waveguide is weak. Figure \[fig3\](upper row) and Fig. \[fig4\] show an example of the SHG process with $u(r)$ corresponding to the point D in Fig. \[fig1\]. A good confinement of both fundamental and SH guided modes can be seen with a very good conversion efficiency and weak distortion of the vortex beam.
However, in the quadratic soliton regime, when $\lambda >0$, the strong parametric interaction between the guided components does not allow good energy conversion between the harmonics. Instead, even for a high-intensity fundamental input, both the fundamental and SH fields approach a stationary state with nonzero but low-amplitude components. The SHG process becomes even worse for the negative phase-matching. Figure \[fig3\](lower row) shows an example of a very strong mode coupling and vortex distortion corresponding to the parameter region $\beta <0$ and $\lambda >0$.
![Examples of SHG in the vortex-induced waveguide with no SH field at the input and the parameters corresponding to the point D (upper row) and point E (lower row) in Fig. \[fig2\] and \[fig1\]. Notice the scale differences between the top and bottom rows.[]{data-label="fig3"}](salgueiroF3.eps){width="3.4in"}
![Grey-scaled images of the vortex waveguide and the guided modes for the SHG process. Initial conditions correspond to a vortex carried by a Gaussian beam and the fundamental wave, both corresponding to the point D in Fig. \[fig1\].[]{data-label="fig4"}](salgueiroF4.eps){width="3.0in"}
If the vortex is removed at the input in the vortex-waveguiding regime ($\lambda <0$), the SHG conversion efficiency drops by at least one order of magnitude or more, and both the strong fundamental and weak SH fields diffract rapidly. In the quadratic soliton regime ($\lambda >0$), the effective self-focusing nonlinearity of the second-order parametric interaction between the fields $w$ and $v$ allows the formation of two-wave parametric solitons even without the vortex component. However, in this case the input power does not transfer into the SH field, it undergoes a redistribution between the harmonics in such a way that both fields either approach a stationary state corresponding to a (2+1)-dimensional quadratic soliton (above the existence threshold), or just diffract (below the threshold). To summarize, our study of SHG in vortex-induced waveguides in different regimes suggests that the enhanced conversion efficiency can be achieved [*only in the vortex-waveguiding regime*]{}.
Possible experimental realizations of the concept of the SHG in vortex-induced waveguides can be achieved in a crystal of Fe:LiNbO$_3$ where phase-matching can be satisfied through the birefringence effect at the angle $\theta = 81^{o}$ with respect to the $z$-axis, provided the four-wave mixing effect is suppressed. The other possibility is to employ photorefractive crystals and the temperature tuning technique, similar to that reported earlier [@OL_lan].
In conclusion, we have analyzed the simultaneous guidance of both the fundamental and second-harmonic waves by an optical vortex soliton. We have described novel classes of three-wave parametric solitons with a vortex-soliton component, and have studied the second-harmonic generation in the vortex-induced waveguides. For the first time to our knowledge, we demonstrated that larger conversion efficiency of the SHG process can be achieved in the vortex-waveguiding regime.
One of the authors (YK) thanks Ming-feng Shih and Solomon Saltiel for useful discussions. The work was partially supported by the Australian Research Council and the Secretaría de Estado de Educación y Universidades of Spain through the European Social Fund.
$^{\dagger}$ Currently at Acreo AB, 16440 Kista, Sweden.
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---
abstract: 'Let $\Gamma_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $\Phi_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $\Phi_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $\Gamma_{g,b}$, in this paper we compute $\Phi_f(G)$ for certain subgroups of $\Gamma_{g,b}$. In particular, we answer Ivanov’s question in the affirmative for these subgroups of $\Gamma_{g,b}$.'
author:
- 'G. Masbaum and A. W. Reid [^1]'
date: 'January 14, 2015'
title: Frattini and related subgroups of Mapping Class Groups
---
To V. P. Platonov on the occasion of his 75th birthday
[^2]
Introduction
============
We fix the following notation throughout this paper: Let $\Gamma_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. When $b=0$ we simply write $\Gamma_g$. In addition, when $b>0$ we let $\P\Gamma_{g,b}$ denote the [*pure Mapping Class Group*]{}; i.e. the subgroup of $\Gamma_{g,b}$ consisting of those elements that fix the punctures pointwise. The Torelli group ${\cal I}_g$ is the subgroup of $\Gamma_g$ arising as the kernel of the homomorphism $\Gamma_g \rightarrow \Sp(2g,{\bf Z})$ coming from the action of $\Gamma_g$ on $H_1(\Sigma_g,{\bf Z})$. As usual $\Out(F_n)$ will denote the Outer Automorphism Group of a free group of rank $n\geq 2$.
For a group $G$, the [*Frattini subgroup*]{} $\Phi(G)$ of $G$ is defined to be the intersection of all maximal subgroups of $G$ (if they exist), otherwise it is defined to be the group $G$ itself. (Here a maximal subgroup is a strict subgroup which is maximal with respect to inclusion.) In addition we define $\Phi_f(G)$ to be the intersection of all maximal subgroups of finite index in $G$. Note that $\Phi(G) < \Phi_f(G)$.[^3]
Frattini’s original theorem is that if $G$ is finite then $\Phi(G)=\Phi_f(G)$ is a nilpotent group (see for example [@Ro Theorem 11.5]). For infinite groups this is not the case: there are examples of finitely generated infinite groups $G$ with $\Phi(G)$ not nilpotent [@H p. 328]. On the other hand, in [@Pl] Platonov showed that if $G$ is any finitely generated linear group then $\Phi(G)$ and $\Phi_f(G)$ are nilpotent.
Motivated by the question as to whether $\Gamma_g$ is a linear group, in [@Lo], Long proved that $\Phi(\Gamma_g)=1$ for $g\geq 3$, and $\Phi(\Gamma_2)={\bf Z}/2{\bf Z}$. This was extended by Ivanov [@Iv1] who showed that (as in the linear case), $\Phi(G)$ is nilpotent for any finitely generated subgroup $G<\Gamma_{g,b}$. Regarding $\Phi_f(G)$, in [@Iv1] (and then again in [@Iv2]), Ivanov asks whether the same is true for $\Phi_f$:
[**Question:**]{} (Ivanov [@Iv1; @Iv2]) Is $\Phi_f(G)$ is nilpotent for every finitely generated subgroup $G$ of $\Gamma_{g,b}$?\
The aim of this note is to prove some results in the direction of answering Ivanov’s question. In particular, the following theorem answers Ivanov’s question in the affirmative for $\Gamma_g$ and some of its subgroups in the case where $g\geq 3$.
\[main1\] Suppose that $g\geq 3$, and that $G$ is either
1. the Mapping Class Group $\Gamma_g$, or
2. a normal subgroup of $\Gamma_g$ (for example the Torelli group ${\cal I}_g$, the Johnson kernel ${\cal K}_g$, or any higher term in the Johnson filtration of $\Gamma_g$), or
3. a subgroup of $\Gamma_g$ which contains a finite index subgroup of the Torelli group ${\cal I}_g$.
Then $\Phi_f(G)=1$.
[**Remarks:**]{} (1) Since $\Phi(G) < \Phi_f(G)$, our methods also give a different proof of Long’s result that $\Phi(\Gamma_g)=1$ for $g\geq 3$. As for the case $g\leq 2$, note that $\Gamma_1$ and $\Gamma_2$ are linear (see [@BB] for $g=2$) and so Platonov’s result [@Pl] applies to answer Ivanov’s question in the affirmative in these cases for all finitely generated subgroups. On the other hand, for $g\geq 3$ the Mapping Class Group is not known to be linear and no other technique for answering Ivanov’s question was known. In fact, as pointed out by Ivanov, neither the methods of [@Lo] or [@Iv1] apply to $\Phi_f$, and so even the case of $\Phi_f$ of the Mapping Class Group itself was not known.\
(2) Note that in Platonov’s and Ivanov’s theorems and in Ivanov’s question, the Frattini subgroup and its variant $\Phi_f$ are considered for finitely generated subgroups. In reference to Theorem \[main1\] above, it remains an open question as to whether the Johnson kernel ${\cal K}_g$, or any higher term in the Johnson filtration of $\Gamma_g$, is finitely generated or not.\
Perhaps the most interesting feature of the proof of Theorem \[main1\] is that it is another application of the projective unitary representations arising in Topological Quantum Field Theory (TQFT) first constructed by Reshetikhin and Turaev [@RT] (although as in [@MR], the perspective here is that of the skein-theoretical approach of [@BHMV]).
We are also able to prove:
\[main1add\] Assume that $b>0$, then $\Phi_f(\P\Gamma_{g,b})$ is either trivial or ${\bf Z}/2{\bf Z}$. Indeed, $\Phi_f(\P\Gamma_{g,b})=1$ unless $(g,b)\in\{(1,b),(2,b)\}$.
The reason for separating out the case when $b>0$ is that the proof does not directly use the TQFT framework, but rather makes use of Theorem \[main1\](i) in conjunction with the Birman exact sequence and a general group theoretic lemma (see §\[sec6\]). We expect that the methods of this paper will also answer Ivanov’s question for $\Gamma_{g,b}$ but at present we are unable to do so. We comment further on this at the end of §\[sec6\].
In addition our methods can also be used to give a straightforward proof of the following.
\[main2\] Suppose that $n\geq 3$, $\Phi(\Out(F_n))=\Phi_f(\Out(F_n))=1$.
Note that it was shown in [@Hu] that $\Phi(\Out(F_n))$ is finite.
As remarked upon above, this note was largely motivated by the questions of Ivanov. To that end, we discuss a possible approach to answering Ivanov’s question in general using the aforementioned projective unitary representations arising from TQFT, coupled with Platonov’s work [@Pl]. Another motivation for this work arose from attempts to understand the nature of the Frattini subgroup and the center of the profinite completion of $\Gamma_g$ and ${\cal I}_g$. We discuss these further in §\[sec7\].\
[**Acknowledgements:**]{} [*The authors wish to thank the organizers of the conference “Braids and Arithmetic” at CIRM Luminy in October 2014, where this work was completed.*]{}
Proving triviality of $\Phi_f$
==============================
Before stating and proving an elementary but useful technical result we introduce some notation. Let $\Gamma$ be a finitely generated group, and let ${\cal S}=\{G_n\}$ a collection of finite groups together with epimorphisms $\phi_n:\Gamma\rightarrow G_n$. We say that $\Gamma$ is [*residually*]{}-${\cal S}$, if given any non-trivial element $\gamma\in\Gamma$, there is some group $G_n\in{\cal S}$ and an epimorphism $\phi_n$ for which $\phi_n(\gamma)\neq 1$. Note that, as usual, this is equivalent to the statement $\bigcap \ker\phi_n = 1$.
\[tool\] Let $\Gamma$ and ${\cal S}$ be as above with $\Gamma$ being residually-${\cal S}$. Assume further that $\Phi(G_n)=1$ for every $G_n\in{\cal S}$. Then $\Phi(\Gamma)=\Phi_f(\Gamma)=1$.
Before commencing with the proof of this proposition, we recall the following property.
\[frattini\_under\_epi\] Let $\Gamma$ and $G$ be groups and $\alpha:\Gamma\rightarrow G$ an epimorphism. Then $\alpha(\Phi(\Gamma)) \subset \Phi(G)$ and $\alpha(\Phi_f(\Gamma)) \subset \Phi_f(G)$.
[**Proof:**]{} We prove the last statement. Let $M$ be a maximal subgroup of $G$ of finite index. Then $\alpha^{-1}(M)$ is a maximal subgroup of $\Gamma$ of finite index in $\Gamma$, and hence $\Phi_f(\Gamma) \subset
\alpha^{-1}(M)$. Thus $\alpha(\Phi_f(\Gamma)) \subset M$ for all maximal subgroups $M$ of finite index in $G$ and the result follows.\
[**Remark:**]{} As pointed out in §1, for a finite group $G$, $\Phi(G)=\Phi_f(G)$.\
[**Proof of Proposition \[tool\]:**]{} We give the argument for $\Phi_f(\Gamma)$, the argument for $\Phi(\Gamma)$ is exactly the same. Thus suppose that $g\in\Phi_f(\Gamma)$ is a non-trivial element. Since $\Gamma$ is residually-${\cal S}$, there exists some $n$ so that $\phi_n(g) \in G_n$ is non-trivial. However, by Lemma \[frattini\_under\_epi\] (and the remark following it) we have:
$$\phi_n(g) \in \phi_n(\Phi_f(\Gamma)) < \Phi_f(G_n)=\Phi(G_n),$$ and in particular $\Phi(G_n)\neq 1$, a contradiction.
The quantum representations and finite quotients
================================================
We briefly recall some of [@MR] (which uses [@BHMV] and [@GM]). As in [@MR] we only consider the case of $p$ a prime satisfying $p\equiv 3 \pmod 4 $.
Let $\Sigma$ be a closed orientable surface of genus $g\geq 3$. The integral $SO(3)$-TQFT constructed in [@GM] provides a representation of a central extension $\widetilde
\Gamma_g$ of $\Gamma_g$ $$\rho_p \,:\, \widetilde
\Gamma_g \longrightarrow \GL(N_g(p),{{\bf{Z}}}[\zeta_p])~,$$ where $\zeta_p$ is a primitive $p$-th root of unity, ${{\bf{Z}}}[\zeta_p]$ is the ring of cyclotomic integers and $N_g(p)$ the dimension of a vector space $V_p(\Sigma)$ on which the representation acts. It is known that $N_g(p)$ is given by a Verlinde-type formula and goes to infinity as $p\rightarrow \infty$. For convenience we simply set $N=N_g(p)$.
As in [@MR] the image group $\rho_p(\widetilde{\Gamma}_g)$ will be denoted by $\Delta_g$. As is pointed out in [@MR], $\Delta_g< \SL(N, {{\bf{Z}}}[\zeta_p])$, and moreover, $\Delta_g$ is actually contained in a special unitary group $\SU(V_p,H_p;{{\bf{Z}}}[\zeta_p])$, where $H_p$ is a Hermitian form defined over the real field ${\bf Q}(\zeta_p+\zeta_p^{-1})$.
Furthermore, the homomorphism $\rho_p$, descends to a projective representation of $\Gamma_g$ (which we denote by $\overline{\rho}_p$):
$$\overline{\rho}_p : \Gamma_g \longrightarrow \PSU(V_p,H_p;{{\bf{Z}}}[\zeta_p]),$$
What we need from [@MR] is the following. We can find infinitely many rational primes $q$ which split completely in ${{\bf{Z}}}[\zeta_p]$, and for every such prime $\tilde q$ of ${{\bf{Z}}}[\zeta_p]$ lying over such a $q$, we can consider the group $$\pi_{\tilde q}(\Delta_g) \subset \SL(N,q),$$ where $\pi_{\tilde q}$ is the reduction homomorphism from $\SL(N,{{\bf{Z}}}[\zeta_p])$ to $\SL(N,q)$ induced by the isomorphism ${{\bf{Z}}}[\zeta_p]/\tilde q\simeq {{\bf{F}}}_q$. As is shown in [@MR] (see also [@Fu]) we obtain epimorphisms $\Delta_g\twoheadrightarrow \SL(N,q)$ for all but finitely many of these primes $\tilde{q}$, and it then follows easily that we obtain epimorphisms $\Gamma_g \twoheadrightarrow \PSL(N,q)$. We denote these epimorphisms by $\rho_{p,\tilde{q}}$. These should be thought of as reducing the images of $\overline{\rho}_{p}$ modulo $\tilde{q}$. That one obtains finite simple groups of the form $\PSL$ rather than $\PSU$ when $q$ is a split prime is discussed in [@MR] §2.2.
\[residual\] For each $g\geq 3$, $\bigcap \ker\rho_{p,\tilde{q}}=1$.
[**Proof:**]{} Fix $g\geq 3$ and suppose that there exists a non-trivial element $\gamma\in \bigcap \ker\rho_{p,\tilde{q}}$. Now it follows from asymptotic faithfulness [@A; @FWW] that $\bigcap \ker\overline{\rho}_p = 1$. Thus for some $p$ there exists $\overline{\rho}_p$ such that $\overline{\rho}_p(\gamma)\neq
1$. Now $\rho_{p,\tilde{q}}(\gamma)$ is obtained by reducing $\overline{\rho}_p(\gamma)$ modulo $\tilde{q}$, and so there clearly exists $\tilde{q}$ so that $\rho_{p,\tilde{q}}(\gamma)\neq 1$, a contradiction.\
Proofs of Theorems \[main1\] and \[main2\]
==========================================
The proof of Theorem \[main1\] for $G=\Gamma_g$ follows easily as a special case of our next result. To state this, we introduce some notation: If $H<\Gamma_g$, we denote by $\widetilde H$, the inverse image of $H$ under the projection $\widetilde{\Gamma}_g \rightarrow \Gamma_g$.
\[saturating\_trivial\_frattini\] Let $g\geq 3$, and assume that $H$ is a finitely generated subgroup of $\Gamma_g$ for which $\rho_p(\widetilde H)$ has the same Zariski closure and adjoint trace field as $\Delta_g$. Then $\Phi(H)=\Phi_f(H)=1$.
[**Proof:**]{} We begin with a remark. That the homomorphisms $\rho_{p,\tilde{q}}$ of §3 are surjective is proved using Strong Approximation. The main ingredients of this are the Zariski density of $\Delta_g$ in the algebraic group $\SU(V_p,H_p)$, and the fact that the adjoint trace field of $\Delta_g$ is the field ${\bf Q}(\zeta_p+\zeta_p^{-1})$ over which the group $\SU(V_p,H_p)$ is defined (see [@MR] for more details). In particular, the proof establishes surjectivity of $\rho_{p,\tilde{q}}$ when restricted to any subgroup $H<\Gamma_g$ equipped with the hypothesis of the proposition.
To complete the proof, the groups $\PSL(N,q)$ are finite simple groups (since the dimensions $N$ are all very large) so their Frattini subgroup is trivial. This follows from Frattini’s theorem, or, more simply, from the fact that the Frattini subgroup of a finite group is a normal subgroup which is moreover a strict subgroup (since finite groups do have maximal subgroups). Hence the result follows from Lemma \[residual\], Proposition \[tool\] and the remark at the start of the proof.\
In particular, $\Gamma_g$ satisfies the hypothesis of Proposition \[saturating\_trivial\_frattini\], and so $\Phi_f(\Gamma_g)=1$. This also recovers the result of Long [@Lo] proving triviality of the Frattini subgroup.\
The proof of Theorem \[main1\] in case (ii), that is, when $G$ is a normal subgroup of $\Gamma_g$, follows from this and the following general fact:
\[fFN\] If $N $ is a normal subgroup of a group $\Gamma$, then $\Phi_f(N) < \Phi_f(\Gamma)$.
This fact is known for Frattini subgroups of finite groups, and the proof can be adapted to our situation. We defer the details to Section \[FFN\].\
In the remaining case (iii) of Theorem \[main1\], $G$ is a subgroup of $\Gamma_g$ which contains a finite index subgroup of the Torelli group ${\cal I}_g$. We shall show that $G$ satisfies the hypothesis of Proposition \[saturating\_trivial\_frattini\], and deduce $\Phi_f(G)=1$ as before.
Consider first the case where $G$ is the Torelli group ${\cal I}_g$ itself. Recall the short exact sequence
$$1 \longrightarrow {\cal I}_g \longrightarrow \Gamma_g \longrightarrow \Sp(2g,{\bf Z})\longrightarrow 1~.$$ We now use the following well-known facts.
1. $\Gamma_g$ is generated by Dehn twists, which map to transvections in $ \Sp(2g,{\bf Z})$.
2. The central extension $\widetilde \Gamma_g$ of $\Gamma_g$ is generated by certain lifts of Dehn twists, and $\rho_p$ of every such lift is a matrix of order $p$.
3. The quotient of $\Sp(2g,{\bf Z})$ by the normal subgroup generated by $p$-th powers of transvections is the finite group $\Sp(2g,{\bf Z}/p{\bf Z})$ (see [@BMS] for example).
It follows that the finite group $\Sp(2g,{\bf Z}/p{\bf Z})$ admits a surjection onto the quotient group $$\Delta_g / \rho_p(\widetilde{\cal I}_g)$$ (recall that $\Delta_g= \rho_p(\widetilde{\Gamma}_g)$) and hence the group $\rho_p(\widetilde{\cal I}_g)$ has finite index in $\Delta_g$. But the Zariski closure of $\Delta_g$ is the connected, simple, algebraic group $\SU(V_p,H_p)$. Thus $\rho_p(\widetilde{\cal I}_g)$ and $\Delta_g$ have the same Zariski closure. Again using the fact that $\SU(V_p,H_p)$ is a simple algebraic group, we also deduce that $\rho_p(\widetilde{\cal I}_g)$ has the same adjoint trace field as $\Delta_g$ (this follows from [@DM] Proposition 12.2.1 for example). This shows that ${\cal I}_g$ indeed satisfies the hypothesis of Proposition \[saturating\_trivial\_frattini\], and so once again $\Phi_f({\cal I}_g)=1$. The same arguments work when $G$ has finite index in ${\cal I}_g$, and also when $G$ is any subgroup of $\Gamma_g$ which contains a finite index subgroup of ${\cal I}_g$. This completes the proof of Theorem \[main1\].\
We now turn to the proof of Theorem \[main2\]. To deal with the case of $\Out(F_n)$, we recall that R. Gilman [@Gil] showed that for $n\geq 3$, $\Out(F_n)$ is residually alternating: i.e. in the notation of §2, the collection ${\cal S}$ consists of alternating groups.\
[**Proof of Theorem \[main2\]:**]{} For $n\geq 3$, the abelianization of $\Out(F_n)$ is ${\bf Z}/2{\bf Z}$ (as can be seen directly from Nielsen’s presentation of $\Out(F_n)$, see [@Vo] §2.1). Hence, $\Out(F_n)$ does not admit a surjection onto $A_3$ or $A_4$. Thus all the alternating quotients described by Gilman’s result above have trivial Frattini subgroups (as in the proof of Proposition \[saturating\_trivial\_frattini\]). The proof is completed using the residual alternating property and Proposition \[tool\].
Proof of Proposition {#FFN}
=====================
Let $\Gamma$ be a group and $N $ a normal subgroup of $\Gamma$. We wish to show that $\Phi_f(N) < \Phi_f(\Gamma)$. We proceed as follows.
First a preliminary observation. Let $K=\Phi_f(N)$. It is easy to see that $K$ is characteristic in $N$ (i.e., fixed by every automorphism of $N$). Since $N$ is normal in $\Gamma$, it follows that $K$ is normal in $\Gamma$. This implies that for every subgroup $M$ of $\Gamma$, the set $$KM=\{km\,|\, k\in K, m\in M\}$$ is a subgroup of $\Gamma$. Moreover, since $K < N$, we have
$$\label{eq1} KM \cap N = KM_1$$
where $M_1= M\cap N$. To see the inclusion $KM \cap N \subset KM_1$, write an element of $KM \cap N$ as $km=n$ and observe that $m\in N$ since $K<N$. Thus $m\in M_1$. The reverse inclusion is immediate.
Now suppose for a contradiction that $K =\Phi_f(N)$ is not contained in $ \Phi_f(\Gamma)$. Then there exists a maximal subgroup $M<\Gamma$ of finite index such that $K$ is not contained in $M$. Write $$M_1=M\cap N$$ as above. Then $M_1$ is a finite index subgroup of $N$. If $M_1=N$ then $N$ is contained in $M$, and hence so is $K$, which is a contradiction. Thus $M_1$ is a strict subgroup of $N$, and since its index in $N$ is finite, $M_1$ is contained in a maximal subgroup $H$, say, of $N$.
The proof is now concluded as follows. By definition, $K=\Phi_f(N)$ is also contained in $H$. Hence the group $KM_1$ is contained in $H$ and therefore strictly smaller than $N$. On the other hand, by the maximality of $M$ in $\Gamma$, we have $KM=\Gamma$, and hence, using (\[eq1\]), we have $$KM_1 = KM \cap N =\Gamma \cap N = N~.$$ This contradiction completes the proof.\
[**Remark:**]{} If we consider the original Frattini group $\Phi$ in place of $\Phi_f$, one can show similarly that $\Phi(N) < \Phi(G)$, provided that every subgroup of $N$ is contained in a maximal subgroup of $N$; e.g. when $N$ is finitely generated.
Proof of Theorem \[main1add\] {#sec6}
=============================
We begin by recalling the Birman exact sequence. Let $\Sigma_{g,b}$ denote the closed orientable surface of genus $g$ with $b$ punctures. If $b=0$ we abbreviate to $\Sigma_g$. There is a short exact sequence (the [*Birman exact sequence*]{}):
$$1\rightarrow \pi_1(\Sigma_{g,(b-1)})\rightarrow \P\Gamma_{g,b}\rightarrow \P\Gamma_{g,(b-1)}\rightarrow 1,$$
where the map $\P\Gamma_{g,b}\rightarrow \P\Gamma_{g,(b-1)}$ is the forgetful map, and the map $\pi_1(\Sigma_{g,(b-1)})\rightarrow \P\Gamma_{g,b}$ the point pushing map (see [@FM] Chapter 4.2 for details). Also in the case when $b=1$, the symbol $\P\Gamma_{g,0}$ simply denotes the Mapping Class Group $\Gamma_g$.
It will be useful to recall that an alternative description of $\P\Gamma_{g,b}$ is as the kernel of an epimorphism $\Gamma_{g,b}\rightarrow S_b$ (the symmetric group on $b$ letters).
The proof will proceed by induction, using Theorem \[main1\](i) to get started, together with the following (which is an adaptation of Lemma 3.5 of [@ABetal] to the case of $\Phi_f$). The proof is included in §\[sec.all\] below.
We introduce the following notation. Recalling §2, let $G$ be a group, say $G$ is [ *residually simple*]{} if the collection ${\cal S}=\{G_n\}$ (as in §2) consists of finite non-abelian simple groups.
\[allenby\] Let $N$ be a finitely generated normal subgroup of the group $G$ and assume that $N$ is residually simple. Then $N\cap \Phi_f(G)=1$. In particular if $\Phi_f(G/N)=1$, then $\Phi_f(G)=1$.
Given this we now complete the proof. In the cases of $(0,1)$, $(0,2)$ and $(0,3)$, it is easily seen that the subgroup $\Phi_f$ is trivial.
Thus we now assume that we are not in those cases. As is well-known, $\pi_1(\Sigma_{g,b})$ is residually simple for those surface groups under consideration, except the case of $\pi_1(\Sigma_1)$ which we deal with separately below. For example this follows by uniformization of the surface by a Fuchsian group with algebraic matrix entries and then use Strong Approximation.
Assume first that $g\geq 3$, then Theorem \[main1\](i), Lemma \[allenby\] and the Birman exact sequence immediately proves that the statement holds for $\P\Gamma_{g,1}$. The remarks above, Lemma \[allenby\] and induction then proves the result for $\P\Gamma_{g,b}$ whenever $g\geq 3$ and $b>0$.
Now assume that $g=0$. The base case of the induction here is $\P\Gamma_{0,4}$. From the above, it is easy to see that $\P\Gamma_{0,3}$ is trivial, and so $\P\Gamma_{0,4}$ is a free group of rank $2$. As such, it follows that $\Phi_f(\P\Gamma_{0,4})=1$. The remarks above, Lemma \[allenby\] and induction then proves the result for $\P\Gamma_{0,b}$ whenever $b>0$.
When $g=1$, $\Gamma_1\cong \Gamma_{1,1}\cong \SL(2,{\bf Z})$ and it is easy to check that $\Phi_f(\SL(2,{\bf Z}))={\bf Z}/2{\bf Z}$ (coinciding with the center of $\SL(2,{\bf Z})$). Now $\P\Gamma_{1,1}=\Gamma_{1,1}$ and so these facts together with Lemma \[allenby\] and induction then prove the result (i.e. that $\Phi_f(\P\Gamma_{1,b})$ is either trivial or ${\bf Z}/2{\bf Z}$).
In the case of $g=2$, by [@BB] $\Gamma_2$ is linear, and so [@Pl] also proves that $\Phi_f(\Gamma_2)$ is nilpotent. We claim that this forces $\Phi_f(\Gamma_2)={\bf Z}/2{\bf Z}$. To see this we argue as follows.
If $\Phi_f(\Gamma_2)$ is finite, it is central by [@Lo] Lemma 2.2. Since $\Phi_f(\Gamma_2)$ contains $\Phi(\Gamma_2)$, which is equal to the center ${\bf Z}/2{\bf Z}$ of $\Gamma_2$ by [@Lo] Theorem 3.2, it follows that $\Phi_f(\Gamma_2) = {\bf Z}/2{\bf Z}$.
Thus it is enough to show that $\Phi_f(\Gamma_2)$ is finite. Assume that it is not. Then by [@Lo] Lemma 2.5, $\Phi_f(\Gamma_2)$ contains a pseudo-Anosov element. Indeed, [@Lo] Lemma 2.6 shows that the set of invariant laminations of pseudo-Anosov elements in $\Phi_f(\Gamma_2)$ is dense in projective measured lamination space. This contradicts $\Phi_f(\Gamma_2)$ being nilpotent (e.g. the argument of [@Lo] p. 86 constructs a free subgroup).
As before, using Lemma \[allenby\] and by induction via the Birman exact sequence, we can now handle the cases of $\Gamma_{2,b}$ with $b>0$.\
[**Remark 1:**]{} Recall that the [*hyperelliptic Mapping Class Group*]{} (which we denote by $\Gamma_g^h$) is defined to be the subgroup of $\Gamma_g$ consisting of those elements that commute with a fixed hyperelliptic involution. It is pointed out in [@BB] p. 706, that the arguments used in [@BB] prove that $\Gamma_g^h$ is linear. Hence once again $\Phi_f(G)$ is nilpotent for every finitely generated subgroup $G$ of $\Gamma_g^h$.\
[**Remark 2:**]{} We make some comments on the case of $\Gamma_{g,b}$ with $b>0$. First, since $\P\Gamma_{g,b}=\ker\{\Gamma_{g,b}\rightarrow S_b\}$ and $\Phi_f(S_b)=1$, if $\P\Gamma_{g,b}$ were known to be residually simple then the argument in the proof of Theorem \[main1add\] could be used to show that $\Phi_f(\Gamma_{g,b})=1$. Hence we raise here:\
[**Question:**]{} [*Is $\P\Gamma_{g,b}$ residually simple?*]{}\
Another approach to showing that $\Phi_f(\Gamma_{g,b})=1$ is to directly use the representations arising from TQFT. In this case the result of Larsen and Wang [@LW] that allows us to prove Zariski density in [@MR] needs to be established. Given this, the proof (for most $(g,b)$) would then follow as above.
Proof of Lemma {#sec.all}
===============
As already mentioned, in what follows we adapt the proof of Lemma 3.5 of [@ABetal] to the case of $\Phi_f$.
We argue by contradiction and assume that there exists a non-trivial element $x\in N\cap
\Phi_f(G)$. By the residually simple assumption, we can find a non-abelian finite simple group $S_0$ and an epimorphism $f:N\rightarrow S_0$ for which $f(x)\neq 1$. Set $K_0=\ker~f$ and let $K_0,K_1\ldots, K_n$ be the distinct copies of $K_0$ which arise on mapping $K_0$ under the automorphism group of $N$ (this set being finite since $N$ is finitely generated). Set $K=\bigcap K_i$, a characteristic subgroup of finite index in $N$. As in [@ABetal], it follows from standard finite group theory that $N/K$ is isomorphic to a direct product of finite simple groups (all of which are isomorphic to $S_0=N/K_0$).
Now $K$ being characteristic in $N$ implies that $K$ is a normal subgroup of $G$. Put $G_1=G/K$ and let $f_1:G\rightarrow G_1$ denote the canonical homomorphism. Also write $N_1$ for $N/K=f_1(N)$. Now $f_1(x) \in N_1$ and $f_1(x)\in f_1(\Phi_f(G))$ which by Lemma \[frattini\_under\_epi\] implies that $f_1(x)\in
\Phi_f(G_1)$. Hence $$f_1(x) \in N_1\cap \Phi_f(G_1)~.$$ Following [@ABetal], let $C$ denote the centralizer of $N_1$ in $G_1$, and as in [@ABetal], we can deduce various properties about the groups $N_1$ and $C$. Namely:
\(i) since $N_1$ is a finite group, its centralizer $C$ in $G_1$ is of finite index in $G_1$ .
\(ii) since $N_1$ is a product of non-abelian finite simple groups it has trivial center, and so $C\cap N_1=1$.
\(iii) since $N_1$ is normal in $G_1$, $C$ is normal in $G_1$.
Using (iii), put $G_2=G_1/C$ and let $f_2:G_1\rightarrow G_2$ denote the canonical homomorphism. Also write $N_2$ for $f_2(N_1)$. Arguing as before (again invoking Lemma \[frattini\_under\_epi\]), we have $$f_2(f_1(x)) \in N_2\cap \Phi_f(G_2)~.$$ Moreover, since $f_1(x)\in N_1$ and $f_1(x)\neq 1$ by construction, we have from (ii) that $f_2f_1(x)\neq 1$. Thus the intersection $$H:= N_2\cap \Phi_f(G_2)$$ is a non-trivial group.
As in [@ABetal], we will now get a contradiction by showing that $H$ is both a nilpotent group and a direct product of non-abelian finite simple groups, which is possible only if $H$ is trivial. Here is the argument. From (i) above we deduce that $G_2$ is a finite group, hence $\Phi_f(G_2)=\Phi(G_2)$ is nilpotent by Frattini’s theorem. Thus $H<\Phi_f(G_2)$ is nilpotent. On the other hand, $N_2$ is a quotient of $N_1$ and hence a direct product of non-abelian finite simple groups. But $H$ is normal in $N_2$ (since $\Phi_f(G_2)$ is normal in $G_2$). Thus $H$ is a direct product of non-abelian finite simple groups.
This contradiction shows that $N\cap\Phi_f(G)=1$, which was the first assertion of the Lemma. The second assertion of the lemma now follows from Lemma \[frattini\_under\_epi\]. This completes the proof.\
Final comments {#sec7}
==============
An approach to Ivanov’s question
--------------------------------
We will now discuss an approach to answering Ivanov’s question (i.e. the nilpotency of $\Phi_f(G)$ for finitely generated subgroups $G$ of $\Gamma_g$) using the projective unitary representations described in §3. In the remainder of this section $G$ is an infinite, finitely generated subgroup of $\Gamma_g$.\
The following conjecture is the starting point to this approach. Recall that a subgroup $G$ of $\Gamma_g$ is [*reducible*]{} if there is a collection of essential simple closed curves $C$ on the surface $\Sigma_g$, such that for any $\beta\in G$ there is a diffeomorphism $\overline{\beta}:\Sigma_g\rightarrow \Sigma_g$ in the isotopy class of $\beta$ so that $\overline{\beta}(C)=C$. Otherwise $G$ is called [*irreducible*]{}. As shown in [@Iv1] Theorem 2 an irreducible subgroup $G$ is either virtually an infinite cyclic subgroup generated by a pseudo-Anosov element, or, $G$ contains a free subgroup of rank $2$ generated by a two pseudo-Anosov elements.\
[**Conjecture:**]{} [*If $G$ is a finitely generated irreducible non-virtually cyclic subgroup of $\Gamma_g$, then $\Phi_f(G)=1$.*]{}\
The motivation for this conjecture is that the irreducible (non-virtually cyclic) hypothesis should be enough to guarantee that the image group $\rho_p(\widetilde{G}) <\Delta_g$ is Zariski dense (with the same adjoint trace-field). Roughly speaking the irreducibility hypothesis should ensure that there is no reason for Zariski density to fail (i.e. the image is sufficiently complicated). Indeed, in this regard, we note that an emerging theme in linear groups is that random subgroups of linear groups are Zariski dense (see [@Ao] and [@Ri] for example). Below we discuss a possible approach to proving the Conjecture.
The idea now is to follow Ivanov’s proof in [@Iv1] that the Frattini subgroup is nilpotent. Very briefly if the subgroup is reducible then we first identify $\Phi_f$ on the pieces and then build up to identify $\Phi_f(G)$. In Ivanov’s argument, this involves passing to certain subgroups of $G$ (“pure subgroups”), understanding the Frattini subgroup of these pure subgroups when restricted to the connected components of $S\setminus C$, and then building $\Phi(G)$ from this information. This uses several statements about the Frattini subgroup, at least one of which (Part (iv) of Lemma 10.2 of [@Iv1]) does not seem to easily extend to $\Phi_f$.\
[**Remark:**]{} As a cautionary note to the previous discussion, at present, it still remains conjectural that the image of a fixed pseudo-Anosov element of $\Gamma_g$ under the representations $\overline{\rho}_p$ is infinite order for big enough $p$ (which was raised in [@AMU]).\
[**An approach to the Conjecture:**]{}\
We begin by recalling that in [@Pl] Platonov also proves that $\Phi_f(H)$ is nilpotent for every finitely generated linear group $H$. Note that if $G$ is irreducible and virtually infinite cyclic then $G$ is a linear group, and so [@Pl] implies that $\Phi_f(G)$ is nilpotent.
Thus we now assume that $G$ is irreducible as in the conjecture. Consider $\rho_p(\widetilde{\Phi_f(G)})$: by Lemma \[frattini\_under\_epi\] above we deduce that $\rho_p(\widetilde{\Phi_f(G)})$ is a nilpotent normal subgroup of $\rho_p(\widetilde{G})$. Now $\overline{\rho}_p(\Gamma_g)<\PSU(V_p,H_p;{{\bf{Z}}}[\zeta_p])$ and it follows from this that (in the notation of §3) $\Delta_g < \Lambda_p=\SU(V_p,H_p;{{\bf{Z}}}[\zeta_p])$. As discussed in [@MR], $\Lambda_p$ is a cocompact arithmetic lattice in the algebraic group $\SU(V_p,H_p)$. Thus $\rho_p(\widetilde{\Phi_f(G)})< \rho_p(\widetilde{G}) < \Lambda_p$. It follows from general properties of cocompact lattices acting on symmetric spaces (see e.g. [@Eb] Proposition 10.3.7) that $\rho_p(\widetilde{\Phi_f(G)})$ contains a maximal normal abelian subgroup of finite index. Now there is a general bound on the index of this abelian subgroup that is a function of the dimension $N_g(p)$. However, in our setting, if the index can be bounded by some fixed constant $R$ independent of $N_g(p)$, then we claim that $\Phi_f(G)$ can at least be shown to be finite. To see this we argue as follows.
Assume that $\Phi_f(G)$ is infinite. Since $G$ is an irreducible subgroup containing a free subgroup generated by a pair of pseudo-Anosov elements, the same holds for the infinite normal subgroup $\Phi_f(G)$ (by standard dynamical properties of pseudo-Anosov elements, see for example [@Lo] pp. 83–84).
Thus we can find $x,y\in \Phi_f(G)$ a pair of non-commuting pseudo-Anosov elements. Also note that $[x^t,y^t]\neq 1$ for all non-zero integers $t$. From Lemma \[frattini\_under\_epi\] we have that $\rho_p(\widetilde{\Phi_f(G)}) < \Phi_f(\rho_p(\widetilde{G}))$ and from the assumption above it therefore follows that $\rho_p(\widetilde{\Phi_f(G)})$ contains a maximal normal abelian subgroup $A_p$ of index bounded by $R$ (independent of $p$). Thus, setting $R_1=R!$, we have $[\overline{\rho}_p(x^{R_1}),\overline{\rho}_p(y^{R_1})]=1$ for all $p$. However, as noted above, $[x^{R_1},y^{R_1}]$ is a non-trivial element of $G$, and by asymptotic faithfulness this cannot be mapped trivially for all $p$. This is a contradiction.\
The profinite completion of $\Gamma_g$
--------------------------------------
We remind the reader that the profinite completion $\widehat{\Gamma}$ of a group $\Gamma$ is the inverse limit of the finite quotients $\Gamma/N$ of $\Gamma$. (The maps in the inverse system are the obvious ones: if $N_1
<N_2$ then $\Gamma/N_1\rightarrow \Gamma/N_2$.) The Frattini subgroup $\Phi(G)$ of a profinite group $G$ is defined to be the intersection of all maximal open subgroups of $G$. Open subgroups are of finite index, and if $G$ is finitely generated as a profinite group, then Nikolov and Segal [@NS] show that finite index subgroups are always open. Hence we can simply take $\Phi(G)$ to be the intersection of all maximal finite index subgroups of $G$.
Now if $\Gamma$ is a finitely generated residually finite discrete group, the correspondence theorem between finite index subgroups of $\Gamma$ and its profinite completion (see [@RZ] Proposition 3.2.2) shows that $\overline{\Phi_f(\Gamma)} < \Phi(\widehat{\Gamma})$.
There is a well-known connection between the center of a group $G$, denoted $Z(G)$ (profinite or otherwise), and $\Phi(G)$. We include a proof for completeness. Note that for a profinite group $\Phi(G)$ is a closed subgroup of $G$, $Z(G)$ is a closed subgroup and by [@NS] the commutator subgroup $[G,G]$ is a closed subgroup.
\[profinitecenter\] Let $G$ be a finitely generated profinite group. Then $\Phi(G) > Z(G) \cap [G,G]$.
[**Proof:**]{} Let $U$ be a maximal finite index subgroup of $G$, and assume that $Z(G)$ is not contained in $U$. Then $<Z(G),U>=G$ by maximality. It also easily follows that $U$ is a normal subgroup of $G$. But then $G/U = Z(G)U/U \cong Z(G)/(U\cap Z(G))$ which is abelian, and so $[G,G]<U$. This being true for every maximal finite index subgroup $U$ we deduce that $\Phi(G) > Z(G) \cap [G,G]$ as required.\
We now turn to the following questions which were also part of the motivation of this note.\
[**Question 1:**]{} [*For $g\geq 3$, is $Z(\widehat{\Gamma}_g)=1$?*]{}
[**Question 2:**]{} [*For $g\geq 3$, is $Z(\widehat{\cal I}_g)=1$?*]{}\
Regarding Question 1, it is shown in [@HM] that the completion of $\Gamma_g$ arising from the congruence topology on $\Gamma_g$ has trivial center. Regarding Question 2, if $Z(\widehat{\cal I}_g)=1$, then the profinite topology on $\Gamma_g$ will induce the full profinite topology on ${\cal I}_g$ (see [@LS] Lemma 2.6). Motivated by this and Lemma \[profinitecenter\] we can also ask:\
[**Question 1’:**]{} [*For $g\geq 3$, is $\Phi(\widehat{\Gamma}_g)=1$?*]{}
[**Question 2’:**]{} [*For $g\geq 3$, is $\Phi(\widehat{\cal I}_g)=1$?*]{}\
Although the results in this paper do not impact directly on Questions 1, 1’,2 and 2’, we note that since $\Gamma_g$ is finitely generated and perfect for $g\geq 3$, it follows that $\widehat{\Gamma}_g$ is also perfect and hence $$Z(\widehat{\Gamma}_g)<\Phi(\widehat{\Gamma}_g)$$ by Lemma \[profinitecenter\]. As remarked above, the correspondence theorem gives $$\overline{\Phi_f(\Gamma_g)} < \Phi(\widehat{\Gamma}_g)~.$$ Thus our result that $\Phi_f(\Gamma_g)=1$ for $g\geq 3$ (which implies $\overline{\Phi_f(\Gamma_g)}=1$) is consistent with triviality of $Z(\widehat{\Gamma}_g)$ (and similarly for $Z(\widehat{\cal I}_g)$).
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Institut de Math[é]{}matiques de Jussieu-PRG (UMR 7586 du CNRS),\
Equipe Topologie et G[é]{}om[é]{}trie Alg[é]{}briques,\
Case 247, 4 pl. Jussieu,\
75252 Paris Cedex 5, France.\
Email: gregor.masbaum@imj-prg.fr\
Department of Mathematics,\
University of Texas\
Austin, TX 78712, USA.\
Email: areid@math.utexas.edu
[^1]: This work was partially supported by the N. S. F.
[^2]: 2000 MSC Classification: 20F38, 57R56
[^3]: We use the notation $G_1<G_2$ to indicate that $G_1$ is a subgroup of $G_2$ (including the case where $G_1=G_2$).
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abstract: 'We give general criteria under which the limit of a system of tropicalizations of a scheme over a nonarchimedean field is homeomorphic to the analytification of the scheme. As an application, we show that the analytification of an arbitrary closed subscheme of a toric variety is naturally homeomorphic to the limit of its tropicalizations, generalizing an earlier result of the third author for quasiprojective varieties.'
address:
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Yale University Mathematics Department\
10 Hillhouse Ave\
New Haven, CT 06511\
U.S.A.
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Fakultät für Mathematik\
Universitätsstr. 1\
40225 DŸsseldorf\
Germany
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Yale University Mathematics Department\
10 Hillhouse Ave\
New Haven, CT 06511\
U.S.A.
author:
- Tyler Foster
- Philipp Gross
- Sam Payne
bibliography:
- 'math.bib'
title: Limits of tropicalizations
---
Introduction
============
Let $K$ be a field that is complete with respect to a nonarchimedean valuation, and let $X$ be a scheme of finite type over $K$. The analytification $X^\an$, in the sense of Berkovich [@Berkovich90], is a locally compact, locally contractible Hausdorff space that admits a natural proper, continuous projection onto the tropicalization of the image of any closed embedding of $X$ into a toric variety. These projections are compatible with maps between tropicalizations induced by toric morphisms. Therefore, for any inverse system $\cS$ of toric embeddings of $X$ there is a natural map to the corresponding limit of tropicalizations $$\pi_\cS : X^\an \rightarrow \varprojlim_{\iota \in \cS} \operatorname{Trop}(X, \iota).$$ Our first main technical result is a sufficient condition on the system $\cS$ for $\pi_\cS$ to be a homeomorphism.
\[thm:sufficient\] Let $\cS$ be a system of toric embeddings that contains finite products. Suppose there is an affine open cover $X = U_1 \cup \cdots \cup U_r$ with the following property, for $1 \leq i \leq r$:
- For any nonzero regular function $f \in K[U_i]$, there is an embedding $\iota \in \cS$ such that $U_i$ is the preimage of a torus invariant open set and $f$ is the pullback of a monomial.
Then $\pi_\cS$ is a homeomorphism.
The closed embeddings of a quasiprojective variety into torus invariant open subsets of projective space satisfy $(\star)$. This is proved using properties of ample line bundles in [@analytification Lemma 4.3]. The main result of that paper, which says that $\pi_\cS$ is a homeomorphism when $X$ is a quasiprojective variety and $\cS$ is the system of all of its closed embeddings into quasiprojective toric varieties, then follows easily. Theorem \[thm:sufficient\] is significantly more general because it applies equally to schemes that are reducible, non reduced, and non quasiprojective, and because it allows some flexibility in the choice of the system $\cS$. For instance, if $X$ is a quasiprojective variety, then one could replace the system of all closed embeddings into quasiprojective toric varieties with a smaller system, such as embeddings into invariant open subsets of products of projective spaces, or a larger system, such as all closed embeddings into toric varieties.
Theorem \[thm:sufficient\] makes precise the idea that the analytification of a scheme with enough toric embeddings should be homeomorphic to the limit of its tropicalizations. Ample line bundles produce enough toric embeddings on quasiprojective varieties, but there are many toric varieties and subschemes of toric varieties that have no nontrivial line bundles at all, for which such methods do not apply. See [@Eikelberg92 Example 3.2], [@Fulton93 pp. 25–26, 72], and the examples studied in [@branchedcovers]. Nevertheless, in Section \[sec:embeddings\] we show that any scheme that admits a single closed embedding into a toric variety also admits enough embeddings to satisfy $(\star)$.
\[thm:subscheme\] Let $X$ be a closed subscheme of a toric variety over $K$. Then the natural map from $X^\an$ to the limit of the inverse system of tropicalizations of all toric embeddings of $X$ is a homeomorphism.
In order to prove Theorem \[thm:subscheme\], we must show that an arbitrary closed subscheme of a toric variety admits many closed embeddings into toric varieties. The subject of closed embeddings in toric varieties has been studied intensively by several authors [@Wlodarczyk93; @Cox95b; @Hausen00; @Hausen02; @BerchtoldHausen02; @HausenSchroer04]. We recall the first fundamental result in this subject.
Let $X$ be a normal variety. Then $X$ admits a closed embedding into a toric variety if and only if any two points of $X$ are contained in an affine open subvariety.
The two point condition in Włodarczyk’s Embedding Theorem is necessary because a toric variety is covered by invariant affine opens and the union of any two invariant affine opens in a toric variety is quasiprojective. Therefore, if $X$ is a closed subscheme of a toric variety then any two points of $X$ are contained in a quasiprojective open subset of $X$, and hence share an affine open neighborhood.
Włodarczyk’s proof of this theorem gives a flexible algorithm to construct embeddings with desirable properties. For instance, he shows that if $X$ is smooth then the embedding can be constructed so that the ambient toric variety is also smooth. However, the theorem does not apply to non normal schemes, and the arguments in the proof do not generalize easily to this case; it remains an open problem to characterize the non normal schemes that admit closed embeddings into toric varieties, even if one allows embeddings into non normal toric varieties.
Our proof of Theorem \[thm:subscheme\] does not apply the theory of toric embeddings to $X$ directly. Instead, given a closed subscheme $X$ of a toric variety $Y$, we apply the algorithm from W[ł]{}odarczyk’s proof of his embedding theorem to the ambient toric variety and show that $(\star)$ holds for the system of toric embeddings of $X$ that factor through closed embeddings of $Y$. The same then holds for any larger system of embeddings that contains finite products, including the inverse system of all toric embeddings of $X$.
Alternative approaches to understanding the topology of analytifications of varieties using limits of polyhedral complexes involve skeletons of semistable or pluristable formal models [@Berkovich04; @KontsevichSoibelman06], or spaces of definable types [@HrushovskiLoeser10]. Such methods produce inverse systems with desirable properties, e.g., in which every map of polyhedra is a strong deformation retraction, but require far more technical machinery.
**Acknowledgments.** We thank Melody Chan and the referee for pointing out an error in an earlier version of this paper. The work of SP is partially supported by NSF DMS-1068689 and NSF CAREER DMS-1149054.
Preliminaries
=============
Throughout, we work over a field $K$ that is complete with respect to a nonarchimedean valuation, which may be trivial. Let $N \cong \Z^n$ be a lattice, let $M = \operatorname{Hom}(N,\Z)$ be its dual lattice, and let $T = \operatorname{Spec}K[M]$ be the torus with character lattice $M$.
Affine toric varieties with dense torus $T$ correspond naturally and bijectively with rational polyhedral cones in the real vector space $N_\RR = N \otimes_\Z \RR$. For such a rational polyhedral cone $\sigma$, we write $S_\sigma$ for the additive monoid of lattice points in $M$ that are nonnegative on $\sigma$. Then the corresponding affine toric variety is $$U_\sigma = \operatorname{Spec}K[S_\sigma].$$ We write $u$ for a lattice point in $M$ and $\bx^u$ for the corresponding regular function and refer to the scalar multiples $a\bx^u$, for $a \in K^*$, as monomials.
The vector space $N_\RR$ is the usual tropicalization of the dense torus $T$. Tropicalizations of more general toric varieties were studied implicitly by Thuillier, as skeletons of analytifications [@Thuillier07], and explicitly by Kajiwara [@Kajiwara08], before being applied to the development of explicit relations between tropical and nonarchimedean analytic geometry in [@analytification; @BPR11; @Rabinoff12], to which we refer the reader for further details. The tropicalization of the affine toric variety $U_\sigma$ is the space of monoid homomorphisms $$N_\R(\sigma) = \operatorname{Hom}(S_\sigma, \R),$$ where $\R$ is the additive monoid $\RR \cup \{ + \infty \}$. This space of monoid homomorphisms is a partial compactification of $N_\RR \cong \operatorname{Hom}(S_\sigma, \RR)$, just as $U_\sigma$ is a partial compactification of $T$.
Recall that the analytification $U^\an$ of an affine scheme over $K$ is the set of ring valuations $$\eta: K[U] \rightarrow \R,$$ that extend the given valuation on $K$, equipped with the subspace topology for the natural inclusion of $U^\an$ in $\R^{K[U]}$. This is the coarsest topology such that the map to $\R$ induced by each function $f$ in $K[U]$ is continuous. The tropicalization map for the affine variety $U_\sigma$ is the natural projection $$\operatorname{Trop}: U_\sigma^\an \rightarrow N_\R(\sigma)$$ taking a valuation $\eta$ to the monoid homomorphism $[u \mapsto \eta(\bx^u)]$. This construction generalizes to arbitrary toric varieties and their subvarieties as follows.
Let $\Delta$ be a fan in $N_\RR$, and let $Y_\Delta$ be the corresponding toric variety with dense torus $T$, so $Y_\Delta$ is the union of the affine toric varieties $U_\sigma$ corresponding to cones $\sigma$ in $\Delta$. Then $Y_\Delta^\an$ is the union of the corresponding subsets $U_\sigma^\an$, which are again open [@Berkovich90 Proposition 3.4.6], and the spaces $N_\R(\sigma)$ glue together to give $N_\R(\Delta)$ with a natural map $$\operatorname{Trop}: Y_\Delta^\an \rightarrow N_\R(\Delta)$$ whose restriction to each $U_\sigma^\an$ is the map $\operatorname{Trop}$ described above. This global tropicalization map is continuous, surjective, and proper in the sense that the preimage of a compact set is compact [@analytification Lemma 2.1]. In particular, it is a closed map. Now, if $X$ is a scheme of finite type over $K$ and $\iota: X \hookrightarrow Y_\Delta$ is a closed embedding, then $\iota^\an: X^\an \hookrightarrow Y_\Delta^\an$ is also a closed embedding [@Berkovich90 Proposition 3.4.7]. Therefore, the image $\operatorname{Trop}(X, \iota)$ of $X^\an$ is closed in $N_\R(\Delta)$.
Tropicalization maps are covariantly functorial; an equivariant morphism of toric varieties $\varphi: Y_\Delta \rightarrow Y_{\Delta'}$ induces a continuous map $\operatorname{Trop}(\varphi): N_\R(\Delta) \rightarrow N_\R(\Delta')$ that forms a commutative diagram with the tropicalization maps and the analytification of $\varphi$. Now, suppose $$\iota: X \hookrightarrow Y_\Delta \mbox{ and } \jmath: X \hookrightarrow Y_{\Delta'}$$ are closed embeddings. We say that $\varphi$ is a morphism of embeddings if $\varphi \circ \iota = \jmath$. In this case, $\operatorname{Trop}(\varphi)$ maps $\operatorname{Trop}(X,\iota)$ onto $\operatorname{Trop}(X, \jmath)$.
A system $\cS$ of toric embeddings of a scheme $X$ of finite type over $K$ is a diagram of closed embeddings of $X$ in toric varieties, with arrows given by morphisms of toric embeddings. By the functoriality properties of tropicalization discussed above, there is a corresponding diagram of tropicalizations, whose objects are the spaces $\operatorname{Trop}(X, \iota)$ for $\iota \in \cS$, and whose arrows are the tropicalizations of the arrows in $\cS$. We say that $\cS$ contains finite products if, for any embeddings $\iota$ and $\iota'$ in $\cS$, the product $\iota \times \iota'$ is in $\cS$, along with the natural projections from $\iota \times \iota'$ to $\iota$ and $\iota'$.
Limits of systems of tropicalizations
=====================================
We now prove Theorem \[thm:sufficient\], which says that a system $\cS$ of toric embeddings of $X$ that contains finite products and satisfies $(\star)$ induces a homeomorphism $\pi_\cS$ from $X^\an$ to the limit of the corresponding system of tropicalizations.
\[prop:surj\] Let $\cS$ be a system of toric embeddings of $X$ that contains finite products. Then $\pi_\cS$ is surjective.
Let $x = (x_\iota)_{\iota \in \cS}$ be a point in the inverse limit over $\iota \in \cS$ of $\operatorname{Trop}(X, \iota)$. We must show that the fiber $$\pi_\cS^{-1}(x) = \bigcap_{\iota \in \cS} \operatorname{Trop}^{-1}(x_\iota)$$ is nonempty.
Fix some $\jmath$ in $\cS$. Then $\operatorname{Trop}^{-1}(x_\jmath)$ is a nonempty compact subset of $X^\an$ that contains $\pi_\cS^{-1}(x)$. For any finite collection of embeddings $\iota_1, \ldots \iota_s$ in $\cS$, the intersection $$\operatorname{Trop}^{-1}(x_{\iota_1}) \cap \cdots \cap \operatorname{Trop}^{-1}(x_{\iota_s}) \cap \operatorname{Trop}^{-1}(x_\jmath)$$ is nonempty, since it contains $\operatorname{Trop}^{-1}(x_{\iota_1 \times \cdots \times \iota_s \times \jmath})$. Since each of these finite intersections inside the compact set $\operatorname{Trop}^{-1}(x_\jmath)$ is nonempty, it follows that the full intersection $\bigcap_{\iota \in \cS} \operatorname{Trop}^{-1}(x_\iota)$ is nonempty, as required.
\[prop:inj\] Let $\cS$ be a system of toric embeddings of $X$ that satisfies $(\star)$. Then $\pi_\cS$ is injective.
Suppose $\cS$ satisfies $(\star)$ and let $\eta$ and $\eta'$ be distinct points in $X^\an$. We must show that there is an embedding $\iota$ in $\cS$ such that the images of $\eta$ and $\eta'$ are distinct in $\operatorname{Trop}(X, \iota)$. Fix an affine open cover $U_1, \ldots, U_r$ of $X$ such that, for $1 \leq i \leq r$ and $f \in K[U_i]$, there is a toric embedding of $X$ such that $U_i$ is the preimage of a torus invariant open set and $f$ is the pullback of a monomial.
Choose $i$ such that $\eta$ is in $U_i^\an$, and suppose $\eta'$ is not in $U_i^\an$. Let $\iota$ be a toric embedding such that $U_i$ is the preimage of a torus invariant open subset $U_{\sigma_1} \cup \cdots \cup U_{\sigma_s}$. Then $\pi_\iota(\eta)$ is in the open subset $N_\R(\sigma_1) \cup \cdots \cup N_\R(\sigma_s)$ of $N_\R(\Delta)$, but $\pi_\iota(\eta')$ is not.
Otherwise, $\eta$ and $\eta'$ are distinct points in $U_i^\an$, corresponding to distinct ring valuations on $K[U_i]$. Let $f$ be a regular function in $K[U_i]$ such that $\eta(f)$ is not equal to $\eta'(f)$. Choose a toric embedding $\jmath$ of $X$ such that $U_i$ is the preimage of an invariant open set and $f$ is the pullback of a monomial. Say $U_\sigma$ is an invariant affine open whose analytification contains the image of $\eta$. If $U_\sigma^\an$ does not contain the image of $\eta'$ then $\pi_\iota(\eta)$ is in $N_\R(\sigma)$ but $\pi_\iota(\eta')$ is not.
It remains to consider the case where $U_\sigma^\an$ contains both $\iota(\eta)$ and $\iota(\eta')$. Then there is a nonzero scalar $a \in K^*$ such that $af$ is the pullback of a character $\bx^u$, and $\pi_\jmath(\eta)$ and $\pi_\jmath(\eta')$ are monoid homomorphisms from $S_\sigma$ to $\R$ that take $u$ to $\eta(f) + \operatorname{val}(a)$ and $\eta'(f) + \operatorname{val}(a)$, respectively. In particular, $\pi_\jmath(\eta)$ and $\pi_\jmath(\eta')$ are distinct, as required.
Finally, we show that existence of finite products and property $(\star)$ are together enough to guarantee that $\pi_\cS$ is a homeomorphism.
Let $\cS$ be a system of toric embeddings of $X$ that contains finite products and satisfies $(\star)$. By Propositions \[prop:surj\] and \[prop:inj\] the induced map $$\pi_\cS: X^\an \rightarrow \varprojlim_{\iota \in \cS} \operatorname{Trop}(X, \iota)$$ is a continuous bijection.
Choose an open cover $U_1, \ldots, U_r$ verifying property $(\star)$, let $f$ be a regular function in $K[U_i]$, and let $\iota$ be a toric embedding such that $U_i$ is the preimage of an invariant open subset $U$ and $f$ is the pullback of a monomial. Say $U$ is the union of invariant affine opens $U_{\sigma_1} \cup \cdots \cup U_{\sigma_s}$, and let $\cU$ be the preimage of $N_\R(\sigma_1) \cup \cdots \cup N_\R(\sigma_s)$ in $\varprojlim \operatorname{Trop}(X, \iota)$. So $\cU$ is exactly the image of $U_i^\an$ under $\pi_\cS$. Now, suppose $f$ is the pullback of the monomial $a\bx^u$. The topology on $N_\R(\sigma_1) \cup \cdots \cup N_\R(\sigma_s)$ is the coarsest such that the map $$\pi_f : N_\R(\sigma_1) \cup \cdots \cup N_\R(\sigma_s) \rightarrow \R$$ taking a monoid homomorphism $\phi: S_{\sigma_i} \rightarrow \R$ to $\phi(u) + \operatorname{val}(a)$ is continuous, for all such $f$. Therefore, the composite map $\cU \rightarrow \R$ is continuous as well. The map $U_i^\an \rightarrow \R$ taking a valuation $\eta$ to $\eta(f)$ is also continuous, and factors through $\pi_f$. Moreover, the topology on $U_i^\an$ is the coarsest such that all such maps to $\R$ are continuous. It follows that $\pi_\cS$ restricts to a homeomorphism from $U_i^\an$ to $\cU$. Since $\pi_\cS$ is bijective and the $U_i^\an$ form an open cover of $X^\an$, it follows that $\pi_\cS$ is a homeomorphism, as required.
Proof of Theorem \[thm:subscheme\] {#sec:embeddings}
==================================
For reducible and non reduced schemes, it is natural to look for a condition such as $(\star)$ phrased in terms of affine covers and regular functions. However, for irreducible varieties it is often easier and more convenient to avoid dealing with open covers and domains of regularity by working directly with rational functions. We will prove Theorem \[thm:subscheme\] using the following observation: when a scheme $X$ is embedded in an irreducible variety $Y$, one can use rational functions on $Y$ to produce embeddings of $X$ that factor through embeddings of $Y$.
\[prop:rationalFunctions\] Let $Y$ be a variety over $K$ and let $\cS$ be a system of toric embeddings of $Y$ that contains finite products. Suppose that for each rational function $f$ in $K(Y)^*$ there is an embedding $\iota$ in $\cS$ such that $f$ is the pullback of a monomial. Then the restriction of $\cS$ to $X$ satisfies $(\star)$.
In particular, when the hypotheses of the proposition are satisfied then the restriction of $\pi_\cS$ to $X^\an$ is a homeomorphism onto $\displaystyle{\varprojlim_{\iota \in \cS}} \operatorname{\bf Trop}(X, \iota)$.
We show that the restriction of $\cS$ to $X$ satisfies $(\star)$. Let $V_1, \ldots, V_r$ be an affine open cover of $Y$, and let $$U_i = X \cap V_i,$$ so $U_1, \ldots, U_r$ is an affine open cover of $X$. Since $X$ is closed in $Y$, the restriction map on coordinate rings $K[V_i] \rightarrow K[U_i]$ is surjective. Let $f \in K[U_i]$ be a regular function, and choose $g \in K[V_i]$ that restricts to $f$. We must show that there is an embedding $\iota \in \cS$ such that $U_i$ is the preimage in $X$ of an invariant open subset $U$ and $f$ is the pullback of a monomial.
For $j \neq i$, choose functions $$g_{j1}, \ldots, g_{js}$$ in $K[V_j]$ that generate the ideal of the closed subvariety $Y \smallsetminus V_i$ on $V_j$. By hypothesis, there exists an embedding $\iota$ in $\cS$ such that $g$ is the pullback of a monomial, and embeddings $\iota_{jk}$ such that $g_{jk}$ is the pullback of a monomial. We consider the product embedding $$\jmath = \iota \times \Big( \prod_{j,k} \iota_{jk} \Big),$$ which is also in $\cS$.
By construction, the image under $\jmath$ of the closed subvariety $Y \smallsetminus V_i$ is cut out by vanishing loci of monomials, and hence is the preimage of a torus invariant closed set. Therefore $V_i$ is the preimage in $Y$ of an invariant open subset $U$, and $g$ is the pullback of a monomial that is regular on $U$. Restricting $\jmath$ to $X$, we see that $U_i$ is the preimage of $U$ and $f$ is the pullback of a monomial. So, the restriction of $\cS$ to $X$ satisfies $(\star)$ and the theorem follows.
To prove Theorem \[thm:subscheme\], we will apply Proposition \[prop:rationalFunctions\] to a system of embeddings of $Y$ in toric varieties that is generated using Włodarczyk’s algorithm. We begin by briefly recalling the outline of this procedure, from [@Wlodarczyk93].
Let $Y$ be a normal variety over $K$, and suppose $\iota: Y \hookrightarrow Y_\Delta$ is a toric embedding whose image meets the dense torus $T \subset Y_\Delta$. Then the pullbacks of the characters on the dense torus form a finitely generated group $\M$ of rational functions on $Y$, and the set of fibers over the generic points of the $T$-invariant divisors of $Y_\Delta$ is a finite collection $\operatorname{Div}(\M)$ of Weil divisors in $Y$ with no common components. The pair $(\M, \operatorname{Div}(\M))$ satsifies four axioms that encode familiar properties satisfied by the monomial functions and intersections with coordinate hyperplanes in any nondegenerate embedding of $Y$ into projective space [@Wlodarczyk93 §3.1]. The first axiom says that the divisor of any function in $\M$ is a $\Z$-linear combination of divisors in $\operatorname{Div}(\M)$. The second and third axioms are a separation and a saturation axiom, respectively. The final axiom says that, for any point $p \in Y$, the subsemigroup $S_p$ of rational functions in $\M$ that are regular at $p$ is finitely generated, that the open set $U_p$ where all functions in $S_p$ are regular is affine, and that $S_p$ generates the coordinate ring $K[U_p]$.
A pair $(\M, \operatorname{Div}(\M))$ satisfying these four axioms is called an *embedding system of functions*. The axioms ensure that the dual cones $\sigma_p \subset \operatorname{Hom}(\M,\mathbb{R})$, consisting of linear functions that are nonnegative on the semigroups $S_p$ for $p \in Y$, form a fan $\Delta$, and that $\M$ is the pullback of the group of characters under a toric embedding $\iota: Y \rightarrow Y_\Delta$ whose image meets the dense torus.
The heart of W[ł]{}odarczyk’s proof is the construction of an embedding system of functions $(\M,\operatorname{Div}(\M))$, which is accomplished through a ten-step procedure. Some steps of this procedure are algorithmic, but others involve choices. In particular, in the first step, one chooses a finite, Zariski open cover $\mathcal{U}$ of $Y$ such that any two points of $Y$ lie in some $U$ in $\mathcal{U}$, and in the second step one chooses a finite generating set for each of the coordinate rings $K[U]$. The flexibility coming from these choices yields the following embedding theorem. This statement is stronger than what is needed to prove Theorem \[thm:subscheme\], but will be useful for intended future applications.
\[thm:strongEmbedding\] Let $Y$ be a normal variety in which any two points are contained in an affine open subvariety. Let $U_1, \ldots, U_r$ be affine open subvarieties of $Y$, and let $R_j$ be a finite collection of regular functions on $U_j$, for $1 \leq j \leq r$. Then there exists a closed embedding $\iota: Y \hookrightarrow Y_{\Delta}$ such that $U_j$ is the preimage of an invariant affine open subset $U_{\sigma_j}$ and each element of $R_j$ is the pullback of a character that is regular on $U_{\sigma_j}$, for $1 \leq j \leq r$.
We apply the ten-step procedure to our variety $Y$. Using W[ł]{}odarczyk’s notation, our initial indexing set $A$ will be the singleton $A=\{a\}$, with $Y_a = Y$. In Step 1 of his algorithm, include $U_1, \ldots, U_r$ among the finite set of affine opens $\{U_{ab}\}_{b \in B}$ such that every pair of points in $Y$ lies in some $U_{ab}$. In Step 2, when choosing a finite set of generators $\{f_{abc}\}_{c \in C}$ for each coordinate ring $K[U_{ab}]$, make sure that the generating set for $K[U_j]$ includes $R_j$, for each $1 \leq j \leq r$. Then upon completing the remaining Steps 3-10 of W[ł]{}odarczyk’s algorithm, one obtains an embedding system of functions $(\M,\operatorname{Div}(\M))$ with the property that each $f$ in $R_j$ appears in $\M$, for each $1 \leq j \leq r$, and each of the affine open subvarieties $U_1, \ldots, U_r$ of $Y$ appears as one of the affine opens $U_{\mathfrak{p}}\subseteq Y$ constructed from $(\M,\operatorname{Div}(\M))$. The embedding $\iota: Y \hookrightarrow Y_{\Delta}$ built from $(\M,\operatorname{Div}(\M))$ is then a closed toric embedding that verifies the theorem.
We now apply Proposition \[prop:rationalFunctions\] to prove Theorem \[thm:subscheme\].
Let $X$ be a closed subscheme of a toric variety $Y$. To prove Theorem \[thm:subscheme\], it is enough to show that the system of embeddings of $X$ that factor through embeddings of $Y$ satisfies $(\star)$. By Proposition \[prop:rationalFunctions\], it is enough to show that for each rational function $f$ in $K(Y)^*$ there is a toric embedding of $Y$ such that $f$ is the pullback of a monomial. Now, let $U$ be an affine open subset of $Y$ on which $f$ is regular. By Theorem \[thm:strongEmbedding\] there is a toric embedding such that $U$ is the preimage of a torus invariant affine open subset and $f$ is the pullback of a monomial, as required.
The proof of Theorem \[thm:subscheme\] can be adapted to give stronger information in various contexts. For instance, the targets of the toric embeddings of $Y$ produced by Włodarczyk’s construction are smooth or $\Q$-factorial if and only if $Y$ is so. It follows that, if $X$ is a scheme that admits a closed embedding into a single toric variety that is smooth or $\Q$-factorial, then its analytification is naturally homeomorphic to the limit of the tropicalizations of all of its embeddings into toric varieties with the same property. Also, from W[ł]{}odarczyk’s Embedding Theorem, we deduce that if $X$ is a normal variety over an algebraically closed field in which any two points are contained in an affine open subvariety, then the analytification of $X$ is naturally homeomorphic to the limit of the tropicalizations of its toric embeddings.
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[**$\,$\
Large BCFT moduli in open string field theory**]{}
1.1cm
[Carlo Maccaferri[^1]$^{(a)}$, Martin Schnabl[^2]$^{(b)}$]{} 1 cm $^{(a)}$[*Dipartimento di Fisica, Universitá di Torino and INFN, Sezione di Torino\
Via Pietro Giuria 1, I-10125 Torino, Italy*]{}\
$^{(b)}$[*[Institute of Physics of the ASCR, v.v.i.]{}\
[Na Slovance 2, 182 21 Prague 8, Czech Republic]{}*]{}
**Abstract**
We use the recently constructed solution for marginal deformations by one of the authors, to analytically relate the BCFT modulus ($\lambda_\BCFT$) to the coefficient of the boundary marginal field in the solution ($\lambda_\SFT$). We explicitly find that the relation is not one to one and the same value of $\lambda_\SFT$ corresponds to a pair of different $\lambda_{\BCFT}$’s: a “small” one, and a “large” one. The BCFT moduli space is fully covered, but the coefficient of the marginal field in the solution is not a good global coordinate on such a space.
Introduction and Conclusion
===========================
Open String Field Theory (OSFT) could provide a complete non-perturbative approach to D-brane physics, once its quantum structure is understood. Still, at the classical level it gives a new perspective and methodology for finding the possible conformal boundary conditions which are consistent with a given bulk two-dimensional CFT. In String Theory language this amounts to the classification of the possible physical D-branes (stable or not) that can be consistently placed in a given closed string background. From the OSFT perspective this means solving the classical equation of motion. The aim of this paper is to provide exact non-perturbative results on the relation between BCFT marginal parameters and the corresponding parameters in OSFT solutions. Remarkably we now know [@EM], that given two generic BCFT’s (sharing a common non-compact time-like factor) it is always possible to explicitly construct an analytic solution relating the two backgrounds, by suitably using a pair of boundary condition changing operators with known OPE. However here we would like to study an example which is not so distant to the available Siegel gauge numerical results, where the time-CFT is only excited through the identity and its descendants. For self-local boundary marginal deformations [@bible], we have an explicit analytic wedge-based solution [@simple-marg], and this is the solution we wish to study in this note.
Given an open string background BCFT$_0$, there is typically a continuous manifold of equally consistent open string backgrounds, connected to BCFT$_0$, forming a moduli space. Such a moduli space is locally spanned by the VEV of the exactly marginal boundary operators that can be switched on in BCFT$_0$. Because of the linear structure of small fluctuations, it appears natural to parametrize the OSFT solutions for marginal deformations by the coefficient of their marginal field. This quantity is typically called $\lambda_{\SFT}$ \_[marg]{}=\_cj(0)0\_[SL(2,R)]{}+. On the other hand, the physical trajectory in moduli space has a more natural coordinate $\lambda_{\BCFT}$, or more succinctly $\lambda$, which corresponds to the strength of the (conformal) boundary interaction that one adds to the sigma-model action to describe the new background S\_[\_]{}=S\_[\_0]{}+\_[M]{} ds j(s). The relation between $\lambda_\SFT$ and $\lambda_\BCFT$ triggered a lot of discussions in the last fifteen years [@large; @toy; @senT; @kurs; @KL1; @TT-branch; @large2], especially because of the following long-standing puzzle. Given an exactly marginal field $j(z)$, a one-parameter family of approximate solutions labeled by $\lambda_\SFT$ was found in Siegel gauge by Sen and Zwiebach [@large]. Evidence was found that this one-parameter family ceases to exist at a finite value of $\lambda_\SFT$, posing the question about the ability of OSFT to cover or not the BCFT moduli space. With the advent of the new analytic methods, beginning with [@Schnabl], new powerful tools have been developed to extract the BCFT data from a given OSFT solution. Notably it has been found how to directly construct the boundary state corresponding to a given solution [@KOZ; @KMS], using a powerful conjecture due to Ellwood [@Ellwood], which relates simple gauge invariants in OSFT to closed string tadpoles in the new open string background defined by a given solution. Using Ellwood conjecture, more recent results for the cosine deformation at the self-dual radius [@large2], showed that the finite critical value at which the solutions of [@large] truncate, correspond to a finite value of $\lambda_\BCFT$, close to the point where the boundary conditions become Dirichlet. After that no further solutions are found. Where is the missing region of the BCFT moduli space?
In this note we propose that such an apparent drawback is simply a consequence of the fact that $\lambda_\SFT$ does not globally parameterize OSFT solutions for marginal deformations which, on the other hand, exist for all physical values of $\lambda_\BCFT$. To do so, we derive, in an explicit computable example, the precise relation between $\lambda_{\SFT}$ and $\lambda_{\BCFT}$, taking advantage of the recently constructed solution for marginal deformations [@simple-marg], which is naturally defined in terms of $\lambda_{\BCFT}$. This allows to $calculate$ $\lambda_{\SFT}$ as a function of $\lambda_{\BCFT}$, by simply computing the coefficient of the marginal field in the solution \_=0|j\_1c\_[-1]{}c\_0|(\_)=f\_(\_).
This computation gives a nice surprise: we find that $\lambda_{\SFT}$, as a function of $\lambda_{\BCFT}$, starts linearly with unit slope and then, after reaching a maximum, it starts decreasing and it eventually relaxes to zero for large values of $\lambda_{\BCFT}$, see figure \[fig:figure 2\]. Therefore, for a given $\lambda_\SFT$ there are typically [*two*]{} values of $\lambda_\BCFT$. This is our main result.
The fine details of the function $\lambda_\SFT(\lambda)$, including the critical value of $\lambda_\BCFT$ at which $\lambda_\SFT$ has a maximum, depend on the gauge freedom in the definition of the OSFT solution, but we find that the relaxation to zero is generic in the whole gauge orbit which we analyze. It is amazing to realize that this is precisely the behavior that Zwiebach conjectured many years ago [@toy], by analyzing a simple field theory model for tachyon condensation. To further confirm Zwiebach’s hypothesis, we also compute the coefficient of the zero momentum tachyon. This time, at large $\lambda$, we find that it asymptotes to a finite positive value. In a particular limit along the gauge orbit the solution localizes to the boundary of the world-sheet, and the above finite positive value agrees with the tachyon coefficient of the tachyon vacuum solution $\Psi_{TV}=\frac1{1+K}c(1+K)Bc$ of [@simple], again in accord with Zwiebach’s picture, see figure \[Fig:large-tach\]. It is tempting to speculate that in fact the whole string field in this limit approaches the tachyon vacuum $\Psi_{TV}$ as has been shown in the case of light-like rolling tachyon in [@HS].
Our simple calculation shows that, at least in this particular example, OSFT does cover the full BCFT moduli space, but such a moduli space cannot be fully described by the coefficient of the marginal field that generates the deformation. At large BCFT modulus it is not the marginal field that drives the marginal flow but it is rather the whole string field with all of its higher level components.
An important question is to what degree is this behavior generic. There are other analytic wedge-based solutions for marginal deformations with singular OPE, which can be constructed systematically at any order in the marginal parameter [@KORZ; @FKP; @KO; @KL]. However their intrinsic perturbative nature is a major obstacle to obtain conclusive results on the issues we are discussing[^3]. A non perturbative treatment of (time-independent) marginal deformations is also clearly provided by the EM solution [@EM]. It is not difficult to see that for this solution we have the exact relation \_[SFT]{}\^[(EM)]{}=\_[BCFT]{},which is in fact common to all SFT solutions which describe marginal deformations with regular OPE [@KORZ; @martin-marg; @KOS]. The reason for this is that the EM solution in this case describes a marginal deformation [@KOS] generated by $j=\frac i{\sqrt2}\del X^0+j^{(c=25)}$, which has regular OPE with itself by construction. Since time is non-compact, the solution only changes the boundary conditions in the $c=25$ part of the initial BCFT.
That said, it seems plausible that the double-valued dependence on $\lambda_{\rm SFT}$ we have found is generic in cases where the solution only excites the matter primaries and descendants generated by the repeated OPE’s of the marginal field. It would be very instructive to “experimentally" confirm this expectation by level truncation computation in the Siegel gauge, and to identify the predicted new branch.
Review of the simple marginal solution
======================================
The solution [@simple-marg] can be constructed from any self-local boundary deformation [@bible], generated by a boundary field $j(x)$ with self-OPE given by[^4] j(x)j(0)\~1[x\^2]{}+(reg). Let us quickly review the structure of the solution, details can be found in [@simple-marg]. It is derived from an identity-based solution, discovered years ago by Takahashi and Tanimoto [@TT], which is used here as an elementary identity-like string field in addition to the well known fields $K,B,c$. Calling $\Phi$ the TT solution [@TT], and defining as in [@id-marg] K’&& K+J= Q\_BQB+\[,B\],\
J&& \[B,\], where $[\cdot,\cdot]$ is the graded commutator, the solution [@simple-marg] can be written as =1[1+K]{}1[1+K’]{}-Q(1[1+K]{}B[1+K’]{}).\[sol\] In the very convenient sliver frame, obtained by mapping the UHP (with coordinate $w$) to a semi-infinite cylinder of circumference $2$ (with coordinate $z$) via the map z=2w, the TT solution is defined as =\_[-i]{}\^[i]{}(f(z)cj(z)+12 f\^2(z)c(z)).\[TTsol\] All the degrees of freedom of the function $f(z)$ are pure gauge except for its zero mode which defines $\lambda_{\BCFT}$ through the relation \_=\_[-i]{}\^[i]{}f(z). In the following we will make the dependence on $\lambda\equiv\lambda_\BCFT$ manifest by defining f(z)&&(z),\
\_[-i]{}\^[i]{}(z)&=&1. The current-like string field $J$ is then given by J=\_[-i]{}\^[i]{}(f(z)j(z)+12 f\^2(z)).
$\lambda_{\SFT}$ vs $\lambda_{\BCFT}$ and the tachyon
=====================================================
Expanding the solution in the Fock space basis, the first components are the zero momentum tachyon and the marginal field =Tc\_10+\_j\_[-1]{}c\_10+ . The coefficient of the marginal field is given by \_&=&0c\_[-1]{}c\_0j\_[1]{}=-\
&=&-\_0\^de\^[-]{}\_0\^dy, while the coefficient of the zero momentum tachyon is T&=&0c\_[-1]{}c\_0=-2\
&=&-2\_0\^de\^[-]{}\_0\^dy. Notice that the BRST exact part of the solution (\[sol\]), does not contribute to these coefficients, as well as to any other coefficient of $c\phi^{(h)}(0)\ket0$, where $\phi$ is a matter primary.\
Let us start with the integrand which defines $\lambda_\SFT$ =ccj(+1/2)(y)e\^[-\_[0]{}\^[y]{}ds J(s)]{} \_[C\_[+1]{}]{}\[corr\]. Here we have defined the world-sheet insertions (y)&& \_[-i]{}\^[i]{}(f(z) cj(z+y)+12 f\^2(z)c(z+y)),\
J(s)&=& \_[-i]{}\^[i]{}(f(z) j(z+s)+12 f\^2(z)). The correlator (\[corr\]) is naturally defined in the cylinder coordinate frame. General correlator of this form on a cylinder of total circumference $L$ is depicted in figure \[Fig:correlator\].
![Graphical presentation of the correlator (\[wedge\]). The two vertical edges are identified to make a cylinder of circumference $L$. In the conventions of [@simple] the coordinate along the boundary increases from right to the left. The shaded region corresponds to the insertion of the boundary interaction spread out to the bulk. On the left border of this region there is an insertion of the Takahashi-Tanimoto identity-like solution $\Phi$. []{data-label="Fig:correlator"}](Marginal_building_block.pdf)
This correlator can be systematically computed by Wick theorem[^5] from the basic current-current correlator j(z) j(w)\_[C\_L]{}=(L)\^21[\^2]{}, and from the standard ghost correlator cc(z) c(w)\_[C\_L]{}=-(L)\^2\^2. In particular, Wick theorem implies that we have e\^[-\_[0]{}\^[y]{}ds J(s)]{} \_[C\_[L]{}]{}=. In computing the above quadruple integral (two integrals along the boundary and two vertical integrals implicit in the $J$’s) one finds out, by a mechanism analogous to [@id-marg], that the contribution from the $f^2$-terms in $J$ precisely cancels with a delta-function contribution coming from the boundary integral of the current correlator. This leaves us with a net result e\^[-\_[0]{}\^[y]{}ds J(s)]{} \_[C\_[L]{}]{}=e\^[-\^2 \_(y,L)]{}. The function(al) $\G_\f$ controls the exponential behavior in $\lambda_\BCFT\equiv\lambda$ and it is given by[^6] \_(y,L)&=&\_[-i]{}\^[i]{} ()\[G\],\
ff()&&\_[-i]{}\^[i]{}f(z-/2)f(z+/2)\[conv1\]. Then, with standard generating function techniques, we can explicitly compute ccj(x)(y)e\^[-\_[0]{}\^[y]{}ds J(s)]{} \_[C\_[L]{}]{}=-[(]{}1+\^2\_(x,y,L)[)]{}e\^[-\^2 \_(y,L)]{}.\[wedge\] The $\lambda^2$ contribution in front of the exponential, which we denote $\F_\f$, is given as a product of two quantities \_(x,y,L)=§\_(x,y,L)¶\_(x,y,L). The first factor accounts for the contraction of $j(x)$, the matter part of the test state, with the exponential interaction and it is given by §\_(x,y,L)=L j(x)\_0\^y ds J(s)\_[C\_L]{}=\_[-i]{}\^[i]{}(z)(+).\[Sf\] The second factor is responsible for the contraction between the current in $\Phi(y)$ and the exponential interaction, as well as the total ghost contribution (which gives an explicit $x$-dependence) ¶\_(x,y,L)&=&\_[-i]{}\^[i]{} \[Pf\],\
(,x,y,L)&&\_[-i]{}\^[i]{}(z-/2)(z+/2)\^2,\[conv2\]\
(,x,y,L)&&\_[-i]{}\^[i]{}(z-/2)(z+/2)12.\[conv3\] The second contribution in $\P_f$, controlled by $(f\!\odot\! f)$, is a residue of a cancelation between the counter-term in the TT solution $\sim \int \frac12 f^2(z) c(z+y)$ and a corresponding term from the contraction of $j(z+y)$ in the TT solution and the exponential interaction. The latter gives rise to a delta function (canceling the TT counter-term) plus a remaining contribution, from which the second term in $\P_\f$ originates[^7].
The basic correlator for the zero momentum tachyon is given by the simpler expression cc(x)(y)e\^[-\_[0]{}\^[y]{}ds J(s)]{} \_[C\_[L]{}]{}=-\^2¶\_(x,y,L)e\^[-\^2 \_(y,L)]{}. The marginal and tachyon coefficients are finally given by the $\f$-dependent functionals \_()&=&\_0\^d e\^[-]{}\_0\^1 d(1+\^2\_(+1/2,,+1))e\^[-\^2 \_(,+1)]{},\
T()&=&\_0\^d(+1) e\^[-]{}\_0\^1 d ¶\_(+1/2,,+1)e\^[-\^2 \_(,+1)]{}, where we introduced $\hy=y/\ell$ for later convenience. Notice that the $\lambda$-dependence is fully manifest.
Explicit results
-----------------
To continue further we choose a family of functions $f_t(z)$, given by the gaussians [@simple-marg] f\_t(z)2te\^[(tz)\^2]{}. As shown in [@simple-marg] the $t$ dependence is just an $L^-$ reparametrization of the TT solution $\Phi$ and it is thus a gauge redundancy. For very large $t$ the gaussian becomes a delta function which localizes the exponential interaction to the boundary, providing a regularization of contact term divergences, alternative to the standard one by Recknagel and Schomerus [@bible]. In our application this choice is particularly fortunate as it allows to perform the convolution-like operations (\[conv1\], \[conv2\], \[conv3\]) analytically. In particular we have f\_t\*f\_t()&=&\^2te\^\[cconv1\],\
f\_tf\_t (,x,y,L)&=&\^2te\^(1-e\^)\[cconv2\],\
f\_tf\_t (,x,y,L)&=&\^2te\^e\^\[cconv3\].
The remaining integrations are performed numerically, except for the second term in $\P_f$ (\[Pf\]) which can be computed analytically \_[-i]{}\^[i]{} (,x,y,L)=e\^.
In figure \[fig:figure 2\] we plot the marginal coefficient as a function of $\lambda$ for a selection of $t$ parameters. The plots reveal a clear peak in $\lambda_\SFT$ and as a result for a given $\lambda_\SFT$ we find two corresponding values of $\lambda_\BCFT$. Had we included smaller values of $t$ (corresponding to less localized gaussians) we would have seen $\lambda_\SFT$ crossing the horizontal axis and approaching zero from below. This implies, in this smaller $t$ regime, a quadruple degeneracy for sufficiently small $\lambda_\SFT$, taking into account also negative values of $\lambda_\BCFT$. For $t\gtrsim 2$ the degeneracy is only two-fold and this is the region that we show in the plots. From the plots it is also quite evident that at large $\lambda$ the marginal field relaxes to zero. Notice that the possibility that the maximum of $\lambda_\SFT$ is reached at the point $\lambda_\BCFT=\frac1{2\sqrt2}$ (which is approximately what happens in Siegel gauge [@large2], and for a range of $t$ parameters also here) is excluded to be true in general. The position of the maximum is not gauge invariant.
In figure \[Fig:large-tach\] we plot the tachyon coefficient. Notice that for large $\lambda$ it tends to a positive constant. This positive constant, for large $t$, approaches the coefficient of the simple tachyon vacuum of [@simple] \[ES-coeff\] T\_[simple]{}=1[4]{}\_0\^de\^[-]{}(+1)\^2(1-)=0.284394.
\[fig:figure1\]
\[fig:figure3\]
Asymptotics for large $\lambda$ and large $t$
----------------------------------------------
The numerical integrations we have just performed suggest that something non-trivial must happen for large $\lambda$ since the marginal coefficient relaxes to zero, while the tachyon coefficient to a positive constant. Naively one would think that both quantities should relax to zero because of the exponential suppression $\sim e^{-\lambda^2 \G_\f}$, but evidently this is not the case. To understand what happens in the $\lambda\to\infty$ limit, we first notice that \_\^2 e\^[-\^2 \_(,+1)]{}=.\[local\] This is the leading term in the asymptotic distributional expansion \^2 e\^[-\^2 f(y)]{}=\_[n=0]{}\^N (-1)\^n+O(\^[-2N-2]{})\[as-dis\]. Using the explicit definitions (\[G\], \[conv1\], \[cconv1\]) this further simplifies to \^2 e\^[-\^2 \_(,+1)]{}= ()+O(1[\^2]{}). This essentially means that the large $\lambda$ behavior is dominated by surfaces where the deformed region has zero width and the exponential term attains unit value
$$e^{-\lambda^2 \G_\f(y\to 0,L)}=1.$$ Therefore the string field coefficients for large $\lambda$ are not necessarily exponentially suppressed. Let us start by looking at the fate of the tachyon coefficient. Using the above results we get \_T()=1[2t]{}\_0\^de\^[-]{}(+1). We can further compute \_[0]{}¶\_(+1/2, , +1)=t(1-e\^). The large $\lambda$ asymptotic value for the tachyon coefficient is thus given by T()=1[4]{}\_0\^de\^[-]{}(+1)\^2(1-e\^)+O(1[\^2]{}), and it is shown in figure \[Fig:large-tach\].
Notice that for very large $t$ this quickly approaches the tachyon coefficient (\[ES-coeff\]) of the simple tachyon vacuum. On the contrary, as expected, the $t\to 0$ limit is very badly behaved which is related to the identity singularities of the TT solution [@simple-marg].
If we apply the same analysis to $\lambda_{\rm SFT}$ we now find \_[SFT]{}()= t \_0\^de\^[-]{}+O(1). Now the $\hy\to0$ limit also includes $\S_\f$ (\[Sf\]), and it is not difficult to see that the limit vanishes as it is the difference of two identical converging integrals. Therefore the $\lambda$-coefficient in the asymptotic expansion vanishes and we are left with[^8] \_[SFT]{}()=O(1),, which is indeed much milder than the naively expected exponential suppression.
At last we would like to extract the $t\to\infty$ limit of our solution at fixed $\lambda$, where the exponential interaction $e^{-\int ds J(s)}$ localizes to the boundary. In order to do so we should note that the $\G$ function (\[G\]), diverges for large $t$ unless $\hy=0$ (which corresponds to a vanishing-width deformed region). To extract the relevant behavior in this limit it is useful to use the asymptotic formula \_[-]{}\^de\^[-b\^2]{}(1+)\~,a0, which allows to extract the small $\hy$ contribution from $\G$ (,+1)\~,0. Then, in the $t\to\infty$ limit we get the same localization mechanism (\[local\]) as in the large $\lambda$ case, where now the role of large $\lambda^2$ is played by large $t$, for fixed $\lambda$ e\^[-\^2(,+1)]{}\~1[\^2 t ]{}(). Following the same steps as for the $\lambda\to\infty$ case, we now find \_[t]{}T()&=&
[ll]{} 1[4]{}\_0\^de\^[-]{}(+1)\^2(1-)=T\_[simple]{},&0,\
\
0,&=0,
.\
\_[t]{}\_[SFT]{}()&=&0.
It is difficult to directly compare these limits with the data because the numerical integrations are very slow in this region, but we have checked that the height and the position of the peaks in $\lambda_\SFT(\lambda,t)$ in figure \[fig:figure 2\] are nicely fitted by \_\^[max]{}(t)&\~& 0.36(1[t]{})\^[0.44]{},\
\_\^[crit]{}(t)&\~& 0.59(1[t]{})\^[0.44]{}, which confirm our analysis for $t\to\infty$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Ted Erler, Matěj Kudrna, Masaki Murata, Yuji Okawa and Barton Zwiebach for discussions. We thank the organizers of “New frontiers in theoretical physics”, Cortona, May 2014 and “String field theory and related aspects”, Trieste, July 2014, where our preliminary results were presented. CM thanks the Academy of Science of Czech Republic for kind hospitality and support during part of this work. The research of CM is funded by a [*Rita Levi Montalcini*]{} grant from the Italian MIUR. The research of MS has been supported by the Grant Agency of the Czech Republic, under the grant 14-31689S.
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[^1]: Email: maccafer at gmail.com
[^2]: Email: schnabl.martin at gmail.com
[^3]: See [@L] for recent developments in this direction.
[^4]: In [@simple-marg] it was further assumed that the current was not only self-local but also chiral, in the sense of [@bible], so that it was guaranteed to be local with respect to all bulk and boundary fields. This was a technical assumption which allowed to easily construct the fluctuations around the new solution and to show that, for chiral marginal deformations, the Hilbert spaces of the undeformed and deformed theory are isomorphic at the level of the operator algebra. This is not necessarily true for generic self-local boundary deformations.
[^5]: See [@ZAMO] for a general discussion.
[^6]: A much quicker way to compute this correlator is to see it as $\langle\sigma_L(y)\sigma_R(0)\rangle_{C_L}$, where the bcc-like operators are given by $\sigma_{L/R}(x)=e^{\mp i\lambda\chi_\f(x)}$, and use Wick theorem directly in terms of $\chi_\f$, see [@simple-marg] for the precise definitions.
[^7]: This can be seen by infinitesimally detaching the TT solution $\Phi$ from the left edge of the exponential interaction.
[^8]: If needed, the precise $t$-dependent coefficient of $\lambda^{-1}$ can be computed by taking into account one subleading correction in (\[as-dis\]).
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---
abstract: 'Jet systems with two unequal components interact with their parent accretion disks through the asymmetric removal of linear momentum from the star-disk system. We show that as a result of this interaction, the disk’s state of least energy is not made up of orbits that lie in a plane containing the star’s equator as in a disk without a jet. The disk’s profile has the shape of a sombrero curved in the direction of acceleration. For this novel state of minimum energy, we derive the temperature profile of thin disks. The flaring geometry caused by the sombrero profile increases the disk temperature especially in its outer regions. The jet-induced acceleration disturbs the vertical equilibrium of the disk leading to mass loss in the form of a secondary wind emanating from the upper face of the disk. Jet time variability causes the disk to radially expand or contract depending on whether the induced acceleration increases or decreases. Jet time variability also excites vertical motion and eccentric distortions in the disk and affects the sombrero profile’s curvature. These perturbations lead to the heating of the disk through its viscous stresses as it tries to settle into the varying state of minimum energy. The jet-disk interaction studied here will help estimate the duration of the jet episode in star-disk systems and may explain the origin of the recently observed one-sided molecular outflow of the HH 30 disk-jet system.'
author:
- Fathi Namouni
title: 'On the flaring of jet-sustaining accretion disks'
---
Introduction
============
Stellar jets interact dynamically with their parent accretion disk in two fundamental ways. First, they extract angular momentum from the inner part of the disk [@b8], a theoretical prediction that has been confirmed recently by the observation of velocity asymmetries in the jet of RW Auriga [@b15; @b7]. Second, a stellar jet has been proposed as the physical carrier of high temperature minerals from the Sun’s vicinity to the outer parts of the early solar system [@b16; @b17; @b18]. The discovery of Calcium-Aluminum rich inclusions in the grains of Comet Wild 2 collected by [Stardust]{} seem to confirm this hypothesis.
In this paper, we study a new dynamical interaction between the disk and jet that arises from the asymmetric removal of linear momentum from the star-disk system. A growing number of stellar jets appear to be asymmetric as the ejection velocities of the jet and counterjet differ by about a factor of 2 [@b10; @b11; @b12; @b14; @b13]. Jet launching regions are confined to the inner part of the disk with estimates from 0.01 AU for the X-wind model to a few AU for disk-wind models (e.g. 1.6 AU for RW Auriga, @b15). The integrated momentum loss over the launching region accelerates the center of mass of the star-disk system which coincides with the star’s center for axisymmetric systems. A gas element in the outer disk sees the matter around which it revolves accelerating and responds dynamically to such change. This response is the object of this work.
An order of magnitude for the acceleration is obtained from the observed mass loss rates and ejection velocities in disk-jet systems as: $$A \sim 10^{-13}\,
\left(\frac{\dot M}{10^{-8} M_\odot\, \mbox{\rm yr}^{-1}}\right)\,
\left(\frac{v_e}{300 \,\mbox{\rm km\,s}^{-1}}\right)\,
\left(\frac{M_\odot}{M}\right) \mbox{\rm km\,s}^{-2}. \label{acc-mag}$$ where $M$ is the star’s mass augmented by that of the jet launching region of the disk, $\dot M$ is the mass loss rate, and $v_e$ is the jet’s ejection velocity [@b9]. This estimate is an instantaneous lower bound on the total momentum loss as we lack long time span observations of stellar jets as well as observations of the jet engine within a few AU from the star. Although small, the acceleration amplitude (\[acc-mag\]) was shown to offer a possible explanation for the large eccentricities and secular resonances of extrasolar planets’ orbits provided the planets formed in a jet-sustaining disk ([@b19] hereafter Paper I). Over the duration of the asymmetric jet episode, the star-disk system acquires a residual velocity, $V$, that must be smaller than the stellar velocity dispersion in the Galaxy, $\left<v_g\right>$. This constraint yields an upper bound on the duration of the asymmetric jet episode, $\tau$, as: $$\tau \leq
10^5
\ \frac{3\times
10^{-12}\,{\rm km\,s}^{-2}}{A}\ \frac{\left<v_g\right>}{10\, {\rm km\, s}^{-1
}} \, {\rm years}, \label{duration}$$ where we used $V\sim A\tau$. Asymmetric momentum removal would contribute at most a few kms$^{-1}$ to the observed velocity dispersion of young stars (of order a few tens of kms$^{-1}$, [@veldisp]) leading to an upper bound on the jet’s lifetime comparable to the viscous time of the disk. The observation of (the same) asymmetric jets since 1994, yields a lower bound of $\sim 10$ years on how long asymmetric jets may be sustained.
Jet asymmetry may have two origins: the ambient medium and/or an asymmetric magnetic field that threads the disk. The former [@assym1] has the advantage of not adding more complex geometries to the magnetic field responsible for jet generation. The latter option of an asymmetric magnetic field has been considered in the context of AGN systems where it has been shown to warp the inner parts of the accretion disks through the asymmetric local magnetic pressure [@referee]. Future observations inside a few AU from the central star will help elucidate the origin of jet asymmetry.
In this work, we show that the acceleration of the star-inner-disk system with respect to the outer disk (section 2) alters the disk’s state of minimum energy from a plane to a sombrero-shaped surface curved along the direction of acceleration (section 3). This flaring geometry has new consequences for the stellar irradiation of the disk (section 4) as well as for its vertical structure (section 5). Jet time-variability is reflected in the varying acceleration imparted locally to the disk leading to the global radial expansion and contraction of the disk (section 6). These results are summed up in section 7 by illustrating the disk’s response as the jet is launched, and discussing their possible application to the HH30 disk-jet system.
Jet-induced acceleration
========================
Jet-induced acceleration arises from the removal of momentum from the star-disk system (Fig. \[f1\]) in an inertial frame. To see this, we divide the star-disk system into three particles: the star, an inner disk materialized by a particle that looses mass and linear momentum located at 0.5 AU, and the outer disk materialized by a constant mass particle located far outside, for instance at 50 AU. This is justified because the jet launching region is confined to the inner part of the disk. Gauss’s theorem (the divergence theorem) for the gravitational potential of the form $r^{-1}$ implies that the outer particle (outer disk) orbits around the combined masses of the star and the inner particle (the inner disk) which form what we shall call the central object. The outer particle is not concerned by the details of mass and momentum losses in the central object around which it revolves, it only responds to them. The central object or more accurately, the center of mass of the star and the inner disk, is loosing mass and linear momentum. It is therefore being accelerated with respect to the outer particle by an amount equal to the ratio of the momentum loss rate from the inner particle divided by the total mass of the central object –note that the inner particle itself is accelerated by the ratio of the momentum loss rate to its mass. The two-body problem, central object-outer particle, being symmetric, this also amounts to saying that the outer particle is accelerated with respect to the central object that remains stationary. This shows that the outer particle evolution may be described by the equation: $$\frac{{\rm d} {\bf v}}{{\rm d} t}=-\frac{GM}{|{\bf x}|^3}\,{\bf
x} +{\bf A}, \label{motion}$$ where ${\bf x}$ and ${\bf v}$ are the position and velocity and $G$ is the gravitational constant. The mass $M$ is that of the star augmented by that of the inner particle and corresponds to the total mass contained in the jet launching region. The acceleration ${\bf A}$ is obtained as the ratio of the total momentum loss to the total mass $M$. In practice, we can neglect the contributions of the disk’s mass and its variations in $M$ for two reasons. First, the mass of the inner disk is small compared to the mass of the star. Second, the mass loss is small and amounts to a correspondingly small dynamical effect given by the Jeans radial migration rate $\dot
r/r=10^{-8}$yr$^{-1}$ for $\dot M=10^{-8} M_\odot$yr$^{-1}$ [@jeans].
From this simple picture, we can understand what goes on in an extended continuous disk. Mass and linear momentum loss modify the disk’s equations of motion by including the disk’s gravitational potential as well as the momentum loss from the disk (in the case of disk winds). The contribution of the disk’s potential is negligible because the disk’s mass is small and the migration resulting from mass loss is also small. As for the momentum loss, it is a function of the radius, $r$, that represents the integrated momentum loss up to $r$. Calling $r_{\rm JLR}$ the outer radius of the jet launching region, then for $r\geq r_{\rm JLR}$, the acceleration is not a function of $r$ as it represents the total momentum loss from the star-disk system. For $r\leq
r_{\rm JLR}$, the variation with $r$ of the momentum loss depends on the model of jet generation. Being interested in the flaring of the disk which mostly concerns its outer parts, we may employ Eq. (\[motion\]) to describe the evolution of a gas element located outside of the jet launching region where $A$ is independent of $r$.
State of minimum energy: The sombrero profile
=============================================
We consider the steady state of the disk and accordingly take ${\bf A}$ to be constant. The dynamical problem described by equation (\[motion\]) has two constants of motion, the energy $E=v^2/2-GM/|{\bf x}|-{\bf A}\cdot {\bf x}$, and the projected component of the angular momentum ${\bf h}={\bf x}\times
{\bf v}$ on the direction of acceleration $h_z={\bf h}\cdot {\bf A}/A$.
Before discussing the situation where the jet is not perpendicular to the disk, we first assume that the acceleration lies along the disk’s plane normal. Least energy orbits make up the invariable surface of the disk – if there is no acceleration, this surface is the disk’s midplane. We determine the orbits of least energy by making use of cylindrical coordinates ($r$, $\theta$, $z$) and by expressing the conservation of the vertical component of angular momentum $h_z=r^2\dot \theta$. The energy equation then reads:
$$E=\frac{\dot r^2+\dot z^2}{2}+\frac{h_z^2}{2r^2}-\frac{GM}{\sqrt{r^2+z^2}}
-Az. \label{energy}$$
Minimizing the energy integral yields $\dot r=\dot z=0$ which corresponds to circular orbits and shows that $r$ and $z$ are not independent as the star’s pull balances the acceleration $A$ according to: $$\frac{GM
z}{(r^2+z^2)^{3/2}}=A.$$ The vertical location of a circular orbit of radius $r$ and its orbital rotation rate, $\Omega=\dot\theta$, are given as: $$r=\left[\left(\frac{GM z}{A}\right)^\frac{2}{3}-z^2\right]^\frac{1}{2},
\ \ \ \Omega=
\left[\frac{GM}{\left(r^2+z^2\right)^{3/2}}\right]^\frac{1}{2}
\label{sombrero}.$$ We see that orbits of least energy are circular but do not lie in the equator plane of the star, instead they hover above it in the direction of acceleration. The locus of circular orbits in the $rz$–plane is shown in Fig. (\[f2\], upper panel). However, not all circular orbits are stable in the long term. There is a natural truncation radius around the star where the star’s gravitational acceleration becomes comparable to $A$ (Paper I). Beyond this radius, the star’s pull is weak and orbits escape its gravity. The location of this radius denoted $a_{\rm kplr}$ is given by the equality of the acceleration excitation time $T_A=2\pi r\Omega/3 |A|$ (Paper I) and the orbital period $T=2\pi/\Omega$. It is given as: $$a_{\rm kplr}=\left[\frac{GM}{3|A|}\right]^\frac{1}{2}\simeq 10^3\,
\left[\frac{2\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}}{|A|}\right]^\frac{1}{2}\, \left[\frac{M}{M_\odot}\right]^\frac{1}{2}\, {\rm AU}.\label{akplr}$$ An accretion disk accelerated by a jet uses the stable orbits of the minimum energy state. The profile thus obtained is that of a sombrero as shown in Fig. (\[f2\], lower panel). We call $z=S(r)$ the stable disk profile obtained by inverting Equation (\[sombrero\]). We will also use the linearized profile, $S(r)$ for $z\ll r$ which reads: $z=Ar^3/GM=r^3/3a_{\rm kplr}^2$. We remark that the sombrero profile coincides with the steady state profile of a thin jet-sustaining disk. For (realistic) thick disks, the sombrero corresponds to the invariable surface, the equivalent of the disk’s midplane if there were no acceleration, as well as to the disk’s surface of maximum density (section 5). In the case where the acceleration is not initially perpendicular to the disk, the state of least energy is still a sombrero whose symmetry axis is that of the acceleration. To reach this state, the disk experiences a transient phase during which it dissipates energy (section 6).
Temperature
===========
The stellar irradiation of a flat thin disk (i.e. a planar disk with zero thickness) with no jet-induced acceleration yields a decreasing temperature profile with distance [@b1]. It is given as $T_{\rm d}\propto T_*
(R_*/r)^{3/4}$ where $T_{\rm d}$ is the disk’s temperature, $T_*$ and $R_*$ are the star’s temperature and radius respectively. For a non-planar disk, the disk’s temperature at distance $r$ from the star is related to the stellar temperature through: $$T^4_{\rm d}=T_*^4
\ \left[\frac{rz^\prime-z}{2r}+\frac{2R_*}{3\pi r}
\right]\ \left[\frac{R_*}{r}\right]^2, \label{eqn11}$$ [@b3] where $z$ is the vertical location of the disk and $z^\prime={\rm d}z/{\rm d}r$. For $z\ll r$, we substitute the linearized sombrero profile $z=S(r)$ into (\[eqn11\]) to find: $$T_{\rm d}= T_* \left[\frac{R_*^2}{3 a_{\rm kplr}^2} +\frac{2R_*^3}{3\pi
r^3}\right]^\frac{1}{4}.$$ The outer regions ($r\gg R_*$) are those that contribute most to the temperature profile. Accordingly, the disk’s temperature is given as: $$T_{\rm d}\simeq 13\, \left[\frac{T_*}{4000\,\mbox{K}}\right]\
\left[\frac{R_*}{2 R_\odot}\right]^\frac{1}{2}\
\left[\frac{500\,\mbox{AU}}{a_{\rm kplr}}\right]^\frac{1}{2} \ \mbox{K}.\label{td}$$ The fundamental state of the disk therefore has a residual temperature regardless of radius. The sombrero profile cancels out the $r^{-2}$ distance factor responsible for the decrease of temperature with distance. Relaxing $z\ll r$ and using the full shape of $S(r)$ leads to a modest temperature increase with distance. The increase and decrease of the slopes of $z/r$ on both sides of the sombrero for a finite thickness disk will make the temperature grow beyond the estimate of Eq. (\[td\]) for the upper side of the disk and exposes the lower side to some irradiation. The unequal heating of jet-sustaining disks therefore sets a local vertical temperature gradient that leads to the stratification of the disk.
Hydrostatic equilibrium and wind generation
===========================================
For a disk with a finite thickness, we may seek the vertical profile obtained from hydrostatic equilibrium. The hydrostatic equilibrium equations are now modified by the presence of the vertical acceleration ${\bf A}$. Assuming an isothermal disk of pressure $p=c^2\rho$ where $\rho$ is the density and $c$ the sound speed, the hydrostatic equilibrium equation reads: $$\frac{{\rm d}p}{{\rm d}z} +\frac{G Mz\rho}{\left(r^2+z^2\right)^{3/2}}-\rho
A=0, \label{hydroeq}$$ whose solution is : $$\rho=\rho_0(r)\exp\frac{1}{c^2} \left[\frac{G M}{\sqrt{r^2+z^2}}+Az\right].\label{hydro}$$ The sombrero profile here corresponds to the surface of maximum density. With respect to this profile and where $z\ll r$, the disk’s thickness, $H=c/\Omega$, is similar to that of an unaccelerated disk. This solution however has a flaw: the linear term related to the acceleration yields an infinite disk mass when $\rho$ is integrated over all space. The finite value of $\rho$ for large positive $z$ is indicative of a matter outflow from the upper side of the disk in response to the lowering of the vertical pull of the star by the acceleration ${\bf A}$. Consequently, Eq. (\[hydroeq\]) can no longer describe the vertical state of the disk. The steady state properties of the new wind component may be described by the unmagnetized Grad-Shafranov equation [@b21] subject to the finite duration acceleration ${\bf A}$. We expect the density and velocity profile of the wind component to bear some resemblance to the hydrodynamic jets generated as force-free vortex funnels in accretion disks [@b21x].
Variability
===========
Vertical evolution
------------------
Stellar jets are time-variable processes [@var1; @var4; @var2; @var3; @b7]. To ascertain the consequences of the jet-induced acceleration’s time variations, we first consider an initially circular planar disk subject to a vertical acceleration of magnitude $A$. This models an instantaneous increase of acceleration from zero to $A$. Solving the vertical equation of motion in (\[motion\]) under the assumption that $z\ll r$ yields: $$z=\frac{Ar^3}{GM}\ (1-\cos\Omega t). \label{vertical}$$
The vertical motion is an oscillation with respect to the linearized sombrero profile $S(r)$ on a timescale equal to the local orbital period in the disk, $T$. The oscillation amplitude is given by the distance from the initial state which in this case is the equator plane to the location of the sombrero profile. If the acceleration grew from zero to $A$ on a timescale, $|A/\dot
A|$, longer than the orbital period, $T$, the disk’s midsurface height $z$ would follow the evolving sombrero profile adiabatically and there will be no oscillation with respect to the surface of least energy. To illustrate the two excitation regimes, sudden and adiabatic, we show in Fig. (\[f3\]) the response of the disk’s midsurface height obtained by numerically integrating the equation of motion (\[motion\]). The acceleration’s time dependence is taken as $A=A_{\rm max}(1-\exp[-t/100\,{\rm yr}])\times W(t)$ where $A_{\rm
max}=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}$, the same magnitude as in Fig. (\[f2\]), and $W(t)$ is a window function that vanishes outside the range \[0,5000yr\] and equals unity inside it. The minimum truncation radius is therefore 500AU. As the launching timescale is $|A/\dot A|=100$yr, the disk evolves adiabatically to the sombrero profile inside 22AU as can be seen on $z(t)/r$ at $r=10$AU ($T=32$yr) in Fig. (\[f3\]). Outside 22AU, the dynamical time $T$ is longer than the launching timescale of 100yr implying that jet launching is perceived as a sudden excitation by most of the outer disk whose typical response is described by Eq. (\[vertical\]) and shown in Fig (\[f3\]) for $r=100$AU.
The acceleration is shut off suddenly after 5000yr. This determines an outer limit, $r\sim 300$AU, in the disk beyond which the state of least energy remains the disk’s initial state (star’s equator plane) as the dynamical time outside this radius is too long to travel all the way to the location of the sombrero profile (Eq. \[vertical\]). Beyond this radius, the jet-induced acceleration episode is felt as a short (with respect to the orbital period) velocity pulse to the outer disk resulting in the excitation of vertical motion with respect to the disk’s initial midplane as seen in Fig. (\[f3\]) for $r=400$AU. This type of excitation has been discussed extensively in Paper I (section 6). The sudden absence of acceleration is felt everywhere in the disk, where for instance at $r=10$AU, the lack of the jet-induced acceleration that balanced the star’s vertical gravitational pull leads to a vertical oscillation with respect to the equator plane and with an amplitude equal to the distance from the sombrero to that plane. This effect is the same as the excitation described by Eq. (\[vertical\]) only this time the invariable surface is the disk’s midplane and the initial state is the sombrero profile. For both the accelerated and unaccelerated phases, the vertical oscillations with respect to the surface of least energy (be it a sombrero if $A\neq 0$ or a plane if $A=0$) are damped by the viscous stresses of the disk. For instance, in a more realistic simulation taking into account the disk’s viscosity, the observed oscillations about the sombrero height $z/r\sim
10^{-2}$ for $r=100$AU, would be erased and the disk would settle into the sombrero profile before the acceleration is shut off at $t=5000$yr. We note that the illustration of the disk’s midsurface height response to the acceleration’s growth shown here is not peculiar to the launching of the jet and equally represents the disk’s response to more general variations of the jet-induced acceleration as those associated with a varying ejection velocity at the source [@var1; @var2].
Horizontal evolution
--------------------
For a time-variable acceleration, the conservation of the specific angular momentum $h_z=r^2\Omega$ forces the mean orbital radius to change in order to compensate for the increase of the midsurface height $z$ to the location of sombrero profile. The rate of this change is found using the conservation of $h_z$ only as:
$$\dot r=3 \ \frac{z}{r}\ \dot z, \label{hz}$$
where we neglected a term of second order in $z/r$. The rate (\[hz\]) is valid regardless of the time dependence of $A$ and whether the initial state of the disk is planar. The change in radius is of second order in $A$ and leads to a radial expansion of the disk if the acceleration increases with time. If however $A$ decreases, the midsurface height $z$ is lowered to the new sombrero location and Eq. (\[hz\]) shows that the disk undergoes a radial contraction. The expansion and contraction occur on the local dynamical time $T$ if it is larger than the variability timescale $|A/\dot A|$ (as in Eq. \[vertical\]). Otherwise, radial motion occurs adiabatically with respect to variability. Outside the radius where $T=|A/\dot A|$, it is the former evolution that takes place implying that contraction and expansion start from the inside of the disk and proceed outward (as $T$ increases with $r$). The local radial displacement $\Delta r$ is given as: $$\frac{|\Delta r|}{r}= \frac{3\epsilon A^2r^4}{G^2M^2}=\frac{\epsilon}{3}\,
\left[\frac{r}{a_{\rm kplr}}\right]^4 \label{exp-con}$$ where $\epsilon=1$ if the acceleration’s change is sudden, $|A/ \dot A|\leq
T$. For an adiabatic change, $|A/\dot A|\gg T$, the averaging of the linearized equations of motion (\[motion\]) yields $\epsilon=2/3$. The outer half of the disk is the most affected by the expansion and contraction: at the disk’s median radius, $r=a_{\rm kplr}/2$, $|\Delta r|/r\sim 2\%$. We also note that this rate is much larger than that of the Jeans migration (section 2).
Figure (\[f4\]) shows the local expansion and contraction of the disk at different radii under the same conditions as those of the previous section and Fig. (\[f3\]). At $r=10$AU the expansion is adiabatic as the launching timescale $|A/\dot A|$ is larger than the dynamical time, while the contraction is sudden as the acceleration is abruptly turned off. Where the dynamical timescale is much larger than the acceleration’s duration as for $r=400$AU, there is a net radial expansion outward as the whole acceleration episode is felt as a velocity kick to the outer disk (Paper I, section 6) — note for instance how for $r=400$ AU, the expansion appears to be delayed until the end of the acceleration phase.
Another consequence of the acceleration’s variability is the local excitation of eccentricity in the disk. Using the linearized equations of motion (\[motion\]) for the radius $r$, it can be shown that the local eccentricity growth rate due to the vertical acceleration is: $$\dot e= \frac{3A^2r^4\Omega}{\pi G^2M^2},\label{ecc-rate}$$ with a secular turnaround time equal to $T_A$. The maximum eccentricity is therefore $ e_{\rm max}= (r/a_{\rm kplr})^2/3$. We note that the eccentricity gradient ${\rm d} (re)/{\rm d}r=(r/a_{\rm kplr})^2$ is always smaller than unity implying that the forced eccentric distortions of the gas streamlines do not make them cross and consequently they do not shock. The estimate of Eq. (\[ecc-rate\]) is valid during the acceleration phase. Outside the location where the dynamical time is smaller than the duration of the acceleration phase, the net effect of acceleration amounts to a small amplitude velocity kick imparted to the disk. To the fluid elements of the disk’s midplane, the velocity kick imparts an eccentricity given as (Paper I, section 6): $$e(r)= \frac{V^2r/GM}{1+V^2r/GM},\label{ecc-rateout}$$ where $V$ is the total velocity obtained by integrating the acceleration over its duration. The global response of the disk’s viscous stresses to the eccentric disturbance (\[ecc-rate\]) depends on the prescription of the viscous stresses that is employed as well as on the density profile [@b51]. We expect the viscous stresses to damp the eccentric distortions as well as the vertical oscillations leading to the heating of the disk. In the outermost part of the accretion disk where the dynamical time is long compared to the acceleration’s duration and where the viscous timescale is similarly long, we expect jet-sustaining accretion disks to exhibit global eccentric distortions.
Synthesis
=========
Stellar jets are ubiquitous processes of star formation and star-disk interaction. The asymmetric removal of linear momentum from the star-disk system accelerates the outer part of the disk with respect the jet launching region. The acceleration amplitude is small (\[acc-mag\]) and can be modeled as a time dependent vector directed along the star’s rotation axis. The state of minimum energy of the disk is a sombrero-shaped surface made up of circular planar orbits that hover above the star’s equator plane (\[sombrero\]) increasingly higher with distance. As the jet is launched, the initially circular equator disk moves towards the state of minimum energy and the truncation radius moves from infinity to the location given by (\[akplr\]). If the disk’s size is larger than the minimum truncation radius over the acceleration phase, the disk will be truncated by the jet-induced acceleration. There is a specific radius in the disk where the orbital period $T$ matches the jet launching timescale $|A/\dot A|$. Inside this radius, the disk moves away from the equator plane to the instantaneous sombrero surface adiabatically. Outside the matching radius, vertical motion (\[vertical\]) as well as eccentricity distortions (\[ecc-rate\]) with respect to the sombrero are excited. The conservation of angular momentum implies that as the jet is launched the disk expands radially (\[hz\],\[exp-con\]). Viscous stresses damp the vertical motion and the eccentricity distortions to heat up the disk especially in its outer regions. The steady state of the disk thus assumes a sombrero profile that exposes the upper face of the disk to the star’s light canceling at first order the irradiation’s decrease with distance (\[td\]). The disk cannot maintain hydrostatic equilibrium under the jet acceleration (\[hydro\]). Instead the disk looses mass from its upper side giving rise to a secondary asymmetric wind component. When the jet-induced acceleration decreases, the sombrero looses curvature, the disk contracts radially at the rate given by (\[exp-con\]) and the truncation radius moves outward (\[akplr\]).
The mechanisms discussed above only require the onset of asymmetric momentum removal in the star-disk system. Whether momentum is removed from the star itself through some peculiar stellar wind component, or more realistically from the disk or the star-disk interface [@assym1], the outer disk will respond to the integrated momentum loss from the matter around which it gravitates. The mechanisms discussed above also do not have a time requirement on the acceleration’s duration. The evolution towards the sombrero, the radial contraction and expansion proceed from the inner part of the disk outwards. Therefore, these mechanisms affect the disk inside the radius where the dynamical time is comparable to the acceleration’s duration. In this sense, disk observations can be used to constrain the duration of the jet episode and/or asymmetric momentum removal by identifying the outermost radius where relaxation toward the sombrero profile has taken place. Asymmetric momentum removal imprints two more signatures on the star-disk system: the presence of a global eccentric distortion in the outermost part of the disk, and an enhancement of the random component of the stellar velocity in the Galaxy. The eccentric distortion results from a perceived sudden pull of the star-inner disk system with respect to the outer disk along the jet system’s axis and should be an increasing function of radius (\[ecc-rateout\]) that does not depend on the acceleration’s timescales. The change in the stellar velocity dispersion may be ascertained by statistically analyzing young stars’ proper motion at different epochs of their evolution.
A good illustration of some of the new properties discussed above is found in the HH 30 disk-jet system. This system [@hh1] is a nearly edge-on $420$ AU disk that appears as two reflection nebulosities separated by a dark lane. The jet and counterjet appear to be asymmetric although both the ejection speed and the mass loss rate remain uncertain [@hh1; @hh6]. The jet exhibits variability in the form of knots with periods of 2.5 years [@hh1]. The disk’s variable lateral asymmetries appear mainly on one side of the disk with possible periods from a few days to a few years [@hh2; @hh7; @hh5]. More recently, [@hh3] reported the observation of a molecular outflow emanating from the upper nebulosity. The outflow was traced back to a scale smaller than the size of the disk while no rotation signatures were detected above 1 kms$^{-1}$ at 200 AU. The variable lateral asymmetries are currently explained by the presence of a single hot spot on the star requiring a complex magnetic field structure that differentiates substantially between the two stellar poles [@hh2]. If the one-sided outflow resulted from the entrainment of surrounding material then the cloud structure would be substantially different on both sides of the disk. In the context of the jet-disk interaction studied here, it is natural to attribute the lack of lateral asymmetries in the lower nebulosity to a change in the disk’s mean profile, and to identify the one-sided molecular outflow as the new secondary wind component. Further modeling of the disk temperature, wind density and velocity profiles produced by the jet-induced acceleration will help determine the properties of the HH 30 disk and its molecular outflow.
The author thanks the referee for constructive comments that helped improve the clarity of the paper and Christiane Froeschlé for stimulating discussions and the careful reading of the manuscript. This work was supported by the Programme National de Planétologie.
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![Schematic of the star-disk system. The momentum loss from the upper component is larger than that of the lower component. The net momentum loss from the jet launching region, which extends to a few astronomical units from the star, modifies the dynamics of the outer disk according to Eq. (\[motion\]).[]{data-label="f1"}](f1.eps){width="125mm"}
![ The locus of circular orbits in the $rz$–plane (AU) for an acceleration $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}$. The truncation radius $a_{\rm
kplr}=500$ AU. The upper panel shows all possible orbits: stable (solid line) and unstable (dashed line). The lower panel is a zoom of the stable orbits inside a radius of 200 AU around the star (dashed circle not to scale).[]{data-label="f2"}](f2a.eps "fig:"){width="127.5mm"}\
![ The locus of circular orbits in the $rz$–plane (AU) for an acceleration $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}$. The truncation radius $a_{\rm
kplr}=500$ AU. The upper panel shows all possible orbits: stable (solid line) and unstable (dashed line). The lower panel is a zoom of the stable orbits inside a radius of 200 AU around the star (dashed circle not to scale).[]{data-label="f2"}](f2b.eps "fig:"){width="127.5mm"}
![Vertical evolution of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the vertical motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times (1-\exp[-t/100\,{\rm
yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq 5000$yr and zero otherwise. The four panels (left scale) illustrate $z(t)/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times
10^{-12}\ {\rm km}\, {\rm s}^{-2}$.[]{data-label="f3"}](f3a.eps "fig:"){width="127.5mm"}\
![Vertical evolution of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the vertical motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times (1-\exp[-t/100\,{\rm
yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq 5000$yr and zero otherwise. The four panels (left scale) illustrate $z(t)/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times
10^{-12}\ {\rm km}\, {\rm s}^{-2}$.[]{data-label="f3"}](f3b.eps "fig:"){width="127.5mm"}\
![Vertical evolution of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the vertical motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times (1-\exp[-t/100\,{\rm
yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq 5000$yr and zero otherwise. The four panels (left scale) illustrate $z(t)/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times
10^{-12}\ {\rm km}\, {\rm s}^{-2}$.[]{data-label="f3"}](f3c.eps "fig:"){width="127.5mm"}\
![Vertical evolution of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the vertical motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times (1-\exp[-t/100\,{\rm
yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq 5000$yr and zero otherwise. The four panels (left scale) illustrate $z(t)/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times
10^{-12}\ {\rm km}\, {\rm s}^{-2}$.[]{data-label="f3"}](f3d.eps "fig:"){width="127.5mm"}
![Radial expansion and contraction of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the radial motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times
(1-\exp[-t/100\,{\rm yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq
5000$yr and zero otherwise. The four panels (left scale) illustrate $\Delta r/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times 10^{-12}\ {\rm km}\, {\rm
s}^{-2}$.[]{data-label="f4"}](f4a.eps "fig:"){width="127.5mm"}\
![Radial expansion and contraction of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the radial motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times
(1-\exp[-t/100\,{\rm yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq
5000$yr and zero otherwise. The four panels (left scale) illustrate $\Delta r/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times 10^{-12}\ {\rm km}\, {\rm
s}^{-2}$.[]{data-label="f4"}](f4b.eps "fig:"){width="127.5mm"}\
![Radial expansion and contraction of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the radial motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times
(1-\exp[-t/100\,{\rm yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq
5000$yr and zero otherwise. The four panels (left scale) illustrate $\Delta r/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times 10^{-12}\ {\rm km}\, {\rm
s}^{-2}$.[]{data-label="f4"}](f4c.eps "fig:"){width="127.5mm"}\
![Radial expansion and contraction of the outer disk. The equations of motion (\[motion\]) were integrated numerically to simulate the radial motion of a gas element initially located in the disk’s midplane before the jet-induced acceleration is activated. The acceleration is modeled as $A=8\times 10^{-12}\ {\rm km}\, {\rm s}^{-2}\times
(1-\exp[-t/100\,{\rm yr}])\times W(t)$ where $W(t)=1$ if $0\leq t\leq
5000$yr and zero otherwise. The four panels (left scale) illustrate $\Delta r/r$ for $r=$10AU, 100AU, 250AU and 400AU –the corresponding dynamical times $T$ are given in each panel. The dashed-line profile (right scale) in the four panels shows the acceleration $A$ normalized to its maximum value $8\times 10^{-12}\ {\rm km}\, {\rm
s}^{-2}$.[]{data-label="f4"}](f4d.eps "fig:"){width="127.5mm"}
|
---
abstract: 'Resource Constrained Project Scheduling Problems (RCPSPs) without preemption are well-known $\mathcal{N}\mathcal{P}$-hard combinatorial optimization problems. A feasible RCPSP solution consists of a time-ordered schedule of jobs with corresponding execution modes, respecting precedence and resources constraints. In this paper, we propose a cutting plane algorithm to separate five different cut families, as well as a new preprocessing routine to strengthen resource-related constraints. New lifted versions of the well-known precedence and cover inequalities are employed. At each iteration, a dense conflict graph is built considering feasibility and optimality conditions to separate cliques, odd-holes and strengthened Chvátal-Gomory cuts. The proposed strategies considerably improve the linear relaxation bounds, allowing a state-of-the-art mixed-integer linear programming solver to find provably optimal solutions for 754 previously open instances of different variants of the RCPSPs, which was not possible using the original linear programming formulations.'
address:
- 'Department of Computing, DECOM, Universidade Federal de Ouro Preto'
- 'Department of Computing and Systems, DECSI, Universidade Federal de Ouro Preto'
- 'Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation, CIRRELT'
- 'Computer Science Department, CODeS, KU Leuven'
- 'Department of Computer Science and Operations Research, Université de Montréal'
- 'Department of Management and Technology, École des Sciences de la Gestion, Université de Québec à Montréal'
author:
- 'Janniele A. S. Araujo'
- 'Haroldo G. Santos'
- Bernard Gendron
- Sanjay Dominik Jena
- 'Samuel S. Brito'
- 'Danilo S. Souza'
bibliography:
- 'rcpsp-preproc-and-cutting-planes.bib'
title: 'Strong Bounds for Resource Constrained Project Scheduling: Preprocessing and Cutting Planes'
---
Resource Constrained Project Scheduling ,Cutting Planes ,Mixed-Integer Linear Programming, Preprocessing
Introduction {#sec:introduction}
============
This paper proposes automatic Mixed-Integer Linear Programming (MILP) reformulation strategies for non-preemptive Resource Constrained Project Scheduling Problems (RCPSPs) and its variants. From a theoretical point of view, RCPSPs are challenging combinatorial optimization problems, classified as $\mathcal{NP}$-hard [@Blazewicz1983; @Garey1979]. These problems cover a wide range of applications, such as particle therapy for cancer treatment in healthcare [@Riedler2017], civil engineering [@Liu2008], manufacturing and assembly of large products [@Liu2014], as well as development and launching of complex systems [@Deumeulemeester2002]. Comprehensive reviews on RCPSPs can be found, for example, in [@Artigues2008; @Deumeulemeester2002; @Schwindt2015].
In this paper, we consider the following problem variants of the RCPSP, from the most specific one to the most generalized version:
SMRCPSP:
: single-mode resource-constrained project scheduling problem;
MMRCPSP:
: multi-mode resource-constrained project scheduling problem;
MMRCMPSP:
: multi-mode resource-constrained multi-project scheduling problem.
The SMRCPSP is the simplest variant and involves only one processing mode for each job. A feasible solution consists in the assignment of jobs at specific time periods over a planning horizon, respecting precedence and resource usage constraints. The resources in the SMRCPSP are renewable at each time period.
In the MMRCPSP, it is possible to choose between different job processing modes, each of them having different durations and consuming different amounts of resources. Two types of resources are available: the renewable resources at each time period and the non-renewable ones available for the entire project execution. While the use of renewable resources only impacts the delay/speedup of the projects, the use of non-renewable resources can produce infeasible solutions.
A generalization of the previous problem variants to handle multiple projects and global renewable resources is the MMRCMPSP. While the previous two problem variants, in their objective (minimization) functions, consider the makespan, i.e., the total length of the schedule to finish all jobs, an important modeling feature of this generalization is an additional objective: the project delays from the Critical Path Duration (CPD). The CPD is a theoretical lower bound on the earliest finishing time ($\breve{e}_p^f$) of project $p$, computed by the Critical Path Method (CPM) [@Kelley1959] and disregarding any resource constraints.
Figure \[fig:gantt\] illustrates the example instance $j102\_4.mm$[^1], a small instance for the MMRCPSP variant. The figure shows the characteristics of this instance and one optimal solution. We will refer to this example instance to explain the cutting planes presented in the next sections.
![An optimal solution for instance $j102\_4.mm$ and its characteristics[]{data-label="fig:gantt"}](j102_4ganttchartnewnew.pdf)
The instance illustrated in Figure \[fig:gantt\] has twelve jobs. The first and the last are artificial jobs representing the start and the end of the project. Each job $j$ can be processed in three different modes $m$. Artificial jobs $0$ and $11$ have only one execution mode that does not consume resources and have duration $0$; thus columns $ m_1 $ and $ m_2 $ are filled with “-". Two renewable resources $\{r_0,r_1\}$ with capacities $\{9,8\}$ and two non-renewable resources $\{k_0,k_1\}$ with capacities $\{35,31\}$ are available. This figure provides information about resource consumption and duration of job $j$ processing on mode $m$. The Gantt chart in the figure shows the starting time allocation and duration of jobs for the optimal solution found for this single project. Arcs represent the precedence relationship between jobs. Values emphasized in grayscale represent the active modes for this solution. The CPD corresponds to value $15$. Note that the earliest starting time ($\breve{e}_j^s$) of job $5$ corresponds to $2$, but due to the availability of resources at this time period, its allocation had to be postponed. The makespan is $18$ time units.
In this paper, we propose a new preprocessing technique to improve resource-related constraints by strengthening their coefficients using known information about different renewable, non-renewable resources and precedence constraints. We also propose a cutting plane algorithm employing five different cut separation algorithms. Conflict-based cuts such as cliques and odd-holes are generated considering an implicit dense conflict graph, which is updated at each iteration considering optimality and feasibility conditions. This conflict graph is also used in a MILP separation of the Chvátal-Gomory cuts to produce stronger inequalities. Furthermore, new lifted versions of the well-known disaggregated precedence cuts and cover cuts are separated. The contribution on the linear relaxation bound provided by each cut family is examined on experiments on a large set of benchmark instances of the three problem variants considered in this paper. Overall, stronger bounds were obtained, allowing a state-of-the-art MILP solver to find provably optimal solutions for 755 previously open instances, which was not possible using the original linear programming formulations.
The paper is organized as follows. Section \[sec:literature\], reviews the literature related to our work. Section \[sec:exactmodel\] introduces the integer programming formulation used along with some simple input data preprocessing routines based on upper bounds. The new strengthening procedure for renewable resources constraints is also presented in this section. Section \[sec:method\] outlines the proposed cutting plane method, followed by a detailed description of each cut generation routine. In Section \[sec:results\], the computational results are presented. Finally, in Section \[sec:conclusion\], we conclude our work and discuss future research directions.
Literature Review {#sec:literature}
=================
Various MILP based formulations have been proposed in the literature to model resource constrained project scheduling problems. @Pritsker1969 proposed the first binary programming formulation where variables $x_{jt}$ indicate whether a job $j$ ends at time $t$ ($x_{jt} = 1$) or not ($x_{jt} = 0$). This formulation is known as the discrete-time or time-indexed formulation. The number of binary decision variables in this formulation is related to an upper bound $\bar{t}$ for the number of time periods required to complete the project. Thus the number of variables is $\mathcal{O}(n\times\bar{t})$, where $n$ is the number of jobs. @Kolisch1996 extended this formulation to handle different execution modes, adding one additional index $m$ to the binary variables and incorporating this index in the resource-related constraints. In @Kone2011, a new formulation was proposed based on events called on/off event-based (OOE) with $\mathcal{O}(n^2)$ variables.
Time-indexed formulations have been extensively studied and applied to the RCPSP (see, for example, [@Baptiste2004; @CHRISTOFIDES1987262; @Demassey2005; @Hardin2008; @Sankaran1999; @deSouza1997]). Six time-indexed formulations were studied in @Artigues2017. These formulations are initially categorized into three groups according to the meaning of variables, as follows:
Pulse:
: pulse discrete time - PDT (the most used) formulation with binary variables $x_{jt} \ \forall j \in \mathcal{J}, \forall t \in \mathcal{T}$, such that $x_{jt} = 1$ if job $j$ starts at time $t$, otherwise $x_{jt} = 0$;
Step:
: step discrete time - SDT formulation with binary variables $y_{jt} \ \forall j \in \mathcal{J}, \forall t \in \mathcal{T}$, such that $y_{jt} = 1$ if job $j$ starts at time $t$ or before, otherwise $y_{jt} = 0$;
On/Off:
: on/off discrete time - OODT (the lesser used) formulation with binary variables $z_{jt} \ \forall j \in \mathcal{J}, \forall t \in \mathcal{T}$ such that $z_{jt} = 1$ if job $j$ is processed at time $t$, otherwise $z_{jt} = 0$.
For each one of these categories, it is possible to model the precedence constraints in a weak or strong way according to the linear programming (LP) relaxation strength.
Aggregated:
: this is a weak way to model precedence constraints, using variable coefficients greater than or equal to $1$; aggregated constraints generate $\mathcal{O}(n^2)$ inequalities, each one with $\mathcal{O}(\bar{t}\times m)$ variables, where $m$ is the maximum number of modes;
Disaggregated:
: this is a stronger way to model precedence constraints, using variable coefficients equal to $\{-1,1\}$; disagregated constraints generate $\mathcal{O}(n^2\times\bar{t})$ inequalities, each one with $\mathcal{O}(\bar{t}\times m)$ variables.
@Artigues2017 concluded that PDDT (pulse disaggregated discrete time), SDDT (step disaggregated discrete time) and OODDT (on/off disaggregated discrete time), which are formulations with disaggregated constraints, are all equivalent in terms of the strength of their LP-relaxations and belong to the family of strong time-indexed formulations. He also concluded that the formulations with their aggregated counterparts, PDT, SDT, and OODT, belong to a family of weak formulations and are also all equivalent in terms of their LP relaxation.
Even though the aggregated constraints are weaker, they are much less dense. For this reason, papers from the literature (see, for example, [@CHRISTOFIDES1987262; @Zhu2006]), begin the formulation with the aggregated constraints and add the disaggregated ones as cutting planes. As in previous works, our algorithm starts with the weak time-indexed formulation pulse discrete time (PDT) based on formulations proposed in [@Kolisch1996; @Pritsker1969] for the MMRCMPSP version. In the next paragraphs, we review some computational approaches to handle MILP formulations for the RCPSPs.
Most of the exact algorithms for the RCPSPs [@Brucker1998272; @Ripon2015; @CHRISTOFIDES1987262; @Demeulemeester1992] are built upon LP based Branch and Bound (B&B) [@CHRISTOFIDES1987262; @Land2010] algorithms. A key component in the design of these algorithms is which formulation is employed: compact formulations, with a polynomial number of variables and constraints, are usually able to quickly provide valid lower bounds since their LP relaxations are easily solved. The LP relaxation bounds are relatively weak but can be significantly improved by adding cutting planes in a Branch-and-Cut (B&C) algorithm.
Cutting planes and strengthening constraints in MILP models are explored for RCPSP and to other variants of scheduling problems. @Applegate1991 proposed cliques cuts, half cuts, and other cuts specific to job-shop scheduling problems (JSSP). @Hardin2008 proposed a lifting procedure to cover-clique inequalities to a resource-constrained scheduling problem with uniform resources (URCSP) requirement. @Cavalcante2001 applied cover cuts to the labor constrained scheduling problem (LCSP), based on practical requirements arising in industry.
@Sankaran1999 proposed a cutting-plane algorithm for the SMRCPSP with minimal cover inequalities and clique inequalities to test problems provided by @Patterson1974. Also, they introduced three preprocessing techniques: reduction of lower and upper bounds; the identification of redundant constraints between resource and precedence constraints, and the coefficient strengthening in constraints [@Johnson1985].
@CHRISTOFIDES1987262 proposed a B&B algorithm that uses disjunction arcs to handle resource conflicts. Four lower bounds are examined: the first one is based on the longest path in the precedence graph; the second one is based on an LP relaxation strengthened with cuts; the third one is based on Lagrangian relaxation and the fourth one is based on disjunctive arcs. Their LP-based method incorporated the dynamic inclusion of disaggregated precedence constraints and resource-based conflict constraints for pairs of jobs. The first lower bound, based on precedence constraints, can be computed quickly and was used in the B&B algorithm with other bounds. The second lower bound was promising, but was not used within the B&B method. The Lagrangian relaxation technique was found to provide less competitive results. They provide additional inequalities for the time-indexed version. Finally, the fourth lower bound performed quite well, especially for problems with tight resource constraints. For the SMRCPSP, @CHRISTOFIDES1987262 were able to prove the optimality of instances involving up to $25$ jobs and $3$ resources.
@Zhu2006 presented a B&C algorithm for the MMRCPSP, including cuts derived from resource conflicts, where all resource constraints are in the form of generalized upper bound (GUB) constraints. Besides, disaggregated cuts from the precedence relationship for pairs of jobs $(j,s)$ in the precedence graph are included. To speedup the solution process, an adaptive branching scheme is developed along with a bound adjustment scheme that is always executed iteratively after branching. To optimize the solutions found in the first stage, the authors use a high-level neighborhood search strategy called Local Branching [@Fischetti2007]. For the MMRCPSP, the authors were able to prove the optimality $554$ of instances with 20-jobs and $506$ instances with 30-jobs.
Different techniques such as Constraint Programming (CP) and Satisfiability Solving (SAT) have also been used to solve the MMRCPSP. In this context, @Demassey2005 uses CP techniques to provide valid inequalities to strengthen LP relaxations. A recent work using cutting planes as valid clauses for SAT is presented in @Schnell2017. They propose three formulations based on Constraint Programming to solve the MMRCPSP, using the G12 CP platform and the Solving Constraint Integer Programs (SCIP) as an optimization framework, both making use of solution techniques combining CP and SAT. They further combine MILP by inserting a new global constraint on the domain of renewable resources for SCIP. The authors achieved the same results with better computational times than @Zhu2006. They also were able to prove the optimality of $1428$ instances with 50-jobs and 100-jobs and to improve various lower and upper bounds.
Integer Programming Formulation {#sec:exactmodel}
===============================
This section introduces the time-indexed formulation based upon the discrete time formulations proposed in [@Kolisch1996; @Pritsker1969]. The most common objective function for the RCPSP is the makespan [@Artigues2008; @Deumeulemeester2002; @Kolisch1996; @Pritsker1969; @Zhu2006] minimizing the total schedule duration required to finish all jobs. @Wauters2016 proposed a hierarchical objective to minimize the Total Project Delay (TPD) and the total makespan (TMS). The former, denoted formally in Eq.(\[eq:tpd\]), is the main objective. The latter, given in Eq.(\[eq:tms\]), is a tiebreaker.
$$\label{eq:tpd}
TPD = \sum_{p \in P}{PD_p}$$
$$\label{eq:tms}
TMS = \max_{p\in \mathcal{P}}{f_p} - \min_{p\in \mathcal{P}}{\sigma_p}$$
The project delay ($PD_p$) for a project $p \in \mathcal{P}$ is defined as the difference between its $CPD_p$ (which does not consider any resource constraints), and its makespan $MS_p = f_p -\sigma_p$, the actual project duration taking into consideration resource constraints, the finishing time $f_p$ and the release date $\sigma_p$ of this project.
$$PD_p = MS_p - CPD_p$$
To work with all three problem variants in a unified way, we always report our results considering the TPD, which is generic enough to handle all problem variants considered in this paper.
Input Data {#subsec:id}
----------
The following notation is used throughout this paper to describe the input data:
[xxxxxxx]{}
set of all projects;
set of all jobs;
set of modes available for job $j \in \mathcal{J}$;
set of jobs belonging to project $p$, such that $\mathcal{J}_{p} \subseteq \mathcal{J} \ \forall p \in \mathcal{P}$;
set of non-renewable resources;
set of renewable resources;
set of direct precedence relationships between two jobs $(j, s) \in \mathcal{J} \times \mathcal{J}$;
set of time periods in the planning horizon for all projects $p \in \mathcal{P}$;
time horizon for each job $j \in \mathcal{J} $ on mode $m \in \mathcal{M}_{j}$, defined after preprocessing;
duration of job $j \in \mathcal{J}$ on mode $m \in \mathcal{M}_{j}$;
required amount of non-renewable resource $k \in \mathcal{K}$ to execute job $j \in \mathcal{J}$ on mode $m \in \mathcal{M}_{j}$;
required amount of renewable resource $r \in \mathcal{R}$ to execute job $j \in \mathcal{J}$ on mode $ m \in \mathcal{M}_{j}$;
available amount of non-renewable resource $k \in \mathcal{K}$;
available amount of renewable resource $r \in \mathcal{R}$;
release date of project $p$;
artificial job belonging to project $p \in \mathcal{P}$, which represents the end of the project.
Preprocessing Input Data {#subsec:pre}
------------------------
An effective way to reduce the search space is by identifying tight time windows in which it is valid to process jobs. A basic technique to define the earliest starting time $\breve{e}_j^s$ for jobs $j \in \mathcal{J}$ consists of computing the CPD using CPM [@Kelley1959] without considering resource constraints. This methods allows to compute the $\breve{e}_j^s$ of all jobs, taking into consideration the precedence relationships. The longest path of a project, also known as the critical path, provides a lower bound for the completion time of each project.
Consider, for each project $p \in \mathcal{P}$, the release date $\sigma_p$, and lower bound based (i.e., the length of the critical path) $\lambda_p$ as input data and the value $\beta_p$, an upper bound for each project $p$, obtained from any feasible solution. Optimality conditions can be used to restrict the set of valid time periods when a job can be allocated. We initially consider the value $\alpha$ computed by Eq.(\[eq:alpha\]), that represents an upper bound to the maximum total project delay allowed. $$\label{eq:alpha}
\alpha = \sum_{p\in \mathcal{P}}{\left(\beta_p-\sigma_p-\lambda_p\right)}$$ Thus, the maximum time period $\breve{t} \in \mathcal{T}$ that needs to be considered int the planning, can be obtained by Eq.(\[eq:MaxT\]). $$\label{eq:MaxT}
\begin{split}
\breve{t} = \max_{p \in P}{ (\sigma_p+\lambda_p+\alpha) } \\
\mathcal{T} = \{ 0,...,\breve{t}\}
\end{split}$$
Analogously, upper bounds can be computed for processing times of jobs. The upper bounds can be strengthened if the selection of modes with different durations is also considered. The upper bounds are used, along with the duration of each job and without considering the resource constraints, to define the latest starting times ($\breve{l}_j^s$) for jobs $j \in \mathcal{J}$. A job $j$ from a project $p$ when processed at mode $m$ will push forward (i.e., postpone) all successor jobs by exactly $d_{jm}$ time units. Consider set $\overline{\mathcal{S}}_j$, containing the entire chain of successors of job $j$ on the longest path from job $j$ to the artificial job $a_p$ (indicating the project completion). Let lower bound $\mathcal{L}_{jm}$ be the total duration in this path, computed considering only the fastest processing modes for each job in this chain. The maximum allocation time or latest starting time $(\breve{l}^{s}_{jm})$ for a job $j$ from a project $p$ when processing on mode $m$ to $\mathcal{T}_{jm}$ is given by Eq.(\[eq:MaxTJM\]). $$\label{eq:MaxTJM}
\begin{split}
\breve{l}^{s}_{jm} = \sigma_p +\lambda_p - \mathcal{L}_{jm} +\alpha \\
\mathcal{T}_{jm} = \{ \breve{e}_{j}^{s},...,\breve{l}^{s}_{jm}\}
\end{split}$$ Similar bounds can be derived for any two jobs in this path also considering the fastest processing modes for all jobs except the first one:
[xxxxxxx]{}
the shortest path in the precedence graph considering the length of the arcs between job $j$ and successor job $s \in \overline{\mathcal{S}}_j$ considering mode $m \in \mathcal{M}_j$;
the shortest path in the precedence graph considering the length of the arcs between job $j$ and successor job $s \in \overline{\mathcal{S}}_j$ considering $j$ fastest mode. \[id:d\]
Formulation {#subsec:form}
-----------
Binary decision variables are used to select the mode and starting times for the jobs. They are defined as follows:
$$x_{jmt} =
\begin{cases}
\; 1 &
\textrm{if job } j \in \mathcal{J} \textrm{ is allocated on mode } m \in \mathcal{M}_{j} \\& \textrm{at starting time } t \in \mathcal{T}_{jm};\\
\; 0 & \textrm{otherwise}.
\end{cases}
\label{eq:eff}$$
We introduce in this formulation on/off discrete time variables studied in @Artigues2017 to allow resources constraints and cutting planes, detailed in the next sections, to be expressed with fewer variables. The following binary decision variables indicate during which time periods jobs are being processed: $$\label{eq:eff2}
z_{jmt} =
\begin{cases}
\; 1 &
\textrm{if the job } j \in \mathcal{J} \textrm{ is allocated on mode } m \in \mathcal{M}_{j} \textrm{ and} \\ & \textrm{ is being processed during time } t \in \mathcal{T}_{jm};\\
\; 0 & \textrm{otherwise}.
\end{cases}$$
The objective function minimizes the total project delay over the project completion times for projects and their critical paths. Consider the following integer variable included in the objective function:
[xxxxxxx]{}
integer variable used to compute the makespan, included in the objective function with a small coefficient $\epsilon$ to break ties.
*Minimize:* $$\begin{aligned}
\label{model:foCons}
\displaystyle \sum_{p\in \mathcal{P}}\sum_{m\in \mathcal{M}_{a_p}}\sum_{t\in \mathcal{T}_{a_pm}}{ \left[t - (\sigma_p + \lambda_p)\right] x_{a_pmt} } + \epsilon h\end{aligned}$$
*subject to:* $$\begin{aligned}
\label{model:maximoUmaVez}
\sum_{m\in \mathcal{M}_{j}}\sum_{t \in \mathcal{T}_{jm}}{ x_{jmt} }
= 1 \ \ \forall j\in \mathcal{J} \\ [0.5cm]
\label{model:recursoNR}
\sum_{j\in \mathcal{J}} \sum_{m\in \mathcal{M}_{j}} \sum_{t \in \mathcal{T}_{jm}} q_{kjm} x_{jmt}
\leq \breve{q}_{k}
\ \ \forall k\in \mathcal{K} \\[0.5cm]
\label{model:recursoR}
\sum_{j\in \mathcal{J}}\sum_{m\in \mathcal{M}_{j}}
{ q_{rjm} z_{jmt}}
\leq \breve{q}_{r}
\ \ \forall r \in \mathcal{R}, \forall t \in \mathcal{T}
\end{aligned}$$ $$\begin{aligned}
\label{model:precedencia1}
\sum_{m\in \mathcal{M}_{j}}\sum_{t \in \mathcal{T}_{jm}}{\left(t+d_{jm}\right) x_{jmt}}\ - \ \sum_{z\in \mathcal{M}_{s}}\sum_{i \in \mathcal{T}_{sz}}{i x_{szi}}
\leq 0 \nonumber \\
\forall j\in \mathcal{J}, \forall s\in \mathcal{S}_j \\ [0.5cm]
\label{model:varlnkxz}
z_{jmt} - \sum_{t^{'} = (t-d_{jm}+1)}^{t}{x_{jmt^{'}}}
= 0 \ \ \forall j \in \mathcal{J}, \forall m \in \mathcal{M}_j, \forall t \in \mathcal{T}_{jm} \\ [0.5cm]
\label{model:ymax}
h - \sum_{m\in \mathcal{M}_{a_{p}}}\sum_{t \in \mathcal{T}_{a_{p}m}}{ t x_{a_{p}mt} }
\geq 0 \ \ \forall p \in \mathcal{P}\\ [0.5cm]
\label{model:varDecx}
x_{jmt} \in \{0,1\} \ \ \forall j \in \mathcal{J}, \forall m \in \mathcal{M}_j, \forall t \\ [0.5cm]
\label{model:varDecz}
z_{jmt} \in \{0,1\} \ \forall j \in \mathcal{J}, \forall m \in \mathcal{M}_j, \forall t \in \mathcal{T}_{jm} \\ [0.5cm]
\label{model:varInth}
h \geq 0 \end{aligned}$$
Constraints (\[model:maximoUmaVez\]) ensure that each job is allocated to exactly one starting time and one mode. Constraints (\[model:recursoNR\]) and (\[model:recursoR\]) are capacity constraints for non-renewable and renewable resources, respectively . Constraints (\[model:precedencia1\]) force precedence relationships to be satisfied. Constraints (\[model:varlnkxz\]) link variables $z$ and variables $x$. Constraints (\[model:ymax\]) compute the total makespan. Finally, constraints (\[model:varDecx\]), (\[model:varDecz\]) and (\[model:varInth\]) respectively ensure that variables $x$ and $z$ can only assume binary values and $h$ can only assume nonnegative values.
Preprocessing MILP Formulation {#subsec:premip}
------------------------------
@Johnson1985 introduce an interesting preprocessing method to strengthen constraint coefficients using the knapsack structure of resource constraints. This preprocessing was used in @Sankaran1999 for the SMRCPSP and analyzes one resource usage constraint at time. In this paper, we propose a preprocessing technique that considers various constraints (precedence and the usage of other renewable and non-renewable resources), besides the renewable resource constraint, which will be strengthened.
The proposed procedure to strengthen resource usage constraints (\[model:recursoR\]) was inspired by Fenchel cutting planes [@Boyd1992; @Boyd1994]. Fenchel cutting planes are based on the enumeration of incidence vectors to find the most violated inequality for a subset of binary variables. In our paper, we enumerate feasible subsets of jobs and modes to create a linear problem to find the best possible strengthening of a given resource constraint.
The strengthening procedure is presented in Algorithm \[alg:strengthening\]. First, it computes, for each $t$, a set $\mathcal{G}_t$ composed of all jobs and modes ($j,m$) available for processing at time $t \in \mathcal{T}_{jm}$ (see Algorithm \[alg:strengthening\], lines \[alg:t\]–\[alg:gt\]). Formally, these sets can be computed as stated in Eq.(\[eq:gt\]). $$\label{eq:gt}
\mathcal{G}_t = \{ \ (j,m) \ \in \ \mathcal{J} \times \mathcal{M}_j \mid \ t \in \mathcal{T}_{jm} \}$$
To illustrate the coefficient strengthening technique, consider for $t=4$, the jobs and modes that make up the set $\mathcal{G}_4 = \{(1,0), (1,1), (1,2), (2,1), (2,2),\\ (3,0), (5,0), (5,1), (5,2), (6,1), (6,2), (9,1), (9,2)\}$. Consider also, the following original constraint restricting the usage of renewable resource $r_0$ at time $t=4$: $$\begin{aligned}
0 z_{1,0,4} + 0 z_{1,1,4} + 0 z_{1,2,4} + 0 z_{2,1,4} + 0 z_{2,2,4} + 3 z_{3,0,4} + 0 z_{5,0,4} + 0 z_{5,1,4} + \\ 2 z_{5,2,4} + 0 z_{6,1,4} + 6 z_{6,2,4} +0 z_{9,1,4} + 9 z_{9,2,4} \leq 9.\end{aligned}$$
The next step is to enumerate all valid combinations of jobs and modes $(j,m)$ that can be processed in parallel at time $t$, i.e., that satisfy all resources constraints and do not have precedence relations among each other. @Mingozzi1998 designated these valid combinations as *feasible subsets*.
Let $\mathcal{E}_t = (\bar{e}_1, \bar{e}_2, \ldots, \bar{e}_n)$ be the set of all these feasible subsets. This set can be computed by using a simple backtracking algorithm that recursively proceeds, from level $0$ to $|\mathcal{G}_{t}|$; tentatively fixing the allocation of each respective job and mode to 0 or 1; proceeding to the next level on the search space only when the partial fixation to the current level does not violate any resource or precedence constraint.
If we enumerate all the possibilities over $\mathcal{G}_4$, we could have $2^{13}=8192$ feasible subsets. However, due to multiple constraints, there are only $51$ feasible subsets in $\mathcal{E}_4$:
`\mathcal{E}_4 = [{(1,0)}, , {(2,2),(1,0)}, {(3,0),(1,0)}, {(1,1)}, {(2,2),(3,0),(1,1)}, {(2,2),(1,1)}, {(3,0),(1,1)}, {(1,2)}, {(2,1),(3,0),(1,2)}, {(2,1),(1,2)}, {(2,2),(3,0),(1,2)}, {(2,2),(1,2)}, {(3,0),(1,2)}, {(2,1)}, {(3,0),(5,2),(2,1)}, {(3,0),(2,1)}, {(5,2),(2,1)}, {(2,2)}, {(3,0),(5,0),(2,2)}, {(3,0),(5,1),(2,2)}, {(3,0),(5,2),(2,2)}, {(3,0),(2,2)}, {(5,0),(2,2)}, {(5,1),(2,2)}, {(5,2),(2,2)}, {(3,0)}, {(5,0),(9,1),(3,0)}, {(5,0),(3,0)}, {(5,1),(9,1),(3,0)}, {(5,1),(3,0)}, {(5,2),(9,1),(3,0)}, {(5,2),(3,0)}, {(6,1),(3,0)}, {(6,2),(9,1),(3,0)}, {(6,2),(3,0)}, {(9,1),(3,0)}, {(5,0)}, {(9,1),(5,0)}, {(9,2),(5,0)}, {(5,1)}, {(9,1),(5,1)}, {(9,2),(5,1)}, {(5,2)}, {(9,1),(5,2)}, {(6,1)}, {(9,2),(6,1)}, {(6,2)}, {(9,1),(6,2)}, {(9,1)}, {(9,2)}]`.
As an example of the impact of considering multiple constraints for reducing the number of valid incidence vectors, if we do not consider precedence constraints and we just consider one renewable resource constraint in the enumeration process, $182$ feasible subsets would be built.For the maximal feasible subset `{(2,2),(3,0),(1,0)}`, highlighted above inside the box, if we do not consider all resources and precedence constraints, it will be extended to `{(2,2),(3,0),(5,2),(1,0)}, {(2,2),(3,0),(6,2),(1,0)}`.
The $i^{th}$ feasible subset $\bar{e}_i$ contains ordered pairs $(j,m) \in \mathcal{G}_t$. For each renewable resource $r$ with capacity $c$ and time $t$ the following linear program ($W_{rt}$) can be used to strengthen constraints (\[model:recursoR\]) if the enumeration process is successful, i.e., a pre-defined maximum number of iterations ($it$) was not reached, (see Algorithm \[alg:strengthening\], lines \[alg:suc\]–\[alg:replace\]). Consider the continuous variables $u_{jm}$, indicating the number of consumed units of resource $r$ by job $j$ at mode $m$ in the strengthened constraint of the following linear programming ($W_{rt}$ model):\
$$\begin{aligned}
\label{eq:miprein1}
\displaystyle \sum_{(j,m) \in \mathcal{G}_t} u_{jm}
\end{aligned}$$ $$\begin{aligned}
\label{eq:miprein2}
\displaystyle \sum_{(j,m) \in \bar{e}} u_{jm} \leq c \ \forall \bar{e} \in \mathcal{E}_{t} \\ [0.5cm]
\label{eq:miprein3}
q_{rjm} \leq u_{jm} \leq c \ \forall (j,m) \in \mathcal{G}_t
\end{aligned}$$
Consider $\forall (r,t) \in \mathcal{R} \times \mathcal{T}: \bar{q}_{rjmt} = u^*_{jm}$ from $W_{rt}$, where $u^*_{jm} $ is the optimal solution in $W_{rt}$. This value is introduced as a new input data for the main formulation:
[xxxxxxx]{}
new values for required amount of renewable resource $r \in \mathcal{R}$ to execute job $j \in \mathcal{J}$ on mode $ m \in \mathcal{M}_{j}$ at time $t$.
Constraints (\[model:recursoRZ\]) are created with the new values $\bar{q}_{rjmt}$, yielding improved capacity constraints for renewable resources.
$$\begin{aligned}
\label{model:recursoRZ}
\sum_{j\in \mathcal{J}}\sum_{m\in \mathcal{M}_{j}} \bar{q}_{rjmt} z_{jmt}
\leq \breve{q}_{r}
\ \ \forall r \in \mathcal{R}, \ \forall t \in \mathcal{T}\end{aligned}$$
Due to the bounds on variables $u$, constraints (\[model:recursoRZ\]) always dominate the original constraints (\[model:recursoR\]), since $\bar{q}_{rjmt} \geq q_{rjm}$. In particular, whenever $u_{jm} > q_{rjm}$ it dominates strictly. The following defines dominance between two generated cuts [@Wolsey1998]:
Let $ c^{1^{T}} x \leq r^{1}$ and $ c^{2^{T}} x \leq r^{2}$ be two inequalities. We say $ c^{1^{T}} x \leq r^{1}$ dominates $ c^{2^{T}} x \leq r^{2}$ if $c_{i}^{1} \geq c_{i}^{2} \ \forall i$ and $r^{1} \leq r^{2}$; if at least one of these inequalities is satisfied as an strict inequality, then there is a strict dominance. \[def:domination\]
An interesting property of this procedure is that it may strengthen constraints of resources that are *not scarce*, given the scarceness of other resources and/or precedence constraints.
#### Example
By solving the MILP model $\mathcal{W}_{0,4}$ to our example introduced above, the coefficient of the emphasized variable $\boldsymbol{z_{5,2,4}}$ corresponding to job $5$ on mode $2$ processed at $t=4$ can be strengthened to $6$, without excluding any feasible integer solution. We can generate the following strengthened constraint: $$\begin{aligned}
0 z_{1,0,4} + 0 z_{1,1,4} + 0 z_{1,2,4} + 0 z_{2,1,4} + 0 z_{2,2,4} + 3 z_{3,0,4} + 0 z_{5,0,4} + 0 z_{5,1,4} + \\ \boldsymbol{6 z_{5,2,4}} + 0 z_{6,1,4} + 6 z_{6,2,4} + 0 z_{9,1,4} + 9 z_{9,2,4} \leq 9.\end{aligned}$$
For the SMRCPSP and MMRCPSP, the original constraints (\[model:recursoR\]) can be replaced by new constraints presented in (\[model:recursoRZ\]) in the case that the latter are stronger. For the MMRCMPSP, the new constraints (\[model:recursoRZ\]) are created to strengthen renewable resources for each project separately, and the original constraints remain in the model.
We consider a time limit $tl$, which will be checked after the subset enumeration procedure for each $t$. If the time limit is reached, the remaining time periods $t$ will be skipped and the algorithm is terminated (see Algorithm \[alg:strengthening\], lines \[alg:time\]–\[alg:timeend\]). We continue to find feasible subsets while it does not reach the last element of $\mathcal{G}_{t}$. If the maximum number of iterations $it$ is reached, we stop the process (by returning $\emptyset$, see Algorithm \[alg:strengthening\], lines \[alg:btsb\]–\[alg:suc\]) and continue to the next value of $t$ on the strengthening algorithm.
The Cutting Plane Algorithm {#sec:method}
===========================
The performance of general purpose MILP solvers on a given formulation strongly depends on how tight is the LP relaxation (dual bound) to the optimal solution [@Wolsey1998]. Cutting planes are commonly used to improve this bound by iteratively adding violated cuts.
The proposed cutting plane algorithm uses traditional RCPSP cuts enhanced with new lifting techniques (see Subsections \[subsec:cover\] and \[subsec:prece\], respectively for cover and precedence cuts), conflict-based cuts (see Subsection \[subsec:conflict\] for clique and odd-holes cuts) and strengthened Chvátal-Gomory cuts (see Subsection \[subsec:cg\]) generated from an implicit dense conflict dynamic graph.
![Execution flow of the proposed cutting plane method[]{data-label="fig:outline"}](paperoutline3.pdf){width="0.9\linewidth"}
An outline of the proposed method is depicted in Figure \[fig:outline\]. After the instance data is read, we create the mathematical programming model (**M**) explained in Section \[sec:exactmodel\] and execute the preprocessing routines. Then, the optimal LP relaxation is computed and if a fractional solution is obtained, different search methods are started to separate violated inequalities. Since the separation of these inequalities may involve the solution of $\mathcal{N}\mathcal{P}$-hard problems, we execute the separation procedures in parallel. This allows us to save some processing time in order to process a larger number of iterations in the cutting plane algorithm within the given time limit.
All generated cuts are inserted into a cut pool where repeated inequalities are discarded. Our algorithm quickly discards repeated cuts by using a hash table. While checking for repeated cuts is very fast, the dominance check is slower since it requires checking the contents of the cuts. We only check the dominance in the pool of cover separation since they generated less cuts compared with other types of cuts.
If new cuts have been found after the separation procedure, a stronger formulation is obtained, and the process is repeated. When the time limit is reached or when no further cut is generated, the strengthened model and its objective function value are returned. In the following, we present the different inequalities that are separated as well as the algorithmic aspects involved in their separation.
Lifted RCPSP Knapsack Cover Cuts - LCV {#subsec:cover}
--------------------------------------
Resource usage constraints present a knapsack structure which was exploited in [@Zhu2006] to generate GUB cover (CV) cuts. Consider the general case of a constraint in the form $ \displaystyle \sum_{j \in \mathcal{N}} b_j z_j \leq c,(b,c)\in {\mathbb{N}^{+}}^{n}\times \mathbb{N}^{+} $ for a set $\mathcal{N}$ of binary variables. The following knapsack problem can be solved to generate a valid inequality that cuts fractional point $z^*$:\
*Minimize* $$\begin{aligned}
\label{eq.coverinit}
\gamma (z^*)=\sum_{j \in \mathcal{N}} (1-z_{j}^{*}) v_j
\end{aligned}$$ *subject to:* $$\begin{aligned}
\sum_{j \in \mathcal{N}}{ b_j v_j} > c\\
v_j\in \{0,1\} \forall j \in \mathcal{N}
\label{eq.coverend}
\end{aligned}$$
Whenever a solution with $\gamma (z^*) < 1$ is found involving a set $\mathcal{V}=\{j \in \mathcal{N} : v_j=1\}$ a violated inequality in the form $\displaystyle \sum_{j \in \mathcal{V}}z_j \leq |\mathcal{V}|-1$ is generated. Since only active variables $(z_j^*>0)$ are considered in this separation, many dominated inequalities can be generated in different iterations.
While general purpose lifting strategies [@Gu2000] can be used, specific problem information can provide an effective procedure to produce lifted cover inequalities. A variable in a traditional cover inequality represents whether a job $j$ and mode $m$ were allocated or not at time $t$. The proposed lifted RCPSP knapsack cover cut (LCV) separation routine may include, for each job $j$, variables of additional processing modes without increasing the right-hand-side, producing much stronger cuts. In the single mode case the strengthening will be similar to the one obtained with traditional lifting (see [@Balas1975; @Balas1978; @Nemhauser1994]) techniques.
The new lifted knapsack cover separation problem for the RCPSP is solved for each renewable resource and each time period. Consider period $t^{*}$, resource $r$ with capacity $\breve{q}_{r}$ and a fractional solution with variable values $z_{jmt^*}^{*}$. The decision variables are:
[xxxxxxxxx]{}
if variable of job $j$ at mode $m$ is selected (1) or not (0);
if at least one mode for job $j$ is included in the cut (1) or not (0);
if job $j$ has $m$ as the selected mode with the smallest resource consumption from the selected ones (1) or not (0);
resource consumption excess if modes with smallest resource consumption are selected.
cut violation.
The LCV separation problem is given by:\
*Maximize* $$\begin{aligned}
\label{model:foCons2}
\displaystyle \omega \overline{v} +\sum_{j\in \mathcal{J}}\sum_{m\in \mathcal{M}_{j}}\mu v_{jm}\end{aligned}$$ *subject to:* $$\begin{aligned}
& & o_{j}=\sum_{m\in \mathcal{M}_{j}}\underline{w}_{jm}\ \ \ \forall j\in J\label{eq:lnkew}\\
& & \underline{e}=\sum_{j\in \mathcal{J}}\sum_{m\in \mathcal{M}_{j}}q_{rjm} \underline{w}_{jm}-\breve{q}_{r}\label{eq:e2}\\
& & v_{jm} \leq \sum_{m'\in \mathcal{M}_{j}:q_{rjm'}\leq q_{rjm}}\underline{w}_{jm'}\ \ \ \forall j\in \mathcal{J},m\in \mathcal{M}_{j}\label{eq:w2}\\
& & \overline{v}=\sum_{j\in \mathcal{J}}\sum_{m\in \mathcal{M}}z_{jmt^*}^{*} v_{jm}-\sum_{j\in \mathcal{J}}o_{j}+1\label{eq:viol}\\
& & 0.005 \leq \overline{v} \leq \infty \label{eq:minViol}\\
& & 1 \leq \underline{e} \leq \infty \label{eq:qfeas2}\\
& & o_{j}, v_{jm}, \underline{w}_{jm} \in \{0,1\} \ \ \ \forall j \in \mathcal{J}, \ \forall m \in \mathcal{M}_j \label{eq:bin}\end{aligned}$$
The objective function (\[model:foCons2\]) maximizes a hierarchical objective function composed of, first, the cut violation and, second, the inclusion of additional jobs and modes in the generated inequality to produce stronger cuts ($\omega \gg \mu$). These weights cannot be too large or too small to prevent numerical instability in the solvers. Constraints (\[eq:lnkew\]) ensure that $o_{j}$ is only activated when some mode is selected for job $j$ as the mode with lower resource usage. Constraints (\[eq:e2\]) and (\[eq:qfeas2\]) ensure that a cover is produced. Constraints (\[eq:w2\]) ensure that only modes with resource usage greater than or equal to the mode with the smallest resource usage selected are allowed for lifting. Equality (\[eq:viol\]) computes the cut violation and constraint (\[eq:minViol\]) ensures that a violated cut is produced. Finally, constraints (\[eq:bin\]) ensure that variables $o_j,v_{jm}$ and $\underline{w}_{jm}$ can only assume binary values.
Let $C$ be the set of selected jobs and modes (with $v_{jm}=1$) in the solution of the problem above. Further, let $o_j$ indicate its corresponding solution values. We may generate they following LCV cut:
$$\label{eq:selectedmodes}
\sum_{(j,m)\in C}z_{jmt}\leq\sum_{j\in J}o_{j}-1 \ \ \ \forall t \in \mathcal{T}$$
For a valid cover cut the value on the right-hand-side would be the size of the set minus $1$, with the lifting strategy whenever it chooses more than one mode per job the sum of the right-hand-side will be strictly smaller than the size of the set minus $1$, generating a stronger cut as in the example below.
#### Example
Consider the following cut for resource $r_0$ on $t=8$, and jobs {4,8} processing respectively on modes {0,2} from Figure \[fig:gantt\], generated using the separation described in (\[eq.coverinit\])-(\[eq.coverend\]): $$\begin{aligned}
(CV) = z_{4,0,8} + z_{8,2,8} \leq 1.\end{aligned}$$
This cut can be lifted since job $8$ has other modes {$0,1$}, each of them consuming $6$ units of resource $r_0$, more than the current mode {$2$} that consumes $3$ units, forming still a valid inequality: $$\begin{aligned}
(LCV)= z_{4,0,8} + \boldsymbol{z_{8,0,8} + z_{8,1,8}} + z_{8,2,8} \leq 1.\end{aligned}$$
It is important to emphasize that the first component of the objective function maximizes the violation considering the consumption of the other modes, and the second component, even for the single mode version, will add additional variables that do not contribute to the cut violation but which contribute to generate a stronger inequality.
Lifted Precedence Based Cuts - LPR {#subsec:prece}
----------------------------------
In addition to the cover cut introduced above, it is possible to further strengthen the formulation by analyzing the precedence between jobs and using precedence (PR) cuts similar to those used by @Zhu2006. Consider a job $j$, its (direct or indirect) successor $s$ and a time $t$. Also, consider the following constants $\displaystyle e_j=\textrm{min}_{m \in \mathcal{M}_j}(\mathcal{T}_{jm})$ and $\displaystyle l_j=\textrm{max}_{m \in \mathcal{M}_j}(\mathcal{T}_{jm})$ to limit the time period. We introduce a new lifted version for the precedence cuts, shown in Eq.(\[eq:ineqprec\]). Consider the lengths of the shortest paths on the precedence graph, $\breve{d}_{jms}$ and $\breve{d}^{*}_{js}$, introduced in Section \[subsec:pre\]. The following inequalities are valid: $$\begin{gathered}
\label{eq:ineqprec}
\sum_{m \in \mathcal{M}_j} \sum_{t'= e_j}^{t+\breve{d}_{js}^{*}-\breve{d}_{jms}} x^*_{jmt'} \geq
\sum_{m \in \mathcal{M}_s} \sum_{t' = e_s}^{min(l_s, t+\breve{d}_{js}^{*})} x^*_{smt'} \\ \forall j \in \mathcal{J}, s \in \overline{\mathcal{S}}_j , t \in \{ \max{(e_j, e_s - \breve{d}^*_{js} )} , \ldots , min(l_j,l_s -\breve{d}^*_{js})\}\end{gathered}$$
#### Example
Consider the following two cuts. The first one, (PR), was generated with the RCPSP precedence cut as proposed in [@Zhu2006]. The second one, (LPR), is the lifted inequality (\[eq:ineqprec\]) to predecessor job $5$ and its successor $10$ at time period $9$: $$\begin{aligned}
(PR) = - x_{5,0,2} - x_{5,0,3} - x_{5,0,4} - x_{5,0,5} - x_{5,0,6} - x_{5,0,7} - x_{5,0,8} - x_{5,0,9} - \\ {\boxed{x_{5,0,10} - x_{5,0,11}}} - x_{5,1,2} - x_{5,1,3} - x_{5,1,4} - x_{5,1,5} - x_{5,1,6} - x_{5,1,7} - \\ x_{5,1,8} - {\boxed{x_{5,1,9} - x_{5,1,10} - x_{5,1,11}}} - x_{5,2,2} - x_{5,2,3} - x_{5,2,4} - x_{5,2,5} - \\ x_{5,2,6} - {\boxed{x_{5,2,7} - x_{5,2,8} - x_{5,2,9}}} + x_{10,0,11} + x_{10,2,11} \leq 0 ;\\
(LPR) = - x_{5,0,2} - x_{5,0,3} - x_{5,0,4} - x_{5,0,5} - x_{5,0,6} - x_{5,0,7} - x_{5,0,8} - x_{5,0,9} - \\ x_{5,1,2} - x_{5,1,3} - x_{5,1,4} - x_{5,1,5} - x_{5,1,6} - x_{5,1,7} - x_{5,1,8} - x_{5,2,2} -\\ x_{5,2,3} - x_{5,2,4} - x_{5,2,5} - x_{5,2,6} + x_{10,0,11} + x_{10,2,11} \leq 0.\end{aligned}$$
Notice the change in the coefficients of the variables on the left-hand-side of the lifted cut (LPR), based on Eq.(\[eq:ineqprec\]) and corresponding to the variables of job $5$. In the case of different durations for processing modes of a job $j$, there will be an increase in the sum of the coefficients of the left-hand-side of (LPR) since we use $\breve{d}_{jms}$ instead of just considering the fastest mode given by $\breve{d}^{*}_{js}$ (as proposed in [@Zhu2006]). The original cut variant (PR) from [@Zhu2006] considers the fastest time. The variables highlighted in the boxes in the (PR) cut are not included in our cut approach (LPR). The lifted cut therefore strictly dominates the original cut, given that the highlighted variables have coefficients $0$ (instead of $-1$) in the cut stated as $\leq$ inequality.
Our separation procedure (see Algorithm \[alg:filtering\]) therefore selects different paths (line \[alg:cp\]) that connect job $j$ and the artificial project completion job $a_{p}$. In particular, it is desirable to avoid redundant constraints, i.e., constraints of predecessor/successor jobs belonging to the same path in the dependency graph with a similar meaning, for jobs on subpaths that have already been used in previously added precedence cuts. Whenever a violated precedence cut is found on this path (lines \[alg:cuts\]–\[alg:endcuts\]), the remaining jobs in this path are skipped (line \[alg:skipp\]). We limit the number $\zeta$ of precedence cuts that can be added per round and only add the most violated cuts (line \[alg:zeta\]).
`sort_cuts_by_highest_violation_and_filters`($\mathcal{C}$, $\zeta$); \[alg:zeta\]\
`return \mathcal{C};`
Conflict-Based Cuts: Cliques (CL) and Odd-Holes (OH) {#subsec:conflict}
----------------------------------------------------
According to @Padberg1973, LP relaxations for problems that mostly contain binary variables linked to generalized upper bound (GUB) constraints can be significantly strengthened by the inclusion of inequalities derived from the set packing polytope (SPP). Generally, clique and odd-holes cuts can be generated using Conflict Graphs ($\mathcal{CG}$). The denser the $\mathcal{CG}$, the more inequalities can be generated. The disadvantage of having dense $\mathcal{CG}$ is that they can be prohibitively large [@atamturk2000], so our algorithm creates the $\mathcal{CG}$ dynamically at each iteration by considering a set ($\mathcal{U}$) that contains the variables of interest, i.e., variables that have a non-zero value in the LP relaxation or variables set to zero but with a small reduced cost.
Several well-known inequalities can be generated considering pairwise conflicts between binary variables stored in $\mathcal{CG}$. Some conflicts can be easily detected by solvers considering constraints such as (\[model:maximoUmaVez\]). Other conflicts can be implied from optimality conditions or by analyzing problem specific structures. Overall, the denser is the $\mathcal{CG}$, stronger are the produced cuts.
The dynamic dense $\mathcal{CG}$ created is used in a separation procedure for inequalities derived from a common class of cuts for the SPP: cliques and odd-holes. A clique inequality for a set $C$ of conflicting variables has the form $\sum_{j \in C} x_j \leq 1$ and an odd-hole inequality for a cycle $C$ can be defined as: $\sum_{j \in C} x_j \leq \lfloor \frac{|C|}{2} \rfloor$ [@Santos2016]. @Santos2016 present a clique separation routine that separates all violated cliques into a conflict subgraph induced by fractional variables. The authors then present a lifting that extends generated cliques considering the original $\mathcal{CG}$. They also present a strengthening of odd-holes inequalities by the inclusion of a so-called wheel center. For an odd-hole with variables $C$ and $W$ being the set of candidates to be included as wheel centers of $C$, the inequality (\[eq:ineqoh\]) is valid:
$$\sum_{j \in W} \lfloor \frac{|C|}{2} \rfloor x_j + \sum_{j \in C} x_j \leq \lfloor \frac{|C|}{2} \rfloor
\label{eq:ineqoh}$$
Our approach presented in Algorithm \[alg:conflictgraph\] considers four conflict types for variables $x_{jmt}$:
1. conflicts between variables of the same job (lines 6–7);
2. conflicts involving jobs that if allocated at the same time exceed the capacity of available renewable resources (lines 8–10);
3. conflicts based on precedence relations, in which the time window between the predecessor job $j$ on mode $m$ and some $s \in \overline{\mathcal{S}}_j$ is smaller than $\breve{d}_{jms}$ (lines 11–14);
4. conflicts considering jobs of different projects, where the sum of the delays generated by allocating these jobs in specific positions implies a total delay greater than $\alpha$ (lines 15–16).
$\mathcal{C}\mathcal{G} \gets \emptyset$;\
**return** $\mathcal{C}\mathcal{G}$;
After the creation of the $\mathcal{CG}$, cuts can be generated considering the current fractional solution. In this paper, we use the routines described in [@Brito2015], where cliques and odd-holes are exactly separated and lifted.
#### Example
Considering the following clique cut: $$\begin{aligned}
(CL) = x_{5,0,12} + x_{5,1,11} + x_{5,2,9} +\\ x_{10,0,11} + x_{10,0,12} + x_{10,0,13} + x_{10,2,11} + x_{10,2,12} + x_{10,2,13} \leq 1.\end{aligned}$$
It is possible to observe, from Figure \[fig:gantt\], that the variables corresponding to jobs $5$ and $10$ have conflicts at different time periods considering different modes, given that they have a precedence relation.
#### Example
Consider the following odd-hole cut: $$\begin{aligned}
(OH) = x_{1,0,0} + x_{1,0,5} + x_{2,1,0} + x_{2,1,3} + x_{5,0,3} \leq 2.\end{aligned}$$
Still referring to job $5$ in Figure \[fig:gantt\], we can observe conflicts from the precedence relationship with job $1$. Also, a low amount of resources available at times when jobs $5$ and $2$ intersect on the variables of the example above reflect conflicts between them. Thus, these three jobs can not be allocated in parallel, considering the amount of resources consumed by their modes.
Strengthened Chvátal-Gomory Cuts - (SCG) {#subsec:cg}
----------------------------------------
Chvátal-Gomory (CG) cuts are well-known cutting planes for MILP models. The inclusion of these cuts allows to significantly reduce the integrality gaps, even when only rank-one cuts are employed, i.e., those obtained from original problem constraints [@Fischetti2007].
Consider the integer linear programming (ILP) problem as min{${\mathbf{c}}^T {\mathbf{x}}: A{\mathbf{x}} \leq {\mathbf{b}}, {\mathbf{x}} \geq 0 $ integer}, where $A \in \mathbb{R}^{m x n}$, $ {\mathbf{b}} \in \mathbb{R}^m $, and $ {\mathbf{c}} \in \mathbb{R}^n $, with the two associated polyhedra $P := \{{\mathbf{x}} \in \mathbb{R}_{+}^{n}: A{\mathbf{x}} \leq {\mathbf{b}} \}$ and $P_{\mathcal{I}} := conv\{{\mathbf{x}} \in \mathbb{Z}_{+}^{n}: A{\mathbf{x}} \leq {\mathbf{b}} \} = conv (P\cap \mathbb{Z}^n) $ with ${\mathbf{x}}$ being integer variables. Consider $\mathcal{I}$ and $\mathcal{H}$ the sets of constraints and variables, respectively.
A Chvátal-Gomory cut [@Chvatal1973] is defined as a valid inequality for $P_{\mathcal{I}}$: $\lfloor {\mathbf{u}}^T A \rfloor {\mathbf{x}}$ $\leq \lfloor {\mathbf{u}}^T{\mathbf{b}} \rfloor$, where ${\mathbf{u}} \in \mathbb{R}_{+}^{m}$ is a multiplier vector. The choice of ${\mathbf{u}} \in \mathbb{R}^{+}$ is crucial to deriving useful inequalities. Fischetti and Lodi [@Fischetti2007] propose the MILP model for Chvátal-Gomory separation. The maximally violated $\sum_{j \in \mathcal{H}(x^*)} {a_j x_j} \leq a_0 $ inequality can be found by optimizing the following separation MILP model:\
*Maximize:* $$\begin{aligned}
\label{eq:cg1}
\sum_{j \in \mathcal{H}(x^*)}{a_j x^*_j} - a_0 \end{aligned}$$ *subject to:* $$\begin{aligned}
\label{eq:cg2}
f_j = {\mathbf{u}}^T A_j - a_j, \ \forall j \in \mathcal{H}(x*) \\
\label{eq:cg3}
f_0 = {\mathbf{u}}^T {\mathbf{b}} - a_0 \\
\label{eq:cg4}
0 \leq f_j \leq 1 - \delta \ \forall j \in \mathcal{H}(x*)\cup\{0\}\\
\label{eq:cg5}
-1+ \delta \leq u_i \leq 1 - \delta \ \forall i = 1, \ldots, m \\
\label{eq:cg6}
a_j \ \in \ \mathbb{Z}, \ \forall j \in \mathcal{H}(x*) \end{aligned}$$ where $\mathcal{H}(x^*) := \{j \in {1, \ldots , \breve{n}} : x^*_j > 0\}$ and $x^*$ are fractional values for all $\breve{n}$ variables of an LP solution for a general problem fixed in the MILP. To strengthen the cut, a penalty term $\sum_{i}w_i u_i$, with $w_i =10^{-4}$ for all $i$, is applied to the objective function. To improve the numerical accuracy of the method, multipliers too close to 1 are forbidden ($u_i \leq 0.99, \forall i$).
#### Example
Consider the following cut generated with the CG for jobs from Figure \[fig:gantt\]: $$\begin{aligned}
(CG) = 3 x_{1,0,8} + x_{5,0,5} + 2 x_{5,0,6} + 4 x_{6,1,4} + 2 x_{6,1,8} + 2 x_{8,1,7} + 2 x_{9,1,7} \leq 5.\end{aligned}$$
On the one hand, the larger the set of non-redundant and tight constraints considered in the Chvátal-Gomory separation the more likely it is that violated inequalities will be found. On the other hand, large separation problems can be hard to solve and the overall performance of the cutting plane method can degrade. The following subsection will therefore consider specific strategies to find suitable sets of constraints.
### Finding a Set of Constraints
In our approach we consider a tuple ($\bar{s},\bar{f}$) to indicate the starting time and the finishing time of a given interval with size $\eta$ to filter the constraints and variables sets. We compute, for different intervals $(\bar{s},\bar{f})$, the summation of all infeasibilities for the integrality constraints for all their variables $x_{jmt}$ where $ \bar{s} \leq t \leq \bar{f}$, $x^*_{jmt}$ are the fractional values for the RCPSP variables. The value $\hat{f}$ is composed of the sum of the nearest integer distance, to indicate how fractional the variables in that interval are.
$$\hat{f}_{(\bar{s},\bar{f})} = \sum_{j \in \mathcal{J}}\sum_{m \in \mathcal{M}_j}\sum_{t=\bar{s}}^{\bar{f}}{|round(x_{jmt}^{*})-x_{jmt}^{*}|} \ \ \ \forall \ (\bar{s},\bar{f}) \in \mathcal{T}$$
To find the most fractional interval $\hat{f}_{(\bar{s},\bar{f})}$ in the scheduling, in which the constraints with their respective variables will be chosen, we start from the beginning of the scheduling, and we slide the interval $(\bar{s},\bar{f})$ until the end of the scheduling (see Algorithm \[alg:constraints\], line \[alg:interval\]). The parameters on the algorithm indicate the interval size $\eta$, the jump size $\iota$ to go to the next interval and a percentage $\zeta$ that allows sliding the interval. The slide only occurs if the violation is greater than the current value, plus the percentage of $\zeta$. Another input data is the dynamic conflict graph $\mathcal{C}\mathcal{G}$.
$\mathcal{V} \gets \emptyset$;\
$(\bar{s},\bar{f}) \gets \texttt{identify\_interval}(x^*,\eta,\iota,\zeta)$;\[alg:interval\]\
\[alg:resREnd\] $ O \gets \texttt{jobs}(\mathcal{V})$; \[alg:modetimeinit\]\
\[alg:modetimeend\] $ \mathcal{K} \gets \texttt{nonrenewable\_resources}(\mathcal{V})$; \[alg:resRNInit\]\
\[alg:resRNend\] $ \mathcal{C} \gets \texttt{pairs\_of\_conflicting\_variables}(\mathcal{C}\mathcal{G},\mathcal{V})$; \[alg:conflictinit\]\
\[alg:conflictend\] **return** $\mathcal{Z}$;
Preliminary experiments showed that cuts separated using constraints from the beginning of the time horizon were more effective for improving the dual bound, jobs allocated in the first time period are responsible for pushing the allocations of the others in the precedence graph. It is therefore desirable to find integer values for the first ones. Once the most fractional interval is found, the most important constraints are identified to compose the set for the Chvátal-Gomory separation.
The renewable resources directly impact the duration of the projects; therefore, all the variables of these constraints into the interval ($\bar{s},\bar{f}$) are considered in the set $\mathcal{V}$ (see Algorithm \[alg:constraints\], lines \[alg:resRInit\]-\[alg:resREnd\]). Further constraints from the original problem are included in the separation problem whenever they are related to the current time window: constraints that restrict the choice of only job (lines \[alg:modetimeinit\]-\[alg:modetimeend\]) and non-renewable resource constraints (lines \[alg:resRNInit\]-\[alg:resRNend\]). Functions `nonrenewable_resources`($\mathcal{V}$) and `jobs`($\mathcal{V}$) returns, respectively, the set of non-renewable resources and jobs where variables of $\mathcal{V}$ appear. A good strategy is to find additional constraints that represent conflicts between the variables of set $\mathcal{V}$. A main conflict is analyzed, whereby the time window comprising variables coming from the conflict graph $\mathcal{C}\mathcal{G}$ generated dynamically at each iteration of the cutting plane received as input parameter (lines \[alg:conflictinit\]-\[alg:conflictend\]). Function `pairs_of_conflicting_variables`($\mathcal{C}\mathcal{G},\mathcal{V}$) returns the pairs of these conflicting variables.
### Strengthening Procedure {#subsubsec:strengthening}
To produce strengthened CG cuts, a strategy similar to the proposal of @Letchford2016 is employed. The key idea is to take violated CG cuts and then strengthen the right-hand-sides ($rhs$). @Letchford2016 solves the maximum weight stable set problem for the conflict graph induced by the binary variables of the CG cut to find a (hopefully smaller) new valid $rhs$. Our approach solves the same problem augmented by additional constraints involving these variables. Consider here that set $\mathcal{H}$ contains all variables of the cut to be strengthened. $A_{\mathcal{H}}$ is the matrix of coefficients of $\mathcal{H}$ including non-renewable and renewable resource constraints, job allocation constraints and conflict constraints, with $rhs$ values specified in a vector ${\mathbf{b}}$. The vector ${\mathbf{c}}$ are the coefficients of the variables that appear in the cut. Consider vector ${\mathbf{x}}$ as variables of $\mathcal{H}$ and the integer linear programming as max{${\mathbf{c}}^{T}{\mathbf{x}}: A_{\mathcal{H}}{\mathbf{x}} \leq {\mathbf{b}}$}. If the optimal solution value of this MILP is smaller than the original $rhs$ of the cut, the original CG can be strengthened with this new value on the $rhs$.
#### Example
Considering the previous example, it is possible to strengthen the Chvátal-Gomory cut by tightening the value on the right side to $4$ based on the MILP presented before. Notice that by applying the strengthening, it was possible to reduce the rhs value by 20%: $$\begin{aligned}
(SCG) = 3 x_{1,0,8} + x_{5,0,5} + 2 x_{5,0,6} + 4 x_{6,1,4} + 2 x_{6,1,8} + 2 x_{8,1,7} + 2 x_{9,1,7} \leq \boldsymbol{4}.\end{aligned}$$
Computational Results {#sec:results}
=====================
This section presents the results of the experiments obtained by the proposed cutting plane algorithm and the preprocessing routine to strengthen resource-related constraints. All computational experiments have been carried out on a computing cluster (Compute Canada) composed by Intel Xeon X5650 Westmere processors with 2,67 GHz and 512 GB of RAM running Scientific Linux release 6.3. All algorithms were coded in `ANSI C 99` and compiled with GCC version 5.4.0, with flags *-Ofast* and solver GUROBI version 8.0.1 [@gurobi].
Benchmark Datasets {#sec:inst}
------------------
In order to facilitate the comparison of algorithmic improvements, several public databases were established, including realistic, but computationally challenging problem instances. In 1996 a public and well-known repository of benchmark datasets, devoted to this class of problems, was established: the Project Scheduling Problem Library - PSPLIB [@Kolisch1996]. In 2013 a new dataset based on PSPLIB instances, considering multiple projects, emerged from the MISTA Challenge [@Wauters2016]. Then, in 2014, another dataset based on PSPLIB was proposed for the multi-mode scheduling problem called MMLIB [@VanPeteghem2014]. For all problem variants, many of those instances are still open. In particular, by the time of writing this paper, we found that 215 of the PSPLIB instances, 2842 of the MMLIB instances, and 27 of the MISTA instances, were still open[^2].
We now define the benchmark used in this work based on [@Schnell2017; @Toffolo2016; @Zhu2006] and on instances from the PSPLIB, MISTA and MMLIB repositories. From PSBLIB, we consider instances for the SMRCPSP and MMRCPSP problem variants. From MISTA, we included all problem instances from the basic dataset A for the MMRCMPSP. For MMRCPSP, we also use all instances from MMLIB in the final experiments.
Table \[tab:benchmarkdataset\] shows, for each library, the total number of problem instances defined for each problem variant, the group name, the number of instances that are used in our benchmark datasets, and the number of instances for which optimality has not yet been proven in the literature.
------- ------------- ------- ------ ------
J60 79 79
J90 105 105
J120 514 514
J30 641 31
MISTA MMRCMPSP/30 A 10 7
J50 540 118
J100 540 176
J50+ 1620 1178
J100+ 1620 1370
------- ------------- ------- ------ ------
: Benckmark datasets
\[tab:benchmarkdataset\]
Cutting Plane Algorithm Experiments {#sec:cpexp}
-----------------------------------
In order to evaluate the performance of the proposed cutting plane in relation to the specific problem cuts and the linear relaxation, preliminary experiments have been performed on the benchmark datasets with $\alpha > 0$[^3] totaling 782 instances from PSPLIB and MISTA. These two sets include instances of the versions SMRCPSP, MMRCPSP and MMRCMPSP. Instances from MMLIB, that contemplate a larger number of varied instances for the MMRCPSP version, were introduced only in experiments carried out using the complete version of our approach. The impact of adding the cuts for instances where $\alpha = 0$ could not be measured, since the LP bound remains at the same value even when the formulation has been improved. We used a time limit of 24 hours for each instance from the MMRCMPSP and 4 hours for each instance from the SMRCPSP and MMRCPSP.
The first experiment was conducted to evaluate the preprocessing MILP formulation by analyzing the LP relaxation (LR) and the strengthened LP relaxation (SLR) using the coefficient strengthening MILP presented in Section \[subsec:premip\] to renewable resources constraints. The reported computing time for the SLR includes the time spent to find the successful feasible sets with parameters to stop the enumeration process as $it=200,000$ and $tl$ as the maximum allowed time defined above to run the approach. Table \[tab:lrlrs\] shows the average integrality gaps [^4] and the average computing times in seconds that have been obtained with the original LP relaxations and the strengthened LR for the instances from PSPLIB and MISTA.
------- ----- ------- ------ ----------- -------
A 10 0.441 85.9 **0.438** 683.6
J30 245 0.663 0.3 **0.658** 2.8
J60 57 0.825 1.0 **0.816** 5.3
J90 80 0.822 2.1 **0.818** 26.9
J120 390 0.840 3.3 **0.835** 37.2
total 782 0.718 18.5 **0.713** 151.2
------- ----- ------- ------ ----------- -------
: Average integrality gaps and the average computing times (sec.) for solving with the original LP relaxations (LR) and the strengthened LR (SLR)
\[tab:lrlrs\]
The results in Table \[tab:lrlrs\], shown in bold numbers, indicate that the strengthening formulation slightly improves the integrality gaps. As expected, solving only the weak initial formulation is faster than solving the formulation with the preprocessing routines. However, we note that even small improvements in the root node can result in a large number of nodes pruned later in the search tree. For this reason, and given that the additional computing times are still quite reasonable, we use this strengthening strategy in all further experiments.
According to @Kolisch1995, network precedence relationships and the factor between availability-consumption of resources are the two critical characteristics of the instances. When pruning is applied considering these two characteristics, we note that even for instances with $120$ jobs, there are time periods for which it is possible to enumerate the feasible subsets. For example, it is possible to enumerate until $t=102$ for the large instance $j1201\_1$. This instance is composed of $120$ jobs and has restricted relationship between the availability and consumption of renewable resources.
### Results for Different Cut Families
In order to devise an effective cutting plane strategy, we next evaluate the different separation strategies. In Section \[sec:cpexp\], it has been shown that using the SLR instead of the original LR strengthens the formulation while only marginally increasing computing times. We now explore the bound improvement when combining the SLR with each of the different cut types. Tables \[tab:results\] and \[tab:resultsA\] show the average integrality gaps and the average computing times for different approaches. In both tables, the first column SLR presents results obtained by the strengthened LP relaxation without cuts. The LCV[^5], LPR, CL, OH ad SCG columns indicate, respectively, results[^6] obtained by combining the SLR with one additional cut type: lift operations to cover and precedence cuts, clique cuts, odd-hole cuts and strengthened Chvátal-Gomory cuts.
------- ----- ------- ------ ------- ------ ----------- ------ ------- ------ ------- ---- ------- -------
J30 245 0.658 3 0.644 6 **0.549** 5 0.567 101 0.657 3 0.551 14200
J60 57 0.816 5 0.814 11 **0.712** 88 0.806 1093 0.816 12 0.812 14401
J90 80 0.818 27 0.816 45 **0.672** 454 0.807 3251 0.818 35 0.815 14400
J120 390 0.835 37 0.820 114 **0.714** 2111 0.817 7265 0.835 56 0.832 14423
total 772 0.781 18.1 0.774 43.9 **0.661** 665 0.749 2928 0.781 27 0.780 14408
------- ----- ------- ------ ------- ------ ----------- ------ ------- ------ ------- ---- ------- -------
\[tab:results\]
The results in Table \[tab:results\] suggest that the best average values (in bold numbers) of the integrality gaps were obtained by adding the lifted precedence cuts. Those cuts offered the best bound improvement (about 15%), while requiring only moderate additional computing time. For the complete cutting plane, in the final experiment, we execute the separation procedures in parallel, but if a hierarchical implementation approach was used, one would add the cut families based the order of their integrality gap improvements: LPR, CL, LCV, SCG and OH.
The results of instances from MMRCMPSP are presented in Table \[tab:resultsA\]. The results summarized indicate that the LPR cuts are also effective for the multi-project problem variant, even for large instances. SCG cuts have been found to be particularly useful for this class of problem. Experiments for instances as from A-7 typically exceeded the given time or memory limits to insertion of cuts such as CL, OH, and SCG.
------- ----------- ----- ----------- ------ ----------- ------ ----------- ------- ----------- ------ ----------- -------
inst.
A-1 1.000 0 1.000 0 0.875 0 0.875 0 1.000 0 **0.000** 1793
A-2 1.000 1 1.000 4 0.987 4 0.989 2 1.000 2 **0.935** 86007
A-3 **0.000** 8 **0.000** 9 **0.000** 15 **0.000** 15 **0.000** 15 **0.000** 62097
A-4 0.414 4 0.411 35 **0.333** 39 0.366 1297 0.414 10 0.396 86005
A-5 0.305 32 0.299 1743 **0.273** 1168 0.301 6157 0.305 254 0.302 86028
A-6 0.428 596 0.427 4118 **0.317** 3303 0.413 34106 0.428 1724 0.424 86005
total 0.5245 221 0.523 985 **0.464** 755 0,491 6930 0.525 334 **0.343** 40983
------- ----------- ----- ----------- ------ ----------- ------ ----------- ------- ----------- ------ ----------- -------
\[tab:resultsA\]
In Table \[tab:lift\] we evaluate the lifting strategies of the traditional RCPSP cuts proposed by [@Zhu2006] and the strengthening strategy for Chvátal-Gomory cuts, also combining with the SLR.
------- ----- ------- ------ ----------- ------- ------- -------- ----------- -------- ----------- ------- ----------- -------
A 6 0.526 59.9 **0.523** 984.8 0.518 555.9 **0.464** 754.8 **0.342** 71825 0.343 67989
J30 245 0.660 0.7 **0.644** 5.6 0.653 3.7 **0.549** 4.9 **0.548** 14092 0.551 14200
J60 57 0.825 2.1 **0.814** 11.2 0.794 30.1 **0.712** 87.5 **0.811** 14402 0.812 14401
J90 80 0.821 4.3 **0.816** 44.9 0.780 131.8 **0.672** 454.3 **0.815** 14402 **0.815** 14400
J120 390 0.839 66.2 **0.820** 113.9 0.809 1516.2 **0.714** 2111.4 **0.832** 14402 **0.832** 14423
total 782 0.734 26.6 **0,723** 232.1 0.711 447.5 **0.622** 682.6 **0.670** 25825 0.671 25083
------- ----- ------- ------ ----------- ------- ------- -------- ----------- -------- ----------- ------- ----------- -------
\[tab:lift\]
The lifting strategies were able to improve the average integrality gaps of all benchmark datasets when comparing with the traditional RCPSP cuts (CV and PR). The results in bold numbers suggest that the lifting strategy is generally successful, substantially improving the average integrality gaps while increasing computing times. Note that even for the benckmark datasets of SMRCPSP, improved lower bounds were achieved, since lifting tends to use more variables in the cut apart from those that contribute to the cover violation. The strengthened CG cuts did not improve upon the original CG cuts, which can be explained by the large computing times spent by the strengthening procedure. When analyzing the average number of iterations for both procedures, the CG without lift did 3118 iterations in the average, while SCG did just 2450 iterations. Even with a reduced number (21%) of iterations it was able to achieve basically the same results.
To analyze the cut generation for each separation strategy, we have computed the number of unique cuts for each strategy to the previous experiment. Figure \[fig:Cuts\] shows boxplots of the number of cuts for different instance types and benchmarks.
![Boxplots of the number of cuts for different types and benchmark datasets to each separation strategy[]{data-label="fig:Cuts"}](BoxPlotEachCutNew.pdf){width=".85\linewidth"}
The lifted precedence strategy LPR finds a reasonable number of cuts for almost all benchmark datasets. The strengthened Chvátal-Gomory SCG strategy seems to find cuts for all the benchmark datasets, including J30 and J90 in which LPR finds less. Another peculiarity of SCG is that although it finds few cuts at each iteration it remains to find cuts for a larger number of iterations. It is also possible to observe that a small number of odd-holes cuts are generated, which explains the little impact on the integrality gaps. Finally, the quantities of both cliques and lifted precedence cuts are relatively high. However, their separation requires more time.
### Results Removing Cut Families
While the previous experiments explored the impact of adding each cut type, we now explore the impact of omitting each cut type. To this end, we use all cut types, separated in parallel in the cutting plane, and then individually remove each type for the benchmark datasets SMRCPSP and MMRCPSP. Table \[tab:resultscp\] summarizes the results of these experiments, where column pair All Cuts represents the strategy using all cut families, and the other column pairs represent the strategy without each of the cut types.
------- ----- ----------- ------- ------- ------- ------- ------- ----------- ------- ----------- ------- ----------- ------
J30 245 **0.458** 13546 0.471 13631 0.485 13640 0.459 13645 0.461 13661 0.518 8
J60 57 **0.702** 14402 0.703 14401 0.798 14401 **0.702** 14401 **0.702** 14402 0.71 107
J90 80 0.666 14403 0.667 14403 0.8 14402 **0.665** 14403 0.666 14403 0.669 545
J120 390 0.668 14409 0.713 14436 0.774 14425 **0.667** 14429 0.67 14429 **0.667** 4779
Total 772 0.624 14190 0.639 14218 0.714 14217 **0.623** 14219 0.625 14224 0.641 1360
------- ----- ----------- ------- ------- ------- ------- ------- ----------- ------- ----------- ------- ----------- ------
\[tab:resultscp\]
By removing the LPR cuts, results worsen by about 13%. The results also get a little worse by removing OH, LCV, and SCG respectively. The results using all cuts performed generally well. In addition, removing the clique cuts further improved the gaps on larger instances. This can be explained by the fact that separating cliques requires more time. When these cuts are removed the numbers of iterations increases, as one can observe in Table \[tab:resultscpround\]. Based on these results, one may define a new hierarchical sequence of the cut types based on the gap increase when the cut type is removed: LPR, SCG, LCV, OH, and CL.
------- ----- ------ ------- ------ ------- ------ ------- ------ ------- ------ ------- ----- ------
J30 245 2077 13546 1946 13631 1796 13640 2486 13645 2035 13661 12 8
J60 57 618 14402 618 14401 877 14401 832 14401 452 14402 16 107
J90 80 294 14403 300 14403 476 14402 401 14403 293 14403 22 545
J120 390 265 14409 271 14436 347 14425 469 14429 284 14429 170 4779
Total 772 814 14190 784 14218 874 14217 1047 14219 766 14224 55 1360
------- ----- ------ ------- ------ ------- ------ ------- ------ ------- ------ ------- ----- ------
\[tab:resultscpround\]
Table \[tab:resultscpA\] shows the results for the same experiments for the MMRCMPSP. Again, column All Cuts represent the strategy were all cut types are used. Instead of reporting the results when each of the other cut types is removed, we only report on removing the cut types that consumed most of the computing time: cliques and odd-holes cuts, or the strengthened CG cuts. The bold numbers show the best average integrality gaps achieved for instances that did not run out of memory. Even though the results are better on average when using all cuts, by removing the cuts, the time was reduced significantly. Even though using all cut types significantly increases the computing times, the gap improvement on some of the instances may still justify their use.
------- ----------- --------- ----------- -------- ----------- --------
A-1 **0.000** 52.5 0.875 0.7 0.875 0.4
A-2 **0.781** 86015.2 0.924 4.2 0.924 5.6
A-3 **0.000** 86368.5 **0.000** 11.0 **0.000** 9.4
A-4 **0.316** 86001.1 0.327 31.1 0.327 104.2
A-5 **0.263** 86010.4 0.267 832.2 0.266 1280.4
A-6 **0.313** 86089.2 0.314 2130.2 0.314 2660.6
Total **0.279** 71756.2 0.451 501,6 0.451 676.8
------- ----------- --------- ----------- -------- ----------- --------
: Average integrality gaps and the average computing times (sec.) obtained by the cutting plane comparing with removing some others cuts with time limit of $24$ hours
\[tab:resultscpA\]
To analyze the number of generated cuts for all separation procedures, we also computed the number of non-repeated cuts to the previous experiments (All Cuts). Figure \[fig:AllCuts\] shows the boxplots representing the number of cuts for different types and benchmarks.
![Boxplots of the number of cuts for different types and benchmark datasets to all separation strategy together[]{data-label="fig:AllCuts"}](BoxPlotAllCut2New.pdf){width=".85\linewidth"}
Figure \[fig:AllCuts\] shows that LPR cuts are effective for all datasets mainly in the case of the SMRCPSP, where it found, on average, more cuts than others. For the MMRCMPSP, the LPR cuts are still the most effective ones. CL and SCG cuts seem to be effective mainly for instances with multiple modes. The average number of generated LCV cuts seems to be stable for all benchmarck dataset. Few OH cuts are found for all benchmarck datasets.
After analyzing the best strategy for the MMRCPSP obtained through experiments on the PSBLIB instances, we run the experiment with all cuts for the instances from MMLIB, comparing the LP relaxation and the strengthened linear relaxation. The results are presented in Table \[tab:resultscpMMLIB\].
---------- ------ ------- ------- ------ ------- ------- ------- ----------- ------- ---------
J50 540 0,750 0,309 0,6 0,749 0,309 6,4 **0,654** 0,367 9669,4
J100 540 0,768 0,293 11,0 0,768 0,293 35,6 **0,688** 0,354 11536,6
JAll50+ 1616 0,586 0,250 6,6 0,584 0,277 30,1 **0,463** 0,227 13746,8
Jall100+ 1489 0,644 0,272 54,8 0,644 0,272 144,0 **0,530** 0,265 14221,9
Total 4185 0,687 0,281 18,2 0,686 0,288 54,0 **0,584** 0,303 12293,7
---------- ------ ------- ------- ------ ------- ------- ------- ----------- ------- ---------
\[tab:resultscpMMLIB\]
The results show that the cutting plane was very effective for all benchmark datasets, improving the average values by approximately 15%. As in earlier comparisons, the SLR slightly improves the integrality gaps for MMLIB instances too. However, 135 of the JA11 instances ran out of memory, which have been removed from the results presented in the table to provide a fair comparison.
Branch & Cut Experiments
------------------------
We now explore how the different cut types can be added dynamically in a Branch-and-Cut manner, to improve the solution process using the general purpose MILP solver Gurobi. Experiments were executed in order to compare the results achieved by solving the model with the inclusion of cuts into the root plus precedence cuts into a callback[^7] procedure when the lower bound is improved and the results achieved by solving the model without cuts just with the preprocessing input data and Gurobi cuts.
Table \[tab:resultsbc\] summarizes the integrality gaps[^8] (average and standard deviation), for open instances with $\alpha > 0$ from PSPLIB and MISTA. The experiments for our approach have been limited to 24 hours of computing time. For Gurobi cuts, the first experiments have been limited to 24 hours and the second have been limited to 48 hours, in order to ensure a fair comparison since we not consider the time spent to insert all cuts into the root node for our approach. Initial solutions available at MISTA website were inserted only for the A dataset.
------- ----- ------- ------- ------- ------- ----------- -----------
group $n$ avg std avg std avg std
A 10 0.210 0.165 0.158 0.139 **0.139** **0.124**
J30 245 0.007 0.022 0.006 0.022 **0.005** **0.020**
J60 57 0.252 0.115 0.237 0.103 **0.184** **0.084**
J90 80 0.268 0.162 0.253 0.139 **0.201** **0.084**
J120 390 0.299 0.253 0.292 0.252 **0.249** **0.224**
------- ----- ------- ------- ------- ------- ----------- -----------
\[tab:resultsbc\]
By analyzing Table \[tab:resultsbc\] we can see that better results are obtained with the introduction of cuts into the root and with the LPR cut into the callback procedure. The average integrality gaps are reduced. Also, for some particular instances like $j12049\_8$, $j12021\_1$, $j12022\_8$, $j3037\_1$, $j3037\_7$, $j6046\_5$, $j6030\_2$, among others, the average optimality gap achieved within $24$ hours of computing time was quite low (0.09%) when adding the cuts. Almost all open instances with $\alpha = 0$ from PSPLIB have been easily solved to optimality, except for some instances in set with $120$ jobs.
We now explore how the two settings compare throughout the optimization process to find feasible solutions and to prove optimality. Table \[tab:resultsbceasy\] summarizes the results for datasets from MISTA and PSPLIB for the three problem variants, also including those instances with $\alpha = 0$. Comparing to @Toffolo2016, we solve to optimality, for the first time, $1$ instance from `A` set. Comparing to @Schnell2017 we solve the `J30` set, for the first time, 13 instances to optimality, and prove infeasibility for 89 of the instances.
------- ----- -------- ----- -------- --------- -------- --------
group $n$ opt fea inf opt fea inf
A 10 **4** 6 0 **4** 6 0
J30 641 530 6 **89** **553** **15** **89**
J60 79 24 0 0 **26** **2** 0
J90 105 **27** 0 0 **27** 0 0
J120 514 160 2 0 **181** **3** 0
------- ----- -------- ----- -------- --------- -------- --------
\[tab:resultsbceasy\]
The results indicate that our approach has been able to prove optimality for more instances than using Gurobi without our cuts. For some specific instances the optimal value is found only when cuts are added to the model. In addition, Gurobi solver proves that some instances are infeasible.
Table \[tab:resultsbcMMLIB\] summarizes the results for our approach on the MMLIB datasets. The table shows the number of instances that have been solved to optimality (opt) and the number of instances for which a feasible solution has been found, but optimality has not been proven (fea). Further, the table shows the number of instances for which our solutions improve upon those reported by the website[^9] (imp) and those reported in [@Schnell2017], as well the number of instances for which optimality has been proven for the first time (new opt).
---------- ------ ----- --------- ----- ----- --------------------
group $n$ opt new opt fea imp imp [@Schnell2017]
J50 540 450 29 32 0 48
J100 540 410 48 43 7 114
JAll50+ 1620 689 252 159 59 401
JAll100+ 1620 482 177 205 174 440
---------- ------ ----- --------- ----- ----- --------------------
\[tab:resultsbcMMLIB\]
Figure \[fig:BoxPlot01GapTime\] presents boxplots of the optimality gaps and computing times for our B&C approach for instances from PSPLIB, MISTA and MMLIB for which feasible or optimal solutions have been found.
{width=".8\linewidth"}
The results presented in the first figure suggest that the gap values equals to 0 are represented by the median in the box plot for almost all datasets, except for the A and Jall100+ datasets. The high outliers for the datasets J120, J50, J100, Jall50+ and Jall100+ indicate that the presented algorithms still require improvement in order to deliver robust results on all instances. Analyzing the computing times it can be observed that our approach quickly proves optimality for all but dataset A. Finally, we also note that some of the instances between the second and third quartile hit the time limit.
In summary, optimality was proven for the first time, for $247$ instances from PSPLIB, for $1$ instance from MISTA and for $506$ instances from MMLIB, totalizing $754$ instances. The LP and solution files for all benchmark datasets are available for download at `http://professor.ufop.br/janniele/downloads`.
Conclusion and Future Works {#sec:conclusion}
===========================
In this paper, new mixed-integer linear programming based methods were proposed to improve the linear programming relaxation of compact formulations for the Resource Constrained Project Scheduling Problem. All methods were extensively evaluated in three RCPSP variants.
An effective preprocessing procedure to strengthen renewable resources constraints was devised. This procedure was capable of improving the lower bounds produced at the root node without any increase in the size of the linear programs.
A parallel cutting plane algorithm was developed including five families of cuts: lifted precedence and cover cuts, cliques, odd-holes and strengthened Chvátal-Gomory cuts. A dense conflict graph, considering feasibility and optimality conditions, was created at each iteration and used by these cut generators in strengthening procedures. All cuts contributed to improve the lower bounds, specially when they are together in the cutting plane. However, the lifted precedence cuts were the most effective for all variants. The strengthened Chvátal-Gomory cuts were specially effective in a group of multi-project instances. These results indicate that an instance feature based tuning of the cut generators may be beneficial.
With the improved linear programming formulations produced with our methods, $754$ open instances from literature were solved for the first time: $247$ instances from PSPLIB, $1$ instance from the MISTA Challenge and for $506$ instances from MMLIB.
The method still has room for improvement. These techniques could be hybridized with constraint propagation, as proposed in previous studies. The separation of some inequalities, such as the strengthened Chvátal-Gomory cuts and the re-optimization of the large linear programs are still quite expensive. The continuous improvements in future MILP solvers will likely speed up these two steps. Finally, the development of a complete B&C with improved branch and node selection strategies for the RCPSP, including our preprocessing and cutting planes routines, is also a promising future path.
**Acknowledgements.** The authors thank UFOP, CNPq (student with scholarship - Brazil), FAPEMIG, and Compute Canada for supporting this research.
**References**
[^1]: http://www.om-db.wi.tum.de/psplib/files/j10.mm.tgz
[^2]: 1339 considering the PSPLIB website and 30 considering the MISTA website. There are no informations about the optimal solutions achieved by [@Schnell2017] and [@Toffolo2016].
[^3]: $\alpha > 0$ is an upper bound to the maximum TPD with value higher then its CPD.
[^4]: given the best known upper bound $\overline{b}$, from PSPLIB and MISTA websites, and an obtained dual bound $\underline{b}$, the integrality gap is computed as follows: $\frac{(\overline{b}-\underline{b})}{\overline{b}}$, $\overline{b}>0$
[^5]: $\omega=100000$ and $\mu=0.1$
[^6]: the maximum number of precedence and clique cuts added to the LP at each iteration corresponds to 20% of the amount of the LP rows
[^7]: a callback is a user function that is called periodically by the Gurobi optimizer in order to allow the user to query or modify the state of the optimization [@gurobi]
[^8]: the integrality gaps is computed as in the previous experiments, since it was not possible to generate an incumbent solution for all instances, so it is not possible to obtain the optimality gap for some instances
[^9]: http://mmlib.eu/solutions.php
|
---
abstract: 'The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras that are useful for the analysis of geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for Fredholmness for geometric operators on suitable non-compact manifolds, as well as to an inductive procedure to study their essential spectrum. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multi-cylindrical ends.'
address:
- 'Universität Mainz. Fachbereich 17-Mathematik, D-55099 Mainz, Germany : Universität Münster, SFB 478, Hittorfstra[ß]{}e 27, D-48149 Münster, Germany'
- 'Pennsylvania State University, Math. Dept., University Park, PA 16802'
author:
- Robert Lauter
- Victor Nistor
title: 'Analysis of geometric operators on open manifolds: A groupoid approach'
---
[ ]{}
[^1]
c Ø[O]{} [ ]{}
Introduction {#introduction .unnumbered}
============
The first half of this paper is a survey of the results from [@LMN; @NistorInt; @NistorIndFam] and [@NWX]. However, there are some new results, including a determination of the essential spectrum of the Laplace operator on complete manifolds with multi-cylindrical ends. This was formulated as a question in [@MelroseScattering] (Conjecture 7.1).
Let us now discuss the contents of this paper. As we mentioned above, the first five sections of this paper are mostly a survey of results on pseudodifferential operators on groupoids. In Section \[Sec.prelim\], we review some definitions involving manifolds with corners and we introduce groupoids. We also define in this section the class of groupoids we are interested in, namely “differentiable groupoids” (Definition \[Almost.differentiable\]), and we recall the definition of the Lie algebroid associated to a differentiable groupoid. (Our “differentiable groupoids” should more properly be called “Lie groupoids.” However, this term was already used for some specific classes of differentiable groupoids.) The construction which associates to a differentiable groupoid ${\mathcal G}$ its Lie algebroid $A({\mathcal G})$ is a generalization of the construction which associates to a Lie group its Lie algebra. This construction is made possible for differentiable groupoids by the fact that the fibers ${\mathcal G}_x :=
d^{-1}(x)$ of the domain map $d : {\mathcal G}\to M$ consist of smooth manifolds without corners (in this paper, a “smooth manifold” will always mean a “smooth manifold without corners,” and a $C^\infty$-manifold with corners will be called simply, a “manifold”). It is convenient to think of ${\mathcal G}$ as a set of arrows between various points, called units, which can be composed according to some definite rules. If $g\in {\mathcal G}$ is such an “arrow,” then $d(g)$ is its domain and $r(g)$ is its range, so ${\mathcal G}_x$ is the set of all arrows starting at (or with domain) $x$.
Section \[Sec.Examples.I\] contains several examples of differentiable groupoids. In Section \[Sec.POOG\], we introduce pseudodifferential operators on groupoids. A pseudodifferential operator $P$ on the differentiable groupoid ${\mathcal G}$ is actually a family $P=(P_x)$ of ordinary pseudodifferential operators $P_x$ on the smooth manifolds without corners ${\mathcal G}_x$. This family is required to be invariant with respect to the natural action of ${\mathcal G}$ by right translations and to be differentiable in a natural sense. Because the family $P$ acts on ${{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$, we can regard $P$ as an operator on this space. To get the functoriality and composition properties right, we also assume that the convolution kernel $\kappa_P$ of $P$ is compactly supported ($\kappa_P$ has been called “the reduced kernel of $P$” in [@NWX]). We denote by ${\Psi^{m}({\mathcal G})}$ the space of such order $m$ (families of) pseudodifferential operators on ${\mathcal G}$. Many properties of the usual pseudodifferential operators on smooth manifolds extend to the operators in ${\Psi^{m}({\mathcal G})}$, most notably, we get the existence of the principal symbol map. At the end of the section, we indicate how to treat operators acting between sections of two vector bundles. Section \[Sec.BR\] deals with the necessary facts about the actions of ${\Psi^{\infty}({\mathcal G})}$ on various classes of functions. It is convenient to present this from the point of view of representation theory (after all, this is the representation theory of the Lie group $G$ if ${\mathcal G}= G$). It is in fact enough to study representations of ${\Psi^{-\infty}({\mathcal G})}$, and this includes the study of boundedness properties of various representations: to check that a representation of ${\Psi^{0}({\mathcal G})}$ consists of bounded operators, it is sufficient to check that its restriction to ${\Psi^{-\infty}({\mathcal G})}$ is bounded. This generalizes the usual boundedness theorems pseudodifferential operators of order $0$.
Motivated by a question of Connes, Monthubert also was lead to define pseudodifferential operators on groupoids in [@Monthubert], independently. Actually, Connes had defined algebras of pseudodifferential operators on the graphs of $C^{0,\infty}$ foliations in [@ConnesF], before. The definitions of pseudodifferential operators on groupoids in [@Monthubert] and [@NWX] is essentially the same as Connes’, but different because we take into account the differentiability in the transverse direction, too, and, most importantly, we allow non-regular Lie algebroids. An approach to operators on singular spaces which is similar in philosophy to ours was outlined in [@MelroseScattering]. There Melrose considers operators whose kernels are defined on compact manifolds with corners and have structural maps that make them “semi-groupoids.” The results of [@NWX] were first presented in July 1996 at the joint SIAM-AMS-MAA Meeting on Quantization in Mount Holyoke.
In addition to the survey of the results from [@LMN; @NistorInt; @NistorIndFam] and [@NWX], the first five sections of the paper also include many examples of groupoids together with the description of the resulting algebras of pseudodifferential operators. Actually, Sections \[Sec.Examples.I\] and \[Sec.Examples.II\] are devoted exclusively to examples, with the hope that this will make the general theory easier to apply. Whenever it was relevant, we have also compared our construction to the classical constructions.
Let us now quickly describe the contents of each of the remaining five sections of this paper. In Section \[Sec.GO\] we show that the geometric differential operators acting on the fibers of the domain map $d:{\mathcal G}\to M$ belong to our algebras ${\Psi^{\infty}({\mathcal G};E)}$, for suitable $E$. To define these geometric operators – except the de Rham operator – we need a metric on ${\mathcal G}_x$, and this will come from a metric on $A({\mathcal G})$, the Lie algebroid of ${\mathcal G}$. Then we need to establish the existence of right invariant connections with the properties necessary to define the geometric operators we are interested in. It needs to be established, for example, that compatible connections exist on Clifford modules, and this is not obvious in the groupoid case. Section \[Sec.Sobolev\] establishes the technical facts needed to define Sobolev spaces in our setting. The reader should find the statements in that section easy to understand and believe (although, unfortunately, not the same thing can be said about their proofs).
Beginning with Section \[Sec.O.M\], we begin to work with groupoids ${\mathcal G}$ of a special kind, which model operators on suitable non-compact Riemannian manifolds $(M_0,g)$. The set of units $M$ of these groupoids is a compactification of $M_0$ to a manifold with corners, and hence we can think of ${\mathcal G}$ as modeling the behavior at $\infty$ of $M_0$ (this approach was inspired in part by Melrose’s geometric scattering theory program). More precisely, we assume that $M_0$ is an open invariant subset of $M$ with the property that ${\mathcal G}_{M_0}$ is the pair groupoid, that is, that for each $x \in M_0$ there exists exactly one arrow to any other point $y \in M_0$, and there exists no arrow to any point not in $M_0$ (see Example \[ex.pair\]). Then the restriction of $A({\mathcal G})$ to $M_0$ identifies canonically with the tangent bundle to $M_0$. For such Riemannian manifolds, the geometric operators on $M_0$ can be recovered from the geometric operators on ${\mathcal G}$. We use these results in Section \[Sec.Examples.III\] to study the Hodge-Laplace operators on suitable non-compact Riemannian manifolds, using the general spectral properties discussed in Section \[Sec.SP\]. We obtain, in particular, criteria for certain pseudodifferential operators on $M_0$ to be Fredholm or compact, similar to the well-known criteria for $b$-pseudodifferential operators to be Fredholm or compact [@Melrose-Nistor1; @mepi92]
### Acknowledgments: {#acknowledgments .unnumbered}
We have benefited from discussions and suggestions from several people: A. Connes, R. Melrose, B. Monthubert, and G. Skandalis, and we would like to thank them. We are grateful to R. Melrose who has instructed us on the $b$-calculus. The first author would like to thank R. Melrose, J. Cuntz, and the SFB 478 [*Geometrische Strukturen in der Mathematik*]{}, for their warm hospitality and useful discussions at MIT and, respectively, at the University of Münster, where part of this work was done.
Manifolds with corners and groupoids\[Sec.prelim\]
==================================================
In the following, we shall consider manifolds with corners. Brief introductions into the analysis on manifolds with corners can be found for instance in [@mepi92], and will be discussed in more detail in a forthcoming book of Melrose. We begin this section with a short discussion of the relevant definitions concerning manifolds with corners. Then we introduce groupoids and the class of [ *differentiable groupoids*]{}. The reader can find more information on groupoids in [@connes; @Renault1].
By a [*manifold*]{} we shall always understand a differentiable manifold possibly [*with corners*]{}, by a [*smooth manifold*]{} we shall always mean a differentiable manifold [*without corners*]{}. By definition, every point $p$ in a manifold with corners $M$ has a coordinate neighborhood diffeomorphic to $[0,\infty)^k \times
{\mathbb R}^{n-k}$ such that the transition functions are smooth (including on the boundary). Moreover, we assume that each boundary hyperface $H$ of $M$ is an embedded submanifold and has a defining function, that is, that there exists a smooth function $x \ge 0$ on $M$ such that $$H = \{ x = 0 \} \text{ and } dx \not = 0 \text{ on }H\,.$$ It follows that if $H_1,\ldots,H_k$ are boundary hyperfaces of $M$ with defining functions $x_1,\ldots,x_k$, then the differentials $dx_{1}, \ldots, dx_k$ are linearly independent at the intersection $H_1 \cap \ldots \cap H_k$.
\[def.1\] A $f : M \to N$, between two manifolds with corners $M$ and $N$, is a differentiable map such that $df$ is surjective at all points and $df(v)$ is an inward pointing tangent vector of $N$ if, and only if, $v$ is an inward pointing vector $M$.
It is not difficult to prove the following lemma.
Let $f : M \to N$ be a submersion of manifolds with corners, $y \in N$ a point belonging to the interior of a face of codimension $k$ and $x_1,x_2,\ldots,x_k$ be the defining functions of the hyperfaces containing $y$. Then $x_1\circ f, x_2\circ f, \ldots,
x_k \circ f$ are defining functions for $k$ distinct hyperfaces of $M$. Let $p \in M$ be such that $f(p) = y$, then all hyperfaces of $M$ containing $p$ are obtained in this way.
Let $p$ and $y$ be as in the statement and $z_j := x_j \circ f$. Because $df$ is surjective, it follows that $dz_j$ are linearly independent at $p$. Each function $z_j$ is the defining function of a hyperface because $f$ must map faces of codimension $k$ to faces of codimension $k$.
For any submersion $f$ as above, it follows that the fibers $f^{-1}(y)$ of $f$ are smooth manifolds (that is [*without*]{} corners). We can see this as follows. Because this property of $f$ is local, we can fix $y$ and replace $M$ and $N$ with some small open neighborhoods of $y$ and $f(y)$, respectively. By decreasing these neighborhoods, we can also assume that they are diffeomorphic to one of the model open sets $[0,1)^k \times {\mathbb R}^{n-k}$. Then we extend $M$ and $N$ to smooth open manifolds without corners, denoted $\tilde M$ and $\tilde N$, and $f$ to a smooth function $\tilde f : \tilde M \to \tilde N$. By decreasing $M$ and $N$ again, if necessary, we can assume that $\tilde f$ is still a submersion, this time in the classical sense, because $\tilde M$ and $\tilde N$ are smooth manifolds. This gives then that $\tilde f^{-1}(y)$ is a smooth submanifold of $\tilde{M}$. Since $f^{-1}(y) = \tilde f^{-1}(y) \cap M$, it is enough to check that this intersection coincides with a component of $\tilde f^{-1}(y)$. Let $x$ be a defining function of $N$ in $\tilde N$ (that is, the defining function of a hyperface of $N$). Then $x \circ f$ is a defining function of some hyperface of $M$. By counting the hyperfaces of $M$ and $N$, we see that we get in this way all defining functions of $M$. Since they all vanish on $y$, $f^{-1}(y) \cap \{x \circ f \ge 0 \} =
f^{-1}(y)$. This completes the argument.
The concept of a “submanifold” means the following in our setting.
\[def.2\] A (or ) $N$ of a manifold with corners $M$ is a submanifold $N \subset M$ such that $N$ is a manifold with corners and each hyperface $F$ of $N$ is a connected component of a set of the form $F' \cap N$, where $F'$ is a hyperface of $M$ intersecting $N$ transversally.
We shall need groupoids endowed with various structures. ([@Renault1] is a general reference for some of what follows.) We recall first that a [*small category*]{} is a category whose class of morphisms is a set. The class of objects of a small category is then a set as well. Here is now a quick definition of groupoids.
\[def.3\] A is a small category ${\mathcal G}$ all of whose morphisms are invertible.
Let us make this definition more explicit. A groupoid ${\mathcal G}$ is a pair $({{\mathcal G}^{(0)}},{{\mathcal G}^{(1)}})$ of sets together with structural morphisms $d,r,\mu,u$, and $\iota$. Here the first set, ${{\mathcal G}^{(0)}}$, represents the objects (or units) of the groupoid and the second set, ${{\mathcal G}^{(1)}}$, represents the set of morphisms of ${\mathcal G}$. Usually, we shall denote the space of units of ${\mathcal G}$ by $M$ and we shall identify ${\mathcal G}$ with ${{\mathcal G}^{(1)}}$. Each object of ${\mathcal G}$ can be identified with a morphism of ${\mathcal G}$, the identity morphism of that object, which leads to an injective map $u : M := {{\mathcal G}^{(0)}} \to {\mathcal G}$, used to identify $M$ with a subset of ${\mathcal G}$. Each morphism $g \in {\mathcal G}$ has a “domain” and a “range.” We shall denote by $d(g)$ the [*domain*]{} of $g$ and by $r(g)$ the [*range*]{} of $g$. We thus obtain functions $$d,r: {\mathcal G}\longrightarrow M:={{\mathcal G}^{(0)}}.$$ The multiplication (or composition) $\mu(g,h)=gh$ of two morphisms $g, h \in {\mathcal G}$ is not always defined; it is defined precisely when $d(g) = r(h)$. The inverse of a morphism $g$ is denoted by $g^{-1}=\iota(g)$.
A groupoid ${\mathcal G}$ is completely determined by the spaces ${{\mathcal G}^{(0)}}$ and ${{\mathcal G}^{(1)}}$ and the structural maps $d,r,\mu,u,\iota$. We sometimes write ${\mathcal G}=({{\mathcal G}^{(0)}},{{\mathcal G}^{(1)}},d,r,\mu,u,\iota)$. The structural maps satisfy the following properties:\
- $r(gh)=r(g)$ and $d(gh)=d(h)$, for any pair $g,h$ with $d(g) = r(h)$;\
- The partially defined multiplication $\mu$ is associative;\
- $d(u(x))=r(u(x))=x$, $\forall x\in {\mathcal G}^{(0)}$, $u(r(g))g=g$, $gu(d(g))=g$, $\forall g\in {\mathcal G}^{(1)}$, and $u:{{\mathcal G}^{(0)}} \to {{\mathcal G}^{(1)}}$ is one-to-one;\
- $r(g^{-1})=d(g)$, $d(g^{-1})=r(g)$, $gg^{-1}=u(r(g))$, and $g^{-1}g=u(d(g))$.\
\[Almost.differentiable\] A is a groupoid $${\mathcal G}=({{\mathcal G}^{(0)}},{{\mathcal G}^{(1)}},d,r,\mu,u,\iota)$$ such that $M:={{\mathcal G}^{(0)}}$ and ${{\mathcal G}^{(1)}}$ are manifolds with corners, the structural maps $d,r,\mu,u,$ and $\iota$ are differentiable, the domain map $d$ is a submersion, and all the spaces $M$ and ${\mathcal G}_x := d^{-1}(x)$, $x\in M$, are Hausdorff.
Note that we [*do not*]{} require ${{\mathcal G}^{(1)}}$ to be Hausdorff. We actually need this generality to treat algebras associated to foliations and other geometric structures in our setting. We also observe that $\iota$ is a diffeomorphism, and hence $d$ is a submersion if, and only if, $r=d\circ \iota$ is a submersion. The reason for requiring $d$ to be a submersion is that then each fiber ${\mathcal G}_x=d^{-1}(x) \subset
{{\mathcal G}^{(1)}}$ is a smooth manifold without corners.
We now recall the definition of the “Lie algebroid” of a differentiable groupoid. Lie algebroids are for differentiable groupoids what Lie algebras are for Lie groups.
\[Lie.Algebroid\] A $A$ over a manifold $M$ is a vector bundle $A$ over $M$, together with a Lie algebra structure on the space $\Gamma(A)$ of smooth sections of $A$ and a bundle map $\varrho: A \rightarrow TM$, extended to a map between sections of these bundles, such that
\(i) $\varrho([X,Y])=[\varrho(X),\varrho(Y)]$, and
\(ii) $[X, fY] = f[X,Y] + (\varrho(X) f)Y$,
for any smooth sections $X$ and $Y$ of $A$ and any smooth function $f$ on $M$. The map $\varrho$ is called the [*anchor*]{}.
Note that we allow the base $M$ in the definition above to be a manifold with corners.
The Lie algebroid associated to a differentiable groupoid ${\mathcal G}$ is defined as follows [@Mackenzie1; @Pradines2]. The vertical tangent bundle (along the fibers of $d$) of a differentiable groupoid ${\mathcal G}$ is, as usual, $$T_{vert} {\mathcal G}= \ker d_*
= \bigcup_{x\in M} T {\mathcal G}_x \subset T{\mathcal G}.$$ Then $A({\mathcal G}) := T_{vert} {\mathcal G}\big |_{M}$, the restriction of the $d$-vertical tangent bundle to the set of units, $M$, determines $A({\mathcal G})$ as a vector bundle.
We now construct the bracket defining the Lie algebra structure on $A({\mathcal G})$. The right translation by an arrow $g \in {\mathcal G}$ defines a diffeomorphism $$R_g:{\mathcal G}_{r(g)}\ni g' \longmapsto g'g \in {\mathcal G}_{d(g)}.$$ A vector field $X$ on ${\mathcal G}$ is called $d$-vertical if $d_*(X(g)) = 0$ for all $g$. The $d$-vertical vector fields are precisely the vector fields on ${\mathcal G}$ that can be restricted to the submanifolds ${\mathcal G}_x$. It makes sense then to consider right–invariant vector fields on ${\mathcal G}$. It is not difficult to see that the sections of $A({\mathcal G})$ are in one-to-one correspondence with $d$–vertical, right–invariant vector fields on ${\mathcal G}$.
The Lie bracket $[X,Y]$ of two $d$–vertical right–invariant vector fields $X$ and $Y$ is also $d$–vertical and right–invariant, and hence the Lie bracket induces a Lie algebra structure on the sections of $A({\mathcal G})$. To define the action of the sections of $A({\mathcal G})$ on functions on $M$, observe that the right invariance property makes sense also for functions on ${\mathcal G}$, and that ${\mathcal{C}^\infty}(M)$ may be identified with the subspace of smooth right–invariant functions on ${\mathcal G}$, because $r$ is a submersion. If $X$ is a right–invariant vector field on ${\mathcal G}$ and $f$ is a right–invariant function on ${\mathcal G}$, then $X(f)$ will still be a right invariant function. This identifies the action of $\Gamma(A({\mathcal G}))$ on ${\mathcal{C}^\infty}(M)$.
We denote by $T_{vert}^* {\mathcal G}$ the dual of $T_{vert} {\mathcal G}$, and by $A^*({\mathcal G})$ the dual of $A({\mathcal G})$. Later on, we shall need the bundle ${{\Omega^{\lambda}_d}}$ of $\lambda$-densities along the fibers of $d$. It is defined as follows. If the fibers of $d$ have dimension $n$, then $${{\Omega^{\lambda}_d}}:=|\Lambda^n T_{vert}^*{\mathcal G}|^\lambda =
\cup_x\Omega^\lambda({\mathcal G}_x).$$ By invariance, these bundles can be obtained as pull-backs of bundles on $M$. For example $T_{vert} {\mathcal G}= r^*(A({\mathcal G}))$ and ${{\Omega^{\lambda}_d}}=r^*({\mathcal D}^\lambda)$, where ${\mathcal D}^\lambda$ denotes ${{\Omega^{\lambda}_d}}\vert_{M}$. If $E$ is a (smooth complex) vector bundle on the set of units $M$ of ${\mathcal G}$, then the pull-back bundle $r^*(E)$ on ${\mathcal G}$ will have right invariant connections obtained as follows. A connection $\nabla$ on $E$ lifts to a connection on $r^*(E)$. Its restriction to any fiber ${\mathcal G}_{x}$ defines a linear connection in the usual sense, which is denoted by $\nabla_{x}$. It is easy to see that these connections are right invariant in the sense that $$\label{eq.invariant.conn}
R_{g}^*\nabla_{x}=\nabla_{y}, \ \ \ \forall g\in {\mathcal G}\mbox{ such that } r(g)=x \mbox{ and } d(g)=y.$$ The bundles considered above will thus have invariant connections.
We observe that in all considerations above, we first use the smooth structure on each ${\mathcal G}_x$ to define the geometric quantities we are interested in: $X(f)$, $[X,Y]$, and so on. We do need then, however, to check that these quantities define global objects on the possibly non-Hausdorff manifold ${\mathcal G}$, more precisely, we need the defined objects to be smooth on ${\mathcal G}$, not just on every ${\mathcal G}_x$. All these global smoothness conditions can be checked on [*smooth*]{} functions on ${\mathcal G}$, as long as we correctly define this concept. For non-Hausdorff manifolds, the correct choice is the one considered by Crainic and Moerdijk in [@cramo00], more precisely, we consider first the spaces $V=\oplus_\alpha {{\mathcal C}^{\infty}_{\text{c}}}(U_\alpha)$ and $W = \oplus_{\alpha,\beta}
{{\mathcal C}^{\infty}_{\text{c}}}(U_\alpha \cap U_\beta)$, where $U_\alpha \subset {\mathcal G}$ are the domains of coordinate charts. Then there is a natural map $\delta:W \to V$, which is the direct sum of the maps $(j,-j) : {{\mathcal C}^{\infty}_{\text{c}}}(U_\alpha \cap
U_\beta) \to {{\mathcal C}^{\infty}_{\text{c}}}(U_\alpha) \oplus {{\mathcal C}^{\infty}_{\text{c}}}(U_\beta)$, with $j$ the natural inclusion, and we define $$\label{eq.Crainic}
{{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G}) = V/\delta(W).$$ A function $f$ on ${\mathcal G}$ is [*smooth*]{} if, and only if, $\phi f \in {{\mathcal C}^{\infty}_{\text{c}}}(U_{\alpha})$ for all $\phi \in {{\mathcal C}^{\infty}_{\text{c}}}(U_\alpha)$.
If $A \to M$ is a given Lie algebroid and ${\mathcal G}$ is a differentiable groupoid whose Lie algebroid is isomorphic to $A$, then we say that ${\mathcal G}$ [*integrates*]{} $A$. Not every Lie algebroid can be integrated (see [@al-mo:suites] for an example). There are many simple-minded Lie algebroids for which the standard integration procedures of [@NistorInt] lead to non-Hausdorff groupoids. For many Lie algebroids (including the tangent bundles to foliations) there are no Hausdorff groupoids integrating them. The integration of Lie algebroids was also studied by Moerdijk in [@moerdijk].
Examples I: Groupoids\[Sec.Examples.I\]
=======================================
We now discuss examples of differentiable groupoids. The examples included in this section are either basic examples or theoretical examples predicted by the general theory of groupoids. The examples that we shall use to study geometric operators on open manifolds will be included in Section \[Sec.Examples.II\]. For each example considered in this section we also describe the corresponding Lie algebroid.
As in the previous section, we shall identify ${\mathcal G}$ with its set of arrows: ${\mathcal G}= {{\mathcal G}^{(1)}}$, and we shall denote by $M$ or by ${{\mathcal G}^{(0)}}$ the set of units of ${\mathcal G}$.
\[ex.manifold\] [*Manifolds with corners: *]{} If $M$ is a manifold with corners, then $M$ is naturally a differentiable groupoid with no arrows other than the units, [[*i.e.,* ]{}]{}we have ${\mathcal G}= {{\mathcal G}^{(1)}} = {{\mathcal G}^{(0)}}=M $, $d=r=id$.
Then $A({\mathcal G}) = 0$, the zero bundle on $M$.
\[ex.Lie\] [*Lie groups: *]{} Every differentiable groupoid ${\mathcal G}$ with space of units consisting of just one point, $M = \{e\}$, is necessarily a Lie group. Conversely, every Lie group $G$ can be regarded as an differentiable groupoid ${\mathcal G}= G$ with exactly one unit $M = \{e\}$, the unit of that group.
In this case $A({\mathcal G}) = Lie (G)$, the Lie algebra of $G$.
The above two sets of examples are in a certain way the two extreme examples of differentiable groupoids. The first set of examples consists of groupoids each of which has the least set of arrows among all groupoids with the same set of units. The second set of examples consists of the differentiable groupoids with the least set of units among all non-empty groupoids. The fact that manifolds and Lie groups are particular cases of groupoids makes them a favorite object of study in non-commutative geometry.
The following example plays a special role in the theory pseudodifferential operators on groupoids.
\[ex.pair\] [*The pair groupoid: *]{} Let $M$ be a smooth manifold (without corners) and let $${\mathcal G}= M \times M \quad \text{and} \quad {{\mathcal G}^{(0)}} = M,$$ with structural morphisms $d(x,y)=y$, $r(x,y)=x$, $(x,y)(y,z)=(x,z)$, $u(x) = (x,x)$ and $\iota(x,y) = (y,x)$. Then ${\mathcal G}$ is a differentiable groupoid, called [*the pair groupoid*]{}.
Denote the pair groupoid with units $M$ by $M \times M$. Then $A(M
\times M) = TM$, the tangent bundle to $M$.
A variant of the above example is the following.
\[ex.fibered.pair\] [*The fibered pair groupoid: *]{} Let $p : Y \to B$ be a submersion of manifolds with corners (see Definition \[def.1\] of the previous section). The fibered pair groupoid ${\mathcal G}$ is obtained as $${\mathcal G}= Y \times_B Y := \{(x,y), p(x)=p(y), x,y \in Y \},$$ with the operations induced from the pair groupoid $Y \times Y$. Its space of units is $Y$.
The Lie algebroid of $Y \times_B Y$ is $A({\mathcal G}) = T_{vert}Y$, the kernel of $TY \to TB$, or, in other words, the vertical tangent bundle to the submersion $p : Y \to B$.
[*Products: *]{} The product ${\mathcal G}_1 \times {\mathcal G}_2$ of two differentiable groupoids ${\mathcal G}_1$ and ${\mathcal G}_2$ is again a differentiable groupoid for the product structural morphisms.
To obtain the Lie algebroid of the product groupoid we use the external product of vector bundles (not the direct sum): $A({\mathcal G}_1 \times {\mathcal G}_2) = A({\mathcal G}_1) \times A({\mathcal G}_2)$, a vector bundle over ${\mathcal G}_1^{(0)}\times {\mathcal G}_2^{(0)}$.
We now include some examples that are suggested by the general theory of differentiable groupoids.
[*Bundles of Lie groups: *]{} In this example ${\mathcal G}$ is a fiber bundle $p:{\mathcal G}\to B$ such that each fiber ${\mathcal G}_b := p^{-1}(b)$ has a Lie group structure, and the induced map $${\mathcal G}\times_B {\mathcal G}:= \{ (g,g') \in {\mathcal G}\times {\mathcal G},
p(g) = p(g') \} \ni (g,g') \mapsto g^{-1}g' \in {\mathcal G}$$ is a smooth map. We define $d = r = p$. The units of the groups ${\mathcal G}_b$ then form a submanifold of ${\mathcal G}$ diffeomorphic to $B$ via $p$. We do not assume that the fibers ${\mathcal G}_b$ are all isomorphic, but this is true in most cases of interest.
Let ${\mathfrak g}$ be the restriction to the space of units of the vertical tangent bundle to the fibration ${\mathcal G}\to B$. Then ${\mathfrak g}$ is a vector bundle over $B$ and $A({\mathcal G}) = {\mathfrak g}$. The fiber of ${\mathfrak g}$ above $b$ is then the Lie algebra of ${\mathcal G}_b$, and ${\mathfrak g}$ is a bundle of Lie algebras. In this example, the anchor map $\varrho:A({\mathcal G}) \to TB$ is the zero map.
[*Fibered products: *]{} Let ${\mathcal G}_1$ and ${\mathcal G}_2$ be two differentiable groupoids with units $M_1$ and $M_2$. We assume that both $M_1$ and $M_2$ come equipped with submersions $p_i: M_i \to B$, $i = 1,2$, for some common manifold with corners $B$. Suppose that for each arrow $g \in M_i$, $p_i(d(g)) = p_i(r(g)) =:p_{i}(g)$. The fibered product of ${\mathcal G}_1$ with ${\mathcal G}_2$ (with respect to $p_i$) is then $${\mathcal G}_1 \times_B {\mathcal G}_2 := \{ (g_1,g_2) \in {\mathcal G}_1 \times_B {\mathcal G}_2 ,
p_1(g_1) = p_2(g_2)\},$$ with product (and structural maps, in general) induced from the product groupoid ${\mathcal G}_1 \times {\mathcal G}_2$.
We get $A({\mathcal G}_1 \times_B {\mathcal G}_2) = A({\mathcal G}_1) \times_B A({\mathcal G}_2)$.
We are particularly interested in the above example when ${\mathcal G}_1 = Y
\times_B Y$ is the fibered pair groupoid of Example \[ex.fibered.pair\] and ${\mathcal G}_2$ is a bundle of Lie groups with base $B$. This situation seems to be fundamental in the study of pseudodifferential operators associated to various Lie algebras of vector fields. It will be used for example in Example \[ex.boundary.f1\]. An index theorem in this framework was obtained in [@NistorIndFam], if the fibers of ${\mathcal G}_2 \to B$ are simply-connected solvable.
We continue with some more elaborate examples.
[*The graph of a foliation: *]{} This groupoid was introduced in [@Winkelnkemper1]. Let $(M,F)$ be a foliated manifold. Thus $F \subset TM$ is an integrable bundle. The graph of $(M,F)$ consists of equivalence classes $[\gamma]$ of paths $\gamma$ which are completely contained in a leaf, with respect to the equivalence relation $[\gamma] = [\gamma']$ if, and only if, $\gamma$ and $\gamma'$ have the same holonomy (this implies, in particular, that they have the same end-points).
The Lie algebroid is $A({\mathcal G}) = F$ in this example.
[*The fundamental groupoid: *]{} Let ${\mathcal G}$ be the fundamental groupoid of a compact smooth manifold $M$ (without corners) with fundamental group $\pi_1(M)=\Gamma$. Recall that if we denote by $\widetilde{M}$ a universal covering of $M$ and let $\Gamma$ act by covering transformations on $\widetilde{M}$, then we have ${{\mathcal G}^{(0)}}=\widetilde{M}/\Gamma = M$, ${\mathcal G}=(\widetilde{M} \times
\widetilde{M})/\Gamma$, and $d$ and $r$ are the two projections. Each fiber ${\mathcal G}_x$ can be identified with $\widetilde{M}$, uniquely up to the action of an element in $\Gamma$.
The Lie algebroid is $TM$, as in the first example.
The following example generalizes the tangent groupoid of Connes; here we closely follow [@connes II,5]. A similar construction was used in [@Skandalis] to define the so called “normal groupoid,” which is, anticipating a little bit, the adiabatic groupoid of a foliation.
\[ex.adiabatic\] [*The adiabatic groupoid: *]{} The adiabatic groupoid ${\mathcal G}_{{\operatorname{ad}}}$ associated to a differentiable groupoid ${\mathcal G}$ is defined as follows. The space of units is $${\mathcal G}_{{\operatorname{ad}}}^{(0)} := [0,\delta) \times {{\mathcal G}^{(0)}}\,, \quad \delta > 0,$$ with the product manifold structure. The set of arrows is defined as the disjoint union $${\mathcal G}_{{\operatorname{ad}}} := A({\mathcal G}) \cup (0,\delta) \times {\mathcal G}.$$ The structural maps are defined as follows. The domain and range are: $$d(t,g)=(t,d(g))\,\quad r(t,g)=(t,r(g))\,,\quad t>0,$$ and $d(v)=r(v)=(0,x)$, if $v \in T_x{\mathcal G}_x$. The composition is $
\mu(\gamma,\gamma')=(t,gg'),
$ if $t >0$, $\gamma=(t,g)$, and $\gamma'=(t,g')$, and $$\mu(v,v')=v +v' \quad \text{ if } v,v' \in T_x{\mathcal G}_x.$$
The smooth structure on the set of arrows is the product structure for $t > 0$. In order to define a coordinate chart at a point $$v \in T_x{\mathcal G}_x = A_x({\mathcal G}) = d^{-1}(0,x),$$ choose first a coordinate system $\psi : U = U_1 \times U_2 \to {\mathcal G}$, $U_1 \subset {\mathbb R}^p$ and $U_2\subset {\mathbb R}^n$ being open sets containing the origin, $U_2$ convex, with the following properties: $\psi(0,0)=x
\in M \subset {\mathcal G}$, $\psi(U) \cap M=\psi(U_1 \times \{0\})$, and there exists a diffeomorphism $\phi : U_1 \to {{\mathcal G}^{(0)}}$ such that $d(\psi(s,y))
= \phi(s)$ for all $y \in U_2$ and $s\in U_{1}$.
We identify, using the differential $\operatorname{D}_2\! \psi$ of the map $\psi$, the vector space $\{s\} \times {\mathbb R}^n$ and the tangent space $T_{\phi(s)}{\mathcal G}_{\phi(s)}=A_{\phi(s)}({\mathcal G})$. We obtain then coordinate charts $ \psi_\varepsilon : A({\mathcal G})\vert_{\phi(U_1)} \times
(0,\varepsilon) \times U_1 \times \varepsilon^{-1}U_2 \to {\mathcal G}, $ $$\psi_\varepsilon(0,s,y)=(0,(\operatorname{D}_2\!\psi) (s,y))
\in T_{\phi(s)}{\mathcal G}_{\phi(s)}=A_{\phi(s)}({\mathcal G})$$ and $\psi_\varepsilon(t,s,y)=(t,\psi(s,ty)) \in (0,1) \times {\mathcal G}$. For $\varepsilon$ small enough, the range of $\psi_\varepsilon$ will contain $v$.
The Lie algebroid of ${\mathcal G}_{{\operatorname{ad}}}$ is the adiabatic Lie algebroid associated to $A({\mathcal G})$, $A({\mathcal G}_{{\operatorname{ad}}})=A({\mathcal G})_{t}$, for all $t$, such that $\Gamma(A({\mathcal G})) \cong t\Gamma(A({\mathcal G}\times [0,\delta)))$.
We expect the above constructions to have applications to semi-classical trace formulæ, see Uribe’s overview [@Uribe]. A variant of the above example can be used to treat adiabatic limits when the metric is blown up in the base. See [@Witten1] for some connections with physics.
More examples are discussed in Section \[Sec.Examples.II\].
Pseudodifferential operators on groupoids\[Sec.POOG\]
=====================================================
We proceed now to define the space of pseudodifferential operators acting on sections of vector bundles on a differentiable groupoid. This construction is the same as the one in [@NWX], but slightly more general because we consider also certain non-Hausdorff groupoids. General reference for pseudodifferential operators on smooth manifolds are, for instance, [@Hormander3] or [@shubin]. We discuss operators on functions, for simplicity, but at the end we briefly indicate the changes necessary to handle operators between sections of smooth vector bundles.
Our construction of pseudodifferential operators on groupoids is obtained considering certain families of pseudodifferential operators on smooth, generally non-compact manifolds. We begin then by recalling a few facts about pseudodifferential operators on smooth manifolds. Let $W \subset {\mathbb R}^N$ be an open subset. Define the space ${\mathcal S}^m(W \times {\mathbb R}^n)$ of [*symbols of order $m\in{\mathbb R}$*]{} on the bundle $W \times {\mathbb R}^n \to W$, as in [@Hormander3], to be the set of smooth functions $a : W \times {\mathbb R}^n \to {\mathbb C}$ such that $$\label{eq.symbol.estimates}
|\partial_y^\alpha \partial_\xi^\beta a(y,\xi)| \leq C_{K,\alpha,\beta}
(1+|\xi|)^{m-|\beta|}$$ for any compact set $K\subset W$ and any multi-indices $\alpha$ and $\beta$, and some constant $C_{K,\alpha,\beta} > 0$. A symbol $a \in {\mathcal S}^m(W \times {\mathbb R}^n)$ is called [*classical*]{} if it has an asymptotic expansion as an infinite sum of homogeneous symbols $a \sim \sum_{k=0}^\infty a_{m-k}$, $a_l$ homogeneous of degree $l$ for large ${\| \xi \|}$, i.e. $a_l(y,t\xi)=t^la_l(y,\xi)$ if $\|\xi\|\geq 1$ and $t \geq 1$. More precisely, $\sim$ means $$a -\sum_{k=0}^{M-1} a_{m-k} \in
{\mathcal S}^{m-M}(W \times {\mathbb R}^n) \quad \text{ for all } M\in{{\mathbb N}_{0}}\,.$$ The space of classical symbols will be denoted by ${\mathcal S}^m_{{\operatorname{cl}}}(W \times {\mathbb R}^n)$. Using local trivializations the definition of (classical) symbols immediately extends to arbitrary vector bundles $\pi:E\longrightarrow M$. We shall consider only classical symbols in this paper. For $a \in {\mathcal S}^m(T^*W) = {\mathcal S}^m(W \times {\mathbb R}^n)$ and $W$ an open subset of ${\mathbb R}^n$, we define an operator $a(y,D_y):{{\mathcal C}^{\infty}_{\text{c}}}(W) \to {\mathcal{C}^\infty}(W)$ by $$a(y,D_y)u(y)=(2\pi)^{-n}\int_{{\mathbb R}^n}e^{i y\cdot \xi}
a(y,\xi)\hat{u}(\xi)d\xi\,,$$ where $\hat{u}$ denotes the Fourier transform of $u$.
Recall that if $M$ is a smooth manifold, a linear map $T:{{\mathcal C}^{\infty}_{\text{c}}}(M) \to
{\mathcal{C}^\infty}(M)$ is called [*regularizing*]{} if, and only if, it has a smooth distributional (or Schwartz) kernel. Also, recall that a linear map $P:{{\mathcal C}^{\infty}_{\text{c}}}(M) \to {\mathcal{C}^\infty}(M)$ is called a [*(classical) pseudodifferential operator of order $m$*]{} if, and only if, for all smooth functions $\phi$ supported in a (not necessarily connected) coordinate chart $W$, the operator $\phi P \phi$ is of the form $a(y,D_y)$ with a (classical) symbol $a$ of order $m$. For a classical pseudodifferential operator $P$ as the one considered here, the collection of all classes of $a$ in ${\mathcal
S}_{{\operatorname{cl}}}^m(T^*W)/{\mathcal S}_{{\operatorname{cl}}}^{m-1}(T^*W)$, for all coordinate neighborhoods $W$, patches together to define a class $\sigma_m(P)\in
{\mathcal S}_{{\operatorname{cl}}}^m(T^*W)/{\mathcal S}_{{\operatorname{cl}}}^{m-1}(T^*W)$, which is called [*the homogeneous principal symbol*]{} of $P$; the latter space can, of course, canonically be identified with $S^{[m]}(T^*M)$, the space of all smooth functions $T^*M\setminus\{0\}\longrightarrow {\mathbb C}$ that are positively homogeneous of degree $m$. We shall sometimes refer to a classical pseudodifferential operator acting on a smooth manifold (without corners) as an [*ordinary*]{} classical pseudodifferential operator, in order to distinguish it from a pseudodifferential operator on a groupoid.
We now begin the discussion of pseudodifferential operators on groupoids. A pseudodifferential operator on a differentiable groupoid ${\mathcal G}$ will be a family $(P_x)$, $x \in M$, of classical pseudodifferential operators $P_x : {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G}_x) \to {\mathcal{C}^\infty}({\mathcal G}_x)$ with certain additional properties that need to be explained.
If $(P_x)$, $x \in M$, is a family of pseudodifferential operators acting on ${\mathcal G}_x$, we denote by $k_{x}$ the distribution kernel of $P_x$ We then define the support of the operator $P$ to be $$\label{support}
{\operatorname{supp}}(P) = \overline{\bigcup_{x\in M} {\operatorname{supp}}(k_{x})}.$$ The support of $P$ is contained in the closed subset $\{(g,g'), d(g)=d(g')\}$ of the product ${\mathcal G}\times {\mathcal G}$.
To define our class of pseudodifferential operators, we shall need various conditions on the support of our operators. We introduce the following terminology: a family $P=(P_x)$, $x \in M$ is [*properly supported*]{} if $p_i^{-1}(K) \cap {\operatorname{supp}}(P)$ is a compact set for any compact subset $K \subset {\mathcal G}$, where $p_1,p_2 :{\mathcal G}\times {\mathcal G}\to {\mathcal G}$ are the two projections. The family $P = (P_x)$ is called [*compactly supported*]{} if its support ${\operatorname{supp}}(P)$ is compact. Finally, $P$ is called [*uniformly supported*]{} if its [*reduced support*]{} ${\operatorname{supp}}_\mu(P):=\mu_1({\operatorname{supp}}(P))$ is a compact subset of ${\mathcal G}$, where $\mu_1(g',g):=g'g^{-1}$. Clearly, a uniformly supported operator is properly supported, and a compactly supported operator is uniformly supported. If the family ${P=(P_x)\,,x \in M,}$ is properly supported, then each $P_x$ is properly supported, but the converse is not true.
Recall that the composition of two ordinary pseudodifferential operators is defined if one of them is properly supported. It follows that we can define the composition $PQ$ of two properly supported families of operators $P=(P_x)$ and $Q=(Q_x)$ acting on the fibers of $d : {\mathcal G}\to M$ by pointwise composition $PQ=(P_xQ_x)$, $x \in M$. The resulting family $PQ$ will also be properly supported. If $P$ and $Q$ are uniformly supported, then $PQ$ will also be uniformly supported.
The action of a family $P=(P_x)$ on functions on ${\mathcal G}$ is defined pointwise as follows. For any smooth function $f \in {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$ denote by $f_x$ its restriction $f\vert_{{\mathcal G}_x}$. If each $f_x$ has compact support, and ${P=(P_x)\,,x \in M,}$ is a family of ordinary pseudodifferential operators, then we define $Pf$ by $(Pf)_x = P_{x}(f_x).$ If $P$ is uniformly supported, then $Pf$ is also compactly supported. However, it is not true that $Pf \in{{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$ if $f \in {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$, in general, so we need some conditions on the family $P$. We shall hence consider uniformly supported families $P = (P_x)$ because this guarantees that $Pf$ has compact support if $f$ does.
A [*fiber preserving diffeomorphism*]{} will be a diffeomorphism $\psi : d(V) \times W \to V$ satisfying $d(\psi(x,w))=x$, where $W$ is some open subset of an Euclidean space of the appropriate dimension. We now discuss the differentiability condition on a family $P= (P_x)$, a condition which, when satisfied, implies that $Pf$ is smooth for all smooth $f \in {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$.
\[Def.Differentiable\] Let ${\mathcal G}$ be a differentiable groupoid with units $M$. A family $(P_x)$ of pseudodifferential operators acting on ${{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G}_x)$, $x \in M$, is called if, and only if, for any fiber preserving diffeomorphism $\psi : d(V) \times W \to V$ onto an open set $V \subseteq {\mathcal G}$, and for any $\phi \in {{\mathcal C}^{\infty}_{\text{c}}}(V)$, we can find $a \in {\mathcal S}_{{\operatorname{cl}}}^m(d(V) \times T^*W)$ such that $\phi P_x \phi$ corresponds to $a(x,y,D_y)$ under the diffeomorphism ${\mathcal G}_x\cap V \simeq W$, for each $x \in d(V)$.
Thus, we require that the operators $P_x$ be given in local coordinates by symbols $a_x$ that depend smoothly on all variables. Note that nowhere in the above definition it is necessary for ${\mathcal G}$ to be Hausdorff. All we do need is that each of ${\mathcal G}_x$ and $M = {{\mathcal G}^{(0)}}$ are Hausdorff.
To define pseudodifferential operators on ${\mathcal G}$ we shall consider smooth, uniformly supported families $P=(P_x)$ that satisfy also an invariance condition. To introduce this invariance condition, observe that right translations on ${\mathcal G}$ define linear isomorphisms $$\label{eq.isomorphism}
U_g:{\mathcal{C}^\infty}({\mathcal G}_{d(g)}) \to {\mathcal{C}^\infty}({\mathcal G}_{r(g)}):
(U_gf)(g')=f(g'g) \,.
$$ A family of operators $P=(P_x)$ is then called [*invariant*]{} if $P_{r(g)}U_g = U_g P_{d(g)}$, for all $g \in {\mathcal G}$. We are now ready to define pseudodifferential operators on ${\mathcal G}$.
\[Main.definition\] Let ${\mathcal G}$ be a differentiable groupoid with units $M$, and let $P=(P_x)$ be a family $P_x : {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G}_x) \to {\mathcal{C}^\infty}({\mathcal G}_x)$ of (order $m\in{\mathbb R}$, ordinary) classical pseudodifferential operators. Then $P$ is an (order $m$) if, and only if, it is
(i) uniformly supported,
(ii) differentiable, and
(iii) invariant.
We denote the space of order $m$ pseudodifferential operators on ${\mathcal G}$ by ${\Psi^{m}({\mathcal G})}$.
We also denote ${\Psi^{\infty}({\mathcal G})} := \cup_{m \in {\mathbb R}}{\Psi^{m}({\mathcal G})}$ and ${\Psi^{-\infty}({\mathcal G})}=\cap_{m \in {\mathbb R}}{\Psi^{m}({\mathcal G})}$. Let us now give an alternative description of ${\Psi^{m}({\mathcal G})}$ that highlights again the conormal nature of kernels of pseudodifferential operators. For $P\in{\Psi^{\infty}({\mathcal G})}$, we call $$\kappa_{P}(g):=k_{d(g)}(g,d(g))\,, g\in{\mathcal G}$$ the [*reduced*]{} or [*convolution kernel*]{} of $P$. Due to the right-invariance of $P$ we expect the reduced kernel to carry all information of the family $P$. For the definition of conormal distributions we refer the reader to [@fio; @Hormander3] in the smooth case.
\[redker\] The map $P\longmapsto \kappa_{P}$ induces an isomorphism $${\Psi^{m}({\mathcal G})}\stackrel{\cong}{\longrightarrow}I_{c}^{m}({\mathcal G},M;d^{*}\mathcal{D})$$ where $I_{c}^{m}({\mathcal G},M;d^{*}\mathcal{D})$ denotes the space of all compactly supported, $d^{*}\mathcal{D}$-valued distributions on ${\mathcal G}$ conormal to $M$. In particular, $P\longmapsto\kappa_{P}$ identifies ${\Psi^{-\infty}({\mathcal G})}$ with the convolution algebra ${{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G},d^{*}\mathcal{D})$. Moreover, we have ${\operatorname{supp}}(\kappa_{P})={\operatorname{supp}}_{\mu}(P)$.
Define ${{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$ as in Section \[Sec.prelim\], then ${\Psi^{m}({\mathcal G})}({{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})) \subset {{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G})$. We obtain in this way a representation $\pi$ of ${\Psi^{\infty}({\mathcal G})}$ on ${{\mathcal C}^{\infty}_{\text{c}}}(M)$, uniquely determined by $$\label{eq.vector.rep}
(\pi(P)f) \circ r = P (f \circ r), \quad P = (P_x) \in {\Psi^{m}({\mathcal G})}.$$ We call this representation acting on any space of functions on $M$ on which it makes sense (${{\mathcal C}^{\infty}_{\text{c}}}(M)$, ${\mathcal{C}^\infty}(M)$, $L^2(M)$, or Sobolev spaces) the [*vector representation*]{} of ${\Psi^{\infty}({\mathcal G})}$.
We now discuss the extension of the principal symbol map to ${\Psi^{m}({\mathcal G})}$. Denote by $\pi:A^*({\mathcal G})\rightarrow M$ the projection. If $P=(P_x)\in {\Psi^{m}({\mathcal G})}$ is an order $m$ pseudodifferential operator on ${\mathcal G}$, then the principal symbol $\sigma_m(P)$ of $P$ will be an order $m$ homogeneous function on $A^*({\mathcal G})\smallsetminus 0$ (it is defined only outside the zero section) such that: $$\label{princ.symb}
\sigma_m(P)(\xi)=\sigma_m(P_x)(\xi) \in {\mathbb C}\quad \text{ if } \xi \in A^*_x({\mathcal G})=T^*_x{\mathcal G}_x.$$ Denote by ${\mathcal S}_{c}^{m} (A^*_x({\mathcal G})) \subset {\mathcal S}_{{\operatorname{cl}}}^{m}
(A^*_x({\mathcal G}))$ the subspace of classical symbol whose support has compact projection onto the space of units $M$. The above definition determines a linear map $$\sigma_m:{\Psi^{m}({\mathcal G})} \to {\mathcal S}_{c}^m(A^*({\mathcal G}))/
{\mathcal S}_{c}^{m-1}(A^*({\mathcal G}))
\cong S^{[m]}_{c}(A^{*}({\mathcal G}))\,,$$ where $S^{[m]}_{c}(A^{*}({\mathcal G}))$ denotes the space of all smooth functions $A^{*}({\mathcal G})\setminus\{0\}\longrightarrow{\mathbb C}$ that are positively homogeneous of degree $m$, and whose support has compact projection onto the space of units $M$. The map $\sigma_{m}$ is said to be the [*principal symbol*]{}. A pseudodifferential operator $P\in{\Psi^{m}({\mathcal G})}$ is called [*elliptic*]{} provided its principal symbol $\sigma_{m}(P)\in S^{[m]}_{c}(A^{*}({\mathcal G}))$ does not vanish on $A^{*}({\mathcal G})\setminus\{0\}$; note, with this definition elliptic operators only exist if the space of units is compact.
The following result extends several of the well-known properties of the calculus of ordinary pseudodifferential operators on smooth manifolds to ${\Psi^{\infty}({\mathcal G})}$. Denote by $\{\; , \; \}$ the canonical Poisson bracket on $A^*({\mathcal G})$.
\[Theorem.Algebra\] Let ${\mathcal G}$ be a differentiable groupoid. Then ${\Psi^{\infty}({\mathcal G})}$ is an algebra with the following properties:
\(i) The principal symbol map $$\sigma_m:{\Psi^{m}({\mathcal G})} \to {\mathcal S}_{c}^m(A^*({\mathcal G}))/
{\mathcal S}_{c}^{m-1}(A^*({\mathcal G}))$$ is surjective with kernel ${\Psi^{m-1}({\mathcal G})}$.
\(ii) If $P \in {\Psi^{m}({\mathcal G})}$ and $Q \in {\Psi^{m'}({\mathcal G})}$, then $PQ \in {\Psi^{m+m'}({\mathcal G})}$ and satisfies $\sigma_{m+m'}(PQ)=\sigma_m(P)\sigma_{m'}(Q)$. Consequently, $[P,Q] \in
{\Psi^{m + m' -1}({\mathcal G})}$. Its principal symbol is given by $\sigma_{m + m'
-1}([P,Q]) = \frac{1}{i}\{ \sigma_m(P), \sigma_{m'}(Q) \}$.
Properly supported invariant differentiable families of pseudodifferential operators also form a filtered algebra, denoted $\Psi_{\prop}^{\infty}({\mathcal G})$. While it is clear that, in order for our class of pseudodifferential operators to form an algebra, we need some condition on the support of their distributional kernels, exactly what support condition to impose is a matter of choice. We prefer the uniform support condition because it leads to a better control at infinity of the family of operators $P=(P_x)$ and allows us to identify the regularizing ideal ([[*i.e.,* ]{}]{}the ideal of order $-\infty$ operators) with the groupoid convolution algebra of ${\mathcal G}$. The choice of uniform support will also ensure that ${\Psi^{m}({\mathcal G})}$ behaves functorially with respect to open embeddings. The compact support condition enjoys the same properties but is usually too restrictive. The issue of support will be discussed again in examples.
We now discuss the restriction of families in ${\Psi^{m}({\mathcal G})}$ to invariant subsets of $M$, or, more precisely, the restriction to ${\mathcal G}_Y$, the “reduction” of ${\mathcal G}$ to the invariant subset $Y$ of $M$. The resulting restriction morphisms $\inn_Y : {\Psi^{\infty}({\mathcal G})} \to
\Psi^\infty({\mathcal G}_Y)$ is the analog in our setting of the indicial morphisms considered in [@mepi92].
We continue to denote by ${\mathcal G}$ a differentiable groupoid with units $M$. Let $Y \subset M$ and let ${\mathcal G}_Y := d^{-1}(Y) \cap r^{-1}(Y)$. Then ${\mathcal G}_Y$ is a groupoid with units $Y$, called the [*reduction of ${\mathcal G}$ to $Y$*]{}. An [*invariant*]{} subset $Y \subset M$ is a subset such that $d(g) \in Y$ implies $r(g) \in Y$. For an invariant subset $Y \subset M$, the reduction of ${\mathcal G}$ to $Y$ satisfies $${\mathcal G}_Y = d^{-1}(Y) = r^{-1}(Y)$$ and is a differentiable groupoid, if $Y$ is a closed submanifold (with corners) of $M$.
If $P = (P_x)$, $x \in M$, is a pseudodifferential operator on ${\mathcal G}$, and $Y \subset M$ is an closed, invariant submanifold with corners, we can restrict $P$ to $d^{-1}(Y)$ and obtain $$\inn_Y (P) := (P_x)_{x\in Y},$$ which is a family of operators acting on the fibers of $d:{\mathcal G}_Y=d^{-1}(Y) \to Y$ and satisfying all the conditions necessary to define an element of $\Psi^{\infty}({\mathcal G}\vert_Y)$. This leads to a map $$\label{eq.indicial}
\inn_Y =\inn_{Y,M}:{\Psi^{\infty}({\mathcal G})} \to \Psi^{\infty}({\mathcal G}\vert_Y),$$ which is easily seen to be an algebra morphism.
Let us indicate now what changes need to be made when we consider operators acting on sections of a vector bundles. Because operators acting between sections of two [*different*]{} vector bundles $E_1$ and $E_2$ can be recovered from operators acting on $E=E_1 \oplus E_2$, we may assume that $E_1 = E_2 = E$ as vector bundles on $M = {{\mathcal G}^{(0)}}$.
Denote by $r^*(E)$ the pull-back of $E$ to ${\mathcal G}= {{\mathcal G}^{(1)}}$. Then the isomorphisms of Equation will have to be replaced by $$U_g:{\mathcal{C}^\infty}({\mathcal G}_{d(g)},r^*(E)) \to {\mathcal{C}^\infty}({\mathcal G}_{r(g)},r^*(E)):
(U_gf)(g')=f(g'g) \in (r^*E)_{g'},$$ which makes sense because of $(r^*E)_{g'}=(r^*E)_{g'g}=E_{r(g')}$. Then, to define ${\Psi^{m}({\mathcal G};E)}$ we consider families $P = (P_x)$ of order $m$ pseudodifferential operators $P_x$, $x \in M$, acting on the spaces ${{\mathcal C}^{\infty}_{\text{c}}}({\mathcal G}_x,r^*(E))$. We require these families to be uniformly supported, differentiable, and invariant, as in the case $E= {\mathbb C}$.
The principal symbol $\sigma_m(P)$ of a classical pseudodifferential operator $P$ belongs then to ${\mathcal
S}_{c}^m(T^*W;{\operatorname{End}}(E))/{\mathcal S}_{c}^{m-1}(T^*W;{\operatorname{End}}(E))$. Finally, the restriction (or indicial) morphism is a morphism. $$\inn_Y :\Psi^{\infty}({\mathcal G};E) \to\Psi^{\infty}
({\mathcal G}\vert_Y ; E\vert_Y).$$ All the other changes needed to treat the case of non-trivial vector bundles $E$ are similar.
There is one particular case of a bundle $E$ that deserves special attention. Let $E := {{\mathcal D}}^{1/2}$ be the square root of the density bundle ${{\mathcal D}}= | \Lambda^n A({\mathcal G})|,$ as before. If $P \in\Psi^{m}({\mathcal G}; {{\mathcal D}}^{1/2})$ consists of the family $(P_x, x\in M)$, then each $P_x$ acts on $V_x = C_c^{\infty}({\mathcal G}_x;r^*({{\mathcal D}}^{1/2})).$ Since $r^*({{\mathcal D}}^{1/2}) = \Omega_{{\mathcal G}_x}^{1/2}$ is the bundle of half densities on ${\mathcal G}_x$, we can define a natural hermitian inner product $(\; ,\; )$ on $V_x$. Let $P = (P_x) \in \Psi^m({\mathcal G};{{\mathcal D}}^{1/2})$, and denote by $P_x^*$ the formal adjoint of $P_x$, that is, the unique pseudodifferential operator on $V_x$ such that $(P_x f, g) = (f, P_x^*g)$, for all $f,g \in V_x$. It is not hard to see that $(P_x^*) \in \Psi^m({\mathcal G};{{\mathcal D}}^{1/2})$. Moreover, $\sigma_m(P^*) = \overline{\sigma_m(P)}$.
If $E$ is the complexification of a real bundle $E_0$: $E \simeq E_0 \otimes {\mathbb C}$, then the complex conjugation operator $J \in {\operatorname{End}}_{\mathbb R}(E)$ defines a real structure on ${\Psi^{*}({\mathcal G};E)}$, that is, a conjugate linear involution on ${\Psi^{*}({\mathcal G};E)}$. In this case, ${\Psi^{*}({\mathcal G};E)}$ is the complexification of the set of fixed points of this involution.
Bounded representations\[Sec.BR\]
=================================
For a smooth, compact manifold $M$ (without corners), the algebra $\Psi^0(M)$ of order zero pseudodifferential operators on $M$ acts by bounded operators on $L^2(M,d\mu)$, the Hilbert space $L^2(M,d\mu)$ being defined with respect to the (essentially unique) measure $\mu$ corresponding to a nowhere vanishing density on $M$. Moreover, this is basically the only interesting $*$-representation of $\Psi^0(M)$ by bounded operators on an infinite-dimensional Hilbert space of functions.
For a differentiable groupoid ${\mathcal G}$ with units $M$, a manifold with corners, it is still true that we can find a measure $\mu$ such that ${\Psi^{0}({\mathcal G})}$ acts by bounded operators on $L^2(M,d\mu)$. However, in this case there may exist natural measures $d\mu$ that are singular with respect to the measure defined by a nowhere vanishing density. Moreover, there may exist several non-equivalent such measures, and these representations may not exhaust all equivalence classes of non-trivial, irreducible, bounded representations of ${\Psi^{0}({\mathcal G})}$.
The purpose of this section is to introduce the class of representations we are interested in, and to study some of their properties. A consequence of our results is that in order to construct and classify bounded representations of ${\Psi^{0}({\mathcal G})}$, it is essentially enough to do this for ${\Psi^{-\infty}({\mathcal G})}$.
We are interested in representations of ${\Psi^{m}({\mathcal G})}$, $m \in
\{0,\pm\infty\}$. We fix a trivialization of ${{\mathcal D}}$, so that we get an isomorphism ${\Psi^{m}({\mathcal G})} \cong \Psi^m({\mathcal G};{{\mathcal D}}^{1/2})$ and hence we have an involution $*$ on ${\Psi^{m}({\mathcal G})}$. Let ${\mathcal H}_0$ be a dense subspace of a Hilbert space $\mathcal{H}$ with inner product $(\; , \;
)$. Recall that a $*$-morphism $\alpha : A \to {\operatorname{End}}(\mathcal H_0)$ from a $*$-algebra $A$ is a morphism such that $(\alpha(P^*)\xi, \eta)
= (\xi, \alpha(P)\eta))$, for all $P \in A$ and all $\xi,\eta \in
\mathcal H_0$.
\[def.bounded.rep\] Let $\mathcal{H}_{0}$ be a dense subspace of a Hilbert space $\mathcal{H}$, and $m = 0$ or $m =
\pm \infty$ be fixed. A of ${\Psi^{m}({\mathcal G})}$ on ${\mathcal H}_0$ is a $*$-morphism $\varrho : {\Psi^{m}({\mathcal G})} \to {\operatorname{End}}( {\mathcal H}_0 )$ such that $\varrho(P)$ extends to a bounded operator on $\mathcal{H}$ for all $P \in {\Psi^{0}({\mathcal G})}$ (for all $P \in {\Psi^{-\infty}({\mathcal G})}$, if $m = -\infty$).
Note that if $\varrho$ is as above and $P$ is an operator of positive order, then $\varrho(P)$ does not have to be bounded. Since $\mathcal{H}_{0}$ is dense in $\mathcal{H}$, each operator $\varrho(P)$ can be regarded as a densely defined operator. The definition implies that $\varrho(P^*) \subset \varrho(P)^*$, so the adjoint of $\varrho(P)$ is densely defined, and hence $\varrho(P)$ is a closable operator. We shall usually make no distinction between $\varrho(P)$ and its closure.
We call a bounded representation $\varrho:{\Psi^{-\infty}({\mathcal G})}\rightarrow{\operatorname{End}}({\mathcal H})$ [ *non-degenerate*]{} provided $\varrho({\Psi^{-\infty}({\mathcal G})})\mathcal{H}$ is dense in $\mathcal{H}$.
The following theorem establishes, among other things, a bijective correspondence between non-degenerate bounded representations of ${\Psi^{-\infty}({\mathcal G})}$ on a Hilbert space $\mathcal{H}$, and bounded representations $\varrho : {\Psi^{\infty}({\mathcal G})} \rightarrow{\operatorname{End}}(\mathcal H_0)$ such that the space $\varrho({\Psi^{-\infty}({\mathcal G})})\mathcal H_0$ is dense in $\mathcal{H}$. We shall need the following slight extension of a result in [@LMN].
\[Theorem.EXT\] Let ${\mathcal H}$ be a Hilbert space and let $\varrho : {\Psi^{-\infty}({\mathcal G})} \to
{\operatorname{End}}({\mathcal H})$ be a bounded representation. Then, to each $P \in {\Psi^{s}({\mathcal G})}$, $s \in {\mathbb R}$, we can associate an unbounded operator $\varrho(P)$, with domain ${\mathcal H_0}:= \varrho({\Psi^{-\infty}({\mathcal G})})
{\mathcal H}$, such that $\varrho(P)\varrho(R) =
\varrho(PR)$ and $\varrho(R) \varrho(P) = \varrho(RP)$, for any $R \in
{\Psi^{-\infty}({\mathcal G})}$.
We obtain in this way an extension of $\varrho$ to a bounded representation of ${\Psi^{0}({\mathcal G})}$ on ${\mathcal H}$ and to a bounded representation of ${\Psi^{\infty}({\mathcal G})}$ on ${\mathcal H_0}:=
\varrho({\Psi^{-\infty}({\mathcal G})}){\mathcal H}$.
We may assume that ${\mathcal H}_0$ is dense in ${\mathcal H}$. Fix $P \in {\Psi^{0}({\mathcal G})}$ and we let $$\varrho(P)\xi=\varrho(PQ)\eta,$$ if $\xi = \varrho(Q)\eta$, for some $Q \in \Psi^{-\infty}({\mathcal G})$ and $\eta
\in {\mathcal H}$. We need to show that $\varrho(P)$ is well-defined and bounded. Thus, we need to prove that $\sum_{k=1}^N \varrho(PQ_k)\xi_k=0$, if $P \in {\Psi^{0}({\mathcal G})}$ and $\sum_{k=1}^N \varrho(Q_k)\xi_k=0$, for some $Q_k
\in {\Psi^{-\infty}({\mathcal G})}$ and $\xi_k \in {\mathcal H}$.
We will show that, for each $P \in {\Psi^{0}({\mathcal G})}$, there exists a constant $k_P>0$ such that $$\| \sum_{k=1}^N\varrho(PQ_k)\xi_k \| \le k_P
\| \sum_{k=1}^N\varrho(Q_k)\xi_k \|.$$ Let $C \ge | \sigma_0(P)| + 1$, and let $$\label{eq.b}
b = (C^2 - |\sigma_0(P)|^2)^{1/2}.$$ Then $b-C$ is in ${{\mathcal C}^{\infty}_{\text{c}}}(S^*({\mathcal G}))$, and it follows from Theorem \[Theorem.Algebra\] that we can find $Q_0 \in{\Psi^{0}({\mathcal G})}$ such that $\sigma_0(Q_0) = b-C$. Let $Q = Q_0 + C$. Using again Theorem \[Theorem.Algebra\], we obtain, for $$R = C^2 - P^*P - Q^*Q \in {\Psi^{0}({\mathcal G})},$$ that $$\sigma_0(R) = \sigma_0(C^2 - P^* P - Q^* Q)= 0,$$ and hence $R \in{\Psi^{-1}({\mathcal G})}$. A standard argument using the asymptotic completeness of the algebra of pseudodifferential operators shows that we can assume that $Q$ has order $-\infty$. Let then $$\label{eq.xieta}
\xi = \sum_{k=1}^N \varrho(Q_k)\xi_k, \;\,
\eta = \sum_{k=1}^N \varrho(PQ_k)\xi_k, \quad \mbox{\rm and } \quad
\zeta= \sum_{k=1}^N \varrho(QQ_k)\xi_k,$$ which gives, $$\begin{gathered}
\label{eq.middle}
\|\eta\|^2=(\eta,\eta)=\sum_{j,k=1}^N
(\varrho(Q_k^*P^*PQ_j)\xi_j,\xi_k)\\ = \sum_{j,k=1}^N \big(C^2
(\varrho(Q_kQ_j)\xi_j,\xi_k) - (\varrho(Q_k^*Q^*QQ_j)\xi_j,\xi_k) -
(\varrho(Q_k^*RQ_j)\xi_j,\xi_k)\big ) \\ = C^2 \|\xi\|^2 -
\|\zeta\|^2 - (\varrho(R)\xi,\xi) \leq (C^2 + \|\varrho(R)\|)
\|\xi\|^2. \end{gathered}$$
The desired representation of ${\Psi^{0}({\mathcal G})}$ on ${\mathcal H}$ is obtained by extending $\varrho(P)$ by continuity to ${\mathcal H}$.
To extend $\varrho$ further to ${\Psi^{s}({\mathcal G})}$, we proceed similarly: we want $$\varrho(P)\varrho(Q)\xi=\varrho(PQ)\xi,$$ for $P \in {\Psi^{\infty}({\mathcal G})}$ and $Q \in {\Psi^{-\infty}({\mathcal G})}$. Let $\xi$ and $\eta$ be as in Equation . We need to prove that $\eta
= 0$ if $\xi = 0$. Now, because ${\mathcal H}_0$ is dense in ${\mathcal H}$, we can find $T_{j}$ in $A_\varrho$ the norm closure of $\varrho({\Psi^{-\infty}({\mathcal G})})$ and $\eta_{j} \in {\mathcal H}$ such that $\eta
= \sum_{j=1}^{N}T_{j}\eta_{j}$. Choose an approximate unit $u_\alpha$ of the $C^*$-algebra $A_\varrho$, then $u_\alpha T_{j} \to T_{j}$ (in the sense of generalized sequences). We can replace then the generalized sequence (net) $u_\alpha$ by a subsequence, call it $u_m$ such that $u_m T_{j} \to T_{j}$, as $m \to \infty$. By density, we may assume $u_m = \varrho(R_m)$, for some $R_m \in {\Psi^{-\infty}({\mathcal G})}$. Consequently, $\varrho(R_m )\eta \to \eta$, as $m \to \infty$. Then $$\eta = \lim \sum_{k = 1}^N \varrho(R_m) \varrho(PQ_k) \xi_k
= \lim \sum_{k = 1}^N \varrho(R_m P)\varrho(Q_k) \xi_k = 0,$$ because $R_mP \in {\Psi^{-\infty}({\mathcal G})}$.
We also obtain, using the above notation, that any extension of $\varrho$ to a representation of ${\Psi^{0}({\mathcal G})}$ is bounded. This extension is uniquely determined if ${\mathcal H_0}$ is dense in ${\mathcal H}$.
Assume that $M$ is connected, so that the manifolds ${\mathcal G}_x$ have the same dimension $n$. We now proceed to define a Banach norm on ${\Psi^{-n-1}({\mathcal G})}$. This norm depends on the choice of a trivialization of the bundle of densities ${{\mathcal D}}$, which then gives rise to a right invariant system of measures $\mu_x$. Then, if $P \in {\Psi^{-n-1}({\mathcal G})}$, we use the chosen trivialization of ${{\mathcal D}}$ to identify the reduced kernel $\kappa_P$, which is a priori a distribution, with a compactly supported, [*continuous function*]{} on ${\mathcal G}$, still denoted by $\kappa_P$. We then define $$\label{eq.norm.one}
\|P\|_1 =\sup_{x \in M}\left\{\int_{{\mathcal G}_x}
|\kappa_P(g^{-1})|d\mu_x(g), \int_{{\mathcal G}_x}
|\kappa_P(g)|d\mu_x(g)\right\}.$$
Some of the most interesting representations of ${\Psi^{\infty}({\mathcal G})}$ are the regular representations $\pi_x$, $x \in M$. These are bounded representations defined as follows; let $x \in M$, then the [ *regular representation*]{} $\pi_x$ associated to $x$ is the natural representation of ${\Psi^{\infty}({\mathcal G})}$ on $C_c^{\infty}({\mathcal G}_x;r^*({{\mathcal D}}^{1/2}))$, that is $\pi_x(P)=P_x$. Moreover, $\|\pi_x(P)\| \le \|P\|_1$, if $P \in
{\Psi^{-n-1}({\mathcal G})}$.
Define now the [*reduced norm*]{} of $P$ by $$\|P\|_r = \sup_{x\in M} \| \pi_x(P)\|=\sup_{x\in M} \|P_x\|\, ,$$ and the [*full norm*]{} of $P$ by $$\|P\| = \sup_\varrho \| \varrho(P)\|,$$ where $\varrho$ ranges through all bounded representations $\varrho$ of ${\Psi^{0}({\mathcal G})}$ satisfying $$\|\varrho(P)\| \le \|P\|_1 \quad \text{ for all } P \in
{\Psi^{-\infty}({\mathcal G})} \, .$$ The above comments imply, in particular, that $\| P \|_r \le
\|P\|$. If we have equality, we shall call ${\mathcal G}$ [*amenable*]{}, following the standard usage.
Denote by ${\mathfrak{A}({\mathcal G})}$ \[respectively, by ${\mathfrak{A}_r({\mathcal G})}$\] the closure of ${\Psi^{0}({\mathcal G})}$ in the norm $\|\;\|$ \[respectively, in the norm $\|\;\|_r$\]. Also, denote by ${C^*({\mathcal G})}$ \[respectively, by ${C^*_r({\mathcal G})}$\] the closure of ${\Psi^{-\infty}({\mathcal G})}$ in the norm $\|\;\|$ \[respectively, in the norm $\|\;\|_r$\]. The principal symbol $\sigma_0$ extends by continuity to ${\mathfrak{A}({\mathcal G})}$ and ${\mathfrak{A}_r({\mathcal G})}$.
Let $S^*({\mathcal G}):=(A^{*}({\mathcal G})\setminus\{0\})/{\mathbb R}_{+}$ be the space of rays in $A^*({\mathcal G})$. (By choosing a metric on $A({\mathcal G})$, we may identify $S^*({\mathcal G})$ with the subset of vectors of length one in $A^*({\mathcal G})$.) Then we have the following two exact sequences of $C^*$-algebras: $$\begin{aligned}
&& 0 \to {C^*({\mathcal G})} \to {\mathfrak{A}({\mathcal G})} \to {\mathcal{C}_0}(S^*({\mathcal G})) \to 0\quad
\text{ and }\\ && 0 \to {C^*_r({\mathcal G})} \to {\mathfrak{A}_r({\mathcal G})} \to
{\mathcal{C}_0}(S^*({\mathcal G})) \to 0\,. \end{aligned}$$ In particular, ${\Psi^{-\infty}({\mathcal G})}$ is dense in ${\Psi^{-1}({\mathcal G})}$.
Let $Y \subset M$ be a closed, invariant submanifold with corners. Then we also have exact sequences $$\begin{aligned}
\label{eq.seq2}
&& 0 \to {C^*({\mathcal G}_{M \setminus Y})} \to {C^*({\mathcal G})} \to
{C^*({\mathcal G}_Y)} \to 0\quad \text{ and }\\
\label{eq.seq2'}
&& 0 \to {\mathfrak{A}({\mathcal G}_{M \setminus Y})} \to {\mathfrak{A}({\mathcal G})} \to
{\mathfrak{A}({\mathcal G}_Y)} \to 0. \end{aligned}$$ (We will not use that, but it is interesting to mentioned that it is known that there are no such exact sequence for reduced $C^*$-algebras, in general.)
All the morphisms of the above four exact sequences are compatible with the complex conjugation on these algebras.
\[def.filtration\] An $Y_0 \subset Y_1 \subset \dots \subset Y_n = M$ is an increasing sequence of closed, invariant subsets of $M$ with the property that the closure $\overline{S}$ of each connected component $S$ of $Y_k
\smallsetminus Y_{k-1}$ is a submanifold with corners of $M$ and that $\overline{S} \smallsetminus S$ is the union of the hyperfaces of $\overline{S}$ (that is, $S = \overline{S} \smallsetminus {\partial}S$).
The exact sequences defined before then give the following result:
\[Theorem.CS\] Let ${\mathcal G}$ be a differentiable groupoid with space of units $M$, and let $\emptyset=:Y_{-1}\subset
Y_0 \subset Y_1 \subset \dots \subset Y_n = M$ be an invariant filtration of $M$. Define ${\mathfrak I}_k := {C^*({\mathcal G}_{M \smallsetminus
Y_{k-1}})}$. Then we get a composition series of closed ideals $$(0) \subset {\mathfrak I}_n \subset {\mathfrak I}_{n-1} \subset
\ldots \subset {\mathfrak I}_0 = {C^*({\mathcal G})} \subset {\mathfrak{A}({\mathcal G})}\,,$$ whose subquotients are determined by $$\begin{aligned}
\sigma_{0}: {\mathfrak{A}({\mathcal G})} /{\mathfrak I}_0
&\overset{\cong}{{\longrightarrow}}&{\mathcal{C}_0}(S^*{\mathcal G})\,, \mbox{ and }\\
{\mathfrak I}_{k}/{\mathfrak I}_{k+1} &\cong&
{C^*({\mathcal G}_{Y_{k} \smallsetminus Y_{k-1}})} \quad
\mbox{ for } 0\leq k\leq n \,. \end{aligned}$$
A completely analogous result holds for the norm closure of the algebra ${\Psi^{0}({\mathcal G};E)}$, for any Hermitian vector bundle $E$. In fact, we can find an orthogonal projection $p_E \in M_N({\mathcal{C}^\infty}(M))$, for some large $N$, such that $E \cong p_E(M \times {\mathbb C}^N)$, and hence ${\Psi^{0}({\mathcal G};E)} \cong p_E M_N({\Psi^{0}({\mathcal G})}) p_E$.
The definition of an invariant filtration given in this paper is slightly more general than the one in [@LMN], however, these definitions are equivalent if each ${\mathcal G}_x$ is connected. Thus, in order to avoid some unnecessary technical complications, [*we shall assume from now on that all the fibers ${\mathcal G}_x := d^{-1}(x)$ of $d$ are connected*]{}. (Recall that a groupoid with this property is called [*$d$-connected*]{}.)
We observe then, that if $(Y_k)_k$ is an invariant filtration, then each connected component of $Y_k \smallsetminus Y_{k-1}$ is an invariant subset of $M$ and $${C^*({\mathcal G}_{Y_k \smallsetminus Y_{k-1}})} \cong
\bigoplus_S {C^*({\mathcal G}_S)}$$ where $S$ ranges through the set of open components of $Y_{k} \smallsetminus Y_{k-1}$. Moreover, a completely similar direct sum decomposition exists for ${\mathfrak{A}({\mathcal G}_{Y_k \smallsetminus Y_{k-1}})}$.
A first consequence of the above theorem is that if each ${\mathcal G}_S$ is an amenable groupoid (that is, ${C^*({\mathcal G}_S)} \cong {C^*_r({\mathcal G}_S)}$), then ${\mathcal G}$ is also amenable. This can be seen as follows. Using an argument based on induction, it is enough to prove that if a groupoid ${\mathcal G}$ has an open invariant subset $ \mathcal O $ such that both ${\mathcal G}_{\mathcal O}$ and ${\mathcal G}_{M \smallsetminus \mathcal O}$ are amenable, then ${\mathcal G}$ is amenable. To prove this, let $I$ be the kernel of the natural map ${C^*({\mathcal G})}\rightarrow{C^*_r({\mathcal G})}$ which is onto because its range is closed and contains the dense subspace ${\Psi^{-\infty}({\mathcal G})}$. Since ${\mathcal G}_{M\setminus \mathcal{O}}$ is amenable, $I$ is in the kernel of the restriction homomorphism ${C^*({\mathcal G})}\rightarrow{C^*({\mathcal G}_{M\setminus\mathcal{O}})}$, [[*i.e.,* ]{}]{}$I$ is a subset of ${C^*({\mathcal G}_{\mathcal{O}})}$. But the maps ${C^*({\mathcal G}_{\mathcal O})} \to {C^*_r({\mathcal G}_{\mathcal O})}$ and ${C^*_r({\mathcal G}_{\mathcal O})} \to {C^*_r({\mathcal G})}$ are both injective. Hence $I = 0$.
The above theorem leads to a characterization of compactness and Fredholmness for operators in ${\Psi^{0}({\mathcal G})}$. This characterization is similar, and it actually contains as a particular case, the characterization of Fredholm operators in the “$b$-calculus” or one of its variants on manifolds with corners, see [@mepi92]. Characterizations of compact and Fredholm operators on manifolds with singularities were, for instance, also obtained in [@defr; @man95; @mame99; @Melrose-Nistor1; @plam86; @plamsen94; @braun; @blau].
The significance of Theorem \[Theorem.CS\] is that often in practice we can find nice invariant stratifications $M = \bigcup S$ for which the subquotients ${C^*({\mathcal G}_S)}$ have a relatively simpler structure than that of ${C^*({\mathcal G})}$ itself. An example is the $b$-calculus and its generalizations, the $c_n$-calculi, which are discussed in Section \[Sec.Examples.III\].
In the following, we shall denote by $\otimes_{min}$ the [*minimal*]{} tensor product of $C^*$–algebras, defined using the tensor product of Hilbert spaces, see [@Sakai]. More precisely, assume that $A_i$, $i = 1,2$, are $C^*$-algebras, which we may assume to be closed subalgebras of the algebras of bounded operators on some Hilbert spaces ${\mathcal H}_i$. Then the algebraic tensor product $A_1
\otimes A_2$ acts on (the Hilbert space completion of) ${\mathcal H}_1
\otimes {\mathcal H}_2$, and $A_1 \otimes_{min} A_2$ is defined to be the completion of $A_1 \otimes A_2$ with respect to the induced norm. The following result is sometimes useful.
\[prop.tens\] If ${\mathcal G}_i$, $i = 0,1$, are two differential groupoids, then $${C^*_r({\mathcal G}_0 \times {\mathcal G}_1)} \simeq
{C^*_r({\mathcal G}_0)} \otimes_{min} {C^*_r({\mathcal G}_1)}.$$
Examples II: Pseudodifferential Operators \[Sec.Examples.II\]
=============================================================
The examples of differentiable groupoids of Section \[Sec.Examples.I\] also lead to interesting algebras of pseudodifferential operators. Many well-known algebras of pseudodifferential operators are in fact (isomorphic to) algebras of the form ${\Psi^{\infty}({\mathcal G})}$. This leads to new insight into the structure of these algebras. In additions to these well-known algebras, we also obtain algebras that are difficult to describe directly, without using groupoids. Moreover, some of these algebras were not considered before the groupoids were introduced into the picture, nevertheless, these algebras are expected to play an important role in the analysis on certain non-compact manifolds.
Our examples will follow in the beginning the same order as the examples considered in Section \[Sec.Examples.I\].
If ${\mathcal G}= M$ is a manifold (possibly with corners), then we have ${\Psi^{\infty}({\mathcal G})} \simeq
{{\mathcal C}^{\infty}_{\text{c}}}(M)$ and $\Psi^\infty_{prop}({\mathcal G}) = {\mathcal{C}^\infty}(M)$.
Denote by $\Psi^m_{\prop}(M)$ the space of [*properly supported*]{} pseudodifferential operators on a smooth manifold $M$.
If ${\mathcal G}=G$ is a Lie group, then ${\Psi^{m}({\mathcal G})} \simeq \Psi^m_{\prop}(G)^G$, the algebra of properly supported pseudodifferential operators on $G$, invariant with respect to right translations. In this example, every invariant properly supported operator is also uniformly supported.
The following example shows that the algebras of pseudodifferential operators (with appropriate support conditions for the Schwartz kernels) on a smooth manifolds without corners can be recovered as algebras of pseudodifferential operators on the pair groupoid. Let $\Psi_{{\operatorname{comp}}}^m(M)$ be the space of pseudodifferential operators on $M$ with compactly supported Schwartz kernels.
Suppose now that ${\mathcal G}= M \times M$, with $M$ a smooth manifold without corners, is the pair groupoid. Then ${\Psi^{m}({\mathcal G})} \cong
\Psi_{comp}^m(M)$ and $\Psi_{prop}^m({\mathcal G}) \cong \Psi_{prop}^m(M)$.
Moreover, the vector representation of ${\Psi^{\infty}({\mathcal G})}$ on ${{\mathcal C}^{\infty}_{\text{c}}}(M)$ recovers the usual action of pseudodifferential operators on functions on $M$. (Recall from , that the vector representation $\pi$ of ${\Psi^{\infty}({\mathcal G})}$ is given by $(\pi(P)f) \circ r
= P (f \circ r)$.)
The fibered pair recovers families of operators.
If ${\mathcal G}= M \times_B M$ is the fibered pair groupoid, for some submersion $M \to B$, then ${\Psi^{m}({\mathcal G})}$ consists of families of pseudodifferential operators along the fibers of $M \to B$ such that their reduced kernels are compactly supported (as distributions on ${\mathcal G}$).
The vector representation $\pi$ of ${\Psi^{\infty}({\mathcal G})}$ on ${{\mathcal C}^{\infty}_{\text{c}}}(M)$ is just the usual action of families of pseudodifferential operators on functions, the action being defined fiberwise.
The following three examples of algebras were probably considered only in the framework of groupoid algebras, although particular cases have been investigated before.
\[prgp\] For a product groupoid ${\mathcal G}= {\mathcal G}_1 \times {\mathcal G}_2$ there is no obvious description of $\Psi^\infty({\mathcal G}_1 \times {\mathcal G}_2)$ in terms of $\Psi^\infty({\mathcal G}_1)$ and $\Psi^\infty({\mathcal G}_2)$, in general. However, when ${\mathcal G}_1 = M_1$ is a manifold with corners (so ${\mathcal G}_1$ has no non-trivial arrows), then $\Psi^m({\mathcal G}_1 \times {\mathcal G}_2)$ consists of families of operators in $\Psi^m({\mathcal G}_2)$ parameterized by $M_1$. For smoothing operators the situation is simpler: $\Psi^{-\infty}({\mathcal G}_1 \times {\mathcal G}_2)$ contains naturally the tensor product $\Psi^{-\infty}({\mathcal G}_1) \otimes \Psi^{-\infty}({\mathcal G}_2)$ as a dense subset.
An interesting particular case is when ${\mathcal G}_1 = M \times M$, the pair groupoid, and ${\mathcal G}_2 = {\mathbb R}^q$ (that is, the groupoid associated to the Lie group ${\mathbb R}^q$), then $\Psi^m({\mathcal G}_1 \times {\mathcal G}_2)$ can be identified with a natural, dense subalgebra of the algebra of $q$-suspended pseudodifferential operators “on” $M$, introduced by Melrose.
\[ex16\] If ${\mathcal G}\to B$ is a bundle of Lie groups, then ${\Psi^{m}({\mathcal G})}$ consists of smooth families of invariant, properly supported, pseudodifferential operators on the fibers of ${\mathcal G}\to B$. For ${\mathcal G}= B \times G$, a trivial bundle of Lie groups, ${\mathcal G}$ is the product (as groupoids) of a smooth manifold $B$, as in Example \[ex.manifold\], and a Lie group $G$, as in Example \[ex.Lie\]. A very important particular case of this construction is when ${\mathcal G}\to B$ is a vector bundle, with the induced fiberwise operations. We shall use this example below several times.
Again, the only general thing that can be said about fibered products is that $\Psi^{-\infty}({\mathcal G}_1) \otimes_{{\mathcal{C}^\infty}(B)}
\Psi^{-\infty}({\mathcal G}_2)$ identifies with a dense subset of $\Psi^{-\infty}({\mathcal G}_1 \times_B {\mathcal G}_2)$. When ${\mathcal G}_1 = M \times_B M$ is a fibered pair groupoid and ${\mathcal G}_2$ is a bundle of Lie groups on $B$, then $\Psi^m({\mathcal G}_1 \times {\mathcal G}_2)$ is an algebra considered in [@NistorIndFam], and consists of smooth families of pseudodifferential operators on $M \times_B {\mathcal G}_2$ invariant with respect to the bundle of Lie groups ${\mathcal G}_2$.
If ${\mathcal G}$ is the holonomy groupoid associated to the foliated manifold $(M,F)$, then $\Psi^*({\mathcal G})$ is the algebra of pseudodifferential operators along the leaves of $(M,F)$, considered first by Connes [@ConnesF]. In fact, our algebra is a little smaller than Connes’ who considered families that are only continuous in the transverse direction. These algebras, however, have the same formal properties as our families.
Let ${\mathcal G}$ be the fundamental groupoid of a compact smooth manifold $M$ with fundamental group $\pi_1(M)=\Gamma$. If $P=(P_x)_{x \in M} \in {\Psi^{m}({\mathcal G})}$, then each $P_x$, $x \in M$, is a pseudodifferential operator on $\widetilde{M}$. The invariance condition applied to the elements $g$ such that $x=d(g)=r(g)$ implies that each operator $P_x$ is invariant with respect to the action of $\Gamma$. This means that we can identify $P_x$ with an operator on $\widetilde M$ and that the resulting operator does not depend on the identification of ${\mathcal G}_x$ with $\widetilde M$. Then the invariance condition applied to an arbitrary arrow $g \in {\mathcal G}$ gives that all operators $P_x$ acting on $\widetilde M$ coincide. We obtain ${\Psi^{m}({\mathcal G})} \simeq
\Psi^m_{\prop}(\widetilde{M})^\Gamma$, the algebra of properly supported $\Gamma$-invariant pseudodifferential operators on the universal covering $\widetilde{M}$ of $M$. An alternative definition of this algebra using crossed products is given in [@Nistor4].
\[exad\] If ${\mathcal G}_{{\operatorname{ad}}}$ is the adiabatic groupoid associated to a groupoid ${\mathcal G}$, then an operator $P \in \Psi^m({\mathcal G}_{{\operatorname{ad}}})$ consists of a family $P=(P_{t,x})$, $t \ge 0$, $x \in M$, ($M$ is the space of units of ${\mathcal G}$), such that if we denote by $P_t$ the family $(P_{t,x})$, for a fixed $t$, then $P_t \in {\Psi^{m}({\mathcal G})}$ for $t > 0$ and $P_{t}$ depends smoothly on $t$ in this range. For $t = 0$, $P_0\in
\Psi^{m}(A({\mathcal G}))$, is a family of operators on the fibers of $A({\mathcal G})
\to M$, translation invariant with respect to the variable in each fiber. Thus, $\Psi^m(A({\mathcal G}))$ is one of the algebras appearing in Example \[ex16\].
In a certain sense $P_t \to P_0$, as $t \to 0$, but this is difficult to make precise without considering the adiabatic groupoid. (Actually, making precise the fact that the family $P_t$ is smooth at $0$ also is precisely the [*raison d’être*]{} for the algebra of pseudodifferential operators on the adiabatic groupoid.)
The best way to formalize this continuity property is the following. Consider the evaluation morphisms $e_t : \Psi^m({\mathcal G}_{{\operatorname{ad}}}) \to
\Psi^m({\mathcal G})$, if $t >0$, and $e_0 : \Psi^m({\mathcal G}_{{\operatorname{ad}}}) \to
\Psi^m(A({\mathcal G}))$. (These morphisms are particular instances of the restriction morphisms defined in Equation ). If $P \in \Psi^0({\mathcal G}_{{\operatorname{ad}}})$, then $\| e_t(P) \|$ and $\|e_t(P)\|_r$ are continuous in $t$. This was proved by Landsman and Ramazan, see [@Landsman; @LandsmanRamazan; @Ramazan].
Some typical operators in $\Psi^m({\mathcal G}_{{\operatorname{ad}}})$ are obtained by rescaling the symbol of a differential operator $D$ on ${\mathcal G}$. To see how this works, note first that there exists a polynomial symbol $a$ on $A^*({\mathcal G})$ such that $q(a) = D$, where $q : {\mathcal S}^m(A^*({\mathcal G}))
\to {\Psi^{m}({\mathcal G})}$ is the quantization map considered in [@NWX]. Let $a_t$ be the symbol $a_t(\xi) = a(t\xi)$, for $t > 0$, and also let $q_{{\operatorname{ad}}} : {\mathcal S}^m(A^*({\mathcal G}_{adb})) \to \Psi^m({\mathcal G}_{{\operatorname{ad}}})$ be the quantization map for the adiabatic groupoid. We can extend $a$ to a symbol on $A^*({\mathcal G}_{{\operatorname{ad}}})$ constant in $t$, then $e_t(q_{{\operatorname{ad}}}(a))
= q(a_t)$, if $t > 0$. For $t = 0$, we obtain that $e_0(q_{{\operatorname{ad}}}(a))$ is isomorphic to the operator of multiplication by $a$, after taking the Fourier transform along the fibers of $A^*({\mathcal G})$.
An important class of examples is obtained by integrating suitable Lie algebras of vector fields on a manifold $M$ with corners. This is related to Melrose’s approach to a pseudodifferential analysis on manifolds with corners [@MelroseScattering], though our techniques are different in the end. We thus start with a Lie subalgebra ${\mathcal{V}}$ of the Lie algebra of all vector fields that are tangent to each boundary hyperface of a given manifold $M$ with corners. The Lie algebra ${\mathcal{V}}$ can be thought of as determining the degeneracies of our operators near the boundary. If ${\mathcal{V}}$ is in addition a projective ${\mathcal{C}^\infty}(M)$-module, then, by the Serre-Swan theorem, there is a smooth vector bundle $A={}^{{\mathcal{V}}}TM\rightarrow M$ together with a smooth map of vector bundles $q:A\longrightarrow TM$ such that ${\mathcal{V}}=q(\Gamma(A))$. (This will be discussed in more detail in a forthcoming book of Melrose on manifolds with corners.)
The next step is to integrate this Lie algebroid $A$, that is, to find a Lie groupoid ${\mathcal G}$ with Lie algebroid $A$. Here, we can follow the general method used in [@NistorInt]. The integration procedure consists in fact of two steps. Let us denote by $A_{S}$ the restriction of $A$ to each open boundary face $S$ of $M$ of positive codimension, suppose that we can find differentiable groupoids ${\mathcal G}_S$ integrating $A_S$, and let ${\mathcal G}=\bigcup{\mathcal G}_S$. By [@NistorInt], there exists at most one smooth structure on ${\mathcal G}$ compatible with the groupoid operations. Whenever such a smooth structure exists, the resulting groupoid satisfies $A({\mathcal G}) = A$. Moreover, if the ${\mathcal G}_S$ are maximal among all $d$-connected groupoids integrating $A_S$, then there is a natural differentiable structure on ${\mathcal G}$ making it into a differentiable groupoid with Lie algebroid $A$. Note that this choice for ${\mathcal G}$ will almost always lead us to non-Hausdorff groupoids and to problems related to the analysis on these spaces. Moreover, the vector representation will not be injective, in general. The reason is that the maximal $d$-connected groupoid integrating a given Lie algebroid is much to big. For instance, for the Lie algebroid $TM \to M$, the maximal $d$-connected groupoid integrating it is the path groupoid [@NWX], not the pair groupoid as expected and usually desired [@Mackenzie1]. In particular cases, however, the given Lie algebroid $A$ can be integrated directly to a Hausdorff differentiable groupoid. These remarks apply to the following two examples. These two examples are essentially due to Melrose [@MelroseScattering] and, respectively, to Mazzeo [@maz91]. A groupoid for a special case of Example \[excn\] (b-calculus on manifolds with corners) was constructed in [@Monthubert], see also [@NWX].
\[excn\] [*The “very small” $c_n$-calculus.*]{} Let $M$ be a compact manifold with corners, and associate to each hypersurface $H \subset M$ an integer $c_H \ge 1$. We also fix a defining function for each hypersurface. Choose also on $M$ a metric $h$ such that each point $p \in F$, belonging to the interior of a face $F\subseteq M$ of codimension $k$, has a neighborhood $V_p \cong
V'_p \times [0,\varepsilon)^k$, with the following two properties: the defining function $x_j$ is obtained as the projection onto the $j$th component of $[0,\varepsilon)^k$ and the metric $h$ can be written as $h = h_F + (dx_1)^2 + \ldots (dx_k)^2$, with $x_1, \ldots,
x_k$ being the defining functions of $F$ and $h_F$ being a two-tensor that does not depend on $x_1,\ldots, x_k$ and restricts to a metric on $F$.
Then, we consider on $M$ the vector fields $X$ that in a neighborhood of each point $p$, as above, are of the form $$X = X_F + \sum_{j=1}^k x_j^{c_j} {\partial}_{x_j},$$ with $c_j$ being the integer associated to the hyperface $\{x_j =
0\}$ and $X_F$ being the lift of a vector field on $F$. The set of all vector fields with these properties forms a Lie subalgebra of the algebra of all vector fields on $M$. We denote this subalgebra by ${\mathcal A}(M,c)$. By the Serre-Swan theorem, there exists a vector bundle $A(M,c)$ such that $\mathcal A(M,c)$ identifies with the space of smooth sections of $A(M,c)$.
We want to integrate $A(M,c)$, and to this end, we shall use the approach from [@NistorInt]. Let $S = int(F)$ be the interior of a face $F \subset M$ of codimension $k$. The restriction of $A(M,c)$ to each open face $S$ is then $TS \times {\mathbb R}^k$, and hence it is integrable; a groupoid integrating this restriction being, for example ${\mathcal G}_S = S \times S \times {\mathbb R}^k$, if $F = \overline{S}$ has codimension $k$.
Define then $${\mathcal G}:= \bigcup_F S \times S \times {\mathbb R}^k,$$ which is a groupoid with the obvious induced structural maps. As a set, ${\mathcal G}$ does not depend on $c$. Because the groupoids ${\mathcal G}_S$ are not $d$-connected, in general, we cannot use the result of [@NistorInt] to prove that it has a natural smooth structure, so we have to construct this smooth structure directly.
The results of [@NistorInt] say that if there exists a smooth structure on ${\mathcal G}$ compatible with its groupoid structure, then it must be obtained using certain coordinate charts defined using the exponential map. In our case, the exponential map amounts to the following.
Let $\psi_l : (0,\infty) \to {\mathbb R}$ be $\psi_l(x) = \ln x$, if $l = 1$ and $\psi_l(x) = x + x^{1-l}/(1-l)$, if $l > 1$. Also, let $\phi_l :
{\mathbb R}\times [0,\infty) \to [0,\infty)$ be defined by $\phi_l(t,0) = 0$ and $\phi_l(t,x) = \psi_l^{-1}( \psi_l(x) + t)$. In particular, $\phi_1(t,x) = e^tx$. Then $\phi_l$ defines a differentiable action of ${\mathbb R}$ on $[0,\infty)$, which hence makes ${\mathbb R}\times [0,\infty)$ a differentiable groupoid denoted ${\mathcal F}_l$. The Lie algebroid of ${\mathcal F}_l$ is generated as a ${\mathcal{C}^\infty}([0,\infty))$-projective module by the infinitesimal generator ${\partial}_t$ of the action of ${\mathbb R}$; note that the action of $\partial_{t}$ on ${\mathcal{C}^\infty}([0,\infty))$ under the anchor map is given by $f(x)x^{l}\partial_{x}$ for some nowhere vanishing (bounded) smooth function $f$. Consequently, $A({\mathcal
F}_l)$ is the projective ${\mathcal{C}^\infty}([0,\infty))$-module generated by $x^l
{\partial}_x$.
Assume now that $M = [0,\infty)$ and fix $l\in {\mathbb N}$. Then ${\mathcal
F}_l$ is a smooth groupoid integrating $A(M,l)$, by the above remarks. Consequently, if $M = [0,\infty)^n$ and $c=(c_1,c_2,\ldots,c_n)$, then ${\mathcal G}:= {\mathcal F}_{c_1} \times {\mathcal F}_{c_2}\times \ldots
\times {\mathcal F}_{c_n}$ satisfies $A({\mathcal G}) = A(M,c)$. To integrate general Lie algebroids of the form $A(M,c)$ we localize this construction. This then gives the following smooth structure on ${\mathcal G}:= \bigcup_S {\mathcal G}_S$.
We now discuss the general case of a manifold with corners. Locally, the smooth structure on ${\mathcal G}$ is given by the discussion above. Since this smooth structure is important in applications, let us try to make it more explicit. Thus, fix an arbitrary point $(p,q,\xi) \in S
\times S \times {\mathbb R}^k$, which we want to include in a coordinate system. By definition, $p,q \in S$. Choose now a small coordinate neighborhood $V_p \cong V'_p \times [0,\varepsilon)^k$ of $p$, with $V'_p$ a small open neighborhood of $p \in S$, as above. Choose $V_q\cong V'_q \times [0,\varepsilon)^k$ similarly. We write $$z = (z', x_1(z), x_2(z), \ldots, x_k(z))$$ for any $z \in V_p \cap V_q$; this is possible since we can assume that $V_p = V_q$ if $p = q$ or that $V_p \cap V_q = \emptyset$ if $p
\not = q$. Fix $R > 2\|\xi\|$ and choose $\delta>0$ so small that $|\phi_l(t, x)| < \varepsilon$ if $|t | \le R$, $x \le \delta$, and $l
= c_j$, for $j = 1, 2, \ldots, k$. Here $c_j$ is the constant associated to the hyperface $\{x_j = 0\}$. Then we define a map $$\begin{aligned}
F : V'_p \times [0,\delta)^k \times V'_q \times
\{ \| \xi \| < R \} &\longrightarrow& {\mathcal G}: \\ \nonumber
(z', y, z'', \xi) &\longmapsto&
(z', y, z'', \Phi(\xi,y), p_y(\xi)) \in {\mathcal G}_{S'} \end{aligned}$$ as follows. Let $y = (y_1, \ldots, y_k)$, $B \subset \{ 1, 2, \ldots,
k \}$ be the subset of those indices $j$ such that $y_j = 0$, and let $p_y : {\mathbb R}^k \to {\mathbb R}^B$ be the corresponding projection. The vector space ${\mathbb R}^B$ identifies naturally with the fiber at $(z',y)$ of the normal bundle to the open face containing $(z',y)$ (this open face was denoted above by $S'=S'(z',y)$). For $y = (y_1, y_2, \ldots, y_k)$ and $\xi = (\xi_1, \xi_2, \ldots, \xi_k)$, the map $\Phi$ is then given by $$\Phi(\xi, y) = ( \phi_{c_1}(\xi_1, y_1), \phi_{c_2}(\xi_2, y_2),
\ldots, \phi_{c_k}(\xi_k, y_k))\,.$$
We shall denote by ${\mathcal G}(M,c)$ the smooth groupoid constructed above.
Fix a face $F \subset M$ of codimension $k$. By construction, $F$ is an invariant subset of $M$ and hence we can consider the restriction maps $\inn_F$ defined in Equation . The range of these restriction (or indicial) maps is in related to the algebras $\Psi^\infty({\mathcal G}(M,c))$. The precise relation is the following.
Each hyperface $H'$ of $F$ is a connected component of $H \cap F$, for a unique hyperface $H$ of $M$. Then we associate to $H'$ the integer $c_H \ge 1$. We denote by $c'$ the collection of integers obtained in this way. Then the restriction of $A(M,c)$ to $F$ is isomorphic to $A(F,c') \times {\mathbb R}^k$. From this we obtain that ${\mathcal G}(M,c)\vert_{F}
\cong {\mathcal G}(F,c') \times {\mathbb R}^k$. The restriction maps thus become $$\inn_F : \Psi^m({\mathcal G}(M,c)) \to \Psi^m({\mathcal G}(F,c') \times {\mathbb R}^k).$$ The right hand side algebras are closely related to the “$k$-fold suspended algebras” of Melrose.
The analytic properties of the algebras $\Psi^\infty({\mathcal G}(M,c))$ will be studied again in Section \[Sec.Examples.III\].
\[ex.boundary.f1\] Let $M$ be a compact manifold whose boundary $\partial M$ is the total space of a locally trivial fibration $p:\partial M \longrightarrow B$ of compact smooth manifolds. A smooth vector field on $M$ is called an [*edge vector field*]{} if it is tangent to the fibers of $p$ at the boundary. The Lie algebra ${\mathcal{V}_{e}(M)}$ of all edge vector fields is a projective ${\mathcal{C}^\infty}(M)$-module, and hence, by the Serre-Swan theorem [@Karoubi], it can be identified with the space of all ${\mathcal{C}^\infty}$ sections of a smooth vector bundle ${{}^{e}TM}\rightarrow M$ that comes equipped with a natural map ${{}^{e}TM}\longrightarrow TM$ [@maz91] making $A:={{}^{e}TM}$ into a Lie algebroid. A pseudodifferential calculus adapted to this setting was constructed by Mazzeo [@maz91], and, in a slightly different way by Schulze [@braun]. To integrate ${{}^{e}TM}$, we shall use the methods of [@NistorInt].
Let $M_0 := M \smallsetminus {\partial}M$ and notice that $A\vert_{M_0}
\cong TM_0$. We can integrate this restriction to the pair groupoid: ${\mathcal G}_{M_0} := M_0 \times M_0$. The restriction of $A$ to the boundary is the crossed product of another Lie algebroid with ${\mathbb R}$: $$A\vert_{{\partial}M} \cong (T_{vert}{\partial}M \times TB) \rtimes {\mathbb R}.$$ It is worthwhile do describe this restriction more precisely. As a vector bundle, $A$ is the direct sum of three vector bundles: $T_{vert}{\partial}M$ (the vertical tangent bundle to the fibers of ${\partial}M \to B$), $p^*(TB)$ (the pull-back of the tangent bundle of $B$), and a trivial, one-dimensional real vector bundle. Thus, every section of $A$ can be represented as a triple $(X,Y,f)$, where $X$ is a vector field on ${\partial}M$, tangent to the fibers of ${\partial}M \to B$, $Y$ is a section of $p^*(TB)$, which is convenient to be represented as a section of the quotient $T {\partial}M /T_{vert} {\partial}M$, and $f \in {\mathcal{C}^\infty}({\partial}M)$. Let $\nabla$ be the Bott connection on $p^*(TB)$. The Lie algebra structure on $\Gamma(A)$ is then $$[(X,Y,f), (X_1,Y_1,f_1)] = ([X,X_1], \nabla_X(Y_1) + fY_1
- \nabla_{X_1}(Y) - f_1Y, 0).$$
Let $G \to B$ be the bundle of Lie groups obtained as the cross-product of the bundle of commutative Lie groups $TB$ with ${\mathbb R}$, the action of $t \in {\mathbb R}$ being as multiplication with $e^t$. The Lie algebroid (or the bundle of Lie groups associated to this bundle of Lie groups) is $A(G) = TB \oplus {\mathbb R}$, with the bracked defined as above: $[(Y,f), (Y_1,f_1)] = (fY_1 - f_1Y, 0)$. Then we can write $A\vert_{{\partial}M} = T_{vert}{\partial}M \times_B A(G)$. This writing immediately leads to a groupoid integrating $A\vert_{{\partial}M}$, namely the fibered product of a groupoid integrating $T_{vert}{\partial}M$ and a groupoid integrating $A(G)$. We can choose these groupoids to be the fibered pair groupoid ${\mathcal G}_1 : = {\partial}M \times_B {\partial}M$ and, respectively, $G$. The resulting groupoid integrating $A\vert_{{\partial}M}$ is then ${\mathcal G}_{{\partial}M} := {\mathcal G}_1 \times_B G \to B$, invariant with respect to the action of $G$ by right translations. The resulting algebra of pseudodifferential operators will be an algebra of smooth families acting on the fibers of ${\partial}M \times_B G$, invariant with respect to $G$.
To obtain a groupoid integrating $A$, it is enough to show that the disjoint union ${\mathcal G}:= {\mathcal G}_1 \cup (M_0 \times M_0)$ has a smooth structure compatible with the groupoid structure. This smooth structure is obtained using the following coordinate charts. Let $x$ be a boundary defining function on $M$, and fix $q \in {\partial}M$ and a neighborhood $V_q \cong V'_q \times [0,\varepsilon)$ such that the defining functions $x$ becomes the second projection on $V_q$ and $V'_q$ is a neighborhood of $q$ in ${\partial}M$. We replace $V_q$ with a smaller neighborhood, if necessary, so that there exists a fiber preserving diffeomorphism $\phi : B_1 \times B_2 \to V'_q$, $\phi(0,0)
= q$, from a product of two small open balls in some Euclidean spaces (so $p$ becomes the first projection with respect to the diffeomorphism $\phi$). Let $q'\in {\partial}M$ be a second point, chosen such that $p(q') = p (q)$, and choose a diffeomorphism $\phi' : B_1
\times B'_2 \to V_{q'}$ as above. We can assume that $p \circ \phi =
p \circ \phi'$ and $\phi = \phi'$, if $q = q'$, or that $V_q$ and $V_{q'}$ are disjoint. Then $\phi$ and $\phi'$ define a diffeomorphism $$\phi \times_B \phi' : B_1 \times B_2 \times B'_2 \to
{\partial}M \times_B {\partial}M,$$ explicitly, $$\phi \times_B \phi'(b_1, b_2, b'_2) = (\phi(b_1,b_2),
\phi'(b_1,b'_2)) \in {\partial}M \times_B {\partial}M \subset {\partial}M \times
{\partial}M.$$
We identify $B_1$ with the fiber of $TB$ at $p(q) = p(q')$ such that $p(q)$ corresponds to $0$, and we let $$\Phi : B_1 \times B_2 \times [0,\delta) \times B_1 \times B_2 \times (-R,R)
\to {\mathcal G}$$ be given by $$\Phi(b_1,b_2, 0, b'_1, b'_2, t) = \big( \phi \times_B
\phi'(b_1, b_2, b'_2), b'_1,t) \in {\partial}M \times_B {\partial}M \times
T_{p(q)}B \times {\mathbb R}\in {\mathcal G}_1$$ or by $$\begin{aligned}
\Phi(b_1, b_2, s, b'_1, b'_2, t) &=& (\phi(b_1,b_2), s,
\phi'(b_1 + sb'_1,b'_2), se^t) \\ &\in& V'_{q} \times
(0,\varepsilon) \times V'_{q'} \times (0,\varepsilon) \subset
M_0 \times M_{0}. \end{aligned}$$
The restriction at the boundary map $\inn_{{\partial}M}$ defined in Equation becomes a map $$\inn_{{\partial}M} : {\Psi^{\infty}({\mathcal G})} \to \Psi^\infty({\mathcal G}_1).$$ The range of this map consists of families of pseudodifferential operators that act on the fibers of ${\partial}M \times_B G \to B$ and are $G$-invariant with respect to the action of $G$ by right translations.
We expect that the above example will be useful for the question from [@FreedWitten] on the Bojarsky additivity formula for the real index of families of elliptic operators. (See also Nicolaescu’s paper [@Nicolaescu].) Also, it will probably be useful for a certain approach in the study of the $S^1$-equivariant Dirac operators [@NistorDIR] and [@Lott].
Geometric operators\[Sec.GO\]
=============================
For two vector bundles $E_{0}, E_{1}$ on $M$, we shall denote by ${{\rm Diff}({\mathcal G};E_0,E_1)}$ the space of differential operators $D :
\Gamma({\mathcal G};r^{*}E_0) \to \Gamma({\mathcal G};r^{*}E_1)$ with smooth coefficients that differentiate only along the fibers of $d: {\mathcal G}\to
M$, and which are right invariant. Thus, ${{\rm Diff}({\mathcal G};E_0,E_1)}$ is exactly the space of differential operators in $\Psi^m({\mathcal G};E_0,E_1)$. The elements of ${{\rm Diff}({\mathcal G};E_0,E_1)}$ will be called [*differential operators on ${\mathcal G}$*]{}.
In this section, we define and study the geometric differential operators on a given differentiable groupoid ${\mathcal G}$. For the definition of most of these operators, we shall need a metric on $A:=A({\mathcal G})$.
To define the de Rham operator, however, we need no metric. Denote then by $\lambda_q=\Lambda^q T^*_{vert} {\mathcal G}$ the $q$th exterior power of the dual of $T_{vert}{\mathcal G}$, the vertical tangent bundle to the fibers of $d : {\mathcal G}\to M$. Recall that ${\mathcal G}_x$ denotes $d^{-1}(x)$ throughout this paper. Then the de Rham differential $d :
\Gamma({\mathcal G}_x,\lambda_q) \to \Gamma({\mathcal G}_x,\lambda_{q+1})$ is invariant with respect to right translations, and hence it defines an operator $d \in \bigoplus_q {{\rm Diff}({\mathcal G};\Lambda^qA^*, \Lambda^{q+1}A^*)} \subset
{{\rm Diff}({\mathcal G}; \Lambda^*A^*)}$.
Let as before $\pi$ be the representation $\pi : \Psi^m({\mathcal G};E_0,E_1)
\to {\operatorname{Hom}}(\Gamma(E_0),\Gamma(E_1))$ given by the formula $\pi(P)(f)
\circ r = P(f \circ r)$. (Recall that we called this representation the [*vector representation*]{}.) The complex determined by the operators $\pi(d)$ then computes the [*Lie algebroid cohomology*]{} of $A$, by definition, see [@Mackenzie1].
We shall use later on the explicit form of $d$ for the adiabatic groupoid associated to ${\mathcal G}$. Recall that an operator $P$ (differential or pseudodifferential) on the adiabatic groupoid ${\mathcal G}_{{\operatorname{ad}}}$ associated to ${\mathcal G}$ consists of a family $P=(P_t)$, such that, in particular, $P_t \in {\Psi^{m}({\mathcal G})}$, for $t >0$. The de Rham operator $d^{{\mathcal G}_{{\operatorname{ad}}}}=(d_t)$ is such that $d_t = td$, for $t > 0$.
Of course, more operators are obtained if we consider a metric on $A:=A({\mathcal G})$. Here, by “metric on $A$” we mean a positive definite bilinear form on $A$, as usual. The metric on $A$ then makes each ${\mathcal G}_x$ a Riemannian manifold, naturally, due to the isomorphisms $T{\mathcal G}_x \cong r^*(A)$ as vector bundles on ${\mathcal G}_x$. Moreover, right translation by an element of ${\mathcal G}$ is an isometric isomorphism. Because of this, every geometric differential operator associated naturally to a Riemannian metric will be invariant with respect to the right translation by an element of ${\mathcal G}$, and hence will define an element in ${{\rm Diff}({\mathcal G};E_0,E_1)}$, for suitable vector bundles $E_i$. We shall not try to formulate this in the greatest generality, but we shall apply this observation to particular operators that appear more often in practice.
For example, the metric allows us to define the Hodge $*$-operator, which then leads to the signature operator $d \pm *d* \in
{{\rm Diff}({\mathcal G},\Lambda^*A)}$. Also, the metric gives rise to an inner product on $\Lambda^*A$ and hence to an adjoint to $d$, denoted $d^*$, which then in turn allows us to define the Euler operator $d + d^* \in
{{\rm Diff}({\mathcal G};\Lambda^*A^*)}$. Similarly, one obtains the Hodge Laplacians $\Delta_p \in {{\rm Diff}({\mathcal G};\Lambda^p A^*)}$, as components of the square of the Euler operator $d + d^*$. We write $\Delta_p^{\mathcal G}$, $d^{\mathcal G}$, ... for these operators when we want to stress their dependence on ${\mathcal G}$.
We now turn to Dirac and generalized Dirac operators. This requires us to introduce the (generalization to groupoids of the) Levi-Civita connection.
For $X \in \Gamma(A)$, we shall denote by $\tilde X$ its lift to a right invariant, $d$-vertical vector field on ${\mathcal G}$. Let $\nabla^x :
\Gamma(T_{vert}{\mathcal G}_x) \to \Gamma(T_{vert}{\mathcal G}_x \otimes
T_{vert}^*{\mathcal G}_x)$ be the Levi-Civita connection associated to the induced metric on ${\mathcal G}_x$. Then for any $X \in \Gamma(A)$, we obtain a smooth, right invariant family of differential operators $$\nabla^x_{\tilde{X}} : \Gamma(T_{vert}{\mathcal G}_x) \to
\Gamma(T_{vert}{\mathcal G}_x).$$ We denote the induced differential operator in ${{\rm Diff}({\mathcal G}, A)}$ simply by $\nabla_X$. For all smooth sections $X$ and $Y$ of $A$, there exists another smooth section $Z$ of $A$ such that $\nabla_X (\tilde
Y) = \tilde Z$.
Suppose now that $A$ is $spin$, that is, that $A$ is orientable and the bundle of orientable frames of $A$ lifts to a principal $Spin(k)$ bundle ($k$ being the rank of $A$). Suppose $k=2l$ is even, for simplicity, and let $S = S_+ \oplus S_-$ be the spin bundle associated to the given spin structure and the spin representation of $Spin(k)$. As in the classical case, the Levi-Civita connection on the frame bundle of $r^*(A)$ lifts to a connection $\nabla^S$ on $S$. Moreover, this connection involves no choices (it is uniquely determined by the spin structure), and hence $\nabla^S$ is right invariant, in the obvious sense. Thus, if $X$ is a section of $A$ and $\tilde X$ is its lift to a right invariant, $d$-vertical vector field on ${\mathcal G}$, then $\nabla^S_{\tilde X}$ is a right invariant differential operator, and hence it is in ${{\rm Diff}({\mathcal G}, S)}$. We denote by ${{\not \!\!D}}^S$ the induced Dirac operator on the spaces ${\mathcal G}_x$, which will then form a right invariant family, and hence ${{\not \!\!D}}^S \in {{\rm Diff}({\mathcal G};S)}$. (We shall write ${{\not \!\!D}}^S_{{\mathcal G}}$ on the few occasions when we shall need to point out the dependence of this operator on the groupoid ${\mathcal G}$.)
Let ${{\rm Cliff}}(A)$ be the bundle of Clifford algebras associated to $A$ and its metric. We shall use the metric to identify $A^*$ with $A$, so that ${{\rm Cliff}}(A^*)$ becomes identified with ${{\rm Cliff}}(A)$. The same construction as above then applies to a ${{\rm Cliff}}(A)$-module $W$ endowed with a right invariant, admissible connection $\nabla^W$ (see below) on each of its restrictions to ${\mathcal G}_x$. Denote by $c : {{\rm Cliff}}(A) \to
End(W)$ the Clifford module structure on $W$. Because $A \subset
{{\rm Cliff}}(A)$, we also obtain a bundle morphism $A \to End(W)$ still denoted $c$. Recall then that $\nabla^W$ is an [*admissible connection*]{} if, and only if, $$\nabla_X^W(c(Y)\xi) = c(\nabla_XY)\xi + c(Y)\nabla^W_X(\xi),$$ for all $\xi \in \Gamma(r^*(W))$ and all $X,Y \in \Gamma(r^*(A))$, the second connection being the Levi-Civita connection discussed above. Then we obtain as in the classical case a Dirac operator ${{\not \!\!D}}^W_x$ on ${\mathcal G}_x$, acting on sections of $r^*(W)$. The right invariance of the connection $\nabla^W$ guarantees that the family ${{\not \!\!D}}^W_x$ is right invariant, and hence that it defines an element in ${{\rm Diff}({\mathcal G};W)}$.
It is a little bit trickier to define the generalized Dirac operator associated to a ${{\rm Cliff}}(A)$-module $W$, if no admissible connection is specified on $W$. This is because it is not clear a priori that right invariant admissible connections exist at all. Our next goal then is to prove that this is always the case, as it is for Clifford modules on Riemannian manifolds.
We shall work with complex ${{\rm Cliff}}(A)$-modules, for simplicity. Also, we assume that $A$ is even dimensional. Cover $M$ with contractible open sets $U_{\alpha}$. Then $A\vert_{U_\alpha}$ has a trivialization $A\vert_{U_\alpha} \simeq U_{\alpha} \times {\mathbb R}^{2l}$, which we can assume to preserve the metric. Then ${{\rm Cliff}}(A)\vert_{U_{\alpha}}
\simeq U_{\alpha} \times M_{2^l}({\mathbb C})$ and $W\vert_{U_\alpha} \simeq
U_{\alpha} \times V \simeq {\mathbb C}^{2^l} \otimes V_0$, with $V_0$ an additional vector bundle, which is acted upon trivially by the Clifford algebra, and hence only serves to encode the local “multiplicity” of the ${{\rm Cliff}}(A)$-module $W$. As in the classical case, we first define the admissible connection locally, using the above trivialization, and then we glue them using a partition of unity. However, in our groupoid setting we need to work a little bit more to make sense of what the “local definition” means. More precisely, all definitions will be given not on $U_\alpha$ itself, but on $r^{-1}(U_\alpha)$. Once we realize this, everything carries over from the case of a Riemannian manifold to that of a differentiable groupoid. For completeness, we now review this construction in our case.
The trivialization of $U_\alpha$ gives an orthonormal family of sections $X_1, X_2,\ldots,X_k$ of $A$ over $U_\alpha$. Then, we obtain smooth functions $\Gamma^a_{bc}$ on $U_\alpha$ such that, working [ *always*]{} over $r^{-1}(U_\alpha)$, $$\nabla_{\tilde X_i} \tilde X_j = \sum_h
(\Gamma^h_{ij} \circ r) \tilde X_h.$$ (Compare with [@LM].) Fix a basis $(e_t)$, $t = 1, \ldots, 2^lm$, of $V$, where $V$ is the vector space appearing in the isomorphism $W\vert_{U_\alpha} \simeq U_\alpha \times V$. We shall denote by $\tilde e_t := e_t \circ r$ the induced basis of $r^*(W)$ on $r^{-1}(U_\alpha)$. The point of these choices is, of course, that the matrix of the multiplication operator $c(\tilde X_j)$ in the basis $\tilde e_t$ consists of [*constant*]{} functions. Using the functions $\Gamma^h_{ij}$ and the Clifford multiplication map $c: A
\to End(W)$, we define a connection $\nabla^{x,W,\alpha}$ on the restriction of $r^*(W)$ to ${\mathcal G}_x \cap r^{-1}(U_\alpha)$ by the formula $$\nabla_{\tilde X_h}^{x,W,\alpha} \tilde e_t := \frac{1}{4}\sum_{a,b}
(\Gamma_{ha}^b \circ r) c(\tilde X_a)c(\tilde X_b) \tilde e_t.$$
Let $\phi_\alpha \in {\mathcal{C}^\infty}(M)$ be a ${\mathcal{C}^\infty}$-partition of unity subordinate to the covering $U_\alpha$. Then $\tilde \phi_\alpha :=
\phi_\alpha\circ r$ is a partition of unity subordinate to $r^{-1}(U_\alpha)$. We define a connection $\nabla^{x,W}$ on the restriction of $W$ to ${\mathcal G}_x$ by the formula $$\nabla^{x,W}_{\tilde X}(\xi) = \sum_{\alpha}
\nabla^{x,W,\alpha}(\tilde X) (\tilde \phi_\alpha \xi).$$ By the definition, $\nabla^{x,W}$ is an admissible connection on the restriction of $r^*(W)$ to ${\mathcal G}_x$.
Let $W \to M$ be a complex vector bundle that is a ${{\rm Cliff}}(A)$-module. Then we can find an admissible connection $\nabla^{x,W}$ on the restriction of $r^*(W)$ to ${\mathcal G}_x$, for any $x \in M$, such that for each $X \in \Gamma(A)$, the operators $\nabla^{x,W}_{\tilde X}$ form a smooth, ${\mathcal G}$-invariant family of differential operators on $r^*(W)$, and hence they define an element $\nabla_X^W$ in ${{\rm Diff}({\mathcal G};W)}$. If $S$ is a spin bundle, then we can take this connection to be the Levi-Civita connection.
This is just the summary of the above discussion.
It follows from the above proposition that if we consider on each ${\mathcal G}_x$ the Dirac operator determined by the connection $\nabla^{x,W}$, then we obtain an invariant family of differential operators, which hence defines an operator ${{\not \!\!D}}^W_{\mathcal G}\in
{{\rm Diff}({\mathcal G};W)}$, the Dirac operators on ${\mathcal G}$ associated to $W$ and the given admissible connection. (When the groupoid ${\mathcal G}$ is clear from the context, we shall drop the subscript ${\mathcal G}$.)
We can also regard the admissible connection on a ${{\rm Cliff}}(A)$-module $W$ as an operator $\nabla^W \in {{\rm Diff}({\mathcal G};W, W\otimes A^*)}$. If we denote by $c \in Hom(W\otimes A^*, W)$ the Clifford multiplication, then, as in the classical case ${{\not \!\!D}}^W = c \circ \nabla^W$. We can also generalize the local description of Dirac operators. Let $M =
\bigcup U_\alpha$ be a covering of $M$ by open subsets which trivializes the bundle $A=A({\mathcal G})$, and choose a partition of unity $\phi_\alpha^2$ subordinate to $U_\alpha$. On each $U_\alpha$, we choose a local [*orthonormal*]{} basis $X_1, \ldots, X_k$ of $A$ and define $X^\alpha_j = \phi_\alpha X_j$. Then $${{\not \!\!D}}^W = \sum_{\alpha,j} c(X^\alpha_j) \nabla^W_{X^{\alpha}_j}.$$
As in the classical case of a Riemannian manifold, the space of ${\mathcal G}$-invariant, admissible connections $\nabla^{x,W}$ on $r^*(W)$ is an affine space with model vector space the space of skew-adjoint elements in the space of ${{\rm Cliff}}(A)$-linear endomorphisms of $W$.
A feature specific to the groupoid case, however, is that all the above constructions and operators are compatible with restrictions to compact, ${\mathcal G}$-invariant subsets of $M$. (Recall that a subset $Y
\subset M$ of the space of units of ${\mathcal G}$ is ${\mathcal G}$-invariant if, and only if, $d^{-1}(Y) = r^{-1}(Y)$.) For instance, consider a ${{\rm Cliff}}(A)$ bundle $W$ on $M$ with admissible connection $\nabla$. Then $W$ restricts to a ${{\rm Cliff}}(A\vert_Y)$ module on $Y$. From this observation we get that the Dirac operator on ${\mathcal G}$ associated to the ${{\rm Cliff}}(A)$-module $W$ will restrict to the Dirac operator on ${\mathcal G}_Y := d^{-1}(Y)$ associated to the ${{\rm Cliff}}(A\vert_Y)$-module $W\vert_Y$. Formally, $$\inn_Y({{\not \!\!D}}^W_{{\mathcal G}}) = {{\not \!\!D}}^{W_Y}_{{\mathcal G}_Y}.$$ Similarly, $$\label{lapl}
\inn_Y(\Delta_p^{\mathcal G}) = \Delta_p^{{\mathcal G}_Y}\,, \quad \inn_Y(d^{\mathcal G})
= d^{{\mathcal G}_Y},$$ and so on. This leads, as we shall see in the following sections, to Fredholmness criteria for these various operators in terms of the invertibility of the corresponding operators associated to proper, invariant, closed submanifolds.
Sobolev spaces\[Sec.Sobolev\]
=============================
Throughout this section, we assume for simplicity, that the space $M$ of units of a given groupoid ${\mathcal G}$ is compact. All the definitions and results extend to the case of sections of a Hermitian vector bundle $E$ and operators acting on sections of $E$. For simplicity, however, we shall discuss in detail only the case where $E$ is the one-dimensional, trivial bundle.
The notation $E$, sometimes is used in this section to denote the identity element of operator algebras, in this section.
Consider a bounded, non-degenerate representation $\varrho:{\Psi^{-\infty}({\mathcal G})}\longrightarrow{\operatorname{End}}({\mathcal{H}})$. Theorem \[Theorem.EXT\] then gives a natural extension of $\varrho$ to a bounded $*$-representation of ${\Psi^{\infty}({\mathcal G})}$. Here “bounded” refers to the fact that the order zero operators act by bounded operators on $\mathcal{H}$, see Definition \[def.bounded.rep\]. Recall that $\varrho$ is non-degenerate if the space ${{\mathcal{H}}_{\infty}}:=\varrho({\Psi^{-\infty}({\mathcal G})}){\mathcal{H}}$ is dense in $\mathcal{H}$. The best we can hope for formally self-adjoint operators $A=A^{*} \in {\Psi^{m}({\mathcal G})}$ is that they are essentially self-adjoint unbounded operators on $\mathcal{H}$. This is in fact the case for elliptic operators; for $m>0$, a formally self-adjoint, elliptic operator $A=A^{*}\in{\Psi^{m}({\mathcal G})}$ leads to densely defined, essentially self-adjoint operator $$\varrho(A):{{\mathcal{H}}_{\infty}}\longrightarrow{\mathcal{H}}.$$ This will allow us to freely use functional calculus for self-adjoint operators later on in this section. Fix a bounded, non-degenerate representation $\varrho$ as above.
Note that under the assumptions above, the unit $E:=({\operatorname{id}}_{{\mathcal G}_{x}})_{x\in M}$ belongs to ${\Psi^{0}({\mathcal G})}$ with $\varrho(E)={\operatorname{id}}_{{\mathcal{H}}}$. Let further ${{\mathcal{H}}_{-\infty}}:={{\mathcal{H}}_{\infty}}^{*}$ be the algebraic dual of ${{\mathcal{H}}_{\infty}}$, and $T:{\mathcal{H}}\hookrightarrow{{\mathcal{H}}_{-\infty}}:h\mapsto
T_{h}$ be the natural, anti-linear embedding. As in the classical case, $\varrho$ induces a multiplicative morphism ${\widetilde{\varrho}}:{\Psi^{\infty}({\mathcal G})}\longrightarrow{\operatorname{End}}({{\mathcal{H}}_{-\infty}})$ by $$[{\widetilde{\varrho}}(A)u](\xi):=u(\varrho(A^{*})\xi)$$ for $A\in{\Psi^{\infty}({\mathcal G})}$, $u\in{{\mathcal{H}}_{-\infty}}$, and $\xi\in{{\mathcal{H}}_{\infty}}$. Chasing definitions yields $$\begin{aligned}
\nonumber
{\widetilde{\varrho}}(A)\circ T= T\circ \varrho(A) &:& {{\mathcal{H}}_{\infty}}\longrightarrow{{\mathcal{H}}_{-\infty}}\mbox{ for } A\in{\Psi^{\infty}({\mathcal G})}\,, \mbox{ and }\\ {\widetilde{\varrho}}(A)\circ T=
\label{gl2} T\circ \varrho(A) &:&
{\mathcal{H}}\hspace{2ex}\longrightarrow{{\mathcal{H}}_{-\infty}}\mbox{ for }
A\in{\Psi^{0}({\mathcal G})}\,. \end{aligned}$$ Recall that an unbounded, closable operator $S:{\mathcal{H}}\supseteq
{\mathcal{D}}(S)\longrightarrow{\mathcal{H}}$ is called [*essentially self-adjoint*]{} if ${\mathcal{D}}(S)$ is dense in ${\mathcal{H}}$, and $\overline{S}=S^{*}$ where $\overline{S}$ denotes the minimal closed extension of $S$, and $S^{*}$ its adjoint in the sense of unbounded operators. The proof of the following proposition is appropriately adapted from [@shubin Theorem 26.2].
\[ess\] Let $m>0$, and $A=A^{*}\in{\Psi^{m}({\mathcal G})}$ be elliptic. Then the unbounded operator $\varrho(A):{\mathcal{H}}\supseteq{{\mathcal{H}}_{\infty}}\longrightarrow{\mathcal{H}}$ is essentially self-adjoint. Moreover, $$\label{domain}
{\mathcal{D}}\left(\overline{\varrho(A)}\right)=
{\mathcal{D}}(\varrho(A)^{*})=
\left\{h\in{\mathcal{H}}:{\widetilde{\varrho}}(A)T_{h}\in T{\mathcal{H}}\right\}\,.$$
For brevity, let ${\mathcal{D}}$ be the space on the right-hand side in . Also, let $h\in{\mathcal{D}}(\varrho(A)^{*})$. Then we get for all $\xi\in{{\mathcal{H}}_{\infty}}$ $${\widetilde{\varrho}}(A)T_{h}(\xi)= \left(\varrho(A)\xi,h\right)
=\left(\xi,\varrho(A)^{*}h\right)= T_{\varrho(A)^{*}h}(\xi)\,,$$ i.e. ${\mathcal{D}}(\varrho(A)^{*})\subseteq{\mathcal{D}}$, and ${\widetilde{\varrho}}(A)T_{h}=T_{\varrho(A)^{*}h}$.
On the other hand, for $h\in{\mathcal{D}}$, there exists $g\in{\mathcal{H}}$ such that for all $\xi\in{{\mathcal{H}}_{\infty}}$ $$\left(\varrho(A)\xi,h\right)=
{\widetilde{\varrho}}(A)T_{h}(\xi)=T_{g}(\xi) =(\xi,g)\,,$$ hence, $h\in{\mathcal{D}}(\varrho(A)^{*})$ which gives the second equality in . By [@shubin Theorem 26.1], it remains to show $$N(\varrho(A)^{*}\pm i {\operatorname{id}}_{{\mathcal{H}}})\subseteq
{\mathcal{D}}\left(\overline{\varrho(A)}\right)\,.$$ Because of $m>0$, $A\pm i E\in{\Psi^{m}({\mathcal G})}$ is elliptic; by the usual symbolic argument we get $B_{\pm}\in{\Psi^{-m}({\mathcal G})}$ satisfying $E-B_{\pm}(A\pm iE)=:R_{\pm}\in{\Psi^{-\infty}({\mathcal G})}$. Furthermore, for $\xi\in N(\varrho(A)^{*}\pm i {\operatorname{id}}_{{\mathcal{H}}})\subseteq{\mathcal{D}}(\varrho(A)^{*})$ another definition chase yields as before $${\widetilde{\varrho}}(A\pm iE)T_{\xi}=
T((\varrho(A)^{*}\mp i {\operatorname{id}}_{{\mathcal{H}}})\xi)=0\,,$$ thus, $$T_{\xi}={\widetilde{\varrho}}(R_{\pm})T_{\xi}=T_{\varrho(R_{\pm})\xi}\in T{{\mathcal{H}}_{\infty}}$$ because of $R_{\pm}\in{\Psi^{-\infty}({\mathcal G})}$ and . Since we have ${{\mathcal{H}}_{\infty}}\subseteq{\mathcal{D}}\left(\overline{\varrho(A)}\right)$, this completes the proof.
Let us now define Sobolev spaces in the setting of groupoids using the powers of an arbitrary positive element $D \in {\Psi^{m}({\mathcal G})}$, $m >0$, as customary. The necessary facts that imply independence of $D$ are contained in the following theorem (and the lemmata leading to its proof). Also, the following theorem will allow us to reduce certain questions about operators of positive order to operators of order zero.
We shall write $P \ge 0$ if $P=P^{*} \in {\Psi^{m}({\mathcal G})}$ is such that $(\varrho(P)\xi,\xi) \ge 0$ for all $\xi \in {{\mathcal{H}}_{\infty}}$ and for every non-degenerate representation $\varrho$ of ${\Psi^{\infty}({\mathcal G})}$ on ${\mathcal H}$. Also, we shall write $ A \ge B$ if $A - B \ge 0$. For $Q \in {\mathfrak{A}({\mathcal G})}$, we write $Q = P^{-1}$, if, and only if, $\varrho(P)\varrho(Q) = {\operatorname{id}}_{{\mathcal{H}}}$ and $\varrho(Q)\varrho(P) \subseteq
{\operatorname{id}}_{{\mathcal{H}}}$, for every non-degenerate, bounded representation $\varrho$. Then, for $s > 0$, $P^{-s}$ stands for $(P^{-1})^{s}$.
\[theorem.red\] Fix a differentiable groupoid ${\mathcal G}$ whose space of units, $M$, is compact. Let $D \in {\Psi^{m}({\mathcal G})}$, $m > 0$, be such that $D \ge E$ and $\sigma_m(D) > 0$. Then $D^{-s} \in {C^*({\mathcal G})}$, for all $s >
0$. Moreover, if $P$ has order $\le k$, then $PD^{-k/m} \in {\mathfrak{A}({\mathcal G})}$.
The proof will consist of a sequence of lemmata.
\[lemma.3\] Fix arbitrarily a metric on $A({\mathcal G})$, and let $B= E + \Delta$, where $\Delta$ is the positive Laplace operator on functions. Then $B$ is invertible in the sense above, and we have $B^{-1} \in {C^*({\mathcal G})}$.
Let $D_{t} = E + t^2 \Delta$, $t>0$. We shall prove first that, for small $t$, there exists $Q_t \in {C^*({\mathcal G})}$ such that $\varrho(Q_t)\varrho(D_t)\subseteq{\operatorname{id}}_{{\mathcal{H}}}$ and $
\varrho(D_t)\varrho(Q_t) = {\operatorname{id}}_{{\mathcal{H}}}$, for all non-degenerate representations $\varrho$ on ${\mathcal{H}}$.
Because the family $(td)$, $t > 0$, extends to a first-order differential operator on ${\mathcal G}_{{\operatorname{ad}}}$, the adiabatic groupoid of ${\mathcal G}$, we obtain that $t^2\Delta = (td)^* (td)$ induces an element in $\Psi^{2}({\mathcal G}_{{\operatorname{ad}}})$, which explains the choice of the power $t^2$.
To be precise, let $E\in{\Psi^{0}({\mathcal G})}$, $E_{{\operatorname{ad}}}\in\Psi^{0}({\mathcal G}_{{\operatorname{ad}}})$, and $E_{0}\in\Psi^{0}(A({\mathcal G}))$ be the identity elements. If $e_t$, $t
\ge 0$, denotes the evaluation map as in Example \[exad\] (so that, in particular $e_{t}: \Psi^{\infty}({\mathcal G}_{{\operatorname{ad}}}) \to
\Psi^{\infty}({\mathcal G})$, $t >0$), then we have $e_{t}(E_{{\operatorname{ad}}})=E$ for $t>0$, and $e_{0}(E_{{\operatorname{ad}}})=E_{0}$. Thus, the family $(D_t)$ leads to an element $D\in\Psi^{2}({\mathcal G}_{{\operatorname{ad}}})$. Choose a quantization map $q$ for ${\mathcal G}_{{\operatorname{ad}}}$ as in [@NWX], and denote by $|\xi|$ the metric on $A^{*}({\mathcal G})$, so that the principal symbol of $\Delta$ is $|\xi|^2$. Then the function $p(t,\xi) := ( 1 + |\xi|^{2})^{-1}$ is an order two symbol on $A^*({\mathcal G}_{{\operatorname{ad}}})$, see Example \[exad\], and $F:=q(p)D\in\Psi^{0}({\mathcal G}_{{\operatorname{ad}}})$ satisfies $e_{0}(F)=E_{0}$. >From the results [@Landsman; @LandsmanRamazan; @Ramazan], we know that the function $t\mapsto\|e_{t}(F - E_{{\operatorname{ad}}})\|$ is continuous at $0$ (in fact everywhere, but that is all that is needed), and hence $e_{t}(F)$ will be invertible in ${\mathfrak{A}({\mathcal G})}$ for $t$ small. We define then $$Q_{t}:=e_{t}(F)^{-1}e_{t}(q(p)) \in {\mathfrak{A}({\mathcal G})} {\Psi^{-2}({\mathcal G})}
\subseteq {C^*({\mathcal G})}\,,$$ and a straight-forward computation gives $\varrho(Q_t) \varrho(D_t)
\xi = \xi$, for $\xi\in{{\mathcal{H}}_{\infty}}$, a dense subspace of ${\mathcal{H}}$, and for $t > 0$ but small. Since $\varrho(D_t)$ is (essentially) self-adjoint, we obtain that $\varrho(Q_t)$ is the inverse of (the closure of) $\varrho(D_t)$. This means $Q_t = D_t^{-1}$, for $t > 0$ but small, according to our conventions.
Let now $h_{t}(y) = (1+t^{2}y)^{-1}$ and $\varepsilon>0$ be arbitrary. Then there exists a continuous function $g_{\varepsilon,t}:[0,1]\longrightarrow[0,1]$ with $g(0)=0$ such that $h_{t}=g_{\varepsilon,t}\circ h_{\varepsilon}$, and we obtain $D^{-1}_{t}=g_{\varepsilon,t}(D_{\varepsilon}^{-1}) \in{C^*({\mathcal G})}$ by the composition property of the functional calculus for continuous functions, for $\varepsilon$ small enough. Because of $B=D_1$ this completes the proof.
\[lemma.4\] Let $D \in {\Psi^{m}({\mathcal G})}$ be elliptic with $\sigma_m(D) > 0$. Then, for each $A \in {\Psi^{m}({\mathcal G})}$ and for each bounded representation $\varrho$ of ${\Psi^{\infty}({\mathcal G})}$ on $\mathcal H$, we can find $C_A \ge 0$ such that $\|\varrho(A)f \| \le C_A(\|f\| + \|\varrho(D) f\|)$, for all $f \in
{{\mathcal{H}}_{\infty}}:=\varrho({\Psi^{-\infty}({\mathcal G})}){\mathcal{H}}$.
The proof is the same as that of the boundedness of operators of order zero, using Hörmander’s trick [@fio]. Let us briefly recall the details.
It suffices to show $\|\varrho(A)f \|^2 \le C( \|f\|^2 + \|\varrho(D)
f\|^2)$, for some constant $C$ independent of $f$. Choose $C_1 > 0$ with $|\sigma_m(A)|^2 \le C_1 |\sigma_m(D)|^2$. This is possible because $\sigma_m(A) \sigma_m(D)^{-1}$ is defined and continuous on the sphere bundle $S^{*}({\mathcal G})$ of $A^*({\mathcal G})$, a compact space. Let $b
> 0$ be smooth with $b^2 = (C_1 + 1) |\sigma_m(D)|^2 -
|\sigma_m(A)|^2$ (this is defined only outside the zero section), and let $B \in {\Psi^{m}({\mathcal G})}$ be an operator with principal symbol $\sigma_{m}(B)=b$. Then $$(C_1 + 1) D^*D - A^*A - B^*B = R,$$ with $R$ of order $l \le 2m - 1$. By replacing $B$ with $B_1$ such that $B_1 - B$ has order $l - m$ and $\sigma_{l-m}(B_1 - B) =
\sigma_l(R)/2b$, we obtain that the order of the operator $(C_1+1)
D^*D - A^*A - B_1^*B_1$ is less than $l$. Continuing in this way, we may assume that $R$ has order $\le 0$, so in particular $\varrho(R)$ is bounded. Then $$\|\varrho(A)f\|^2 \le (C_1+1) (\varrho(D^*D)f,f) - (\varrho(R)f,f) \le
C(\|\varrho(D)f\|^2 + \|f\|^2)$$ for $C := \max\{ \|\varrho(R)\|, C_1 + 1\}$.
\[lemma.5\] Let $D =D^* \in {\Psi^{m}({\mathcal G})}$, be elliptic with $\sigma_m(D) > 0$. Then we can find $C \ge 0$ such that, for any bounded representation $\varrho$ of ${\Psi^{\infty}({\mathcal G})}$ on $\mathcal H$ we have $(\varrho(D) f,f) \ge -C(f,f)$, for all $f \in
{{\mathcal{H}}_{\infty}}:=\varrho({\Psi^{-\infty}({\mathcal G})}){\mathcal{H}}$.
The statement follows from the boundedness of $D$, if $m \le 0$, so assume that $m > 0$. Then the proof is the same as that of the previous lemma if in the proof of that lemma we replace $D^*D$ with $D$ and take $A = 0$.
For the rest of the proof of Theorem \[theorem.red\], we shall fix a non-degenerate representation $\varrho$ of ${\Psi^{\infty}({\mathcal G})}$ on ${\mathcal H}$, and we shall identify the elements of ${\Psi^{\infty}({\mathcal G})}$ with unbounded operators with common domain ${{\mathcal{H}}_{\infty}}$, and the elements of ${\mathfrak{A}({\mathcal G})}$ with bounded operators on $\mathcal H$.
\[cor.1\] If $D = D^* \in {\Psi^{m}({\mathcal G})}$, $m > 0$, is such that $\sigma_m(D) > 0$, then there exists $C \geq 0$ such that $D + C E \ge E$. For any such $C
\geq 0$ and any $A\in{\Psi^{m}({\mathcal G})}$, the operator $A(D + C)^{-1}$ extends uniquely to a bounded operator on ${\mathcal{H}}$.
The first statement follows from the previous lemma. Fix $C \geq 0$ such that $D + CE \geq E$. Lemma \[lemma.4\] gives $\|Af\| \le C_1(\| f
\| + \| (D + CE) f \|),$ for some $C_1 > 0$ and all $f \in {{\mathcal{H}}_{\infty}}$ Consequently, there is $C_{2}>0$ with
$$\label{eq.ineq}
\|Af\| \le C_2 \| (D + C)f \|,$$
for all $f\in{{\mathcal{H}}_{\infty}}$. Since $D+CE$ is essentially self-adjoint by Proposition \[ess\] and $D+CE\geq E$, its range ${\mathcal{H}}_{1}:=(D+CE){\mathcal{H}}$ is dense by [@resi2 Theorem X.26]. By , we obtain for $g=(D+CE)f\in{\mathcal{H}}_{1}$ $$\| A(D + CE)^{-1}g \| \le C_2 \| g \|\,,$$ which completes the proof.
\[cor.2\] Consider now two self-adjoint, elliptic elements $D_1, D_2 \in
{\Psi^{m}({\mathcal G})}$, $m > 0$, with $D_i \ge E$ and $\sigma_m(D_i) > 0$, $i =
1,2$. Then $D_1D_2^{-1}$ extends uniquely to a bounded invertible operator.
By the previous corollary, both $D_1D_2^{-1}$ and $D_2D_1^{-1}$ extend to bounded operators.
\[lemma.6\] Let $D \in {\Psi^{m}({\mathcal G})}$, $m >0$, be such that $D \ge E$, $\sigma_m(D) >
0$, and $D^{-1} \in {C^*({\mathcal G})}$. Then we have $PD^{-k},D^{-k}P \in
{\mathfrak{A}({\mathcal G})}$, if $P$ has order $\le km$. Moreover, we have $\sigma_{0}(PD^{-k})=\sigma_{km}(P)\sigma_{m}(D)^{-k}$, and $\sigma_{0}(D^{-k}P)=\sigma_{m}(D)^{-k}\sigma_{km}(P)$.
We notice that if $D$ satisfies the assumptions of the lemma, then $D^k$ satisfies them as well. We can assume then that $k = 1$.
We shall check only that $PD^{-1} \in {\mathfrak{A}({\mathcal G})}$. The relation $D^{-1}P
\in {\mathfrak{A}({\mathcal G})}$ can be proved in the same way or follows from the first one by taking adjoints.
Let $A \in {\Psi^{-m}({\mathcal G})}$ be with $AD -E = R \in {\Psi^{-\infty}({\mathcal G})}$, $B_n \in
{\Psi^{-\infty}({\mathcal G})}$ be a sequence converging to $D^{-1}\in{C^*({\mathcal G})}$, and define $A_n := A - RB_n \in{\Psi^{-m}({\mathcal G})}$. Then we have $A_n - D^{-1} =
R(D^{-1} - B_n)$, thus, $PD^{-1}=PA_{n}-PR(D^{-1}-B_{n})$ first defined on the dense subspace $D{{\mathcal{H}}_{\infty}}$, has a unique bounded extension with $PD^{-1}\in{\mathfrak{A}({\mathcal G})}$ because of $PA_n \in {\Psi^{0}({\mathcal G})}$ and $$\|PA_n - PD^{-1}\| \le \|PR\| \| D^{-1} - B_n\| \to 0\,, \quad
n \to \infty.$$ Since $\sigma_{0}(PD^{-1})$ is the limit of $\sigma_{0}(PA_{n})=\sigma_{m}(P)\sigma_{-m}(A)=
\sigma_{m}(P)\sigma_{m}(D)^{-1}$, we obtain the formula for the principal symbol as well.
\[lemma.7\] Let $D \in {\Psi^{m}({\mathcal G})}$ be with $D \ge E$ and $\sigma_m(D) > 0$. Then $D^{-1} \in {C^*({\mathcal G})}$.
From Lemma \[lemma.3\] we know that $(E + \Delta)^{-m}$ is in ${C^*({\mathcal G})}$. By Corollary \[cor.2\], applied to $D_{1}=(E +
\Delta)^{m}$ and $D_{2}=D^{2}$, we have $(E +
\Delta)^{m}D^{-2}=D_{1}D_{2}^{-1}\in{\mathfrak{A}({\mathcal G})}^{-1}$, thus, $D^{-2} = (E
+ \Delta)^{-m}(E + \Delta)^{m}D^{-2} \in {C^*({\mathcal G})}$. Taking square roots completes the proof.
We now complete the proof of Theorem \[theorem.red\].
Assume first that $m = 1$. Then the theorem follows from Lemma \[lemma.6\] and Lemma \[lemma.7\]. For arbitrary $m$, $D^{-1}\in {C^*({\mathcal G})}$, and hence we get $D^{-s} \in {C^*({\mathcal G})}$ by using functional calculus with continuous functions. A look at Lemma \[lemma.6\] completes the proof.
We now obtain some corollaries of Theorem \[theorem.red\].
For the following results, we need to define Sobolev spaces. Fix a metric on $A({\mathcal G})$. Let then $\Delta:=\Delta_0 \in {{\rm Diff}({\mathcal G})}$ be the Hodge-Laplacian acting on functions, and $\varrho$ be a non-degenerate, bounded representation as above. Then $D:=\varrho(E+\Delta)$ is essentially self-adjoint and strictly positive, hence we can define $D^s$, for each $s \in {\mathbb R}$, using the functional calculus for essentially self-adjoint operators. Then $H^s(\mathcal H,\varrho)$, [*the $s$th Sobolev space of $(\mathcal
H,\varrho)$*]{}, is by definition, the domain of $D^{s/2}$ with the graph topology, if $s \ge 0$, or its dual if $s < 0$.
\[cor.Sobolev\] The spaces $H^s(\mathcal H,\varrho)$ do not depend on the choice of the metric on $A({\mathcal G})$, and every pseudodifferential operator $P \in {\Psi^{m}({\mathcal G})}$ gives rise to a bounded map $H^{s}(\mathcal H ,\varrho) \to H^{s - m}(\mathcal H ,\varrho)$.
If we change the metric on the compact space $M$, we obtain a new Laplace operator, and $D$ will be replaced by a different operator $D_1$. However, by Corollary \[cor.2\], $D^s
D_1^{-s}$ and $D_1^s D^{-s}$ are bounded for all even integer $s$. By interpolation, they are bounded for all $s$. This proves the independence of the Sobolev space on the choice of a metric on $M$.
The last claim follows from Lemma \[lemma.6\] if $s$ is an integer. Let $H^{\infty} := \bigcap H^{k}(\mathcal H ,\varrho)$. Then $\varrho(P)(H^{\infty})\subseteq H^{\infty}$. Using this fact and applying the Phragmen-Lindelöf principle to $s \mapsto
(\varrho(D)^{s} P \varrho(D)^{-s} \xi,\xi')$, with $\xi,\xi' \in
H^{\infty}$, we obtain the desired result for all $s$.
Similarly, we prove the following corollary.
\[coriso\] Let $A\in{\Psi^{k}({\mathcal G})}$ be elliptic. Then $\Lambda:=\varrho(E+A^{*}A)$ is essentially self-adjoint, and $\Lambda^{t}$ induces for all $s,t\in{\mathbb R}$ isomorphisms $$\Lambda^{t}:H^{s}({\mathcal{H}},\varrho)\longrightarrow H^{s-2kt}({\mathcal{H}},\varrho)\,.$$
Another corollary is related to the Cayley transform.
\[cor.Cayley\] If $A=A^{*} \in {\Psi^{m}({\mathcal G})}$, $m >0$, is elliptic, then the Cayley transform $(A + iE)(A - iE)^{-1}$ of $A$ belongs to $ {\mathfrak{A}({\mathcal G})}$. Moreover, we have $$\sigma_{0}((A+iE)(A-iE)^{-1})=\sigma^{m}(A)\sigma_{m}(A)^{-1}=1\,,$$ where the last equality holds in the scalar case only.
We have $(A + iE)(A - iE)^{-1} = (A + iE)^2 (A^2 + E)^{-1} \in
{\mathfrak{A}({\mathcal G})}$, by Theorem \[theorem.red\], because $A^2 + E \ge E$ and $\sigma_{2m}(A^2 + E) > 0$. The identity for the principal symbol follows from the corresponding one in Lemma \[lemma.6\].
The Cayley transform of $A$ will be denoted in the following sections simply by $(A + iE)(A - iE)^{-1}$, because no more confusions can arise.
Operators on open manifolds\[Sec.O.M\]
======================================
One of the main motivations for studying algebras of pseudodifferential operators on groupoids is that they can be used to analyze geometric operators on certain complete Riemannian manifolds $(M_0,g)$ (without corners). The groupoids ${\mathcal G}$ used to study these geometric operators will be of a particular kind. They will have as space of units a compactification $M$ of $M_0$ to a manifold with corners such that $M_0$ will be an open invariant subset of $M$ with the property that the reduction of ${\mathcal G}$ to $M_0$ is the product groupoid. If $M_0$ happens to be compact, then $M = M_0$, and our results simply reduce to the usual “elliptic package” for compact smooth manifolds without corners. Our results thus can be viewed as a generalization of the classical elliptic theory from compact manifolds to certain non-compact, complete Riemannian manifolds.
We now make explicit the hypothesis we need on the groupoid ${\mathcal G}$.
[**Assumptions.**]{} [*In this and the following sections, $M_0$ will be a smooth manifold without corners which is diffeomorphic to (and will be identified with) an open dense subset of a compact manifold with corners $M$, and ${\mathcal G}$ will be a differentiable groupoid with units $M$, such that $M_0$ is an invariant subset and*]{} $${\mathcal G}_{M_0}\cong M_0 \times M_0.$$
The above assumptions have a number of useful consequences for ${\mathcal G}$, $M$, and $M_0$, and we shall use them in what follows, without further comment.
Let $A = A({\mathcal G})$. First of all, $A\vert_{M_0} \cong TM_0$. Fix a metric on $A$. The metric on $A$ then restricts to a metric on $M_0$, so $M_0$ is naturally a Riemannian manifold such that the map $r:{\mathcal G}_x
\to M_0$ is an isometry for any $x \in M_0$. Moreover, because $M$ is compact, all metrics on $M_0$ obtained by this procedure will be equivalent: if $g_1$ and $g_2$ are metrics on $M_0$ obtained from metrics on $A$, then we can find $C,c > 0$ such that $cg_1 \le g_2 \le
Cg_1$ (this is of course not true for any two metrics on the non-compact smooth manifold $M_0$). The same result holds true for the induced smooth densities (or measures) on $M_0$, and hence all the spaces $L^2(M_0)$ defined by these measures actually coincide.
Let $\pi$ be the vector representation of ${\Psi^{\infty}({\mathcal G})}$ on ${\mathcal{C}^\infty}(M)$ (uniquely determined by $(\pi(P)f) \circ r = P (f \circ r)$, see Equation ). Then $\pi({\Psi^{\infty}({\mathcal G})})$ maps ${{\mathcal C}^{\infty}_{\text{c}}}(M_0)$ to itself. Fix $x \in M_0$. The regular representation $\pi_x : {\Psi^{\infty}({\mathcal G})} \to End(C_c^{\infty}({\mathcal G}_x))$ is equivalent to $\pi$ via the isometry $r: {\mathcal G}_x \to M_0$, and hence $\pi$ is a bounded representation of ${\Psi^{\infty}({\mathcal G})}$ on $L^2(M_0)$.
We now relate the geometric operators on $M_0$, defined using a metric induced from $A$, and the geometric operators on ${\mathcal G}$, ${\mathcal G}$ as above (${\mathcal G}_{M_0} \cong M_0 \times M_0$).
We start with a ${{\rm Cliff}}(A)$-module $W$ on $M$ together with an admissible connection $\nabla^W \in {{\rm Diff}({\mathcal G};W,W\otimes A^*)}$, defined as an invariant family of differential operators on ${\mathcal G}_x =
d^{-1}(x)$. Fix $x \in M_0$ arbitrary. Then the restriction of $r^*(W)$ to ${\mathcal G}_x$ is a Clifford module on ${\mathcal G}_x$, which hence can be identified with a Clifford module $W_0$ on $M_0$, using the isometry ${\mathcal G}_x \cong M_0$.
Let ${{\not \!\!D}}^{W} \in \Psi^{1}({\mathcal G};W)$ be the Dirac operator on ${\mathcal G}$ associated to $W$ and its admissible connection, and let ${{\not \!\!D}}^{W_0}$ be the Dirac operator on $M_0$ associated to $W_0$ and its admissible connection obtained by pulling back the connection on ${\mathcal G}_x$. These operators are related as follows.
\[Theorem.Image\] The Dirac operator ${{\not \!\!D}}^W$ on ${\mathcal G}$ acts in the vector representation as ${{\not \!\!D}}^{W_0}$, the Dirac operator on $M_0 \subset M$ defined above. More precisely, $$\pi({{\not \!\!D}}^{W}) = {{\not \!\!D}}^{W_0}.$$
By construction, ${{\not \!\!D}}^{W_0}$ is, up to similarity, the restriction of ${{\not \!\!D}}^W$ to one of the fibers ${\mathcal G}_x$, with $x \in M_0$.
At first sight, the above theorem applies only to a very limited class of (admissible) Dirac operators on $M_0$, the ones coming from ${{\rm Cliff}}(A)$-modules. Not every Dirac operator on a Clifford module on $M_0$ can be obtained in this way. However, as we shall see in a moment, if we are given a Clifford module on $M_0$, we can always adjust our compatible connection so that the resulting Dirac operator comes from a Dirac operator on ${\mathcal G}$ (corresponding to a ${{\rm Cliff}}(A)$-module).
\[Theorem.Geometric\] Suppose there exists a compact subset $M_1 \subset M_0$ which is a deformation retract of $M$. Let $W_0$ be a Clifford module on $M_0$. Then we can find an admissible connection on $W_0$ such that the associated admissible Dirac operator ${{\not \!\!D}}^{W_0}$ is (conjugate to) $\pi({{\not \!\!D}}^{W})$, for some ${{\rm Cliff}}(A)$-module $W$, ${{\not \!\!D}}^W$ being the Dirac operator on ${\mathcal G}$ associated to $W$.
If $W_0$ is a spin bundle, then we can choose this connection to be the Levi-Civita connection on $W_0$.
Using the deformation retract $f : M \to M_1$, we define (up to isomorphism) $W = f^*(W_0)$. Then $W\vert_{M_0} \simeq W_0$, the isomorphism being uniquely determined up to homotopy. Moreover, we have a (non-canonical) isomorphism $A \simeq f^*(TM_0)$ of vector bundles, which allows us to define a ${{\rm Cliff}}(A)$-module structure on $W$. By replacing $W_0$ with an isomorphic bundle, we can assume then that $W_0 = W\vert_{M_0}$, as Clifford modules. Choose an admissible connection on $W$. Theorem \[Theorem.Image\] then gives that $\pi({{\not \!\!D}}^{W}) = {{\not \!\!D}}^{W_0}$.
Spectral properties\[Sec.SP\]
=============================
We shall use now the results of the previous section to study operators on suitable Riemannian manifolds. We are interested in spectral properties, Fredholmness, and compactness for these operators. The results of this section extend essentially without change to the case of families of such manifolds.
[*We fix, throughout this section, a groupoid ${\mathcal G}$ satisfying the assumptions of Section \[Sec.O.M\].*]{} In particular, $M_0$ is an open invariant subset of $M$ and ${\mathcal G}_{M_0}\cong M_0 \times M_0$.
We denote as before by ${\mathfrak{A}({\mathcal G})}$ the closure of ${\Psi^{0}({\mathcal G})}$ in the norm $\|\;\cdot\|$ and by ${C^*({\mathcal G})}$ the closure of ${\Psi^{-\infty}({\mathcal G})}$ in the same norm. Our analysis of geometric operators on $M_0$ depends on the structure of the algebras ${\mathfrak{A}({\mathcal G})}$ and ${C^*({\mathcal G})}$. The results of Section \[Sec.BR\] applied to our groupoid ${\mathcal G}$ (satisfying the assumptions of Section \[Sec.O.M\]) give the following. Let ${\mathfrak I}=
{C^*({\mathcal G}_{M_0})}$, then ${\mathfrak I}$ is isomorphic to $\mathcal{K}(L^2(M_0))$, the algebra of compact operators on $L^2(M_0)
= L^2(M)$, the isomorphism being induced by the vector representation $\pi$, or by any of the representations $\pi_x$, $x \in M_0$, and the isometry ${\mathcal G}_x \simeq M$. Otherwise, if $x \notin M_0$, then $\pi_x$ descends to a representation of $Q({\mathcal G}) : = {\mathfrak{A}({\mathcal G})}/{\mathfrak I}$.
We shall study various spectra, for this purpose, the results of Section \[Sec.Sobolev\] will prove indispensable.
We denote by $\sigma(P)$ the spectrum of an element $P \in {\mathfrak{A}({\mathcal G})}$ and by $\sigma_{Q({\mathcal G})}(P)$ the spectrum of the image of $P$ in $Q({\mathcal G}) := {\mathfrak{A}({\mathcal G})}/{\mathfrak I}= {\mathfrak{A}({\mathcal G})}/{C^*({\mathcal G}_{M_0})}$. These definitions extend to elliptic, self-adjoint elements $P \in {\Psi^{m}({\mathcal G})}$, $m >0$, using the Cayley transform, as follows. Let $f (t) = (t +
i)/(t -i)$ and $f(P):=(P + i)(P - i)^{-1} \in {\mathfrak{A}({\mathcal G})}$ be its Cayley transform, which is defined by Corollary \[cor.Cayley\]. We define then $$\sigma(P) := f^{-1}(\sigma(f(P)))\,, \quad
\mbox{ and } \quad
\sigma_{Q({\mathcal G})}(P) := f^{-1}(\sigma_{Q({\mathcal G})}(f(P))).$$ We observe that if $P$ is identified with an unbounded, self-adjoint operator on a Hilbert space, then the relation $\sigma(P) :=
f^{-1}(\sigma(f(P)))$ is automatically satisfied, by the spectral mapping theorem.
The spectrum and essential spectrum of an element $T$ acting as an unbounded operator on a Hilbert space will be denoted by $\sigma(T)$ and, respectively, by $\sigma_{ess}(T)$. (We shall do that for an operator of the form $T = \pi(P)$, with $\pi$ the vector representation and $P \in {\Psi^{m}({\mathcal G};F)}$, $m > 0$, elliptic, self-adjoint.)
We shall formulate all results below for operators acting on vector bundles. Fix an elliptic operator $A \in {\Psi^{m}({\mathcal G};F)}$, $m >0$, then for any $P \in {\Psi^{k}({\mathcal G};F)}$, we have $P_1 := P (E + A^*A)^{-k/2m} \in {\mathfrak{A}({\mathcal G};F)}$, by Theorem \[theorem.red\].
Let us notice that if $\pi$ is the vector representation of ${\Psi^{0}({\mathcal G})}$ on $L^2(M) = L^2(M_0)$ (see Equation \[eq.vector.rep\] for the definition of the vector representation), then the spaces $H^s(M) =
H^s(L^2(M),\pi)$ are the usual Sobolev spaces associated to the Riemannian manifold of bounded geometry ${\mathcal G}_x \simeq M_0$ [@roe1; @shubin92]. If we are working with sections of a Hermitian vector bundle $F$, then we write $H^s(M;F) := H^s(L^2(M;F),\pi)$.
\[theorem.I\] Let $M_0 \subset M$, the groupoid ${\mathcal G}$, and $A \in {\Psi^{k}({\mathcal G};F)}$, elliptic, be as above.
(i) If $P \in {\Psi^{m}({\mathcal G};F)}$ is such that the image of $P_1:=P(E +
A^*A)^{-m/2k} \in {\mathfrak{A}({\mathcal G})}$ in $Q({\mathcal G}) : = {\mathfrak{A}({\mathcal G})}/ {C^*({\mathcal G}_{M_0})}$ is invertible, then $\pi(P)$ extends to a Fredholm operator $H^{m}(M;F) \to L^2(M;F)$.
(ii) If $P_{1}:=P(E + A^*A)^{-m/2k}$ maps to zero in $Q({\mathcal G})$, then $\pi(P)$ is a compact operator $H^{m}(M;F) \to L^2(M;F)$.
(iii) If $P \in {\Psi^{0}({\mathcal G};F)}$ or $P \in {\Psi^{m}({\mathcal G};F)}$, $m > 0$, is self-adjoint, elliptic, then $\sigma(\pi(P)) \subseteq \sigma(P)$ and $\sigma_{ess}(\pi(P)) \subseteq \sigma_{Q({\mathcal G})}(P)$.
Let $P_1 := P(E + A^*A)^{-m/2k}$, as above.
(i) Choose $Q_1 \in {\mathfrak{A}({\mathcal G})}$ such that $$Q_1P_1 - E, P_1 Q_1 - E \in {\mathfrak I}:= {C^*({\mathcal G}_{M_0})},$$ and define $Q = \pi ((E + A^*A)^{-m/2k}Q_1)$. Then $ \pi(P) Q -
{\operatorname{id}}_{L^{2}(M)} \in \pi({\mathfrak I}) = \mathcal K$. Similarly, we find a right inverse for $\pi(P)$ up to compact operators. Thus, $\pi(P)$ is Fredholm.
(ii) The operator $\pi(P) : H^{m}(M;F) \to L^2(M;F)$ is the product of the bounded operator $\pi(E + A^*A)^{m/2k} : H^{m}(M;F) \to
L^2(M;F)$ and of the compact operator $\pi(P_1)$.
(iii) For $P$ in a $C^*$-algebra $A_0$ and $\varrho$ a bounded $*$-representation of $A_0$, the spectrum of $\varrho(P)$ is contained in the spectrum of $P$ (we do not exclude the case where they are equal). If $P \in {\Psi^{0}({\mathcal G};F)}$, this gives (iii), by taking $A_0 = {\mathfrak{A}({\mathcal G})}$ or $A_0 = Q({\mathcal G})$. If $P \in {\Psi^{m}({\mathcal G};F)}$, $m >0$, is self-adjoint, elliptic, then we use the result we have just proved for $f(P) =(P +
iE) (P - iE)^{-1}$, the Cayley transform of $P$.
It is interesting to observe the following. Both (i) and (ii) can be proved using (iii). However, because (i) and (ii) are more likely to be used, we also included separate, simpler proofs of (i) and (ii). A derivation of (i) and (ii) from (iii) can be obtained as in the proof of the following theorem.
Let $\pi:{\Psi^{\infty}({\mathcal G})}\longrightarrow {\operatorname{End}}({{\mathcal{C}^\infty}_{c}}(M))$ be the vector representation. Then the homogeneous principal symbol $\sigma_{0}(P)$ of $P\in{\Psi^{0}({\mathcal G})}$ can be recovered from the action of $\pi(P)$ on ${\mathcal{C}^\infty}(M)$ by oscillatory testing as in the classical case. Indeed, let $x\in M$ and $\xi\in A_{x}^{*}({\mathcal G})=T^{*}_{x}{\mathcal G}_{x}$ be arbitrary. Then we have $$\label{psy}
\sigma_{0}(P)(\xi)= \lim_{t\to\infty}\big[
e^{-itf}\pi(P)\varphi e^{itf}\big](x)$$ for all $\varphi\in{{\mathcal{C}^\infty}_{c}}(M)$ with $\varphi=1$ near $x$, and all $f\in{\mathcal{C}^\infty}(M,{\mathbb R})$ with $d(f\circ r)\neq 0$ on ${\operatorname{supp}}(\varphi\circ r)$, and $d(f\circ r)|_{x}=\xi$. The proof of uses the fact that the homogeneous principal symbol ($\sigma_{0}(P)(\xi)=\sigma_{0}(P_{x})(\xi)$) as well as the action of $\pi(P)$ on ${{\mathcal{C}^\infty}_{c}}(M)$ ($(\pi(P)h)(x)=P_{x}(h\circ r)|_{{\mathcal G}_{x}}(x)$) are defined using the manifold ${\mathcal G}_{x}$ only. Thus, the classical result applies.
Suppose now that the vector representation $\pi : {C^*({\mathcal G})} \to
\mathcal{B}(L^2(M))$ is injective. Then the above result can be sharpened to a necessary and sufficient condition for Fredholmness, respectively for compactness. We first note that since the principal symbol of a pseudodifferential operator can be determined from its action on functions, the representation $\pi : {\mathfrak{A}({\mathcal G})} \to
\mathcal{B}(L^2(M))$ is also injective. Indeed, this follows from formula .
\[theorem.II\] Assume that the vector representation $\pi$ is injective on ${C^*({\mathcal G})}$. Using the notation from the above theorem, we have.
(i) If $P \in {\Psi^{m}({\mathcal G};F)}$ is such that $\pi(P)$ defines a Fredholm operator $H^{m}(M;F) \to L^2(M;F)$ then the image of $P(E +
A^*A)^{-m/2k} \in {\mathfrak{A}({\mathcal G})}$ in $Q({\mathcal G}) : = {\mathfrak{A}({\mathcal G})}/ {C^*({\mathcal G})}$ is invertible.
(ii) If $\pi(P)$ defines a compact operator $H^{m}(M;F) \to
L^2(M;F)$, then the image of $P(E + A^*A)^{-m/2k}$ in $Q({\mathcal G})$ vanishes.
(iii) If $P \in {\Psi^{0}({\mathcal G};F)}$ or $P \in {\Psi^{m}({\mathcal G};F)}$, $m > 0$, is self-adjoint, elliptic, then $\sigma(\pi(P)) =\sigma(P)$ and $\sigma_{ess}(\pi(P)) =
\sigma_{Q({\mathcal G})}(P)$.
An injective representation $\pi$ of $C^*$-algebras preserves the spectrum, and in particular, $a$ is invertible if, and only if, $\pi(a)$ is invertible.
Denote by $\mathcal B$ the algebra of bounded operators on $L^2(M;F)$. The morphism $\pi':Q({\mathcal G}) \to \mathcal{B/K}$ induced by $\pi$ is also injective. Fix $P_0 \in {\mathfrak{A}({\mathcal G})}$. Then $P_0$ is invertible if, and only if, $\pi(P_0)$ is invertible. By replacing $P_0$ with $P_0 -
\lambda E$, we obtain $\sigma(P_0) = \sigma(\pi(P_0))$. We see then that $P_0$ is invertible modulo ${C^*({\mathcal G}_{M_0})}$ if, and only if, $\pi(P_0)$ is invertible modulo compact operators. This gives $\sigma_{Q({\mathcal G})}(P_0) = \sigma_{ess}(\pi(P_0))$. We thus obtain (iii) if we take $P_0 = P$ or $P_0 = f(P)$, the Cayley transform of $P$.
\(i) By definitions, $\pi(P_1)$ is Fredholm if, and only if, $\pi(P)$ defines a Fredholm operator $H^{m}(M;F) \to L^2(M;F)$. Then $$\begin{aligned}
\pi(P_1) \text{ is Fredholm } &\iff &0
\notin\sigma_{ess}(\pi(P_1)) \\ &\iff& 0 \notin
\sigma_{Q({\mathcal G})}(P_1) \\ &\iff& P_1 \text{ is invertible in }
Q({\mathcal G})\,. \end{aligned}$$
For (ii), a similar reasoning holds: $$\begin{aligned}
\pi(P_1) \text{ is compact }& \iff &
\sigma_{ess}(\pi(P_1^*P_1)) = \{0\} \\ &\iff &
\sigma_{Q({\mathcal G})}(P_1^{*}P_{1}) = \{0\} \\ &\iff & P_1 = 0 \in
Q({\mathcal G})\,. \end{aligned}$$
The criteria in the above theorems can be made even more explicit in particular examples.
\[theorem.III\] Suppose the restriction of ${\mathcal G}$ to $M \smallsetminus M_0$ is amenable, and the vector $\pi$ representation is injective. Then,
(i) $P : H^s(M;F) \to L^2(M;F)$ is Fredholm if, and only if, $P$ is elliptic and $\pi_x(P): H^s({\mathcal G}_x,r^*F) \to L^2({\mathcal G}_x,r^*F)$ is invertible, for any $x \not \in M_0$.
(ii) $P : H^s(M;F) \to L^2(M;F)$ is compact if, and only if, its principal symbol vanishes, and $\pi_x(P) = 0$, for all $x \not \in
M_0$.
(iii) For $P \in {\Psi^{0}({\mathcal G};F)}$, we have $$\sigma_{ess}(\pi(P)) = \bigcup_{x
\not \in M_0}\sigma(\pi_x(P)) \cup
\bigcup_{\xi\in S^{*}{\mathcal G}}{\rm spec}(\sigma_{0}(P)(\xi))\,,$$ where ${\rm spec} (\sigma_{0}(P)(\xi))$ denotes the spectrum of the linear map $\sigma_{0}(P)(\xi):E_x\rightarrow E_x$.
(iv) If $P \in {\Psi^{m}({\mathcal G};F)}$, $m >0$, is self-adjoint, elliptic, then we have $\sigma_{ess}(\pi(P)) = \cup_{x \not \in M_0}\sigma(\pi_x(P))$.
Again, (i) and (ii) follow from (iii) and (iv). The assumption ${\mathfrak{A}({\mathcal G})} = {\mathfrak{A}_r({\mathcal G})}$ implies ${\mathfrak{A}({\mathcal G})}/{\mathfrak I}= {\mathfrak{A}_r({\mathcal G})}/{\mathfrak I}$. Because the groupoid obtained by reducing ${\mathcal G}$ to $M \smallsetminus M_0$ is amenable, the representation $\varrho := \prod \pi_x$, $x \not \in
M_0$ is injective on $Q({\mathcal G})$. This gives $\sigma_{Q({\mathcal G})}(T) = \cup_x
\sigma(\pi_x(T))$, $x \not \in M_0$, for all $T \in {\mathfrak{A}({\mathcal G})}$.
Another explicit criterion is contained in the theorem below.
\[theorem.IV\] Suppose the vector representation $\pi$ is injective and $M \setminus
M_0$ can be written as a union $\bigcup_{j=1}^rZ_j$ of closed, invariant manifolds with corners $Z_j \subset M$.
(i) Let $P\in {\Psi^{m}({\mathcal G};F)}$, then $P : H^s(M) \to L^2(M)$ is Fredholm if, and only if, it is elliptic and $\inn_{Z_j}(P) : H^s(Z_j) \to
L^2(Z_j)$ is invertible, for all $j$.
(ii) Let $P\in {\Psi^{m}({\mathcal G};F)}$, then $P : H^s(M) \to L^2(M)$ is compact if, and only if, its principal symbol vanishes and $\inn_{Z_j}(P) = 0$, for all $j$.
(iii) For $P \in {\Psi^{0}({\mathcal G};F)}$, we have $$\sigma_{ess}(\pi(P)) =
\bigcup_{j=1}^r\sigma(\inn_{Z_j}(P)) \cup
\bigcup_{\xi\in S^{*}{\mathcal G}}{\rm spec}(\sigma_{0}(P)(\xi))\,.$$ (iv) Suppose $P \in {\Psi^{m}({\mathcal G};F)}$, $m >0$, is self-adjoint, elliptic. Then we have $\sigma_{ess}(\pi(P)) = \cup_{j=1}^r\sigma(\inn_{Z_j}(P))$.
The representation ${\mathfrak{A}({\mathcal G})}/{\mathfrak I}\to \oplus_j {\mathfrak{A}({\mathcal G}_{Z_j})}\oplus\mathcal{C}(S^{*}{\mathcal G};{\operatorname{End}}(F))$ given by the restrictions $\inn_{Z_{j}}$ and the homogeneous principal symbol is injective. This gives (iii) and (iv). For $m>0$ note that we have $\sigma_{0}(f(P))={\operatorname{id}}_{F}$ for the Cayley transform $f(P)=(P+iE)(P-iE)^{-1}\in{\mathfrak{A}({\mathcal G})}$ of $P$, and $f^{-1}(1)=\{\infty\}$.
To obtain (i) and (ii) from (iii) as above, it is enough to observe that the operator $P_1 = P(E +
A^*A)^{-m/2k}$, (with $A$ elliptic of order $m$, fixed) belongs to ${\mathfrak I}=
{C^*({\mathcal G}_{M_0})}$ if, and only if, $\inn_{Z_j}(P_1) = 0$ for all $j$, and that $\inn_{Z_j}(P_1) = 0$ if, and only if, $\inn_{Z_j}(P) = 0$.
In Section \[Sec.Examples.III\], we shall see examples of groupoids for which the conditions of the above theorem are satisfied. Having this natural characterization of Fredholmness, it is natural to ask for an index formula for these operators, at least in the case when the restriction of ${\mathcal G}$ to each component $S$ of is such that $r({\mathcal G}_x)$ has constant dimension for all $x$ in a fixed component $S$ of $Y_k \smallsetminus Y_{k-1}$ (that is, when the restriction of $A({\mathcal G})$ to each component $S$ of $Y_k \smallsetminus
Y_{k-1}$ is a regular Lie algebroid). The results of [@NistorIndFam] deal with a particular case of this problem, when $M = Y_n$, the induced foliation on $M$ is a fiber bundle, and the isotropy bundle can be integrated to a bundle of Lie groups that consists either of compact, connected Lie groups or of simply-connected, solvable Lie groups.
Examples III: Applications\[Sec.Examples.III\]
==============================================
For geometric operators $P$, the operators $\pi_x(P)$ and $\pi_Z(P)$ appearing in the statements of the above theorems are again geometric operators of the same kind (Dirac, Laplace, ...). This leads to very explicit criteria for their Fredholmness and to the inductive determination of their spectrum.
If ${\mathcal G}= M \times M$ is the pair groupoid, then ${C^*({\mathcal G})} \cong \mathcal K = \mathcal K(L^2(M))$ and all the results stated above were known for these algebras. In particular, the exact sequence $$0 \to \mathcal K \to {\mathfrak{A}({\mathcal G})} \to \mathcal{C}(S^*M) \to 0$$ is well-known. Moreover, the criteria for compactness and Fredholmness are part of the classical elliptic theory on compact manifolds. There is no need for an inductive determination of the spectrum in this case.
We are now going to apply the results of the previous section to the $c_{n}$-calculus considered in Example \[excn\]. The main result being an inductive method for the determination of the essential spectrum of Hodge-Laplace operators. Because the $b$-calculus corresponds to the special case $c_{H}=1$ for all boundary hyperfaces $H$ of $M$, we in particular answer an question of Melrose on the essential spectrum of the $b$-Laplacian on a compact manifold $M$ with corners [@MelroseScattering Conjecture 7.1].
Let ${\mathcal G}(M,c)$ be the groupoid constructed in Example $\ref{excn}$ for an arbitrary system $c=(c_{H})$.
\[lemma7\] The groupoid ${\mathcal G}(M,c)$ is amenable and the vector representation of ${\mathfrak{A}({\mathcal G}(M,c))}$ is injective.
It is enough to prove that the representation $\pi$ is injective on ${C^*({\mathcal G}(M,c))}$, because we can recover the principal symbol of a pseudodifferential operator from its action on functions, as explained in the previous section.
The groupoid ${\mathcal G}$ is amenable because the composition series of Theorem \[Theorem.CS\] are associated to the groupoids $(S \times S)
\times {\mathbb R}^k$, which are amenable groupoids.
It is then enough to prove that each representation of the form $\pi_x$ is contained in the vector representation. Let $x \in F$ be an interior point. By considering a small open subset of $x$, we can reduce the problem to the case when the manifold $M$ is of the form $[0,\infty)^k \times {\mathbb R}^{n-k}$. Then the result is reduced to the case $k=n=1$ using Proposition \[prop.tens\]. But for this case ${C^*({\mathcal G})}$ is isomorphic to the crossed product algebra $C_0({\mathbb R}\cup
\{-\infty\}) \rtimes {\mathbb R}$ and the vector representation corresponds to the natural representation on $L^2({\mathbb R})$ in which $C_0({\mathbb R}\cup
\{-\infty\})$ acts by multiplication and ${\mathbb R}$ acts by translation. This representation is injective (it is actually often used to define this crossed product algebra). From this the result follows.
The algebra $C_0({\mathbb R}\cup \{-\infty\}) \rtimes {\mathbb R}$ is usually called the algebra of [*Wiener-Hopf operators*]{} on ${\mathbb R}$, for which it is well-known that the vector representation is injective.
Fix now a metric $h$ on $A=A({\mathcal G}(M,c))$, and let $\Delta_p^{c}:=\Delta_{p}^{{\mathcal G}(M,c)}$ be the corresponding Hodge-Laplacian acting on $p$-forms. Note that each boundary hyperface $H$ of $M$ is a closed, invariant submanifold with corners, whereas the interior $M_{0}:=M\setminus\partial M = M\setminus\bigcup H$ is invariant and satisfies ${\mathcal G}_{M_0}\cong M_{0}\times M_{0}$. We are in position then to use the results of the previous section.
First we need some notation. For each hyperface $H$ of $M$, we consider the system $c^{(H)}$ determined by $c^{(H)}_{F}=c_{F'}$ for all boundary hyperfaces $F'$ of $M$ with $F:=H\cap F'\neq \emptyset$, as in the Example \[excn\]. By the construction of the groupoid ${\mathcal G}(M,c)$, $${\mathcal G}(M,c)_H \cong {\mathcal G}(H,c^{(H)}) \times {\mathbb R}.$$ It will be convenient to use the Fourier transform to switch to the dual representation in the ${\mathbb R}$ variable, so that the action of the group by translation becomes an action by multiplication. Then pseudodifferential operators on ${\mathcal G}(M,c)_H$ become families of pseudodifferential operators on ${\mathcal G}(H,c^{(H)})$ parametrized by ${\mathbb R}$. Using also , this reasoning then gives $$\begin{gathered}
\label{ide}
\inn_{H}(\Delta_{p}^{c})=\Delta_{p}^{{\mathcal G}(M,c)_{H}} =
\begin{cases} \lambda^{2} + \Delta_{0}^{ c^{(H)} }, & \text{
if } p = 0, \\ \big (\lambda^{2} + \Delta_{p}^{ c^{(H)} } \big
)\oplus \big ( \lambda^{2} + \Delta_{p-1}^{ c^{(H)} } \big ),
& \text{ if } p > 0, \end{cases} \end{gathered}$$ because for $p > 0$ the space of $p$-forms on the product with $[0,\infty)$ splits into the product of the spaces of $p-1$ and $p$ forms that contain, respectively, do not contain, $dt$, $t \in
[0,\infty)$.
Denote by $m_H^{(p)} = \min \sigma(\Delta_{p}^{c^{(H)}})$ and by $m^{(p)} = \min_H m_H^{(p)}$. Then $m^{(p)}\geq0$ because the Hodge-Laplace operators $\Delta_{p}^{c^{(H)}}$ are positive operators.
On the other hand, note that $\pi(\Delta_p^c)$ is (conjugated to) $\Delta_p$, the Hodge-Laplace operator acting on $p$-forms on the complete manifold $M_0 := M \smallsetminus {\partial}M$, with the induced metric from $A(M,c)$.
\[mainth\] Consider the manifold $M_0$, which is the interior of a compact manifold with corners $M$, with the metric induced from $A(M,c)$. Then the essential spectrum of the (closure of the) Hodge-Laplacian $\Delta_{p}$ acting on $p$-forms on $M_0$ is $[m,\infty)$, with $m = m^{(0)}$, if $p = 0$, or $m = \min \{ m^{(p)}
, m^{(p-1)}\}$, if $p>0$, using the notation explained above.
In particular, the spectrum of $\Delta_{p}$ itself is the union of $[m,\infty)$ and a discrete set consisting of eigenvalues of finite multiplicity.
We are going to apply Theorem \[theorem.IV\] (iv), with $Z_j$ ranging through the set of hyperfaces of $M$; this is possible because of Lemma \[lemma7\]. Furthermore, note that by the definition of the groupoid structure on ${\mathcal G}(M,c)$ in Example \[excn\], the boundary hyperfaces $H$ of $M$ are closed, invariant submanifolds with $M\setminus M_{0}=\bigcup_{H}H$.
For each boundary hyperface $H$ of $M$, we have by $$\sigma(\inn_{H}(\Delta_{p}^{c})) =
\bigcup_{\lambda\in{\mathbb R}}\left(
\lambda^{2}+\left(\sigma(\Delta_{p}^{c^{(H)}})
\cup\sigma(\Delta_{p-1}^{c^{(H)}})\right)\right) =
[\min\{m_{H}^{(p-1)},m_{H}^{(p)}\},\infty)\,,$$ where $\lambda^{2}+\sigma(\Delta_{p-1}^{c^{(H)}})$ is missing if $p=0$. Since $\Delta_{p}$ is essentially self-adjoint and elliptic, Theorem \[theorem.IV\] (iv) completes the proof.
Using an obvious inductive procedure, we then obtain the following more precise result on the spectrum of the Laplace operator acting on functions. [^2]
Let $M_0$ be as above, then the spectrum of the (closure of the) Laplace operator $\Delta_0$ on $M_0$ is $\sigma(\Delta_0) = [0,\infty)$, and hence it coincides with its essential spectrum.
Let $F $ be a minimal face of $M$ (that is, not containing any other face of $M$). Then $F$ is a compact manifold without corners and hence the Laplace operator on $F$ contains $0$ in its spectrum. The above theorem then shows that $[0,\infty) \subset
\sigma_{ess}(\Delta_0)$. On the other hand, $\Delta_0$ is positive, and hence $\sigma(\Delta_0) \subset [0,\infty)$. This completes the proof.
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[^1]: Lauter was partly supported by a scholarship of the German Academic Exchange Service (DAAD) within the [*Hochschulsonderprogramm III von Bund und Ländern*]{}, and the Sonderforschungsbereich 478 [*Geometrische Strukturen in der Mathematik*]{} at the University of Münster. Nistor was partially supported an NSF Young Investigator Award DMS-9457859 and NSF Grant DMS-9971951. [ **http:[//]{}www.math.psu.edu[/]{}nistor[/]{}**]{} .
[^2]: This refinement of our theorem was suggested to us by a comment of Richard Melrose during the talk of the first named author at the Oberwolfach meeting [*Geometric Analysis and Singular Spaces*]{}, June 2000.
|
---
abstract: 'The “optical springs” regime of the signal-recycled configuration of laser interferometric gravitational-wave detectors is analyzed taking in account optical losses in the interferometer arm cavities. This regime allows to obtain sensitivity better than the Standard Quantum Limits both for a free test mass and for a conventional harmonic oscillator. The optical losses restrict the gain in sensitivity and achievable signal-to-noise ratio. Nevertheless, for parameters values planned for the Advanced LIGO gravitational-wave detector, this restriction is insignificant.'
author:
- 'F. Ya. Khalili, V. I. Lazebny, S. P. Vyatchanin'
title: 'Sub-SQL Sensitivity via Optical Rigidity in Advanced LIGO Interferometer with Optical Losses '
---
Introduction
============
Signal-recycled “optical-springs” topology of interferometric gravitational-wave detectors [@Buonanno2001; @Buonanno2002] is now considered as a likely candidate to design the second generation of these detectors, such as Advanced LIGO [@WhitePaper1999; @Fritschel2002]. This topology offers an elegant way to overcome the Standard Quantum Limit (SQL) for a free mass — a characteristic sensitivity level when the measurement noise of the position meter is equal to its back-action noise [@67a1eBr; @92BookBrKh; @03a1BrGoKhMaThVy].
This method is based on the use of optical pondermotive rigidity which exists in detuned electromagnetic cavities [@67a1eBrMa; @70a1eBrMa]. It turns the test masses in a gravitational-wave detector into harmonic oscillators, thus providing a resonance gain in test masses displacement signal [@99a1BrKh; @01a1BrKhVo]. In the signal recycling topology, the pondermotive rigidity can be created relatively easy by adjusting position of the signal recycling mirror. In this case, only the signal (anti-symmetric) optical mode eigenfrequency is changed. The arm cavities and the power (symmetric) mode remain resonance-tuned causing, therefore, no additional problems with the pumping power.
In large-scale optical systems having bandwidth equal to or smaller than the signal frequency $\Omega$, the optical rigidity has a complicated frequency dependence [@97a1BrGoKh; @01a2Kh; @Buonanno2002; @05a1LaVy]. This feature allows to obtain not one but two mechanical resonances and consequently two minima in the noise. Alternatively, these minima can be placed close to each other or even superimpose, thus providing a single wider “well” in the noise spectral density.
The former regime was explored in detail in several articles [@Buonanno2001; @Buonanno2002; @Buonanno2003; @Harms2003; @Buonanno2004; @05a1LaVy]. The latter was examined rather briefly in papers [@01a2Kh; @05a1LaVy]. At the same time, it looks very promising for detection narrow-band gravitational-wave signals with known frequencies, in particular, those from the neutron stars. In this article we present an in depth analysis of this regime.
In Sec.\[sec:theor\_intro\] we analyze dynamic properties of the frequency-dependent pondermotive rigidity in no optical loss case, including, in particular, an instability inherent to the electromagnetic rigidity. This section is based, in part, on the results obtained in the articles [@01a2Kh; @Buonanno2003; @05a1LaVy]
In Sec.\[sec:comp\] we compare the optical rigidity-based scheme with other methods of circumvent the SQL for a free mass and show that it has a significant advantage, namely, it is much less vulnerable to optical losses.
In Sec.\[sec:LIGO\] we calculate the sensitivity of the Advanced LIGO gravitational-wave detectors in the double-resonance optical springs regime.
To clarify this consideration we try to avoid bulky calculations in the main text. In Appendix \[app:analysis\] we give detailed analysis and calculations of sensitivity and signal-to-noise ratio for Advanced LIGO interferometer. In Appendices \[app:xi\] and \[app:snr\] we provide calculations for simplified model without optical losses.
Frequency-dependent optical rigidity. No optical losses {#sec:theor_intro}
=======================================================
“Conventional” v.s. “double” resonances {#ord_vs_dbl}
---------------------------------------
In a single Fabry-Perot cavity, the pondermotive rigidity is equal to [@01a2Kh; @05a1LaVy]: $$\label{K_single}
K(\Omega) = \frac{2\omega_p{\cal E}}{L^2}\,
\frac{\delta}{{\cal D}(\Omega)}\,,\quad
{\cal D}(\Omega) = (-i\Omega+\gamma)^2+\delta^2 \,,$$ where $\Omega$ is the observation (side-band) frequency, $\omega_p$ is the pumping frequency, $\delta=\omega_p-\omega_o$ is the detuning, $\omega_o$ is the cavity eigenfrequency, $L$ is its length, ${\cal E}$ is the optical energy stored in the cavity, and $\gamma$ is the cavity half-bandwidth.
In articles [@Buonanno2003; @05a1LaVy] the signal-recycled gravitational-wave detectors topology was considered in detail and its equivalence to a single cavity was shown. It was shown, in particular, that Eq.(\[K\_single\]) is valid for this topology too, with obvious substitution of $\gamma$ and $\delta$ by the anti-symmetric optical mode half-bandwidth $\gamma_0$ and detuning $\delta_0$ \[see Appendix \[app:analysis\] and Eqs.(\[gamma\_0load\]),(\[delta\_0\])\]. Energy ${\cal E}$ in this case is equal to the total optical energy stored in both interferometer arms: $${\cal E} = \frac{4I_cL}{c} \,,$$ where $I_c$ is the power circulating in each arm. For the sake of consistency with other parts of the paper, notations $\delta_0$ and $\gamma_0$ will be used throughout this paper.
It should be noted that for the narrow-band regimes which we dwell on, $\gamma_0$ have to be small: $$\label{nb}
\gamma_0 \ll \delta_0 \sim \Omega \,.$$ Therefore, we neglect for a while term $\gamma_0$, setting $$\label{K_simple}
K(\Omega) = \frac{K_0\delta_0^2}{\delta_0^2-\Omega^2} \,,$$ where $K_0$ is the rigidity value at zero frequency: $$K_0 = \frac{2\omega_p{\cal E}}{L^2\delta_0}\,.$$ A detailed analysis is provided in \[sec:instab\] and \[sec:LIGO\].
Consider a harmonic oscillator having mass $m$ and rigidity (\[K\_single\]). We can write the equation of motion as follows: $$\label{eq_motion}
m\frac{d^4x(t)}{dt^4} + m\delta_0^2\frac{d^2x(t)}{dt^2}
+ \delta_0^2K_0\,x(t)
= \frac{d^2F(t)}{dt^2} + \delta_0^2F(t) \,,$$ where $x$ is the oscillator coordinate and $F$ is an external force acting on it.
In general the system has two resonances with frequencies $$\label{Omega_pm}
\Omega_\pm = \sqrt{\frac{\delta_0^2}{2}
\pm\sqrt{\frac{\delta_0^4}{4}-\frac{\delta_0^2K_0}{m}}} \,.$$ In articles [@Buonanno2001; @Buonanno2002; @Buonanno2003] they are referred to as “mechanical” and “optical”, because in asymptotic case $K_0\to 0$, the frequencies are equal to: $$\begin{aligned}
\Omega_+ &= \delta_0 \,, & \Omega_- &=
\sqrt{\frac{K_0}{m}} \,,\end{aligned}$$ Therefore, the high-frequency resonance can be readily interpreted as a result of the optical power sloshing between the optical cavity and detuned pumping field, and the low-frequency one — as a conventional resonance of mechanical oscillator with $K_0$ rigidity.
The second-order pole [@01a2Kh; @05a1LaVy], or double resonance case takes place if these two frequencies are equal to each other: $$\label{dbl_res}
\Omega_+ = \Omega_- = \Omega_0 = \frac{\delta_0}{\sqrt{2}}\,,$$ [*i.e.*]{} if $$\label{K_0}
K_0 = \frac{m\delta_0^2}{4} \quad\Leftrightarrow\quad
{\cal E} = {\cal E}_{\rm crit} \,,$$ where $$\label{E_crit}
{\cal E}_{\rm crit} = \frac{mL^2\delta_0^3}{8\omega_0}$$ is the critical energy. In this case, the equation of motion (\[eq\_motion\]) has the following form: $$\label{eq_motion_double}
m\left(\frac{d^2}{dt^2} + \Omega_0^2\right)^2x(t)
= \frac{d^2F(t)}{dt^2} + 2\Omega_0^2F(t) \,.$$
It is useful to consider action of resonance force $F(t)=F_0\cos\Omega_0t$ on this system. Solving Eq.(\[eq\_motion\_double\]), we obtain: $$x(t) = \frac{F_0}{8m}\left(-t^2\cos\Omega_0t
+ \frac{t\sin\Omega_0t}{\Omega_0}\right) \,.$$ The leading term in amplitude of $x(t)$ grows with time as $t^2$. At the same time, for a conventional oscillator we have $$x(t) = \frac{F_0t\sin\Omega_0t}{2m\Omega_0}\,,$$ and for a free mass, $$x(t) = \frac{F_0\big(1-\cos\Omega_0t\big)}{m\Omega_0^2}\,.$$ Therefore, response of the “double resonance” oscillator on the resonance force is $(\Omega_0t/4)$ times stronger than that of a conventional harmonic oscillator, and $(\Omega_0t/4)^2$ times stronger than that of a free mass one.
It was shown in [@01a2Kh; @05a1LaVy], that due to this feature, the “double resonance” oscillator has much smaller value of the Standard Quantum Limit for narrow-band signals, than both free mass and conventional harmonic oscillators. It is convenient to express this gain in terms of dimensionless parameter $$\label{xi_def}
\xi^2 = \frac{S_h(\Omega)}{ h_{\rm SQL}^2(\Omega)} \,,$$ where $S_h$ is the single-sided spectral density of detector noise normalized as the equivalent fluctuational gravitation wave $h$, $$\label{h_SQL}
h_{\rm SQL}^2(\Omega) = \frac{8\hbar}{mL^2\Omega^2}$$ is the value of $S_h$ corresponding to SQL. For conventional SQL-limited gravitation wave detectors (with free test masses), $\xi\ge
1$.
It is shown in Appendix \[app:xi\], that in case of a conventional first-order resonance, the equivalent noise curve has a “well” at resonance frequency $\Omega_0$, which provides the gain in sensitivity: $$\label{xi_oscill}
\xi_{\rm oscill} = \sqrt{\frac{\Delta\Omega}{\Omega_0}} \,,$$ where $\Delta\Omega$ is the bandwidth where this gain is provided. In particular, this gain can be obtained by using either the “mechanical” or “optical” resonance of optical rigidity).
In the case of “double resonance” oscillator, this gain can be substantially more significant: $$\label{xi_dbl}
\xi_{\rm dbl} = \frac{\Delta\Omega}{\Omega_0}
= \xi_{\rm oscill}\sqrt{\frac{\Delta\Omega}{\Omega_0}} \,.$$ Even better result can be obtained if optical energy is slightly smaller than the critical value (\[E\_crit\]): $$\label{E_subcrit}
K_0 = \frac{m\delta_0^2}{4}(1-\eta^2) \quad\Leftrightarrow\quad
{\cal E} = {\cal E}_{\rm crit}(1-\eta^2)\,, \quad \eta\ll 1\,.$$ In this case, function $\xi(\Omega)$ has two minima, which correspond to two resonance frequencies $$\Omega_\pm = \Omega_0\sqrt{1\pm\eta}
\approx \Omega_0\left(1\pm\frac{\eta}{2}\right) \,,$$ and a local maximum at frequency $\Omega_0$. If parameter $\eta$ is equal to the optimal value $$\eta_c = \xi(\Omega_0) \,,$$ then the bandwidth is $\sqrt{2}$ times wider than in pure double resonance case for the same value of $\xi$:
$$\label{xi_enh_dbl}
\xi_{\rm enh\,dbl} = \frac{\xi_{\rm dbl}}{\sqrt{2}}
= \frac{\Delta\Omega}{\sqrt{2}\,\Omega_0}\,.$$
Above considerations give an important result concerning the signal-to-noise ratio values for different regimes [@Chen_private].
The signal-to-noise ratio is equal to: $$\label{snr}
{\rm SNR} = \frac{2}{\pi}\int_0^{\infty}
\frac{|h(\Omega)|^2}{S_h(\Omega)}\,d\Omega \,,$$ where $h(\Omega)$ is the gravitation wave signal spectrum. It is shown in Appendix \[app:snr\], that for a conventional resonance-tuned interferometer (without optical springs), $${\rm SNR}_{\rm no\ springs} = {\cal N}\times\frac{|h(\Omega_0)|^2\Omega_0}
{h^2_{\rm SQL}(\Omega_0)} \,,$$ where factor ${\cal N}$ for wide band signal (with bandwidth $\Delta\Omega_{\rm signal}\ge \Omega_0$) is about unity: for short pulse ${\cal N}\approx 2.0$ and for step-like signal ${\cal N}\approx
0.7$. For narrow-band signals with $\Delta\Omega_{\rm
signal}\ll\Omega_0$ we have ${\cal N} \sim \Delta\Omega_{\rm
signal}/\Omega_0$.
In the narrow-band cases described above the signal-to-noise ratio can be estimated as follows: $$\label{snr_nb}
{\rm SNR} \sim \frac{|h(\Omega_0)|^2\Delta\Omega}{S_h(\Omega_0)}
= \frac{\Delta \Omega}{\xi^2}\,
\frac{|h(\Omega_0)|^2}{h_{\rm SQL}^2(\Omega_0)} \,.$$ (we omit here a numeric factor of the order of unity). In case of a conventional harmonic oscillator, it can be shown using Eqs.(\[xi\_oscill\],\[snr\_nb\]), that: $$\label{snr_oscill}
{\rm SNR}_{\rm oscill}
\approx \frac{2|h(\Omega_0)|^2\Omega_0}{h_{\rm
SQL}^2(\Omega_0)}$$ (numeric factors in this formula and in Eq.(\[snr\_dbl\]) below are obtained through rigorous integration in Eq.(\[snr\]), see Appendix \[app:snr\]). This value [*does not depend*]{} on $\Delta\Omega$. Thus conventional harmonic oscillator provide arbitrary (in the case of zero losses) high sensitivity at resonance frequency, and also gain in signal-to-noise ratio equal to $\sim \Omega_0/\Delta\Omega_{\rm signal}$ for narrow-band signals as compared with conventional interferometer. At the same time, the value (\[snr\_oscill\]) is fixed and can not be increased by improving the meter parameters, and there is no gain in signal-to-noise ratio for wide-band signals.
On the other hand, in case of double-resonance oscillator, it stems form Eqs.(\[xi\_dbl\], \[snr\_nb\]), that: $$\label{snr_dbl}
{\rm SNR}_{\rm dbl}
\approx \frac{\sqrt 2 \Omega_0}{\Delta\Omega}\times
\frac{|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}
\approx \frac{ \Omega_0}{\sqrt 2\Delta\Omega} \times
{\rm SNR}_{\rm oscill}\,.$$
Therefore, the double resonance oscillator allows to increase both the resonance frequency sensitivity and the signal-to-noise ratio by decreasing the bandwidth $\Delta\Omega$.
The signal-to-noise ratio for Advanced LIGO interferometer with optical rigidity and influence of optical losses is considered in Sec.\[sec:snr\].
Dynamic instability {#sec:instab}
-------------------
It is well known [@67a1eBrMa; @70a1eBrMa] that [*positive*]{} pondermotive rigidity is accompanied by [*negative*]{} dumping, [*i.e.*]{} a pondermotive rigidity-based oscillator is always unstable. In case of frequency-depended rigidity, this instability was calculated in article [@97a1BrGoKh] (see also [@Buonanno2001; @Buonanno2002]).
If the eigenfrequencies are separated well apart from each other, $\Omega_+-\Omega_-\gg\gamma_0$, then the characteristic instability time equals to $$\tau_{\rm instab} \approx \left(
\frac{\gamma_0\Omega_0^2}{\Omega_+^2-\Omega_-^2}
\right)^{-1}
\sim \gamma_0^{-1} \sim 0.1\div 1{\rm s}\,.$$ The instability becomes more strong, if $\Omega_+-\Omega_-\to
0$. In double resonance case ($\Omega_+=\Omega_-$), the instability time is equal to $$\tau_{\rm instab}\approx
\frac{2}{\sqrt{\gamma_0\Omega_0}}\,.$$ However, inequality $\Omega_0\tau_{\rm instab}\gg 1$ holds in this case too.
It has to be noted that, in principle, any instability can be dumped without affecting signal-to-noise ratio, if the feedback system sensor sensitivity is only limited by quantum noises [@Buonanno2002] but its implementation is a separate problem which we do not discuss here..
Comparison with other methods to overcome SQL and influence of optical losses {#sec:comp}
=============================================================================
“Real” vs. “virtual” rigidities
-------------------------------
From the Quantum Measurements Theory point of view, a laser interferometric gravitational-wave detector can be considered as a meter which continuously monitors position $\hat x(t)$ of a test mass $m$ [@92BookBrKh; @Buonanno2002; @03a1BrGoKhMaThVy]. Output signal of this meter is equal to: $$\label{tilde_x}
\tilde x(t) = \hat x(t) + \hat x_{\rm meter}(t) \,,$$ where $\hat x_{\rm meter}(t)$ is the measurement noise and $\hat x(t)$ is the “real” position of the test mass. It includes its responses on the signal force $$\label{F_signal}
F_{\rm signal}(t) = \frac{mL\ddot h(t)}{2}$$ and on the meter back-action force $\hat F_{\rm meter}(t)$: $$\hat x(t) = {\bf Z}^{-1}[F_{\rm signal}(t) + \hat F_{\rm
meter}(t)]$$ (the terms containing the initial conditions can be omitted, see article [@03a1BrGoKhMaThVy]). Here ${\bf Z}$ is a differential operator describing evolution of the test object. For a free test mass ([*i.e.*]{} for the initial LIGO topology), $$\label{Z_fm}
{\bf Z} = m\frac{d^2}{dt^2} \,,$$ If rigidity $K$ (including the frequency-dependent pondermotive one) is associated with the test mass, then $$\label{Z_oscill}
{\bf Z} = m\frac{d^2}{dt^2} + K$$ Therefore, the signal force estimate is equal to $$\label{tilde_F}
\tilde F(t) \equiv {\bf Z}\tilde x(t)
= F_{\rm signal}(t) + \hat F_{\rm sum\,noise}(t)\,,$$ where $$\label{F_noise}
\hat F_{\rm sum\,noise}(t)
= \hat F_{\rm meter}(t) + {\bf Z}\hat x_{\rm meter}(t)\,.$$ is the meter total noise.
The back-action noise $\hat F_{\rm meter}(t)$ is proportional to the amplitude quadrature component of the output light. The measurement noise $\hat x_{\rm meter}(t)$, in the simplest case of SQL-limited detector, is proportional to the phase quadrature component. In this case the spectral densities of these noises satisfy the following uncertainty relation (see e.g. [@92BookBrKh]): $$\label{SxSF}
S_xS_F\ge\hbar^2 \,.$$
In more sophisticated schemes which allow to overcome SQL, the noises $F_{\rm meter}(t)$ and $\hat x_{\rm meter}(t)$ correlate with each other. In this case, the back-action noise can be presented as follows: $$\label{VM}
F_{\rm meter}(t) = F_{\rm meter}^{(0)}(t) + {\cal K}\hat x_{\rm meter}(t)
\,,$$ where $F_{\rm meter}^{(0)}(t)$ is the back-action noise component non-correlated with the measurement noise and coefficient $\cal K$ can be referred to as “virtual” rigidity.
Note that it is precisely the idea of quantum variational measurement [@95a1VyZu; @96a1eVyMa; @96a2eVyMa; @98a1Vy; @02a1KiLeMaThVy]. In conventional optical position meters, including the LIGO interferometer, one measures the phase quadrature component in output wave. This component contains both the measurement noise ($x_\text{meter}$) produced by phase fluctuations in input light wave and the back action noise ($F_{\rm meter}$) caused by amplitude fluctuations (optimization of the sum of these two uncorrelated noises produces SQL). However, using homodyne detector one can measure tuned mix of the phase and amplitude quadratures of output waves and this mix can be selected in such a way that the back action noise [*can be compensated*]{} by the noise of the amplitude quadrature. It can be considered as introduction of [*correlation*]{} between the back action and the measurement noises as presented in Eq.(\[VM\]). We see that the “virtual” rigidity only relates to the measurement procedure (homodyne angle).
Substituting Eq (\[VM\]) into Eq.(\[tilde\_F\]), we obtain, that $$\label{tilde_F_1}
\hat F_{\rm sum\,noise}(t)
= \hat F_{\rm meter}^{(0)}(t) + {\bf Z}_{\rm eff}\hat x_{\rm meter}(t)\,,$$ where $$\label{two_K}
{\bf Z}_{\rm eff} = {\bf Z} + {\cal K}
= m\frac{d^2}{dt^2} + K_{\rm eff} \,,$$ and $$K_{\rm eff} = K + {\cal K} \,.$$ is the [*effective rigidity*]{}.
Thus, [*in the lossless case, the total meter noise (\[tilde\_F\_1\]) only contains the sum of real rigidity $K$ and virtual one ${\cal K}$, and replacement of any one of them by another one does not change the total noise spectral density and the signal-to-noise ratio*]{} [@94a1eKhSy].
Spectral densities of the noises $\hat x_{\rm meter}(t)$ and $\hat F_{\rm
fluct}^{(0)}(t)$ also satisfy the uncertainty relation $$\label{SxSF0}
S_xS_F^{(0)}\ge\hbar^2 \,,$$ which does not permit simultaneously making both noise terms in Eq.(\[tilde\_F\_1\]) arbitrary small. However, factor ${\bf Z}_{\rm eff}$ can be made equal to zero by setting $$\label{K_res}
K_{\rm eff} = m\Omega^2 \,.$$ In this case, only noise $\hat F_{\rm meter}^{(0)}(t)$ remains in Eq.(\[tilde\_F\_1\]). In principle this noise can alone be made arbitrary small, thus providing arbitrary high sensitivity.
Both “real” and “virtual” (created by the noise correlation) can be used for this purposes. Applying the idea of variational measurement a simple frequency-independent cross-correlation (and thus frequency-independent virtual rigidity ${\cal K}$) can be created relatively easily by using a homodyne detector. In this case, Eq.(\[K\_res\]) is fulfilled at some given frequency, creating resonance gain in sensitivity similar to one provided by a conventional harmonic oscillator.
A frequency-dependent cross-correlation and thus a frequency-dependent virtual rigidity ${\cal K}$ can be induced through modification of the input and/or output optics of the gravitation-wave detectors using additional large-scale filter cavities [@02a1KiLeMaThVy]. In this case, condition (\[K\_res\]) can be fulfilled and thus the sensitivity better than SQL is obtained in any given frequency range.
Consider now the real pondermotive rigidity. It is evident that it can not be tuned in such a flexible way as the virtual one. In the double-resonance case, graphics of $K(\Omega)$ and $m\Omega^2$ touch each other only at frequency $\Omega_0$. However, around this point, $$\label{Kcross}
\big(K_{\rm eff}(\Omega)\big) - m\Omega^2 \sim
(\Omega-\Omega_0)^2 \,,$$ instead of $\Omega-\Omega_0$ for the case of a conventional frequency-independent rigidity. As it was shown in Sec.\[sec:theor\_intro\], slightly better results can be obtained by using sub-critical pumping ${\cal E}<{\cal E}_{\rm crit}$, which provide two closely placed first-order resonances, see Fig.\[fig:regimes\](b).
\[cc\]\[cc\][$(\Omega/\Omega_0)^2$]{} \[ct\]\[cb\][$K(\Omega)/m\Omega_0^2$]{} ![Plots of the frequency-dependent rigidity $K(\Omega)$ as a function of $\Omega^2$ (solid line); dashed line corresponds to $m\Omega^2$; (a) — critical pumping (the double resonance regime); (b) — sub critical pumping.[]{data-label="fig:regimes"}](regime_crit.eps "fig:"){width="45.00000%" height="26.00000%"} ![Plots of the frequency-dependent rigidity $K(\Omega)$ as a function of $\Omega^2$ (solid line); dashed line corresponds to $m\Omega^2$; (a) — critical pumping (the double resonance regime); (b) — sub critical pumping.[]{data-label="fig:regimes"}](regime_sub.eps "fig:"){width="45.00000%" height="26.00000%"}
Influence of optical losses {#sec:losses}
---------------------------
It is possible to conclude from above considerations that the virtual rigidity provides a more promising solution than the real pondermotive one. However, as we show in this subsection, the pondermotive rigidity has one important advantage: it is much less vulnerable to losses in optical elements. Our consideration will be based on the following statement, which follows from Eqs.(\[ae\_to\_pq\]): [*a lossy optical position meter is equivalent to the similar lossless one with gray filter attached to its signal port.*]{} This filter transmittance has to be equal to: $$T_{\rm equiv}^2 = \frac{\gamma_0^{\rm load}}{\gamma_0}\,,$$ where $$\label{gamma0}
\gamma_0 = \gamma_0^{\rm load} + \gamma_0^{\rm loss}\,,$$ $\gamma_0^{\rm load}$ is the term describing signal mode half-bandwidth $\gamma_0$ describing coupling with photodetector and $\gamma_0^{\rm loss}$ is the term describing optical losses.
Term “similar” means, that (i) both meters have same dynamic parameters, [*i.e.*]{} bandwidths and eigenfrequencies of the cavities, etc. In particular, half-bandwidth of the lossless meter has to be equal to $\gamma_0$ and (ii) optical energies stored in the lossless meter and the lossy one have to be equal to each other.
Therefore, the lossy meter output signal can be presented in the following form (compare with Eq.(\[tilde\_x\])): $$\tilde x_{\rm lossy}(t) = \hat x(t) + \hat x_{\rm
meter}(t)
+ |{\cal A}_0|\hat x_{\rm loss}(t) \,,$$ where $$\label{A0}
|{\cal A}_0| = \frac{\sqrt{1-T_{\rm equiv}^2}}{T_{\rm
equiv}}
= \sqrt{\frac{\gamma_0^{\rm loss}}{\gamma_0^{\rm load}}}$$ is the effective loss factor and $\hat x_{\rm loss}(t)$ is the additional noise produced by the optical losses uncorrelated with $x_\text{meter}$. Spectral density of this noise is also equal to $S_x$, because both $\hat x_{\rm loss}(t)$ and $\hat x_{\rm loss}(t)$ are in essence zero-point fluctuations normalized by the transfer function of measurement system. The key point is that while the back-action noise $\hat F_{\rm meter}(t)$ can correlate with the measurement noise $\hat x_{\rm meter}(t)$, it [*can not correlate*]{} with additional noise $\hat x_{\rm loss}(t)$: $$\label{tilde_F_loss}
\hat F_{\rm sum\,noise}(t) = \hat F_{\rm meter}^{(0)}(t)
+ {\bf Z}_{\rm eff}\hat x_{\rm meter}(t)
+ |{\cal A}_0|{\bf Z}\hat x_{\rm loss}(t)$$ \[compare with Eq.(\[tilde\_F\_1\])\].
It follows from this equation that in the lossy meter case there is no symmetry of the real and virtual rigidities: the term proportional to the meter noise $\hat x_{\rm meter}(t)$ still depends on the effective rigidity, but the new losses term only depends on the real rigidity.
Compare now too particular cases of “pure real” and “pure virtual” rigidities.
In the first case, ${\bf Z}_{\rm eff}={\bf Z}$, and $$\hat F_{\rm sum\,noise}(t) = \hat F_{\rm meter}^{(0)}(t)
+ {\bf Z}\big[\hat x_{\rm meter}(t)
+ |{\cal A}_0|\hat x_{\rm loss}(t)\big]\,.$$ Therefore, both terms proportional to $\hat x_{\rm meter}(t)$ and $\hat
x_{\rm loss}(t)$ can be canceled by setting ${\bf Z}=0$, thus providing arbitrary high sensitivity at least for one given frequency.
In the second case, $$\hat F_{\rm sum\,noise}(t) = \hat F_{\rm meter}^{(0)}(t)
+ {\bf Z}_{\rm eff}\hat x_{\rm meter}(t)
- m\Omega^2|{\cal A}_0|\hat x_{\rm loss}(t)\,.$$ Suppose that ${\bf Z}_{\rm eff}$ is canceled using some QND procedure. In this case, the sum noise will still consist of two non-correlated parts proportional to $\hat F_{\rm meter}^{(0)}(t)$ and $\hat x_{\rm
loss}(t)$, and its spectral density will be equal to: $$S_{\rm sum}(\Omega) = S_F^{(0)} + m^2\Omega^4|{\cal A}_0|^2S_x \,.$$ Taking into account uncertainty relation (\[SxSF0\]), it is easy to see that $$S_{\rm sum}(\Omega) \ge 2|{\cal A}_0|\hbar m\Omega^2 \,,$$ or $$\label{xi_min_loss}
\xi \equiv \sqrt{\frac{S_{\rm sum}(\Omega)}{S_{\rm SQL}(\Omega)}}
\ge \sqrt{|{\cal A}_0|}
= \left(\frac{\gamma_0^{\rm loss}}{\gamma_0^{\rm
load}}\right)^{1/4}\,,$$ where $$\label{S_SQL}
S_{\rm SQL}(\Omega) = \frac{m^2L^2\Omega^4}{4}\,h_{\rm SQL}^2(\Omega)
= 2\hbar m\Omega^2$$ \[see Eqs.(\[h\_SQL\],\[F\_signal\])\].
The restriction (\[xi\_min\_loss\]) shows that the use of “pure virtual” rigidity (variational measurement) is very sensitive to losses — for parameters from Table \[tab1\] we have $\big(\gamma_0^\text{loss}/\gamma_0^\text{load}\big)^{1/4}\simeq 0.7$ (!).
For conventional (resonance-tuned) Advanced LIGO topology, $\gamma_0\simeq\Omega\simeq 2\pi\times 100{\rm s}^{-1}\gg \gamma_0^{\rm
loss}$ and “virtual” rigidity can be introduced by means of the variational measurement. In this case the optical losses restrict the sensitivity by the following value: $$\label{xi_min_std}
\xi\simeq \left(\frac{\gamma_0^{\rm loss}}{\Omega}\right)^{1/4}
\simeq 0.2$$ (for the value of $\gamma_0^{\rm loss}$, see Table\[tab1\]).
In general both real and virtual rigidities exist in the signal-recycled configuration of the interferometric gravitation-wave detectors as well as for that of a single detuned cavity. However, in the narrow-band regimes (\[nb\]), the real rigidity dominates: it is approximately $\delta_0/\gamma_0$ times stronger than the virtual one (compare Eqs.(\[K\])) and (\[K\_eff\]), or (\[Z\_approx\]) and (\[Z\_approx\_eff\]), which differ by terms of the order of magnitude $\lesssim\gamma_0/\delta_0$ only). Due to this reason, this regime is free from limitation (\[xi\_min\_loss\]).
It is also interesting to note that while both $K$ and ${\cal K}$ contain, in general, imaginary parts, in effective rigidity $K_{\rm
eff}$ these imaginary parts exactly compensate each other.
-------------------------------------------- ---------------------------------------------
Transmissivity of SR mirror $T_s^2=0.05$
Transmissivity of input mirrors in arms $T^2=0.005$
Loss coefficient of each mirror in arms $ {\cal
A}_1^2={\cal A}_2^2=
1.5\times 10^{-5}$
Length of interferometer arm $L=4$ km
Effective loss factor $|{\cal A}_0|^2= 0.24$
Relaxation rate of difference mode $\gamma_{0}=2.9\ \text{s}^{-1}$
“Intrinsic” relaxation rate $\gamma_0^\text{loss} =
0.56\ \text{s}^{-1}$
Mean frequency of gravitational wave range $\Omega_{0}=2\pi\times 100\ \text{s}^{-1}$
-------------------------------------------- ---------------------------------------------
: Parameters planned to use in Advanced LIGO [@WhitePaper1999]. In estimates of $|{\cal A}_0|^2$ and $\gamma_0$ using formulas (\[gamma0\],\[A0\],\[gamma\_0load\]) we assume that $|1+R_se^{2i\phi}|^2\simeq 2$ []{data-label="tab1"}
Advanced LIGO sensitivity in double-resonance regime {#sec:LIGO}
====================================================
The sum noise spectral density
-------------------------------
Sensitivity of the signal-recycled Advanced LIGO topology in the narrow-band approximation (close to the double resonance) is calculated in Appendix \[app:analysis\]. It is shown, that in this case \[see Eq. (\[xi(nu)\])\] $$\label{xi_real}
\xi^2(\Omega) \approx \xi_0^2+
\frac{|{\cal A}_\alpha|^2}{4\gamma_0^{\rm loss}\Omega_0^3}
\left(4\nu^2-\eta_\alpha^2\Omega_0^2\right)^2$$ where $\nu=\Omega-\Omega_0$, $$\label{C}
\xi_0^2=\frac{\gamma_0^{\rm loss}}{\Omega_0|{\cal A}_\alpha|^2}\,C\,,\quad
C = 1 + \frac{3}{\sqrt{2}}\,|{\cal A}_\alpha|^2 + |{\cal A}_\alpha|^4\,,$$ and $\eta_\alpha,\ |{\cal A}_\alpha|$ are basically parameters $\eta,\
|{\cal A}_0|$ corrected to take into account virtual rigidity ([*i.e.*]{} homodyne angle $\alpha$), see Eqs.(\[eta\_A\_alpha\]).
It follows from Eq.(\[xi\_real\]) that the single minimum dependence of sensitivity $\xi(\Omega)$ takes place if $\eta_\alpha=0$ with bandwidth $\Delta \Omega$ equal to $$\Delta \Omega=\sqrt{2\gamma_0^\text{loss}\Omega_0
\frac{\sqrt C}{|{\cal A}_\alpha|^2}\, }.$$ On Figs.\[oneA\], \[oneB\] we present the sensitivity plots for various sets of parameters which allows to realize single minimum dependence of $\xi$ on different frequencies $\Omega_0$ and with different depths for the case when effective loss factor $|{\cal A}_0|^2=0.24$ (as planned in Advanced LIGO, solid curves) and for no losses case (dotted curves). We see that the sensitivity degradation due to the optical losses is negligibly small. Possibility of scaling the frequency $\Omega_0$ is also demonstrated in this plots.
\[rc\]\[cl\][$0.1$]{} \[ct\]\[cb\][$1$]{} \[ct\]\[cb\][$0.4$]{} \[ct\]\[cb\][$0.6$]{} \[ct\]\[cb\][$0.8$]{} \[ct\]\[cb\][$2$]{} \[lt\]\[rb\][$\Omega/\Omega_{00}$]{} \[cc\]\[lc\][$\frac{\sqrt S_h}{h_{SQL}}$]{} \[cc\]\[lc\][$\frac{Z(\Omega)}{m\Omega_{00}^2}$]{} \[cb\]\[ct\][$\alpha=0$,$\eta=0$, $\gamma_{0}/\Omega_0=0.01$]{} \[cb\]\[ct\][$\alpha=\pi/2$, $\eta=0$, $\gamma_{0}/\Omega_0=0.01$]{} \[cb\]\[ct\][$\eta=0$, $\gamma_{0}/\Omega_0=0.01$]{}\
\
\[rc\]\[cl\][$0.1$]{} \[ct\]\[cb\][$1$]{} \[cc\]\[lc\][$10$]{} \[ct\]\[cb\][$0.7$]{} \[ct\]\[cb\][$0.9$]{} \[ct\]\[cb\][$0.8$]{} \[ct\]\[cb\][$2$]{} \[lt\]\[rb\][$\Omega/\Omega_{0}$]{} \[cc\]\[lc\][$\xi$]{} \[cc\]\[lc\][$\frac{Z(\Omega)}{m\Omega_{oo}^2}$]{} \[cb\]\[ct\][$\alpha=0$,$\eta=0$]{} \[cb\]\[ct\][$\alpha=\pi/2$,$\eta=0$]{}
\[rc\]\[cl\][$0.1$]{} \[rc\]\[cl\] \[ct\]\[cb\][$1$]{} \[ct\]\[lb\][$5$]{} \[ct\]\[cb\][$0.4$]{} \[ct\]\[cb\][$0.6$]{} \[ct\]\[cb\][$0.7$]{} \[ct\]\[cb\][$0.8$]{} \[ct\]\[cb\][$2$]{} \[lt\]\[rb\][$\Omega/\Omega_{0}$]{} \[cc\]\[lc\][$\xi$]{} \[cc\]\[lc\][$\frac{Z(\Omega)}{m\Omega_{oo}^2}$]{} \[cb\]\[ct\][$\alpha=0,\quad \eta=\eta_c$]{} \[cb\]\[ct\][$\alpha=\pi/2,\quad
\eta=\eta_c,$]{}
For $\eta >0$ we have two minima dependence of sensitivity $\xi$ at two different frequencies $\Omega_\pm\simeq \Omega_0\sqrt{1\pm \eta}$. With increase of $\eta$ the distance between minima also increases. Comparing values $\xi(\Omega_\pm)$ with $\xi(\Omega_0)$ we can introduce a “characteristic” value $\eta_{\alpha\,c}$ when $\sqrt
2\xi(\Omega_\pm) = \xi(\Omega_0)$: $$\eta_{\alpha\,c} = \frac{\xi_0}{C^{1/4}} \,.$$ In this case $$\xi_0^2 = \frac{2\gamma_0^{\rm loss}}{\Omega_0|{\cal A}_\alpha|^2}\,C
= \frac{\sqrt C}{2}\,\frac{\Delta\Omega^2}{\Omega_0^2} \,.$$ \[compare with Eq.(\[xi\_enh\_dbl\])\]. The sensitivity plots for this case (again for loss factor $|{\cal A}_0|^2=0.24$) are given in Fig.\[twoA\].
\[rc\]\[cl\][$0.1$]{} \[rc\]\[cl\] \[ct\]\[cb\][$1$]{} \[ct\]\[lb\][$5$]{} \[ct\]\[cb\][$0.4$]{} \[ct\]\[cb\][$0.6$]{} \[ct\]\[cb\][$0.7$]{} \[ct\]\[cb\][$0.8$]{} \[ct\]\[cb\][$2$]{} \[lt\]\[rb\][$\Omega/\Omega_{0}$]{} \[cc\]\[lc\][$\xi$]{} \[cc\]\[lc\][$\frac{Z(\Omega)}{m\Omega_{oo}^2}$]{} \[cb\]\[ct\][$\alpha=0,\, \pi/2, \quad \eta=2\eta_c$]{} \[cb\]\[ct\][$\alpha=\pi/2,\quad
\eta=\eta_c,$]{}
On Fig. \[wide\] we also present sensitivity curves for well separated minimums ($\eta=2\eta_c$) and slightly less relaxation rate $\gamma_0=0.03\Omega_0$ fot different values of homodyne angle $\alpha=0,\,
\pi/2$. It is close to regime considered in [@Buonanno2001] including the negligible role of optical losses (compare with plots on Fig. 8 in [@Buonanno2001]).
Minimum of the spectral density
-------------------------------
Suppose that almost monochromatic signal has to be detected and we are interested in the minimum of normalized spectral density $\xi$ at some fixed frequency. First of all we have to remind that $\gamma_0^\text{loss}$ is fixed (it depends on mirror’s absorption only) whereas the effective loss factor $|{\cal
A}_0|^2$ (as well as $|{\cal A}_\alpha|^2$) can be modified by variation of mirrors transmissivities.
It follows from Eq.(\[xi\_real\]) that $\xi(\Omega)$ reaches its minimum at $\nu=\pm\eta_\alpha\Omega_0/2$ and $|{\cal A}_\alpha|=1$, and this minimum is equal to $$\xi_{\rm min} = \sqrt{\frac{\gamma_0^{\rm loss}}{\Omega_0}
\left(2+\frac{3}{\sqrt{2}}\right)}\simeq 0.06$$ \[compare with Eq.(\[xi\_min\_std\])\]. Here for the estimate we used parameters from Table \[tab1\].
Unfortunately, this value can not be obtained for planned Advanced LIGO parameters. Indeed, it follows from Eqs.(\[gamma\_0load\]),(\[delta\_0\]), that $$\delta_0<\frac{2\gamma_0^{\rm load}}{T_s^2} ,.$$ (The physical sense of $\gamma_0^{\rm load}/T_s^2$ is quite clear — it is relaxation rate of a single FP cavity in one arm.) In the double resonance regime we need to have $\delta_0\approx\sqrt{2}\Omega_0\approx
10^3\ \text{s}^{-1}$. However for Advanced LIGO parameters (Table \[tab1\]) $$\label{large_gamma}
\frac{2\gamma_0^\text{load}}{T_s^2}\simeq 100\
\text{s}^{-1}$$ It is less by one order of magnitude than required. The problem can be solved through modifying Advanced LIGO parameters, namely, by decreasing the signal recycling mirror transmittance $T_s^2$ by approximately one order of magnitude and corresponding increase in the arm cavities input mirrors transmittance by the same value.
Signal-to-noise ratio {#sec:snr}
---------------------
As was mentioned above, the double-resonance regime allows to obtain signal-to-noise ratio better than SQL even for [*wide-band*]{} signals. It was shown in [@05a1LaVy] that for no loss case the gain in signal-to-noise ratio, in principle, can be arbitrary high. Here we show that optical losses restrict this gain. The details of calculations are presented in Appendix\[app:analysis:snr\].
As an example of wide band signal we consider the perturbation of metric having the shape of a step function in time domain and the Fourier transform equal to $$\label{step}
h(\Omega)=\text{const}/\Omega.$$ It is worth to underline that the result practically does not depend on the shape of wide band signal spectrum (as alternative example one could consider a short pulse (delta-function) — its Fourier transform is a constant).
To demonstrate the gain we also calculate the signal-to-noise ratio $\text{SNR}_{\rm conv}$ for conventional LIGO interferometer [@02a1KiLeMaThVy] without signal recycling mirror with registration of phase quadrature and take the quotient ${\cal P}$ of $\text{SNR}$ by $\text{SNR}_{\rm conv}$ in order to characterize the gain in signal-to-noise ratio (the value $\text{SNR}_{\rm conv}$ we calculated numerically): $$\begin{aligned}
{\cal P}&= \frac{\text{SNR} }{\text{SNR}_{\rm conv} },\quad
\text{SNR}_{\rm conv}\simeq 0.7\times
\frac{h(\Omega_0)^2\Omega_0}{ h^2_{SQL}(\Omega_0)}\end{aligned}$$
\[cb\]\[ct\][$\alpha=0$, $\gamma_{0}/\Omega_0=0.0046$]{} \[cb\]\[ct\][$\alpha=\pi/2$, $\gamma_{0}/\Omega_0=0.0046$]{} \[tc\]\[bc\][$0$]{} \[tc\]\[bc\] \[tc\]\[bc\][$0.1$]{} \[tc\]\[bc\] \[tc\]\[bc\][$0.2$]{} \[tc\]\[bc\] \[tc\]\[bc\][$0.3$]{} \[tc\]\[bc\] \[tc\]\[bc\][$0.4$]{} \[tc\]\[bc\] \[tc\]\[bc\][$0.5$]{} \[cc\]\[lc\][$2$]{} \[cc\]\[lc\][$4$]{} \[cc\]\[lc\][$5$]{} \[cc\]\[lc\][$6$]{} \[tc\]\[bc\] \[cc\]\[lc\][$8$]{} \[tc\]\[bc\] \[cc\]\[cc\][$10$]{} \[tc\]\[bc\] \[cc\]\[cc\][$15$]{} \[tc\]\[bc\] \[cc\]\[cc\][$20$]{} \[cc\]\[cc\][$25$]{} \[cc\]\[cc\][$30$]{} \[cc\]\[cc\]\
Using the accurate formulas (\[xi\]) we numerically calculated plots (see Fig. \[B2\]) for gain $\cal P$ as function of $\eta$ at fixed ratio $\gamma_{0}/\Omega_0= 0.0046$ and loss factor factor $|{\cal A}_0|^2= 0.24$ corresponding to Advanced LIGO parameters (see Table \[tab1\]). We see that the degradation due to losses is not large as compared with no loss case.
The gain $\cal P$ decreases with increase in $\eta$ bigger than optimal — i.e. when double-resonance ($\eta=0$) transforms to two well separated first-order resonances ($\eta\gg \eta_c$).
We see from plots in Fig. \[B2\] that there is an optimal value of parameter $\eta$ when gain has maximum. Analysis presented in Appendix \[app:analysis:snr\] shows that gain $\cal P$ reaches its maximum for optimal values of $\eta_\alpha$ and $|{\cal A}|_\alpha$ and it is equal to $$\label{Pmax}
{\cal P}_\text{max}\simeq
0.6\times \sqrt \frac{\Omega_0}{\gamma_0^\text{loss}}\simeq 20$$ where estimates given for parameters listed in Table \[tab1\].
Note that this gain may be achieved at Advanced LIGO parameters — in bottom plot on Fig. \[B2\] the maximum of dotted curve is quite close to ${\cal
P}_\text{max}$.
It is worth noting that in the “pure” double-resonance regime ($\eta_\alpha=0$) the gain in the signal-to-noise ratio only slightly differs from the maximum gain: $$\frac{{\cal P}_\text{max}}{{\cal P}(\eta_\alpha=0,\,
|{\cal A}|_\alpha^\text{opt}) }=
\frac{3^{3/4}}{2}\simeq 1.14$$
Using the signal-to-noise ratio for a conventional oscillator (\[snr\_oscill\]), one can calculate the gain $$\label{Posc}
{\cal P}_\text{osc}=\frac{ \text{SNR}_\text{oscill} }{
\text{SNR}_\text{conv}}\simeq 2.8$$ Comparing (\[Pmax\]) and (\[Posc\]) we see that “double resonance” regime provides gain in signal-to-noise ratio about $7$ times larger than conventional oscillator.
Squeezing in output wave
------------------------
\[cb\]\[ct\][No losses: $|r_0|=0,\quad
\eta=0,\quad \gamma_0^\text{no losses}/\Omega_0=0.01$]{} \[cb\]\[ct\][Optical losses: $|r_0|=1,\quad \eta=0,
\quad
\gamma_{0+}/\Omega_0=0.01$]{} \[lt\]\[rb\][$\Omega/\Omega_0$]{} \[cc\]\[lc\][$\cal X$]{}\
It is also useful to find how optical losses affect output wave squeezing. Using Eq.(\[bzeta\]) one can calculate the squeezing factor $\cal X$ (for the coherent quantum state, ${\cal X}=1$): $$\begin{aligned}
{\cal X}(\Omega)&\equiv \sqrt{\frac{\langle
b_\zeta b_\zeta^+ + b_\zeta^+ b_\zeta
\rangle}{\langle
a_\text{vac} a_\text{vac}^+ + a_\text{vac}^+
a_\text{vac}\rangle} }\end{aligned}$$ The result of this calculation yields squeezing factor $\cal
X$ as a function of $\Omega$. For particular case $\eta=0$ the profile is given in Fig.\[squ\]. The top plot correspond to the lossless case (${\cal
A}_0=0$), the bottom one — to the case when $|{\cal
A}_0|=1$.
We see that in both cases the squeezing monotonously increases with increase in homodyne angle. More important is the fact that the values of squeezing factor are close to one. It confirms our assumption that it is the optical rigidity rather than pondermotive nonlinearity, [*i.e.*]{} the meter noises cross-correlation (as source of squeezing) that produces the major input into the sensitivity gain.
Conclusion
==========
The optical rigidity which can be created in the signal-recycled configuration of laser interferometric gravitational-wave detectors turns the detector test masses into oscillators and thus allows to obtain narrow-band sensitivity better than the Standard Quantum Limit for a free test mass. This method of circumnebting the Standard Quantum Limit does not rely on squeezed quantum states of the optical filed and due to this it is much less vulnerable to optical losses.
Moreover, sophisticated frequency dependence of this rigidity makes it possible to implement the “double resonance” regime which provides narrow-band sensitivity better than the Standard Quantum Limits for both a free test mass and an conventional harmonic oscillator.
The “double resonance” regime may be useful to detect narrow band gravitational waves, e.g. from pulsars. Knowing pulsar parameters one can tune the bandwidth and sensitivity in the optimal way. It is important that this tuning may be produced “on line” by varying signal recycling mirror position and adjusting circulating power.
Another advantage of the “double resonance” regime is its better sensitivity to [*wide-band*]{} signals. While an conventional harmonic oscillator provides approximately the same value of the the signal-to-noise ratio as a free test mass, in the case of a “double resonance” oscillator this parameter is limited only by the optical and mechanical losses and other noise sources of non-quantum origin. Estimates based on the Advanced LIGO parameters values shown that the “double resonance” regime can provide more than tenfold increase of the wide-band signal-to-noise ratio.
We would like to extend our gratitude to V.B. Braginsky and Y. Chen for stimulating discussions. This work was supported by LIGO team from Caltech and in part by NSF and Caltech grant PHY-0353775, as well as by the Russian Foundation of Fundamental Research, grant No. 03-02-16975-a.
Analysis of Advanced LIGO interferometer. {#app:analysis}
=========================================
\[lc\]\[lb\][$x_E$]{} \[lc\]\[lb\][$y_E$]{} \[lc\]\[lb\][$x_N$]{} \[lc\]\[lb\][$y_N$]{}
\[cb\]\[lb\][$i$]{} \[lc\]\[lb\][$-i$]{}
\[cb\]\[lb\][$a_P$]{} \[ct\]\[lb\][$b_P$]{} \[ct\]\[lb\][$a_W$]{} \[cb\]\[lb\][$b_W$]{}
\[rc\]\[lb\][$a_D$]{} \[lc\]\[lb\][$b_D$]{} \[lc\]\[lb\][$a_s$]{} \[rc\]\[lb\][$b_s$]{}
\[cb\]\[lb\][$a_E$]{} \[ct\]\[lb\][$b_E$]{} \[cc\]\[cc\][$a_{E1}$]{} \[cc\]\[cc\][$b_{E1}$]{} \[cc\]\[cc\][$a_{E2}$]{} \[ct\]\[ct\][$b_{E2}$]{}
\[rc\]\[lb\][$a_N$]{} \[lc\]\[lb\][$b_N$]{} \[lc\]\[lc\][$a_{N1}$]{} \[rc\]\[cc\][$b_{N1}$]{} \[rc\]\[cc\][$a_{N2}$]{} \[lc\]\[lt\][$b_{N2}$]{} \[lc\]\[rc\][$T, \,r_1$]{} \[cc\]\[cc\][$r_2$]{}
![Scheme of Advanced LIGO[]{data-label="extraintra"}](SrLigoLoss.eps){width="3.5in"}
Notations and approximation
---------------------------
We consider Advanced LIGO interferometer with signal recycling (SR) mirror and power recycling (PR) mirror as shown in Fig.\[extraintra\]. SR, PR mirrors and beam splitter are immobile and have no optical losses. We assume that both Fabry-Perot (FP) cavities in the east and north arms are identical: each input mirror has transmittance $T\ll 1$ and reflectivity $R_1=\sqrt{1-T^2-{\cal A}_1^2}$, each end mirror has reflectivity $R_2=\sqrt{1-{\cal A}_2^2}$, where ${\cal A}_1,\,{\cal A}_2$ are optical loss coefficients for input and end mirrors correspondingly. The end mirrors and input mirrors have equal masses $m$ and they can move as free masses.
The contribution of losses in SR mirror and in beam splitter is small as compared with contribution of losses in mirrors in arms which is larger by factor $\sim 1/T^2$ — consequently we assume in our analysis that SR mirror and beam splitter have no optical losses.
We consider “dark port” regime when no regular optical power comes to detector in the signal port. In this case the power recycling mirror is used only to increase the mean power delivered to the beam splitter and no fluctuational fields from the laser (west arm) access detector in the south arm.
Upper letters denote the mean complex amplitudes, lower case letters denote operator describing fluctuations and signal. south arm, $b_D$ is the operator of Fig.\[extraintra\]). The electric field $E$ in propagating wave can be written as sum of large mean field and small component (see details in [@02a1KiLeMaThVy]):
$$\begin{aligned}
E & \simeq \sqrt{\frac{2\pi \, \hbar \omega_o}{Sc}
}\,e^{-i\omega_o t}
\left( A +\int_{-\infty}^\infty
a\,e^{-i\Omega t} \frac{d\Omega}{2\pi}\right)+
\{\mbox{h.c.} \},\\
I & = \hbar \omega_p|A|^2,\quad
a\equiv a(\omega_o+\Omega),\quad a_-\equiv
a(\omega_o-\Omega)\\
& \big[a(\omega_o\pm\Omega),\,a^+(\omega_o\pm
\Omega')\big]=
2\pi\,\delta_0(\Omega -\Omega'),\end{aligned}$$
where $A$ is the complex amplitude, $S$ is the cross section of the light beam, $c$ is the velocity of light, $I$ is the mean power of the light beam, $a$ and $a^+$ are the annihilation and creation operators. We consider sidebands $\omega_o\pm \Omega$ about carrier $\omega_o$ with side band frequencies $\Omega$ in gravitational wave range ($\Omega/2\pi \in 10\dots 1000$ Hz). Detailed notations of wave amplitudes and mirror displacements are given on Fig.\[extraintra\].
Arm cavities
------------
First we consider the fields propagating in FP cavity in the east arm shown in dashed box on Fig.\[extraintra\] (formulas for FP cavity in the north arm are the same with obvious substitutions in subscripts $E\to N$). Below we assume that the following conditions are fulfilled: $$\begin{aligned}
\label{smallt}
\frac{L\Omega}{c}\ll 1,\quad T,\ {\cal A}_1, \ {\cal
A}_2\ll 1,\end{aligned}$$ where $L$ is the distance between the mirrors in arms ($4$ km for LIGO).
We start with a set of equations for mean amplitudes $$\begin{aligned}
B_{E1}&= iT A - R_1 A_{E1},\quad
B= iT A_{E1} - R_1 A_E,\\
A_{E1}&= B_{E2}\, e^{i\omega_o L/c} =-R_2 B_{E1}\,
e^{2i\omega_o L/c}.
\nonumber\end{aligned}$$ Below we assume that the carrier frequency of the incident wave $\omega_o$ is equal to the eigenfrequency of each FP cavity, i.e. $e^{i\omega_o
L/c}=1$. So we obtain $$\begin{aligned}
B_{E1}& = {\cal T} A_E,\quad B_{E2}\simeq -B_{E1}, \quad
B_E= {\cal R} A_E,\\
{\cal T} &\equiv\frac{iT}{1-R_1R_2}\simeq\frac{2i\gamma_{\rm
load}}{T\gamma},\\
{\cal R}&\equiv
\frac{-R_1+R_2(1-r_1^2)}{1-R_1R_2}\simeq
\frac{\gamma_-}{\gamma},\\
\gamma &= \gamma_{\rm load} + \gamma_{\rm loss}, \quad
\gamma_- = \gamma_{\rm load} - \gamma_{\rm loss}, \nonumber
\\
\gamma_{\rm load} &= \frac{cT^2}{4L}, \quad
\gamma_{\rm loss} = \frac{c{\cal A}^2}{4L} \,, \nonumber \\
{\cal A} &= \sqrt{{\cal A}_1^2+{\cal A}_2^2} \,. \nonumber\end{aligned}$$
To calculate fluctuational components of field we start from the set of equations: $$\begin{aligned}
\label{fl2a}
b_{E1}&= iT a_E - R_1 a_{E1} +i{\cal A}_1 e_{E1}
+R_1A_{E1}\,2iky_E,\\
\label{fl2b}
b_E&= iT a_{E1} - R_1 a_E -R_1A_E\, 2iky_E +i{\cal A}_1
e_{E1},\\
\label{fl2c}
a_{E1}&=\theta R_2\left(
-a_{2E}\, -A_{2E}\,2ikx_E\right) +i{\cal
A}_2e_{2E}.\end{aligned}$$ Here $e_{1E,}\ e_{2E}$ is vacuum fields appearing due to losses. We denote $\theta=e^{i(\omega_o+\Omega) L/c}$ using obvious approximation $\theta \simeq 1+i\Omega L/c$,
Substituting (\[fl2c\]) into (\[fl2a\]) and using conditions (\[smallt\]) we obtain: $$\begin{aligned}
\label{b2rr}
b_{E1}&\simeq {\cal T}_{\Omega}\, a_E+ e_E r\,{
\cal T}_\Omega+
\frac{{\cal T}_{\Omega}\,B_{E1}\, 2ik(x_E-y_E)}{iT} ,\\
b_{2E}&\simeq -b_{E1}, \quad a_{2E}\simeq b_{E1},\quad
a_{E1}\simeq - b_{E1},\\
{\cal T}_{\Omega} &\equiv\frac{iT}{1-R_1R_2\theta^2}\simeq
\frac{2i\gamma_{\rm
load}}{T\big(\gamma-i\Omega\big)},\\
e_E&=\frac{e_{E1}{\cal A}_1- e_{2E}{\cal A}_2}{{\cal A}},
\nonumber\end{aligned}$$ Here we introduced operator $e_E$ of vacuum fluctuations, which fulfill usual commutator relation $\big[e_E(\omega_o+\Omega), \,
e_E^+(\omega_o+\Omega')\big]=2\pi\, \delta_0(\Omega-\Omega')$.
Substitution of (\[b2rr\]) into (\[fl2b\]) under conditions (\[smallt\]) allows us to get the formula for reflected field: $$\begin{aligned}
\label{2Fe}
b_E &= a_E\,{\cal R}_{\Omega} - ierT\,{\cal T}_\Omega -
A{\cal T}{\cal
T}_\Omega \, 2ik(x_e-y_E),\\
{\cal R}_{\Omega} &=
\frac{-R_1+R_2(1-{\cal
A}_1^2)\theta^2}{1-R_1R_2\theta^2}\simeq
\frac{\gamma_-+i\Omega}{\gamma-i\Omega}\end{aligned}$$
Beam splitter
-------------
We assume that lossless beam splitter has transmittance and reflection factors equal to $i/\sqrt2$ and $-1/\sqrt2$, correspondingly. The additional phase shift $\pi/2$ is added to the west arm and $-\pi/2$ to the south one. It is convenient to introduce new variables $$a_{(\pm)} = \frac{a_E\pm a_N}{\sqrt2} \,, \qquad
b_{(\pm)} = \frac{b_E\pm b_N}{\sqrt2} \,.$$ For these variables beam splitter equations read:
\[eil\_bs\] $$\begin{gathered}
a_W = -b_{(+)} \,, \qquad
a_s = -b_{(-)} \,, \\
a_{(+)} = -b_W \,, \qquad
a_{(-)} = -b_s \,.
\end{gathered}$$
These equations are valid both for the zero and the first approximations as the beam splitter is a linear system.
Now we can consider the fields in entire interferometer. For our mean amplitudes of $(\pm)$ variables we have $$\begin{aligned}
B_{(\pm)}&={\cal R} A_{(\pm)}
\end{aligned}$$ It is easy to note that the set of equations splits into two independent sets: one for $W$ and $(+)$ and second for $S$ and $(-)$. The “$S,(-)$” set has only trivial zero solution since $A_D=0$. Therefore,
\[eil\_0\_symm\] $$\begin{aligned}
A_E &= A_N = \frac{A}{\sqrt2} \,, \quad
B_E = B_N = \frac{{\cal R} A}{\sqrt2} \,, \\
A_{E1} &= A_{N1} =
\frac{-\Theta^2B_1}{\sqrt 2} \,, \quad
B_{E1} = B_{N1} = \frac{B_1}{\sqrt2}\,,
\end{aligned}$$
>From this point on we omit indices $+$ for the sake of simplicity, $A$ is the complex mean amplitude left in the beam splitter.
Output field
------------
The signal wave, registered by detector in the south arm (“$S,\
(-)$” mode) is coupled with the differential motion of the mirrors, i.e. it has a part proportional to differential coordinate $$x=\frac{(x_E-y_N)-(x_N-y_N)}{2}$$ coupled with the gravitation-wave signal. We are interested in “$S,\,
(-)$” mode and below we omit subscripts $_{(-)}$ for $a_{(-)},\ b_{(-)} $.
>From equations for the south arm
\[eil\_s\_1s\] $$\begin{gathered}
b_D = -R_s a_D + iT_s\theta_s a_s \,, \\
b_s = -R_s\theta_s^2 a_s + iT_s\theta_s a_D \,.\end{gathered}$$
(here $\theta_s=e^{i(\omega_o+\Omega)l/c}$, $l$ is the optical length between SR mirror and input mirror in the arm) one can obtain the following formulas for fluctuational component of the output field: $$\begin{aligned}
\label{bD1}
b_D &\simeq {\cal R}_s\,a_D - {\cal T}_s{\cal T}_\Omega
(2ikB_1x + i{\cal A}\,e),\\
e & =\frac{e_E-e_N}{\sqrt 2}\end{aligned}$$ Here we introduced the “generalized” transparency ${\cal
T}_s$ and reflectivity ${\cal R}_s$ of “$S,\ (-)$” mode of interferometer: $$\begin{aligned}
\label{eil_1s_4}
{\cal T}_s &= \frac{-iT_s\theta_s}{ 1+{\cal R}_\Omega
R_s\theta_s^2},\quad
{\cal R}_s = \frac{-\big(R_s +{\cal
R}_\Omega\theta_s^2\big)}{ 1+
{\cal R}_\Omega R_s\theta_s^2},\end{aligned}$$ Due to small value of $l\ll L$ we consider below $\theta_s$ a constant not depending on $\Omega$.
It is useful to rewrite formulas (\[eil\_1s\_4\]) in approximation (\[smallt\]) denoting $\theta_s=i\,e^{i\phi}$: $$\begin{aligned}
{\cal R}_s &\simeq
G\times\frac{\Gamma_-^*+i\Omega}{\Gamma-i\Omega}, \quad
G=\frac{R_s+e^{2i\phi}}{1+R_s e^{2i\phi}}\, \\
\Gamma &= \gamma_0-i\delta_0,\quad
\Gamma_- = \gamma_{0-}-i\delta_0, \\
\label{gamma0p}
\gamma_0 &= \gamma_0^{\rm load} + \gamma_0^{\rm loss}, \quad
\gamma_{0-}=\gamma_0^{\rm load} - \gamma_0^{\rm loss}, \\
\label{gamma_0load}
\gamma_0^{\rm load}
&= \frac{\gamma_{\rm load}\,T_s^2}{\big|1+R_s e^{
2i\phi}\big|^2}
= \frac{c\,T^2T_s^2}{4L\big|1+R_s e^{ 2i\phi}\big|^2}\,,
\\
\label{gamma_0loss}
\gamma_0^{\rm loss} &= \gamma_{\rm loss}=\frac{c{\cal
A}^2}{4L}, \\
\label{delta_0}
\delta_{0} &= \frac{2R_s\gamma_{\rm load}\sin 2\phi}{
\big|1+R_s e^{2i\phi}\big|^2}=
\frac{2R_s\gamma_0^{\rm load}\sin 2\phi}{
T_s^2}\\
{\cal T}_s &= \frac{-iT_s\theta_s}{ 1+{\cal R}_\Omega
R_s\theta_s^2}=
\frac{T_s\,e^{i\phi}
\big[\gamma-i\Omega\big]}{
\big[1+R_s\,e^{2i\phi}\big]\big[\Gamma-i\delta_0\big]}\,
.\nonumber\end{aligned}$$ Here $\delta_0$ is the detuning, $\gamma_0$ is the relaxation rate of our difference mode (“$S,\,(-) $” mode) which can be presented as a sum of “loaded” and “intrinsic” rates. “Intrinsic” relaxation rate $\gamma_0^{\rm loss}$ is exclusively provided by intrinsic losses in mirrors in arms while “loaded” relaxation rate $\gamma_0^{\rm load}$ is provided only by transparencies of SR and input mirrors. Note that for Advanced LIGO relaxation rate $\gamma_0\simeq 2\
\text{s}^{-1}$ — it is much less than the mean frequency of gravitational wave range $\Omega \sim
2\pi\times 100\ \text{s}^{-1}$.
Pondermotive forces
-------------------
To calculate pondermotive force acting on movable mirrors in interferometer arms we should write down the equation for differential field $b_1=\big(b_{E1}-b_{N1})/\sqrt 2$ in approximation (\[smallt\]): $$\begin{aligned}
b_1&\simeq {\cal T}_s{\cal T}_\Omega\left(
a_D +\frac{\big(1+R_se^{2i\phi}\big) }{T_s
e^{i\phi}}
\left[re+\frac{2B_1kz}{T}\right]\right)\end{aligned}$$
The incident wave acts on the mirror with force proportional to square of amplitude module; and we keep only the cross term of this square. $$\begin{aligned}
{\cal F}&\simeq \int_{-\infty}^\infty F(\Omega)
\, e^{-i\Omega t}\, \frac{d\Omega}{2\pi},
\quad
F(\Omega)=\hbar k\,\big(A^*a +A a^+_-\big) \nonumber \\end{aligned}$$
First we write down the formulas for forces acting on the back mirrors. The difference between the forces acting on the east and north back mirrors is equal to: $$\begin{aligned}
F_2 & \simeq 2\hbar k\,\big( A^*_{2}a_{2}+A_{2}a^+_{2-}+
B^*_{2}b_{2}+B_{2}b^+_{2-}\big),\end{aligned}$$ where $A_2=(A_{E2}-A_{N2})/\sqrt 2, \
a_2=(a_{E2}-a_{N2})/\sqrt 2 $ and so on (recall that we continue considering “$S\, (-)$” mode). One can extract two terms in formula for $F_2$: $$F_2 = F_{\rm meter} + F_{\rm rigid} \,,$$ where the first term corresponds to fluctuational component (back action) and the second one — to the regular force depending on mirrors positions (optical rigidity). The formula for $F_{\rm meter}$ is the following $$\begin{gathered}
\label{faD2a}
F_{\rm meter} \simeq \frac{2i\hbar\omega_o T_sT\,\,B_1^*
\, e^{i\phi}
\big(a_{D} +{\cal A}_0e\big)}{
\big(1+R_s\,e^{2i\phi}\big)\big(\Gamma-i\Omega\big)} \\
- \frac{2i\hbar\omega_o T_sT\,\,B_1\,e^{-i\phi}
\big(a_{D-}^+ +{\cal A}_0^*e^+_-\big)}{
\big(1+R_s\,e^{-2i\phi}\big)\big(\Gamma^*-i\Omega\big)} \,,\end{gathered}$$ where $$\label{r0}
{\cal A}_0 = \frac{{\cal
A}\big(1+R_s\,e^{2i\phi}\big)}{TT_se^{i\phi}}$$ is the effective loss factor.
The equation for optical rigidity $K$ has the following shape $$\begin{aligned}
\label{K}
K&\equiv \frac{-F_\text{rigid}}{x}=
\frac{16\hbar k^2 \, \gamma_{\rm
load}\delta_{0}|B_1|^2}{T^2\cal D}=
\frac{8k\,I_c \delta_{0}
}{L{\cal D}},\\
{\cal D}&= (\Gamma-i\Omega)(\Gamma^*-i\Omega), \quad
I_c =\frac{\hbar \omega_o}{2}|B_1|^2,\end{aligned}$$ We see that optical rigidity $K$ is proportional to detuning $\delta_0$ of the difference mode of the interferometer. This detuning can be only introduced by displacement of SR mirror (while Fabry-Perot cavities in arms remain in optical resonance).
In approximation (\[smallt\]) the forces acting on the back mirrors are approximately equal (with negative sign) to the forces acting on input mirrors — the difference is negligible. So for the difference coordinate $x$ we have the following equation: $$\label{zZa}
Z(\Omega)x = F_{\rm meter} + F_{\rm signal} \,,$$ where $$Z(\Omega) = -m\Omega^2 + K$$ and $$F_{\rm signal} = \frac{m\Omega^2 Lh}{2}$$ is the signal force due to action of gravitational wave, $h$ — is the dimensionless gravitational-wave signal.
Output signal
-------------
We see from formula (\[faD2a\]) that the back action force is produced by sum of fluctuational fields: $a_D$ from signal port and $e$ due to losses in mirrors. It is useful to introduce a new pair of independent fluctuational fields $p$ and $q$ (the new basis) as following:
\[ae\_to\_pq\] $$\begin{aligned}
p &= G\,\frac{\Gamma^*+i\Omega}{\Gamma-i\Omega}
\frac{a_D + {\cal A}_0e}{\sqrt{1+|{\cal A}_0|^2}} \,, \\
q &= \frac{-{\cal A}_0^*a_D+e}{\sqrt{1+|{\cal A}_0|^2}} \,.
\end{aligned}$$
These two bases are equivalent — both pairs $a_D,\ e$ and $p,\ q$ describe vacuum fluctuations (we do not consider possible squeezing of field $a_D$).
Then we can rewrite formulas for the fluctuational force $F_{fl}$ and output field $b_D$ in a more compact form:
$$\begin{aligned}
F_{fl} & \simeq\frac{8i\hbar kB_1^*\gamma_0
\big(1+R_se^{2i\phi}\big)}{TT_s\, e^{i\phi}
\big(\Gamma^*+i\Omega\big)}\times
\frac{p}{\sqrt{1+|{\cal A}_0|^2}}+
\big\{\mbox{h.c.}\big\}_{-\Omega} \,, \\
b_D &= \frac{p}{ \sqrt{1+|{\cal A}_0|^2}}+
\frac{q\, {\cal A}_0^*}{\sqrt{1+|{\cal A}_0|^2}}-
{\cal T}_s {\cal T}_\Omega B_1 ikz. \label{bDa}
\end{aligned}$$
Calculating the difference position $x$ from (\[zZa\]) and substituting it into (\[bDa\]) one can obtain the final formula for the output field $b_D$: $$\begin{aligned}
\label{bD2a}
b_D& = \frac{q\, {\cal A}_0^*}{\sqrt{1+|{\cal A}_0|^2}} +
\frac{-m\Omega^2}{Z(\Omega)\sqrt{1+|{\cal
A}_0|^2}}\times\\
& \times\left\{
p\left(
1-\frac{iJ\Gamma^*}{\Omega^2\big((\Gamma^*)^2
+\Omega^2\big)}
\right)\right.+\nonumber\\
&\quad \left. + p_-^+\left(
\frac{-iJ\gamma_0G}{\Omega^2\big(\Gamma^2
+\Omega^2\big)}\right)
+
\frac{ih\sqrt{2J\gamma_0G}}{\Omega\,h_{SQL}\big(\Gamma-i\Omega\big)}
\right\} \,,\nonumber\end{aligned}$$ where $$J = \frac{8kI_c}{mL} \,.$$
This formula has two terms. The first one ($\sim q$) is proportional to the effective loss factor ${\cal A}_0$ and appears due to the optical losses. The second term has the same form as for no losses case — compare with Eq.(16) in [@05a1LaVy] — with weighted multiplier $1/\sqrt{1+|{\cal A}_0|^2}$ and substituted damping rate $\gamma_0$ accounting for losses in [@05a1LaVy]. It is the background to interpret formula (\[bD2a\]) as field reflected from lossless interferometer (with damping rate $\gamma_{0}$) and then passed through grey filter with total loss factor $|{\cal A}_0|$ which decreases our field and adds fluctuations (the first term).
It is important that the second term has multiplier $
m\Omega^2/Z(\Omega)$ which increases the relative contribution of the second term at frequencies close to mechanical resonance (when $|Z(\Omega)|\ll m\Omega^2$) while the first term does not depend on frequency — it demonstrates the advantage of “real” rigidity and may explain the relatively weak degradation of sensitivity due to optical losses.
Sensitivity
-----------
In gravitational wave antenna one registers the quadrature component $b_\zeta$ of output fields using the balanced homodyne scheme (not shown in Fig.\[extraintra\]):
\[bzeta\] $$\begin{aligned}
b_\zeta
&=\frac{b_De^{-i\zeta}+b_{D-}^+e^{i\zeta}}{\sqrt2}
\nonumber\\
& = \frac{-m\Omega^2}{(Z(\Omega)\sqrt {2(1+|{\cal
A}_0|^2}}
\left\{
A_q\left(q{\cal A}_0^*e^{-i\zeta} +
q_-^+{\cal
A}_0e^{i\zeta}\right)\right.\nonumber\\
&\qquad \left. + A_p\,p +A_p^*\,p^+_-
+ A_s\frac{h_s}{h_{SQL}(\Omega_0)}
\right\}\,, \label{bzeta1} \\
A_q &= \frac{-m\Omega^2+K}{-m\Omega^2}
= 1-\frac{J\delta_0}{\Omega^2{\cal D}}\,,\\
A_p &= \frac{e^{-i\zeta\left[
(\Gamma^*_+)^2+\Omega^2 -iY\Gamma^*_+ +
iY\gamma_{0+}e^{2i\alpha}
\right]}}{(\Gamma^*_+)^2+\Omega^2}\,, \\
A_s &= \frac{\sqrt{J\gamma_0G}}{\Omega_0}
\left(\frac{ie^{-i\alpha}}{\Gamma_+-i\Omega}-
\frac{ie^{i\alpha}}{\Gamma_+^*-i\Omega}\right)=
\nonumber\\
&= \frac{\sqrt{8J\gamma_0}}{\Omega_0}
\frac{(\gamma_0-i\Omega)\sin\alpha-\delta_0\cos\alpha}{{\cal D}}
\,,\\
\alpha &= \zeta-\phi \,.
\end{aligned}$$
Here and below we normalize dimensionless metric $h_s$ by SQL sensitivity $h_{SQL}(\Omega_0)$ at some frequency $\Omega_0$.
Now we can write down one-sided spectral noise density $S_h$ recalculated to variation of dimensionless metric $h_s$ and normalize it to SQL sensitivity $h_{SQL}(\Omega_0)$:
\[xi\] $$\begin{aligned}
\xi^2(\Omega)&= \frac{S_h(\Omega)}{h_{SQL}(\Omega_0)^2}=
\frac{2|A_q|^2+2|A_p|^2}{|A_s|^2}=\frac{P_1+P_2+P_3}{Q}, \\
P_1&= \Big[ \Omega^4 -\Omega^2(\delta_0^2-\gamma_{0+}^2)
+ J\big(\delta_0 - \gamma_0\sin 2\alpha\big)\Big]^2 \,, \\
P_2&=\gamma_0^2\big(2\delta_0\Omega^2 - J(1- \cos
2\alpha)\big)^2,\\
P_3&= |{\cal
A}_0|^2\Big\{\big[\Omega^4-(\delta_0^2+\gamma_0^2)\Omega^2
+
J\delta_0\big]^2 + 4\gamma_0^2\Omega^6\Big\}, \\
Q&=\frac{4J\gamma_0\Omega^4}{\Omega_0^2}
\big|(\gamma_0-i\Omega)\sin\alpha -
\delta_0\cos\alpha\big|^2,\end{aligned}$$
Note that without signal recycling mirror SQL sensitivity in LIGO lossless interferometer can be achieved at working frequency $\Omega_0=\gamma$ if the optical power $I_c$ is equal to the optimal one $I_{SQL}(\Omega_0)$ [@02a1KiLeMaThVy; @Buonanno2001]: $$\label{ISQL}
I_{SQL}(\Omega_0)=\frac{m\Omega_0^3 Lc}{8\omega_o},\quad
\text{or}\
\frac{J}{\Omega_0^3}=1.$$
Although presentation (\[xi\]) is compact and convenient for numeric estimates, it can mask the physical structure of the noise. Due to this reason, we provide a more transparent form of this equation: $$S_h(\Omega) = \frac{4}{m^2L^2\Omega^4}\left[
\frac{\hbar^2}{S_x} + |Z_{\rm eff}|^2S_x + |Z|^2S_{\rm
loss}
\right] \,,$$ where $$S_x = \frac{\hbar}{2mJ\gamma_0}\frac{|{\cal D}|^2}
{\bigl|(\gamma_0-i\Omega)\sin\alpha -
\delta_0\cos\alpha\bigr|^2}$$ is the measurement noise, $$S_{\rm loss} = |{\cal A}_0|^2S_x$$ is the noise created by the optical losses, $$Z_{\rm eff} = K_{\rm eff} - m\Omega^2 \,,$$ and $$\begin{gathered}
\label{K_eff}
K_{\rm eff} = \frac{m J}{|{\cal D}|^2}\,\Bigl[
\delta_0(\gamma_0^2+\delta_0^2-\Omega^2) \\
+ \gamma_0(\gamma_0^2-\delta_0^2+\Omega^2)\sin2\alpha
- 2 \delta_0\gamma_0^2\cos2\alpha
\Bigr] \,.\end{gathered}$$ is the effective rigidity.
Narrow-band case
----------------
Suppose that the observation frequency is close to the double resonance frequency $\Omega_0=\delta_0/\sqrt{2}$ and the pumping power is close to the critical value:
$$\begin{gathered}
\label{Jinit}
\Omega = \Omega_0 + \nu \,, \\
J = \frac{\Omega_0^3(1-\eta^2)}{\sqrt 2}
\end{gathered}$$
where $|\nu|\ll 1$, $\eta^2 \ll 1$.
In this approximation,
$$\begin{aligned}
P_1 &\approx \Omega_0^4\left(
4\nu^2-\eta^2\Omega_0^2
- \frac{\gamma_0\Omega_0}{\sqrt{2}}\sin2\alpha
\right)^2 \,, \\
P_2 &\approx 2\gamma_0^2\Omega_0^6(1+\cos^2\alpha)^2\,, \\
P_3 &\approx |{\cal A}_0|^2\Omega_0^4
\left[(4\nu^2-\nu^2\Omega_0^2)^2 + 4\gamma_0^2\Omega_0^2\right]\,,\\
Q &\approx 2\sqrt{2}\gamma_0\Omega_0^7(1+\cos^2\alpha)\,,
\end{aligned}$$
and
$$\begin{aligned}
Z &\approx m(4\nu^2-\eta^2\Omega_0^2-2i\Omega_0\gamma_0)\,,
\label{Z_approx}\\
Z_{\rm eff}& \approx m\left(
4\nu^2-\eta^2\Omega_0^2
- \frac{\Omega_0\gamma_0}{\sqrt{2}}\sin2\alpha
\right) \,, \label{Z_approx_eff} \\
S_x &\approx \frac{\hbar}{\sqrt{2}m\gamma_0\Omega_0(1+\cos^2\alpha)}\,.
\end{aligned}$$
In this approximation we have the following formula for the sensitivity $\xi$: $$\begin{gathered}
\label{xi2nb}
\xi^2(\nu) \approx
\frac{\gamma_0}{\Omega_0}\,\frac{(1+\cos^2\alpha)}{\sqrt{2}}
+ \frac{1}{2\sqrt{2}\gamma_0\Omega_0^3(1+\cos^2\alpha)} \\
\times\biggl\{
\left(
4\nu^2-\eta^2\Omega_0^2
- \dfrac{\gamma_0\Omega_0}{\sqrt{2}}\sin 2\alpha
\right)^2 \\
+ |{\cal A}_0|^2
\left[
\left(4\nu^2-\eta^2\Omega_0^2\right)^2
+ 4\Omega_0^2\gamma_0^2
\right]
\biggr\}\,.\end{gathered}$$ Using the following notations:
\[eta\_A\_alpha\] $$\begin{gathered}
\eta_\alpha^2 = \eta^2 + \dfrac{\gamma_0\sin 2\alpha}
{\sqrt{2}\Omega_0(1+|{\cal A}_0|^2)}\,,\\
|{\cal A}_\alpha|^2 = \frac{\sqrt{2}|{\cal A}_0|^2}{1+\cos^2\alpha} \,,
\end{gathered}$$
and taking into account that $$\gamma_0=\gamma_0^\text{loss}\left(1+\frac{1}{|{\cal A}_0|^2}\right)\,,$$ Eq.(\[xi2nb\]) can be rewritten in a more compact form: $$\label{xi(nu)}
\xi^2(\nu) \approx
\frac{\left(4\nu^2-\eta_\alpha^2\Omega_0^2\right)^2}{4g\Omega_0^4}
+ gC \,,$$ where $$\begin{gathered}
g = \frac{\gamma_0^{\rm loss}}{\Omega_0|{\cal A}_\alpha|^2} \,, \\
C = 1 + \frac{3}{\sqrt{2}}\,|{\cal A}_\alpha|^2 + |{\cal A}_\alpha|^4 \,.\end{gathered}$$
Signal-to-noise ratio {#app:analysis:snr}
---------------------
\[rc\]\[rc\][$k(|{\cal A}_\alpha|)$]{} ![Dependence of the signal-to-noise ratio on $|{\cal A}_\alpha|$ at optimal $\eta_\alpha^2$ (\[eta\_opt\]).[]{data-label="fig:snrdbl"}](snr.eps "fig:"){width="44.00000%" height="33.00000%"}
In the narrow-band case, the main contribution into the integral of the signal-to-noise ratio is produced in vicinity of $\Omega_0$. In this case we can use approximation (\[xi(nu)\]) and expand the limits of integration over $\nu$ from $-\infty$ to $\infty$ and making substitution $x=\sqrt 2\nu/\Omega_0\sqrt g\, C^{1/4}$ with notation $a=\eta^2_{\alpha}/2g\sqrt C$: $$\begin{gathered}
{\rm SNR} = \frac{2}{\pi}\int_0^{\infty}
\frac{|h(\Omega)|^2\,d\Omega}{S_h(\Omega)}
\approx
\frac{2}{\pi}\,\frac{|h(\Omega_0)|^2}{h_{\rm SQL}^2(\Omega_0)}
\int_{-\infty}^\infty\frac{d\nu}{\xi^2(\nu)} \\
= \frac{2|h(\Omega)|^2\Omega_0}{\pi h_{\rm SQL}^2(\Omega_0)} \times
\frac{1}{\sqrt{2g}C^{3/4}} \int_{-\infty}^\infty
\frac{dx}{1+(x^2+a^2)^2}\\
=\frac{|h(\Omega)|^2\Omega_0}{ h_{\rm SQL}^2(\Omega_0)} \times
\frac{1}{\sqrt{2g}\, C^{3/4}}\left(
\frac{1}{\sqrt{a+i}} +
\frac{1}{\sqrt{a-i}}\right)=\\
= \frac{|h(\Omega)|^2\Omega_0}{ h_{\rm SQL}^2(\Omega_0)} \times
\frac{1}{\sqrt{2g}\, C^{3/4}}\times
\frac{\sqrt{a+\sqrt{a^2+1}}}{\sqrt 2\sqrt{a^2+1}}\end{gathered}$$ The maximum of this expression is achieved, if $$\label{eta_opt}
a=\frac{1}{\sqrt 3},\quad \text{or}\quad
\eta_\alpha^2 = \frac{2g\sqrt{C}}{\sqrt{3}}\,,$$ and it is equal to: $${\rm SNR} = k(|{\cal A}_\alpha|)\times
\frac{|h(\Omega)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}\,
\sqrt{\frac{\Omega_0}{\gamma_0^{\rm loss}}} \,,$$ where $$k(|{\cal A}_\alpha|) = \frac{3^{3/4}|{\cal A}_\alpha|}{2C^{3/4}}$$ is dimensionless function plotted in Fig.\[fig:snrdbl\]. It attains the maximum at $$|{\cal A}_\alpha| = \sqrt{\frac{\sqrt{73}-3}{8\sqrt{2}}} \approx 0.61$$ and it is equal to $$\frac{2^{7/4}\sqrt{\sqrt{73}-3}}{(23+3\sqrt{73})^{3/4}}
\approx 0.43 \,.$$
Gain in spectral density for simplified model without optical losses {#app:xi}
====================================================================
Starting with this Appendix we simplify the model system. In particular, we assume here, that there is no optical losses in the system we examine. When considering the frequency-dependent rigidity based system, the approximate formula (\[K\_simple\]) is used.
Conventional harmonic oscillator
--------------------------------
Spectral density of the total meter noise (\[F\_noise\]) is equal to: $$\label{app_S_sum}
S_{\rm sum}(\Omega) = S_F(\Omega) + |Z(\Omega)|^2S_x(\Omega) \,,$$ where $Z(\Omega)$ is the spectral image of the differential operator ${\bf
Z}$. In case of a conventional harmonic oscillator, $$\label{app_Z_oscill}
|Z(\Omega)|^2 = (-m\Omega^2 + K)^2 \,,$$ and in close vicinity of the resonance frequency $\Omega_0=\sqrt{K/m}$, $$\label{app_Z_oscill_nu}
|Z(\Omega)|^2 \approx 4m^2\Omega_0^2\nu^2 \,,$$ and $$\xi^2(\Omega)\equiv\frac{S_{\rm sum}(\Omega)}{2\hbar m\Omega^2}
\approx \xi_{\rm min}^2 + \dfrac{\nu^2}{\xi_{\rm min}^2\Omega_0^2} \,,$$ where $$\begin{gathered}
\nu = \Omega-\Omega_0 \,, \quad |\nu| \ll \Omega_0 \,,\label{def_nu}\\
\xi_{\rm min}^2 = \frac{S_F(\Omega_0)}{2\hbar m\Omega_0^2} \,,\end{gathered}$$ see Eqs(\[SxSF\], \[S\_SQL\]).
Let us require that $\xi(\Omega)$ does not exceed a given value $\xi_0$ within as wide a frequency band $\Delta\Omega$ as possible. It is easy to show that this requirement is met if $$\xi_{\rm min} = \frac{\xi_0}{\sqrt{2}} \,,$$ and $$\Delta\Omega = 2\xi_{\rm min}^2\Omega_0 \,.$$ Therefore, $$\xi_0^2 = \frac{\Delta\Omega}{\Omega_0} \,.$$
Frequency-dependent rigidity
----------------------------
### Double-resonance case
Consider now an oscillator with the frequency-dependent rigidity (\[K\_simple\]). In this case, $$|Z(\Omega)|^2 = \frac
{m^2(\Omega_+^2 - \Omega^2)^2(\Omega_-^2 - \Omega^2)^2}
{(\Omega_+^2 + \Omega_-^2 -\Omega^2)^2}$$ \[see Eq.(\[Omega\_pm\])\].
If the double-resonance condition (\[dbl\_res\]) is fulfilled, then in close vicinity of the resonance frequency $\Omega_0=\delta_0/\sqrt{2}$, $$\begin{gathered}
|Z(\Omega)|^2 \approx 16m^2\nu^4 \,, \\
\xi^2(\Omega) \approx \xi_{\rm min}^2
+ \dfrac{4\nu^4}{\xi_{\rm min}^2\Omega_0^4} \,.\end{gathered}$$ Performing again the same optimization as in previous subsection, we can obtain that again $\xi_{\rm min} = \xi_0/\sqrt{2}$, and $$\xi_0 = \frac{\Delta\Omega}{\Omega_0} \,.$$
### Two close resonances
In the sub critical pumping case (\[E\_subcrit\]), $$\begin{gathered}
|Z(\Omega)|^2 \approx m^2(4\nu^2-\Omega_0^2\eta^2)^2\,,\label{Z_fdrigid}\\
\xi^2(\Omega) \approx \xi_{\rm min}^2
+ \dfrac{(4\nu^2-\Omega_0^2\eta^2)^2}{4\xi_{\rm min}^2\Omega_0^4} \,.\end{gathered}$$ These functions has a local maximum at $\nu=0$ and two minima at $\nu=\pm\Omega_0\eta/2$.
The same optimization as in two previous cases gives, that $$\begin{gathered}
\xi(\Omega_0) = \xi_0 = \sqrt{2}\xi_{\rm min} \,, \\
\xi_0 = \frac{1}{\sqrt{2}}\,\frac{\Delta\Omega}{\Omega_0} \,,\end{gathered}$$ and the optimal value of parameter $\eta$ is equal to $$\eta_c = \xi_0\,,$$
Gain in signal-to-noise ratio for simplified model without optical losses {#app:snr}
=========================================================================
Free test masses interferometer
-------------------------------
Rewrite the signal-to-noise ratio (\[snr\]) as follows: $$\label{snrF}
{\rm SNR} = \frac{2}{\pi}\int_0^\infty
\frac{|F_{\rm signal}(\Omega)|^2\,d\Omega}{S_{\rm sum}(\Omega)} \,.$$ For conventional interferometer (without optical springs), the total meter noise spectral density is equal to (see [@02a1KiLeMaThVy]): $$S_{\rm sum}(\Omega) = \frac{\hbar m}{2}\left[
\frac{2\Omega_0^4}{\Omega_0^2+\Omega^2}
+ \frac{\Omega^4(\Omega_0^2+\Omega^2)}{2\Omega_0^4}
\right] \,.$$ Therefore, in this case, $$\begin{gathered}
{\rm SNR} = \frac{8}{\pi\Omega_0^2h_{\rm SQL}^2(\Omega_0)}
\int_0^\infty
\frac{|h(\Omega)|^2\,d\Omega}{
\dfrac{2\Omega_0^4}{\Omega^4(\Omega_0^2+\Omega^2)}
+ \dfrac{\Omega_0^2+\Omega^2}{2\Omega_0^4}
} \\
= {\cal N}\times\frac{|h(\Omega_0)|^2\Omega_0}{h^2_{SQL}(\Omega_0)} \,,\end{gathered}$$ where $${\cal N} = \frac{8}{\pi\Omega_0^3|h(\Omega_0)|^2}
\int_0^\infty
\frac{|h(\Omega)|^2\,d\Omega}{
\dfrac{2\Omega_0^4}{\Omega^4(\Omega_0^2+\Omega^2)}
+ \dfrac{\Omega_0^2+\Omega^2}{2\Omega_0^4}
} \,.$$ is the numeric factor depending on the gravitation-wave signal shape $h(\Omega)/|h(\Omega_0)|$.
Conventional harmonic oscillator
--------------------------------
Substituting Eqs.(\[app\_S\_sum\]) and (\[app\_Z\_oscill\]) into Eq.(\[snrF\]), we obtain, that for a conventional harmonic oscillator the signal-to-noise ratio is equal to: $${\rm SNR} = \frac{2}{\pi}
\int_0^\infty\frac{|F_{\rm signal}(\Omega_0)|^2\,d\Omega}
{S_F(\Omega) + m^2S_x(\Omega)(\Omega_0^2-\Omega^2)^2} \,.$$ In the narrow-band case \[see Eqs.(\[SxSF0\], \[def\_nu\])\], this equation can be presented as the following: $$\begin{gathered}
{\rm SNR} \approx \frac{2}{\pi}|F_{\rm signal}(\Omega_0)|^2
\int_{-\infty}^\infty\frac{d\nu}
{S_F(\Omega_0) + 4m^2\Omega_0^2S_x(\Omega_0)\nu^2} \\
= \frac{|F_{\rm signal}(\Omega_0)|^2}{\hbar m\Omega_0}
= \frac{2|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)} \,.\end{gathered}$$
Frequency-dependent rigidity
----------------------------
In similar way, using Eqs.(\[app\_S\_sum\],\[Z\_fdrigid\]) and (\[snrF\]), we obtain, that for optical spring based oscillator, $$\begin{gathered}
{\rm SNR}
\approx \frac{2}{\pi}|F_{\rm signal}(\Omega_0)|^2 \\
\times\int_{-\infty}^\infty\frac{d\nu}
{S_F(\Omega_0) + m^2S_x(\Omega_0)(4\nu^2-\Omega_0^2\eta^2)^2} \\
= \frac{\sqrt{2}|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}\,
\frac{\xi_0^2}{\sqrt{
\bigl(\xi_0^4+\eta^4\bigr)\bigl(\sqrt{\xi_0^4+\eta^4}-\eta^2\bigr)
}}\,.\end{gathered}$$
In a pure double resonance case ($\eta=0$), $${\rm SNR} = \frac{\sqrt{2}}{\xi_0}\,
\frac{|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}\,.$$ Slightly better result can be obtained for the case of two optimally placed resonances. If $$\eta^2 = \frac{\eta_c^2}{\sqrt{3}} \equiv \frac{\xi_0^2}{\sqrt{3}}\,,$$ then $${\rm SNR} = \frac{3^{3/4}}{\sqrt{2}\xi_0}
\frac{|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}\,,$$
If the separation between the two resonance frequencies is too big, $\eta\gg \xi_0$ (but still $\eta\ll 1$), then $${\rm SNR} = \frac{2}{\eta}\,
\frac{|h(\Omega_0)|^2\Omega_0}{h_{\rm SQL}^2(\Omega_0)}\,.$$
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Hadronic properties such as the lightness of the pion masses and the absence of parity doublets, strongly indicate that chiral symmetry is broken spontaneously. In QCD, a great deal of insight in such nonperturbative phenomena has been obtained from extensive lattice QCD simulations [@DeTar; @Ukawa]. This did not go without a significant amount of effort. One of the difficulties is that the order parameter of the chiral phase transition, $\langle\bar\psi\psi\rangle$, can be obtained only after a complicated limiting procedure: the thermodynamic limit, the chiral limit, and the continuum limit. In addition, in the chiral limit it is extremely costly to take into account the effect of the fermion determinant. Since the fermion determinant can be expressed as a product over the Dirac eigenvalues, this alone warrants a detailed study of the QCD Dirac spectrum. Moreover, $\langle\bar\psi\psi\rangle$ is directly related to the QCD Dirac spectrum through the Banks-Casher relation [@Bank80], $$\label{eq1}
\langle\bar\psi\psi\rangle=\lim_{m\rightarrow 0}
\lim_{V\rightarrow\infty}\frac{\pi}{V}\rho(0).$$ Here, $m$ is the quark mass, $V$ is the volume of space-time, and $\rho(\lambda)=\langle\sum_n\delta(\lambda-\lambda_n)\rangle$ is the eigenvalue density of the Euclidean Dirac operator, $iD=i\gamma_\mu\partial_\mu+\gamma_\mu A_\mu$, averaged over gauge field configurations. We observe that the average position of the smallest eigenvalues is determined by the chiral condensate. In this letter we focus on fluctuations of the smallest eigenvalues about their average position. It should be clear that such fluctuations affect the fermion determinant and are important for the understanding of finite size effects [@Gock]. The hope is that they are given by universal functions which can be obtained analytically. This analytical information could then be used to facilitate extrapolations to the thermodynamic and chiral limits.
A similar situation arises in mesoscopic physics [@HDgang]. In these studies, it was shown that for a sufficient amount of disorder, spectral correlations are universal and can be obtained from a Random Matrix Theory (RMT) with only the basic symmetries included. On the other hand, the average spectral density is non-universal and requires specific knowledge of the dynamics of the system. In this letter we investigate the question whether a similar separation of scales takes place in QCD. Does the disorder of lattice QCD gauge field configurations result in universal fluctuations of the small Dirac eigenvalues?
According to the Banks-Casher relation, the low-lying Dirac eigenvalues are spaced as $1/V$ for $\langle\bar\psi\psi\rangle \ne
0$. Recent work by Leutwyler and Smilga [@Leut92] shows that this part of the spectrum is related to the pattern of chiral symmetry breaking by means of a class of sum rules for the inverse Dirac eigenvalues. It is natural to magnify the spectrum near $\lambda= 0$ by a factor of $V$. This leads to the introduction of the microscopic spectral density [@Shur93] defined by $$\label{eq2}
\rho_s(z)=\lim_{V\rightarrow\infty}\frac{1}{V\Sigma}
\rho\left(\frac{z}{V\Sigma}\right) \:,$$ where $\Sigma$ is the absolute value of $\langle\bar\psi\psi\rangle$. Based on the analysis of the Leutwyler-Smilga sum rules, it was conjectured [@Shur93] that this distribution is universal and only determined by the global symmetries of the QCD partition function, the number of flavors, and the topological charge. If that is the case it can be obtained from a much simpler theory with only the global symmetries as input. Such a theory is chiral RMT which will be discussed below. Whether or not QCD is in this universality class is a dynamical question that can only be answered by lattice QCD simulations. The investigation of this question is the main purpose of this letter. At this moment it can only be addressed on relatively small lattices where our results are consistent with zero topological charge. The pertinent question of what happens in the continuum limit has to be postponed to future work. In this limit we expect zero modes to become important. $\rho_s(z)$ is then different in different sectors of topological charge. However, on present day lattices with staggered fermions there seems to be no evidence of a “zero-mode zone” [@Kogu97], and the situation is controversial at best.
There are already several pieces of evidence supporting the conjecture that $\rho_s$ is universal: (1) The moments of $\rho_s$ generate the Leutwyler-Smilga sum rules [@Verb93]. (2) $\rho_s$ is insensitive to the probability distribution of the random matrix [@Brez96; @Nish96]. (3) Lattice data for the valence quark mass dependence of the chiral condensate could be understood using the analytical expression for $\rho_s$ [@Chan95; @Verb96a]. (4) The functional form of $\rho_s$ does not change at finite temperature [@Jack96b]. (5) The analytical result for $\rho_s$ is found in the Hofstadter model for universal conductance fluctuations [@Slev93]. (6) For an instanton liquid $\rho_s$ shows good agreement with the random-matrix result [@Verb94b]. However, a direct demonstration for lattice QCD was missing.
An analysis of Dirac spectra on the lattice was performed in Ref. [@Hala95] where it was shown that the spectral fluctuations in the bulk of the spectrum on the scale of the mean level spacing are universal and described by RMT. This showed that the eigenvalues of the Dirac operator are strongly correlated. Only few configurations were available in this study, but spectral ergodicity allowed to replace the ensemble average by a spectral average. However, spectral averaging is not possible for $\rho_s$ since only the first few eigenvalues contribute. Therefore, a large number of configurations is essential.
We briefly summarize the main ingredients of chiral RMT. In a random-matrix model, the matrix elements of the operator under consideration are replaced by the elements of a random matrix with suitable symmetry properties. Here, the operator is the Euclidean Dirac operator $iD$ which is hermitian. Because $\gamma_5$ anti-commutes with $iD$ the eigenvalues occur in pairs $\pm\lambda$. In a chiral basis, the random-matrix model has the structure [@Shur93] $$iD+im \rightarrow \left[\matrix{im&W\cr W^\dagger&im}\right]\:,$$ where $W$ is a matrix whose entries are independently distributed random numbers. In full QCD with $N_f$ flavors, the weight function used in averaging contains the gluonic action in the form $\exp(-S_{\rm gl})$ and $N_f$ fermion determinants. In RMT, the gluonic part of the weight function is replaced by a Gaussian distribution of the random matrix $W$. The symmetries of $W$ are determined by the anti-unitary symmetries of the Dirac operator. Depending on the number of colors and the representation of the fermions the matrix $W$ can be real, complex, or quaternion real [@Verb94a]. The corresponding random-matrix ensembles are called chiral Gaussian orthogonal (chGOE), unitary (chGUE), and symplectic (chGSE) ensemble, respectively. The microscopic spectral density has been computed analytically for all three ensembles [@Verb93; @Verb94d; @Naga95].
We have performed numerical simulations of lattice QCD with staggered fermions and gauge group SU(2) for couplings $\beta=4/g^2 = 2.0$, 2.2, and 2.4 on lattices of size $V=L^4$ with $L=8$, 10, and 16. This range of lattice parameters covers the crossover region from strong to weak coupling of SU(2) [@Creu80]. The boundary conditions are periodic for the gauge fields and periodic in space and anti-periodic in Euclidean time for the fermions. In this work, we only study the quenched approximation using a hybrid Monte Carlo algorithm [@Meye90]. This made it possible to generate a large number of independent configurations (indicated in the figures). The analysis of unquenched data with 4 dynamical flavors is in progress.
In SU(2) with staggered fermions, every eigenvalue of $iD$ is twofold degenerate due to a global charge conjugation symmetry. In addition, the squared Dirac operator $-D^2$ couples only even to even and odd to odd lattice sites, respectively. Thus, $-D^2$ has $V/2$ distinct eigenvalues. We use the Cullum-Willoughby version of the Lanczos algorithm [@Stoe93] to compute the complete eigenvalue spectrum of the sparse hermitian matrix $-D^2$ in order to avoid numerical uncertainties for the low-lying eigenvalues. There exists an analytical sum rule, ${\rm tr}(-D^2) = 4V$, for the distinct eigenvalues of $-D^2$ [@Kalk95]. We have checked that this sum rule is satisfied by our data, the largest relative deviation was $\sim 10^{-8}$. We have also made a detailed study to determine the optimal acceptance rates and trajectory lengths [@Berb97]. The integrated autocorrelation times are in the range of 1 to 4. The chiral condensate was obtained by fitting the spectral density and extracting $\rho(0)$ and is given in Table \[table1\] below.
The overall spectral density of the Dirac operator cannot be obtained in a random-matrix model since it is not a universal function. The lattice result for $\rho(\lambda)$ is displayed in Fig. \[fig1\] for $\beta=2.0$, $V=10^4$ and $\beta=2.4$, $V=16^4$, respectively. Note the strong decrease in $\langle\bar\psi\psi\rangle$ (in lattice units) for $\beta=2.4$, cf. Eq. (\[eq1\]) and Table \[table1\].
We are particularly interested in the region of small eigenvalues to check the predictions from chiral RMT. According to Ref. [@Verb94a], staggered fermions in SU(2) have the symmetries of the chGSE. Analytical expressions can be obtained in the framework of RMT for the microscopic spectral density and the distribution of the smallest eigenvalue by slight modifications of results computed for Laguerre symplectic ensembles [@Naga95; @Forr93]. Incorporating the chiral structure of the Dirac operator, we obtain from Ref. [@Naga95] $$\begin{aligned}
\label{eq3}
\rho_s(z)=2z^2\int_0^1duu^2\int_0^1dv&&[J_{4a-1}(2uvz)J_{4a}(2uz)
\nonumber \\ &&-vJ_{4a-1}(2uz)J_{4a}(2uvz)] \end{aligned}$$ with $4a=N_f+2\nu+1$, where $N_f$ is the number of massless flavors and $\nu$ is the topological charge. For our quenched data, $4a=1$ since $\nu=0$ as explained in the introduction. According to Eq. (\[eq2\]), lattice data for $\rho_s(z)$ are constructed from the numerical eigenvalue density using a scale $V\langle\bar\psi\psi\rangle$. This scale is determined by the data, hence the random-matrix predictions are parameter-free. Similarly, the distribution of the smallest eigenvalue for $N_f=\nu=0$ follows from Ref. [@Forr93], $$\label{eq4}
P(\lambda_{\rm min})=\sqrt{\frac{\pi}{2}}c(c\lambda_{\rm min})^{3/2}
I_{3/2}(c\lambda_{\rm min})e^{-\frac{1}{2}(c\lambda_{\rm min})^2} \:,$$ where $c=V\langle\bar\psi\psi\rangle$ is the same scale as above.
In Fig. \[fig2\] we have plotted the lattice results for $\rho_s(z)$ and $P(\lambda_{\rm min})$ together with the analytical results of Eqs. (\[eq3\]) and (\[eq4\]) for four different combinations of $\beta$ and lattice size. The agreement between lattice data and the parameter-free RMT predictions is impressive. Note that the RMT results were derived in the limit $V\to\infty$. Clearly, the agreement improves as the physical volume increases, i.e., with larger lattice size and smaller $\beta$. From the results for $\beta =2.0$ we observe that the agreement with RMT improves with increasing lattice size while the value of condensate remains the same. This suggests that a similar improvement will occur for $\beta =2.2$ and $\beta = 2.4$. These values are just below the $\beta$-value above which $\langle
\bar \psi \psi \rangle$ approaches zero, where the above RMT results are inapplicable, and an increased sensitivity to the size of the lattice is expected. Since $P(\lambda_{\rm min})$ for these couplings agrees with the RMT distribution for zero topological charge we expect that the discrepancy for $\rho_s(z)$ is not due to a superposition of configurations with different topological charge. We hope that future work will clarify this issue.
Related quantities testing similar properties are the higher-order spectral correlation functions, in particular the two-point function which enters in the computation of scalar susceptibilities. The $n$-point correlation function $R_n(x_1,\ldots,x_n)$ is defined as the probability density of finding a level (regardless of labeling) around each of the points $x_1,\ldots,x_n$. The two-level cluster function $T_2(x,y)$, which contains only the non-trivial correlations, is defined by $T_2(x,y)=-R_2(x,y)+R_1(x)R_1(y)$, i.e., the disconnected part is subtracted. For the chGUE, there are analytical arguments [@Guhr97] that the microscopic correlations are universal, and the same is expected for the chGSE. In this case, the predictions from RMT can again be obtained from the results of Ref. [@Naga95], but we do not write down the explicit expressions here. In Fig. \[fig3\], we have plotted data for $\rho_s(x,y)$ for $\beta=2.0$ on an $8^4$ lattice as a function of $x$ for some fixed value of $y$, along with the analytical random-matrix prediction. Clearly, the statistics are not as good as for the one-point function, but the agreement is still quite impressive.
Finally, we have checked the Leutwyler-Smilga sum rule $\langle\sum_n\lambda_n^{-2}\rangle/V^2=\langle\bar\psi\psi\rangle^2/2$ appropriate for the chGSE [@Leut92; @Verb94c]. The numerical results are compared with the analytical predictions in Table \[table1\]. Again, the agreement improves with physical volume.
--------------- ----- ------------------------------ ------------------------------------------ ----------------------------------
$\beta$ $L$ $\langle\bar\psi\psi\rangle$ $\langle\sum_n\lambda_n^{-2}\rangle/V^2$ $\langle\bar\psi\psi\rangle^2/2$
\[0.5mm\] 2.0 8 0.1228(25) $8.20(20)\!\cdot\!10^{-3}$ $7.54(31)\!\cdot\!10^{-3}$
2.0 10 0.1247(22) $7.97(30)\!\cdot\!10^{-3}$ $7.78(27)\!\cdot\!10^{-3}$
2.2 8 0.0556(19) $1.67(03)\!\cdot\!10^{-3}$ $1.55(11)\!\cdot\!10^{-3}$
2.4 16 0.00863(48) $3.97(14)\!\cdot\!10^{-5}$ $3.72(42)\!\cdot\!10^{-5}$
--------------- ----- ------------------------------ ------------------------------------------ ----------------------------------
: Chiral condensate and a comparison of lattice data and analytical predictions for the Leutwyler-Smilga sum rule for $\lambda_n^{-2}$.
\[table1\]
In summary, we have performed a high-statistics study of the eigenvalue spectrum of the lattice QCD Dirac operator with particular emphasis on the low-lying eigenvalues. In the absence of a formal proof, our results provide very strong and direct evidence for the universality of $\rho_s$. In the strong coupling domain, the agreement with analytical predictions from random matrix theory is very good. On the scale of the smallest eigenvalues, agreement is found even in the weak-coupling regime. Furthermore, we found that the microscopic two-level cluster function agrees nicely with random-matrix predictions and that the Leutwyler-Smilga sum rule for $\lambda_n^{-2}$ is satisfied more accurately with increasing physical volume. We predict that corresponding lattice data for SU(2) with Wilson fermions and for SU(3) with staggered and Wilson fermions will be described by random matrix results for the GOE, chGUE, and GUE, respectively [@Verb94a]. (The U$_{\rm A}$(1) symmetry is absent for the Hermitean Wilson Dirac operator.) The identification of universal features in lattice data is both of conceptual interest and of practical use. In particular, the availability of analytical results allows for reliable extrapolations to the chiral and thermodynamic limits. In future work we hope to analyze the fate of the fermionic zero modes in the approach to the continuum limit, and we expect random-matrix results to be a useful tool in the analysis.
It is a pleasure to thank T. Guhr and H.A. Weidenmüller for stimulating discussions. This work was supported in part by DFG and BMBF. SM and AS thank the MPI für Kernphysik, Heidelberg, for hospitality and support. The numerical simulations were performed on a CRAY T90 at the Forschungszentrum Jülich and on a CRAY T3E at the HWW Stuttgart.
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|
---
abstract: 'The extender based Magidor-Radin forcing is being generalized to supercompact type extenders.'
address: |
School of Computer Science\
Tel-Aviv Academic College\
Rabenu Yeroham St.\
Tel-Aviv 68182\
Israel
author:
- Carmi Merimovich
date: 'August 1, 2016'
title: 'Supercompact Extender Based Magidor-Radin Forcing'
---
Introduction
============
This work[^1] continues the project of of generalizing the extender based Prikry forcing [@GitikMagidor1992] to larger and larger cardinals. In [@Merimovich2003; @Merimovich2011b] the methods introduced in [@GitikMagidor1992] (which generalized Prikry forcing [@Prikry1968] from using a measure to using an extender), were used to generalize the Magidor [@Magidor1978] and Radin [@Radin1982] forcing notions to use a sequence of extenders. In a different direction [@Merimovich2011c] used the methods of [@GitikMagidor1992] to define the extender based Prikry forcing over extenders which have higher directedness properties than their critical point. Such extenders give rise to supercompact type embeddings. Generalization of Prikry forcing to fine ultrafilters yielding supercompact type embeddings appeared in [@Magidor1977I]. Extending this forcing notion to Magidor-Radin type forcing notions were done in [@ForemanWoodin1991] and [@Krueger2007]. In the current paper we use extenders with higher directedness properties to define the extender based Magidor-Radin forcing notion. All of the forcing notions mentioned above are of course of Prikry type. For more information on Prikry type forcing notions one should consult [@Gitik2010].
Before stating the theorem of this paper we need to make some notions precise. Assume $E$ is an extender. We let $j_E{\mathrel{:}}V \to M \simeq \operatorname{Ult}(V,E)$ be the natural embedding of $V$ into the transitive collpase of the ultrapower $\operatorname{Ult}(V,E)$. We denote by $\operatorname{crit}E$ the critical point of the embedding $j_E$. In principle, an extender is a directed family of ultrafilters and projections. We denote by ${\lambda}(E)$ a degree of directedness holding for the extender $E$. We do not require ${\lambda}(E)$ to be optimal, i.e., ${\lambda}(E)$ is not necessarily the minimum cardinal for which $E$ is not ${\lambda}(E)^+$-directed. Note $M \supseteq {{\vphantom{M}}^{<{\lambda}(E)}{M}}$.
A sequence of extenders $\Vec{E} = {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$, all with the same critical point $\operatorname{crit}E_{\xi}$ and the same directedness size ${\lambda}(E_{\xi})$, is said be Mitchell increasing if for each ${\xi}< \operatorname{o}(\Vec{E})$ we have ${\ensuremath{{\ensuremath{\langle E_{{\xi}'} \mid {\xi}' < {\xi}\rangle}}}} \in M_{\xi}\simeq \operatorname{Ult}(V,E_{\xi})$. We will denote by $\operatorname{crit}(\Vec{E})$ and ${\lambda}(\Vec{E})$ the common values of $\operatorname{crit}E_{\xi}$ and ${\lambda}(E_{\xi})$, respectively.
If $\Vec{E} = {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$ is a Mithcell increasing sequence of extenders and ${\alpha}\in [\operatorname{crit}{\Vec{E}}, j_{E_0}({\kappa}))$ then ${\Bar{E}}= {\ensuremath{\langle {\alpha}, \Vec{E} \rangle}}$ is said to be an extender sequence. Hence an extender sequence is an ordered pair with the first coordinate being an ordinal and the second coordinate being a Mitchell increasing sequence of extenders. Note that an empty sequence of extenders is legal in an extender sequence, e.g., ${\ensuremath{\langle {\alpha}, {\ensuremath{\langle \rangle}} \rangle}}$ is an extender sequence. Let $\operatorname{ES}$ be the collection of extender sequences. If ${\Bar{E}}$ is an extender sequence then we denote the projections to the first and second coordinates by $\mathring{{\Bar{E}}}$ and $\accentset{\mid}{{\Bar{E}}}$, respectively. The ordinals at the first coordinate of an extender sequence induce an order $<$ on $\operatorname{ES}$ by setting ${\Bar{{\nu}}}< {\Bar{{\mu}}}$ if $\mathring{{\Bar{{\nu}}}} < \mathring{{\Bar{{\mu}}}}$. We lift the functions defined on the Mitchell increasing sequence of extenders to extender sequences in the obvious way, i.e., $\operatorname{o}({\Bar{E}}) = \operatorname{o}(\accentset{\mid}{{\Bar{E}}})$ and ${\lambda}({\Bar{E}})={\lambda}(\accentset{\mid}{{\Bar{E}}})$. We will also abuse notation by writing ${\Bar{E}}_{\xi}$ for the extender $E_{\xi}$. There are two restrictions we have on ${\lambda}(\Vec{E})$. The first one seems a bit technical. We demand ${\lambda}({\Bar{E}})^{<\operatorname{crit}{\Bar{E}}} = {\lambda}({\Bar{E}})$ due to limitations we encountered in \[GetGoodPair\]. The second one is more substantial. We demand ${\lambda}({\Bar{E}}) \leq j_{E_0}(\operatorname{crit}(E_0))$. (It seems this last demand can be removed for the special case $\operatorname{o}(\Vec{E}) = 1$.) With all these preliminaries at hand we can write the theorem proved in this paper.
Assume the GCH. Let $\Vec{E}$ be a Mitchell increasing sequence such that ${\lambda}({\Vec{E}}) < j_{E_0}(\operatorname{crit}(\Vec{E}))$ and ${\mu}^{<\operatorname{crit}{\Vec{E}}} < {\lambda}({\Vec{E}})$ for each ${\mu}< {\lambda}({\Vec{E}})$. Furthermore, assume ${\epsilon}\leq j_{E_0}({\kappa})$. Then there is a forcing notion ${\mathbb{P}}(\Vec{E}, {\epsilon})$ such that the following hold in $V[G]$, where $G \subseteq {\mathbb{P}}(\Vec{E}, {\epsilon})$ is generic. There is a set $G^{\kappa}\subseteq \operatorname{ES}$ such that $G^{\kappa}{\cup}{\ensuremath{\{ {\ensuremath{\langle \operatorname{crit}\Vec{E}, \Vec{E} \rangle}} \}}}$ is increasing and for each ${\Bar{{\nu}}}\in G^{\kappa}{\cup}{\ensuremath{\{ {\ensuremath{\langle \operatorname{crit}\Vec{E}, \Vec{E} \rangle}} \}}}$ such that $\operatorname{o}({\Bar{{\nu}}}) > 0$ the following hold:
1. ${\ensuremath{\{ \operatorname{crit}{\Bar{{\mu}}}\mid {\Bar{{\mu}}}\in G^{\kappa}, {\Bar{{\mu}}}< {\Bar{{\nu}}}\}}} \subseteq \mathring{{\Bar{{\nu}}}}$ is a club.
2. $\operatorname{crit}{\Bar{{\nu}}}$ and ${\lambda}({\Bar{{\nu}}})$ are preserved in $V[G]$, and $(\operatorname{crit}{\Bar{{\nu}}}^+ = {\lambda}({\Bar{{\nu}}}))^{V[G]}$.
3. If $\operatorname{o}({\Bar{{\nu}}})<\operatorname{crit}{\Bar{{\nu}}}$ is $V$-regular then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= \operatorname{cf}\operatorname{o}({\Bar{{\nu}}})$ in $V[G]$.
4. (Gitik) If $\operatorname{o}({\Bar{{\nu}}}) \in [\operatorname{crit}{\Bar{{\nu}}}, {\lambda}({\Bar{{\nu}}}))$ and $\operatorname{cf}(\operatorname{o}({\Bar{{\nu}}})) \geq \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= {\omega}$ in $V[G]$.
5. If $\operatorname{o}({\Bar{{\nu}}}) \in [\operatorname{crit}{\Bar{{\nu}}}, {\lambda}({\Bar{{\nu}}}))$ and $\operatorname{cf}(\operatorname{o}({\Bar{{\nu}}})) < \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= \operatorname{cf}\operatorname{o}({\Bar{{\nu}}})$ in $V[G]$.
6. If $\operatorname{o}({\Bar{{\nu}}})= \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= {\omega}$ in $V[G]$.
7. If $\operatorname{o}({\Bar{{\nu}}})= {\lambda}({\Bar{{\nu}}})$ then $\operatorname{crit}{\Bar{{\nu}}}$ is regular in $V[G]$.
8. If $\operatorname{o}({\Bar{{\nu}}})= {\lambda}({\Bar{{\nu}}})^{++}$ then $\operatorname{crit}{\Bar{{\nu}}}$ is measurable in $V[G]$.
9. $2^{\operatorname{crit}{\Bar{{\nu}}}} = \max {\ensuremath{\{ {\lambda}({\Bar{{\nu}}}), {\lvert{\epsilon}\rvert} \}}}$.
Thus for example, if we assume ${\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< {\omega}_1 \rangle}}}}$ is a Mitchell increasing sequence of extenders on ${\kappa}$ giving rise to a $<{\kappa}^{++}$-closed elementary embeddings (and no more), then in the generic extension ${\kappa}$ will change its cofinality to ${\omega}_1$, and ${\kappa}^+$ would be collapsed. Moreover, there is a club of ordertype ${\omega}_1$ cofinal in ${\kappa}$, and for each limit point ${\tau}$ in this club ${\tau}^+$ of the ground model is collapsed. The GCH would be preserved, and no other cardinals are collapsed.
As another example, assume ${\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< {\omega}_1 \rangle}}}}$ is a Mitchell increasing sequence of extenders on ${\kappa}$ giving rise to a $<{\kappa}^{++}$-close elementary embeddings which are also ${\kappa}^{+3}-strong$ (and no more), then in the generic extension ${\kappa}$ will change its cofinality to ${\omega}_1$, and ${\kappa}^+$ would be collapsed. Moreover, there is a club of ordertype ${\omega}_1$ cofinal in ${\kappa}$, and for each limit point ${\tau}$ in this club ${\tau}^+$ of the ground model is collapsed. In this case we get $2^{\kappa}= {\kappa}^{++}$ and $2^{\tau}= {\tau}^{++}$ for the limit points of the club. In fact we have $2^{\kappa}= ({\kappa}^{+3})_V$ and $2^{\tau}= ({\tau}^{+3})_V$, and we see only gap-2 in the generic extension since ${\kappa}^+$ of the ground mode gets collapsed as do all the ${\tau}^+$ of the ground model. No other cardinal get collapsed. The structure of the work is as follows. In section \[sec:ExtendersAndNormality\] a formulation of extenders useful for ${\lambda}$-directed extenders is presented, and an appropriate diagonal intersection operation is introduced. In section \[sec:Forcing\] the forcing notion is defined and the properties of it which do not rely on understanding the dense subsets of the forcing are presented. In section \[sec:Dense\] claims regarding the dense subsets of the forcing notion are presented. This section is highly combinatorial in nature. In section \[sec:KappaProperties\] the influence of $\operatorname{o}({\Vec{E}})$ on the properties of ${\kappa}$ in the generic extension is shown. The claims here rely on the structure of the dense subsets as analyzed in section \[sec:Dense\].
This work is self contained assuming large cardinals and forcing are known.
${\lambda}$-Directed Extenders and Normality
============================================
Assume the GCH. Let ${\Vec{E}}= {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$ be a Mitchell increasing sequence of ${\lambda}$-directed extenders such that ${\lambda}\leq j_{E_0}({\kappa})$ is regular and ${\lambda}^{<{\kappa}} = {\lambda}$, where ${\kappa}= \operatorname{crit}\Vec{E}$. For each ${\xi}< \operatorname{o}(\Vec{E})$ let $j_{E_{\xi}} {\mathrel{:}}V \to M_{\xi}\simeq \operatorname{Ult}(V, E_{\xi})$ be the natural embedding. Assume $d \in [{\epsilon}]^{<{\lambda}}$ and ${\lvertd\rvert}+1 \subseteq d$. We let $\operatorname{OB}(d)$ be the set of functions ${\nu}{\mathrel{:}}\operatorname{dom}{\nu}\to \operatorname{ES}$ such that ${\kappa}\in \operatorname{dom}{\nu}\subseteq d$, and if ${\alpha}, {\beta}\in \operatorname{dom}{\nu}$ and ${\alpha}< {\beta}$ then ${\mathring{{\nu}}}({\alpha}) < {\mathring{{\nu}}}({\beta})$. Define an order on $\operatorname{OB}(d)$ by saying for each pair ${\nu},{\mu}\in \operatorname{OB}(d)$ that ${\nu}< {\mu}$ if $\operatorname{dom}{\nu}\subseteq \operatorname{dom}{\mu}$, ${\lvert{\nu}\rvert} < {\mathring{{\mu}}}({\kappa})$, and for each ${\alpha}\in \operatorname{dom}{\nu}$, ${\nu}({\alpha}) < {\mathring{{\mu}}}({\kappa})$.
For ${\xi}< \operatorname{o}({\Vec{E}})$ and a set $d \in [{\epsilon}]^{<{\lambda}}$ define the measure $E_{\xi}(d)$ on $\operatorname{OB}(d)$ as follows: $$\begin{aligned}
X \in E_{\xi}(d) \iff {\ensuremath{\{ {\ensuremath{\langle j_{E_{\xi}}({\alpha}), {\ensuremath{\langle {\alpha}, {\ensuremath{{\ensuremath{\langle E_{{\xi}'} \mid {\xi}' < {\xi}\rangle}}}} \rangle}} \rangle}} \mid {\alpha}\in d \}}} \in j_{E_{\xi}}(X).
\end{aligned}$$ For a set $d \in [{\epsilon}]^{<{\lambda}}$ let $\Vec{E}(d) = {\bigcap}{\ensuremath{\{ E_{\xi}(d \mid {\xi}< \operatorname{o}({\Bar{E}}) \}}}$. It is clear $E_{\xi}(d)$ is a ${\kappa}$-complete ultrafilter over $\operatorname{OB}(d)$ and ${\Vec{E}}(d)$ is a ${\kappa}$-complete filter over $\operatorname{OB}(d)$. In addition to this, the filter ${\Vec{E}}(d)$ has a useful normality property with a matching diagonal intersection soon to be introduced.
If $S \subseteq \operatorname{OB}(d)$, ${\nu}^* \in j_{E_{\xi}}(S)$, and ${\nu}^* < \operatorname{mc}_{\xi}(d)$, then there is ${\nu}\in S$ such that ${\nu}^* = j_{E_{\xi}}({\nu})$.
Assume $S \subseteq \operatorname{OB}(d)$ and for each ${\nu}\in S$ there is a set $X({\nu}) \subseteq \operatorname{OB}(d)$. Define the diagonal intersection of the family ${\ensuremath{\{ X({\nu}) \mid {\nu}\in S \}}}$ as follows: $$\begin{aligned}
\operatorname*{\triangle}_{{\nu}\in S}X({\nu}) = {\ensuremath{\{ {\nu}\in \operatorname{OB}(d) \mid \forall {\mu}\in S\ ({\mu}< {\nu}\implies {\nu}\in X({\mu})) \}}}.\end{aligned}$$
Assume $S \subseteq \operatorname{OB}(d)$, and for each ${\nu}\in S$, $X({\nu}) \in {\Vec{E}}(d)$. Then $X^* = \operatorname*{\triangle}_{{\nu}\in S}X({\nu}) \in {\Vec{E}}(d)$.
We need to show for each ${\xi}< \operatorname{o}({\Vec{E}})$, $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(X^*)$. I.e., we need to show $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(X)({\nu}^*)$ for each ${\nu}^* \in j_{E_{\xi}}(S)$ such that ${\nu}^* < \operatorname{mc}_{\xi}(d)$. Fix ${\nu}^* \in j_{E_{\xi}}(S)$ such that ${\nu}^* < \operatorname{mc}_{\xi}(d)$. There is ${\nu}\in S$ such that ${\nu}^* = j_{E_{\xi}}({\nu})$. Hence $j_{E_{\xi}}(X)({\nu}^*) = j_{E_{\xi}}(X({\nu}))$. Since $X({\nu}) \in E_{\xi}(d)$ we get $\operatorname{mc}_{\zeta}(d) \in j_{E_{\zeta}}(X({\nu}))$, by which we are done.
The diagonal intersection above can be generalized to work with more than one measure in the following way. A set $T \subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$, where $n<{\omega}$, is said to be a tree if the following hold:
1. Each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in T$ is increasing.
2. For each $k<n$ and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in T$ we have ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_k \rangle}} \in T$.
Assume $T \subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$ is a tree and ${\ensuremath{\langle {\nu}\rangle}} \in T$. Set $T_{{\ensuremath{\langle {\nu}\rangle}}}= {\ensuremath{\{ {\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-2} \rangle}} \mid {\ensuremath{\langle {\nu}, {\mu}_0, \dotsc, {\mu}_{n-2} \rangle}} \in T \}}}$. Denote the $k$-level of the tree $T$ by $\operatorname{Lev}_k(T)$, i.e., $\operatorname{Lev}_k(T) = T {\cap}{{\vphantom{\operatorname{OB}(d)}}^{k+1}{\operatorname{OB}(d)}}$. We will use ${{\Vec{{\nu}}}}$ as a shorthand for ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}$. For each ${{\Vec{{\nu}}}}\in T$ we define the successor level of ${{\Vec{{\nu}}}}$ in $T$ by setting $\operatorname{Suc}_T({{\Vec{{\nu}}}}) = {\ensuremath{\{ {\mu}\mid {{\Vec{{\nu}}}}{\mathop{{}^\frown}}{\mu}\in T \}}}$. A tree $S\subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$, with all maximal branches having the same finite height $n<{\omega}$, is said to be an ${\Vec{E}}(d)$-tree if the following hold:
1. There is ${\xi}< \operatorname{o}({\Vec{E}})$ such that $\operatorname{Lev}_0(S) \in E_{\xi}(d)$.
2. For each ${{\Vec{{\nu}}}}{\mathop{{}^\frown}}{\mu}\in S$ there is ${\xi}< \operatorname{o}({\Vec{E}})$ such that $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{\xi}(d)$.
If $S$ is a tree of finite height $n<{\omega}$ then we write $\operatorname{Lev}_{\max} S$ for $\operatorname{Lev}_{n-1}S$.
Assume $S$ is an ${\Vec{E}}(d)$-tree, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$ there is a set $X({{\Vec{{\nu}}}}) \subseteq \operatorname{OB}(d)$. By recursion define the diagonal intersection of the family ${\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}}$ by setting $\operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}} =
\operatorname*{\triangle}{\ensuremath{\{ X^*({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^* \}}}$, where $S^* = S {\cap}{{\vphantom{\operatorname{OB}(d)}}^{n-1}{\operatorname{OB}(d)}}$ and $X^*({{\Vec{{\mu}}}}) = \operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\mu}}}}{\mathop{{}^\frown}}{\ensuremath{\langle {\nu}\rangle}}) \mid {\nu}\in S_{{\ensuremath{\langle {{\Vec{{\mu}}}}\rangle}}} \}}}$. The following is immediate.
Assume $S$ is an ${\Vec{E}}(d)$-tree, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$ there is a set $X({{\Vec{{\nu}}}}) \in {\Vec{E}}(d)$. Then $\operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}} \in {\Vec{E}}(d)$.
The Forcing Notion {#sec:Forcing}
==================
A finite sequence ${\ensuremath{\langle {\Bar{{\nu}}}_0, \dotsc, {\Bar{{\nu}}}_k \rangle}} \in {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ is said to be $\operatorname{o}$-decreasing if it is increasing and ${\ensuremath{\langle \operatorname{o}({\Bar{{\nu}}}_0), \dotsc, \operatorname{o}({\Bar{{\nu}}}_k \rangle}})$ is non-increasing.
A condition $f$ is in the forcing notion ${\mathbb{P}}^*_f({\Vec{E}}, {\epsilon})$ if $f$ is a function $f {\mathrel{:}}d \to {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ such that:
1. $d \in [{\epsilon}]^{<{\lambda}}$.
2. $d \supseteq ({\lvertd\rvert}+1)$.
3. For each ${\alpha}\in d$, $f({\alpha})$ is $\operatorname{o}$-decreasing.
Assume $f, g \in {\mathbb{P}}_f^*({\Vec{E}}, {\epsilon})$ are conditions. We say $f$ is an extension of $g$ ($f \leq^*_{{\mathbb{P}}_f^*({\Vec{E}},{\epsilon})} g$) if $f \supseteq g$.
For a condition $f \in {\mathbb{P}}_f^*(\Vec{E})$ we will write $E_{\xi}(f)$ and ${\Vec{E}}(f)$ instead of $E_{\xi}(\operatorname{dom}f)$ and $\Vec{E}(\operatorname{dom}f)$, respectively. If $T \subseteq \operatorname{OB}(e)$ and $d \subseteq e$ then $T {\mathrel{\restriction}}d = {\ensuremath{\{ {\nu}{\mathrel{\restriction}}d \mid {\nu}\in T \}}}$.
A condition $p$ is in the forcing notion ${\mathbb{P}}^*(\Vec{E}, {\epsilon})$ if $p$ is of the form ${\ensuremath{\langle f^p, T^p \rangle}}$, where $f^p \in {\mathbb{P}}^*_f(\Vec{E}, {\epsilon})$, $T^p \in {\Vec{E}}(f^p)$, and for each ${\nu}\in T^p$ and each ${\alpha}\in \operatorname{dom}{\nu}$, $\max \mathring{f}^p({\alpha}) < {\mathring{{\nu}}}({\kappa})$.
Assume $p, q \in {\mathbb{P}}^*(\Vec{E}, {\epsilon})$ are conditions. We say $p$ is a direct extension of $q$ ($p \leq^*_{{\mathbb{P}}^*(\Vec{E}, {\epsilon})} q$) if $f^p \supseteq f^q$ and $T^p {\mathrel{\restriction}}\operatorname{dom}f^q \subseteq T^q$. We say $p$ is a strong direct extension of $q$ ($p \leq^{**}_{{\mathbb{P}}^*(\Vec{E}, {\epsilon})} q$) if $p$ is a direct extension of $q$ and $f^p =f^q$.
Since ${\epsilon}$ and the sequence ${\Vec{E}}$ are fixed througout this work we designate ${\mathbb{P}}^*(\Vec{E}, {\epsilon})$ by ${\mathbb{P}}^*$.
A condition $p$ is in the forcing $\Bar{{\mathbb{P}}}$ if $p = {\ensuremath{\langle p_0, \dotsc, p_{n^p-1} \rangle}}$, where $n^p < {\omega}$, there is a sequence ${\ensuremath{{\ensuremath{\langle \Vec{E}^p_i \mid i < n^p \rangle}}}}$ such that each $\Vec{E}^p_i$ is a Mitchell increasing sequence of extenders, ${\ensuremath{\langle \operatorname{crit}(\Vec{E}^p_0), \dotsc, \operatorname{crit}(\Vec{E}^p_{n^p-1}) \rangle}}$ is strictly increasing, $\sup {\ensuremath{\{ j_{E^p_{i,{\xi}}}(\operatorname{crit}\Vec{E}^p_i) \mid {\xi}< \operatorname{o}(\Vec{E}^p_i) \}}} < \operatorname{crit}\Vec{E}^p_{i+1}$, ${\lambda}(\Vec{E}^p_i) < \operatorname{crit}(\Vec{E}^p_{i+1})$, and for each $i < n^p$, $p_i \in {\mathbb{P}}^*(\Vec{E}^p_i, {\epsilon}^p_i)$.
Assume $p, q \in \Bar{{\mathbb{P}}}$ are conditions. We say $p$ is a direct extension of $q$ ($p \leq^*_{\Bar{{\mathbb{P}}}} q$) if $n^p = n^q$ and for each $i < n^p$, $p^i \leq^* q^i$. We say $p$ is a strong direct extension of $q$ ($p \leq^{**}_{\Bar{{\mathbb{P}}}} q$) if $n^p = n^q$ and for each $i < n^p$, $p^i \leq^{**} q^i$.
The following sequence of definitions leads to the definition of the order $\leq_{\Bar{{\mathbb{P}}}}$ (which is somewhat involved, hence the breakup to several steps). If ${\nu}\in \operatorname{OB}(d)$ we let $\operatorname{o}({\nu}) = \operatorname{o}({\nu}({\kappa}))$.
Assume $f {\mathrel{:}}d \to {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ is a function, ${\nu}\in \operatorname{OB}(d)$, and for each ${\alpha}\in \operatorname{dom}{\nu}$, $\max \mathring{f}({\alpha}) < {\mathring{{\nu}}}({\kappa})$. Define $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ and $f_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ as follows.
1. If $\operatorname{o}({\nu})=0$ then $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ is the function $g$, where:
1. $\operatorname{dom}g = \operatorname{ran}{\mathring{{\nu}}}$.
2. For each ${\alpha}\in \operatorname{dom}{\nu}$, $g({\mathring{{\nu}}}({\alpha})) = {\ensuremath{\langle {\Bar{{\tau}}}_{k+1}, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$, where $f({\alpha}) = {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$ and $k<n$ is maximal such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$. Set $k = -1$ if there is no $k < n$ such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$.
2. Define $f_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ to be the function $g$ where:
1. $\operatorname{dom}g = \operatorname{dom}f$.
2. For each ${\alpha}\in \operatorname{dom}{\nu}$, $g({\alpha})= {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{k} \rangle}}$, where $f({\alpha}) = {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$ and $k<n$ is maximal such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$. Set $k = -1$ if there is no $k < n$ such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$.
The following definitions show how to reflect down a function ${\mu}\in \operatorname{OB}(d)$ using a larger function ${\nu}\in \operatorname{OB}(d)$.
1. Assume ${\mu}, {\nu}\in \operatorname{OB}(d)$, ${\mu}< {\nu}$, and $\operatorname{o}({\mu}) < \min(\operatorname{o}({\nu}), {\mathring{{\nu}}}({\kappa}))$. Define the function ${\tau}= {\mu}\downarrow {\nu}\in \operatorname{OB}(\operatorname{ran}{\mathring{{\nu}}})$ by:
1. $\operatorname{dom}{\tau}= {\ensuremath{\{ {\mathring{{\nu}}}({\alpha}) \mid {\alpha}\in \operatorname{dom}{\mu}{\cap}\operatorname{dom}{\nu}\}}}$.
2. For each ${\xi}\in \operatorname{dom}{\tau}$, ${\tau}({\xi}) = {\mu}({\alpha})$, were ${\xi}= {\mathring{{\nu}}}({\alpha})$.
2. Assume $T \subseteq \operatorname{OB}(d)$ and ${\nu}\in \operatorname{OB}(d)$. If $\operatorname{o}({\nu}) = 0$ then set $T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}= {\ensuremath{\{ {\mu}\downarrow {\nu}\mid {\mu}\in T,\ {\mu}< {\nu},\ \operatorname{o}({\mu}) < \min(\operatorname{o}({\nu}), {\mathring{{\nu}}}({\kappa})) \}}}$.
Assume $p \in {\mathbb{P}}^*({\Vec{E}})$ and ${\nu}\in T^p$. We define $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ as follows. If $\operatorname{o}({\nu}) = 0$ then $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ is the condition $q \in {\mathbb{P}}^*({\accentset{\mid}{{\nu}}})$ defined by setting $f^q = f^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ and $T^q = T^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$. Define $p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ to be the condition $q \in {\mathbb{P}}^*({\Vec{E}})$, where $f^q = f^p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ and $T^q = T^p_{{\ensuremath{\langle {\nu}\rangle}}}$. Finally set $p_{{\ensuremath{\langle {\nu}\rangle}}} = {\ensuremath{\langle p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}, p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow} \rangle}}$.
Of course for the above definition to make sense $T^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \in {\accentset{\mid}{{\nu}}}(\operatorname{ran}{\nu})$ should hold, which we prove in \[MakesSense\]. If $T \subseteq \operatorname{OB}(d)$ hen we let ${{{\vphantom{T}}^{<{\omega}}{T}}}= {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_n \rangle}} \mid n<{\omega},\ {\nu}_0, \dotsc, {\nu}_n \in T,\ {\nu}_0 < \dotsb < {\nu}_n \}}}$.
Assume $p, q \in \Bar{{\mathbb{P}}}$. We say $p$ is an extension of $q$ ($p \leq_{\Bar{{\mathbb{P}}}} q$) if the following hold:
1. $n^p \geq n^q$.
2. ${\ensuremath{\{ \Vec{E}^q_j \mid j < n^q \}}} \subseteq {\ensuremath{\{ \Vec{E}^p_i \mid i < n^p \}}}$ and ${\Vec{E}}^q_{n^q-1} = {\Vec{E}}^p_{n^p-1}$.
3. For each $i < n^q$ there is ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{q_i}$ such that ${\ensuremath{\langle p_{j_0+1}, \dotsc, p_{j_1} \rangle}} \leq^* q_{i{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$, where $i$, $j_0$ and $j_1$, are being set as follows. Let $j_1 < n^p$ satisfy $\Vec{E}^p_{j_1} = \Vec{E}^q_{i}$. If $i = 0$ then set $j_0 = -1$. If $i > 0$ then let $j_0 < j_1$ satisfy $\Vec{E}^p_{j_0} = \Vec{E}^q_{i-1}$.
Finally we give the definition of the forcing notion we are going to work with:
${\mathbb{P}}(\Vec{E}, {\epsilon}) = {\ensuremath{\{ q \leq_{\Bar{{\mathbb{P}}}} p \mid p \in {\mathbb{P}}^*(\Vec{E}, {\epsilon}) \}}}$. The partial orders $\leq_{{\mathbb{P}}(\Vec{E}, {\epsilon})}$ and $\leq^*_{{\mathbb{P}}(\Vec{E}, {\epsilon})} $ are inherited from $\leq_{\Bar{{\mathbb{P}}}}$ and $\leq^*_{\Bar{{\mathbb{P}}}}$.
Since ${\epsilon}$ and the sequence ${\Vec{E}}$ are fixed throughout this work we will write ${\mathbb{P}}$ instead of ${\mathbb{P}}(\Vec{E}, {\epsilon})$ throughout this paper. is needed in order to show the forcing notion defined above makes sense.
If $T \in {\Vec{E}}(d)$ then $X = {\ensuremath{\{ {\nu}\in T \mid T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \in {\accentset{\mid}{{\nu}}}(\operatorname{ran}{\mathring{{\nu}}}) \}}} \in {\Vec{E}}(d)$.
We need to show $X \in {\Vec{E}}(d)$. I.e., we need to show for each ${\xi}< \operatorname{o}(\Vec{E})$, $X \in E_{\xi}(d)$. Fix ${\xi}< \operatorname{o}(\Vec{E})$. We need to show $\operatorname{mc}_{\xi}(d_{\xi}) \in
j_{E_{\xi}}(X)$. Hence it is enough showing $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(T)$ and $ j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$. Since $T \in \Vec{E}(d)$ we have $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(T)$. So we are left with showing $j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$. From the definition of the operation $\downarrow$ we get $$\begin{aligned}
j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow}= \begin{aligned}[t]
{\ensuremath{\{ {\mu}\downarrow \operatorname{mc}_{\xi}(d) \mid {\mu}\in j_{E_{\xi}}(T), \ {\mu}< \operatorname{mc}_{\xi}(d),\
\operatorname{o}({\mu}) < \min({\kappa},{\xi}) \}}}.
\end{aligned}\end{aligned}$$ Consider ${\mu}\in j_{E_{\xi}}(T)$ such that ${\mu}< \operatorname{mc}_{\xi}(d)$. There is ${\mu}^* \in T$ such that ${\mu}= j_{E_{\xi}}({\mu}^*)$. Since for each ${\mu}^* \in T$ such that $\operatorname{o}({\mu}^*) < {\xi}$ we have $j_{E_{\xi}}({\mu}^*) \downarrow \operatorname{mc}_{\xi}(d) = {\mu}^*$, we get $j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} =
{\ensuremath{\{ {\mu}\in T \mid \operatorname{o}({\mu}) < {\xi}\}}} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$.
For each condition $p \in {\mathbb{P}}$ let ${\mathbb{P}}/p = {\ensuremath{\{ q \in {\mathbb{P}}\mid q \leq p \}}}$. It is immediate from the definitions above that for each $0<i<n^p-1$ the forcing notion ${\mathbb{P}}/p$ factors to $P_0 \times P_1$, where $P_0 = {\ensuremath{\{ q^0 \leq p^0 \mid q^0 {\mathop{{}^\frown}}p^1 \in {\mathbb{P}}\}}}$, $P_1 = {\ensuremath{\{ q^1 \leq p^1 \mid p^0 {\mathop{{}^\frown}}q^1 \in {\mathbb{P}}\}}}$, $p^0 = {\ensuremath{\langle p_0, \dotsc, p_{i-1} \rangle}}$, and $p^1 = {\ensuremath{\langle p_i, \dotsc, p_{n^p-1} \rangle}}$. Together with the Prikry property (\[PrikryProperty\]) and the closure of the direct order, one can analyze the cardinal structure in $V^{{\mathbb{P}}}$ straightforwardly.
If $e\supseteq d$ we define ${\pi}^{-1}_{e,d}$ to be the inverse of the operation ${\mathrel{\restriction}}d$, i.e., for each $X \subseteq \operatorname{OB}(d)$ we let ${\pi}_{e,d}^{-1}(X) = {\ensuremath{\{ {\nu}\in \operatorname{OB}(e) \mid {\nu}{\mathrel{\restriction}}d \in X \}}}$. If $f, g \in {\mathbb{P}}^*_f$ are conditions then we write ${\pi}_{f,g}^{-1}$ for ${\pi}_{\operatorname{dom}f, \operatorname{dom}g}^{-1}$. We end this section with the analysis of the cardinal structure above ${\kappa}$ in the generic extension: The cardinals between ${\kappa}$ and ${\lambda}$ are collapsed, and ${\lambda}$ and the cardinals above it are preserved. The properties of cardinals up to ${\kappa}$ will be dealt with in later sections.
${\mathbb{P}}$ satisfies the ${\lambda}^{+}$-cc.
Begin with a family of conditions ${\ensuremath{{\ensuremath{\langle p^{\xi}\mid {\xi}< {\lambda}^{+} \rangle}}}}$. Without loss of generality we can assume $n^{p^{{\xi}_0}} = n^{p^{{\xi}_1}}$ for each ${\xi}_0, {\xi}_1 < {\lambda}^{+}$. Without loss of generality we can assume ${\ensuremath{\langle p^{{\xi}_0}_0, \dotsc, p^{{\xi}_0}_{n^{p^{{\xi}_0}}-2} \rangle}} = {\ensuremath{\langle p^{{\xi}_1}_0, \dotsc, p^{{\xi}_1}_{n^{p^{{\xi}_1}}-2} \rangle}}$ for each ${\xi}_0, {\xi}_1 < {\lambda}^{+}$. Thus, without loss of generality, we can assume $n^{p^{{\xi}}} = 1$ for each ${\xi}< {\lambda}^{+}$. By the ${\Delta}$-system lemma we can assume ${\ensuremath{\{ \operatorname{dom}f^{p^{{\xi}}} \mid {\xi}< {\lambda}^+ \}}}$ is a ${\Delta}$-system with kernel $d$. Since ${\lvertd\rvert} < {\lambda}$ we can assume that for each ${\xi}_0, {\xi}_1 < {\lambda}^+$ and ${\alpha}\in d$, $f^{p^{{\xi}_0}}({\alpha}) = f^{p^{{\xi}_1}}({\alpha})$. Fix ${\xi}_0 < {\xi}_1 < {\lambda}^+$. Set $f = f^{p^{{\xi}_0}}{\cup}f^{p^{{\xi}_1}}$, $T = {\pi}^{-1}_{f,f^{p^{{\xi}_0}}} T^{p^{{\xi}_0}}{\cap}{\pi}^{-1}_{f,f^{p^{{\xi}_1}}}T^{p^{{\xi}_1}}$, and let $p = {\ensuremath{\langle f, T \rangle}}$. Then $p \leq p^{{\xi}_0}, p^{{\xi}_1}$.
${\mathrel\Vdash}{{}\text{``} \text{There are no cardinals between ${\kappa}$ and ${\lambda}$} {}\text{''}}$.
Fix a $V$-regular cardinal ${\tau}\in ({\kappa}, {\lambda})$. Fix a condition $p \in {\mathbb{P}}$ such that $\operatorname{dom}f^{p_{n^p-1}} \supseteq {\tau}\setminus {\kappa}$ will hold. Let $G \subseteq {\mathbb{P}}$ be generic such that $p \in G$. Set $C = {\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T^{p_{n^p-1}}}}^{<{\omega}}{T^{p_{n^p-1}}}} \mid p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \in G \}}}$. Then $\sup {\ensuremath{\{ \sup({\tau}{\cap}{\bigcup}\operatorname{dom}{{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in C \}}} = {\tau}$. Since ${\mathrel\Vdash}{{}\text{``} {\lvertC\rvert} \leq {\kappa}{}\text{''}}$ we get $p {\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\tau}\leq {\kappa}{}\text{''}}$.
Preservation of ${\lambda}$ will be proved by a properness type argument (\[WeAreProper\]) for which we need some preparation.
We say the elementary substructure $N {\prec}H_{\chi}$, where ${\chi}$ is large enough, is ${\kappa}$-internally approachable if there is an increasing continuous sequence of elementary substructures ${\ensuremath{{\ensuremath{\langle N_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ such that $N = {\bigcup}{\ensuremath{\{ N_{\xi}\mid {\xi}<{\kappa}\}}}$, for each ${\xi}< {\kappa}$, $N_{\xi}{\prec}H_{\chi}$, ${\lvertN_{\xi}\rvert} < {\lambda}$, $N_{{\xi}+1} \supseteq \operatorname{{\mathcal{P}}}_{{\kappa}}({\lvertN_{{\xi}}\rvert})$, $N_{\xi}{\cap}{\lambda}\in {\ensuremath{\text{On}}}$, ${\mathbb{P}}^*_f \in N_{\xi}$, $N_{{\xi}+1} \supseteq {{\vphantom{N_{{\xi}+1}}}^{<{\kappa}}{N_{{\xi}+1}}}$, and ${\ensuremath{{\ensuremath{\langle N_{{\xi}'} \mid {\xi}'<{\xi}\rangle}}}} \in N_{{\xi}+1}$.
We say the pair ${\ensuremath{\langle N, f \rangle}}$ is a good pair if $N {\prec}H_{\chi}$ is a ${\kappa}$-internally approachable elementary substructure and there is a sequence ${\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_{\xi}, f_{\xi}\rangle}} \mid {\xi}<{\kappa}\rangle}}}}$ such that ${\ensuremath{{\ensuremath{\langle N_{\xi}\mid {\xi}< {\kappa}\rangle}}}}$ witnesses the ${\kappa}$-internal approachablity of $N$, $f = {\bigcup}{\ensuremath{\{ f_{\xi}\mid {\xi}< {\kappa}\}}}$, ${\ensuremath{{\ensuremath{\langle f_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ is a $\leq^*$-decreasing continuous sequence in ${\mathbb{P}}^*_f$, and for each ${\xi}< {\kappa}$, $f_{{\xi}} \in {\bigcap}{\ensuremath{\{ D \in N_{\xi}\mid D \text{ is a dense open subset of }
{\mathbb{P}}^*_f \}}}$, $f_{\xi}\subseteq N_{{\xi}+1}$, and $f_{\xi}\in N_{{\xi}+1}$. Note that if $N {\prec}H_{\chi}$ is an elementary substructure such that ${\lvertN\rvert} < {\lambda}$, $N \supseteq {{\vphantom{N}}^{<{\kappa}}{N}}$, ${\mathbb{P}}_f^* \in N$, $f \in {\bigcap}{\ensuremath{\{ D \in N \mid D \text{ is a dense open subset of }{\mathbb{P}}_f^* \}}}$, $f \subseteq N$, and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in N{\cap}\operatorname{OB}(\operatorname{dom}f)$, then $f_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} } \in
{\bigcap}{\ensuremath{\{ D \in N \mid D \text{ is a dense open subset of }{\mathbb{P}}_f^* \}}}$.
Hence if ${\ensuremath{\langle N, f \rangle}}$ is a good pair and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in N {\cap}\operatorname{OB}(\operatorname{dom}f)$, then ${\ensuremath{\langle N, f_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}} \rangle}}$ is a good pair also. The following is immediate.
For each set $X$ and $f \in {\mathbb{P}}^*_f$ there is a good pair${\ensuremath{\langle N, f^* \rangle}}$ such that $ f^* \leq^* f$ and $X,f \in N$.
Assume ${\chi}$ is large enough and $N {\prec}H_{\chi}$ is an elementary substructure such that ${\mathbb{P}}\in N$. We say the condition $p \in N$ is $N$-generic if for each dense open subset $D \in N$ of ${\mathbb{P}}$ we have $p {\mathrel\Vdash}{{}\text{``} {\Check{{\mathbb{P}}}} {\cap}{\underset{\widetilde{}}{G}} {\cap}{\Check{N}} \neq \emptyset {}\text{''}}$.
We say the forcing notion ${\mathbb{P}}$ is ${\lambda}$-proper if for an unbounded set of structures $N {\prec}H_{\chi}$ such that ${\mathbb{P}}\in N$ and ${\lvertN\rvert} < {\lambda}$, and for each condition $p \in {\mathbb{P}}{\cap}N$ there is a stronger $N$-generic condition. The followig lemma shows a property stronger than properness.
Let $N{\prec}H_{\chi}$ be a ${\kappa}$-internally approachable structure, ${\mathbb{P}}\in N$, and $p \in N {\cap}{\mathbb{P}}$ a condition. Then there is a direct extension $p^* \leq^* p$ such that for each dense open subset $D \in N$ of ${\mathbb{P}}$ the set ${\ensuremath{\{ s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow} \in D \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}^{p^*},\
s \leq^* p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow} \}}}$ is predense below $p^*$. Moreover, if $s \leq^* p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow}$ and $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow}
\in D$ then there is a weaker condition $q \geq^* p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}q \in D {\cap}N$.
Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $f^* \leq^* f^{p_{n^p-1}}$. Choose a set $T \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T \rangle}} \leq^* p_{n^p-1}$. Let ${\ensuremath{{\ensuremath{\langle D_{\alpha}\mid {\alpha}< {\lvertN\rvert} \rangle}}}}$ be an enumeration of the dense open subsets of ${\mathbb{P}}$ appearing in $N$. Let ${\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_{\iota},f_{\iota}\rangle}} \mid {\iota}< {\kappa}\rangle}}}}$ be a sequence witnessing ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair. For each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}$ construct the set $T^{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$ as follows.
Fix ${{\Vec{{\nu}}}}= {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}$.
Let ${\mathcal{D}}= {\ensuremath{\{ D_{\alpha}\mid {\alpha}\in \operatorname{dom}{\nu}_{k-1} \}}}$. Note ${\mathcal{D}}\in N$ since ${\lvert{\nu}_{k-1}\rvert} < {\kappa}$ and $N \supseteq {{\vphantom{N}}^{<{\kappa}}{N}}$. For each $s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1})$ and $D \in {\mathcal{D}}$ define the sets $D^\in_{{{\Vec{{\nu}}}},s,D}$, $D^{\perp}_{{{\Vec{{\nu}}}},s,D}$, and $D^*_{{{\Vec{{\nu}}}},s,D}$, as follows: Let $g \in D^\in_{{{\Vec{{\nu}}}},s,D}$ if $g \leq f^{p_{n^p-1}}$, $\operatorname{dom}g \supseteq \operatorname{dom}{\nu}_{k-1}$, and $s {\mathop{{}^\frown}}{\ensuremath{\langle g_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},T' \rangle}} \in D$ for some $T' \in {\Vec{E}}(g)$. Let $h \in D^{\perp}_{{{\Vec{{\nu}}}},s,D}$ if $h {\perp}g$ for each $g \in D^\in_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}},s,D}$. Set $D^*_{{{\Vec{{\nu}}}},s,D} =
D^\in_{{{\Vec{{\nu}}}},s,D} {\cup}D^{\perp}_{{{\Vec{{\nu}}}},s,D}$. It is immediate $D^\in_{{{\Vec{{\nu}}}},s,D}$ and $D^{\perp}_{{{\Vec{{\nu}}}},s,D}$ are open subsets of ${\mathbb{P}}^*_f$ below$f^{p_{n^p-1}}$. Thus $D^*_{{{\Vec{{\nu}}}},s,D}$ is a dense open subset of ${\mathbb{P}}^*_f$ below $f^{p_{n^p-1}}$. Set $D^*_{{{\Vec{{\nu}}}}} =
{\bigcap}{\ensuremath{\{ D^*_{{{\Vec{{\nu}}}},s,D} \mid s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1}),\ D \in {\mathcal{D}}\}}}$. Note $D^*_{{{\Vec{{\nu}}}}} \in N$ is a dense open subset of ${\mathbb{P}}^*_f$ below $f^{p_{n^p-1}}$. Let ${\iota}< {\kappa}$ be minimal such that ${{\Vec{{\nu}}}},{\mathcal{D}}, D^*_{{{\Vec{{\nu}}}}} \in N_{{\iota}}$. Then $f_{{\iota}} \in D^*_{{{\Vec{{\nu}}}}}{\cap}N_{{\iota}+1}$. Thus for each $s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1})$ and $D \in {\mathcal{D}}$ either there is a set $T^{{{\Vec{{\nu}}}},s,D} \in {\Vec{E}}(f_{{\iota}}) {\cap}N_{{\iota}+1}$ such that $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},
T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in D$ or $s {\mathop{{}^\frown}}{\ensuremath{\langle h,T'' \rangle}}\notin D$ for each $h \leq^* f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ and $T''\in {\Vec{E}}(h)$. Set $T^{\nu}= {\bigcap}{\ensuremath{\{ T^{{{\Vec{{\nu}}}},s,D} \mid s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1}),\ D \in {\mathcal{D}},\
s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},T^{{\nu},s,D} \rangle}}\in D \}}}$.
Set $T^* = \operatorname*{\triangle}{\ensuremath{\{ {\pi}^{-1}_{f^*,f_{{\iota}({{\Vec{{\nu}}}})}}T^{{\Vec{{\nu}}}}\mid {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{<{\omega}}{T}} \}}}$. Set $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^*$ satisfies the lemma. To show this fix a dense open subset $D \in N$ and a condition $q \leq p^*$.
Let ${\alpha}< {\lvertN\rvert}$ be such that $D = D_{\alpha}$. Without loss of generality assume $q \in D$, $q_{n^q-1} \leq p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}\uparrow}$, ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}^*$, and ${\alpha}\in \operatorname{dom}{\nu}_{k-1}$. Set $s = q {\mathrel{\restriction}}n^q - 1$. Thus $q = s {\mathop{{}^\frown}}{\ensuremath{\langle f^{q_{n^q-1}}, T^{q_{n^q-1}} \rangle}} \in D$. Let ${\iota}<{\kappa}$ be minimal such that ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}, {\ensuremath{\{ D_{\alpha}\mid {\alpha}\in \operatorname{dom}{\nu}_{k-1} \}}}\in N_{\iota}$. Since $f^{q_{n^q-1}} \leq^* f_{{\iota}{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$ we must have $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}, T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in D$, hence $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow} \in D$. It is clear $q$ and $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ are compatible. In addition $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}, T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in N$, thus we are done.
${\mathbb{P}}$ is ${\lambda}$-proper.
${\mathrel\Vdash}{{}\text{``} {\lambda}\text{ is a cardinal} {}\text{''}}$.
Dense open sets and measure one sets
====================================
In order to reduce clutter later on, given a condition $p \in {\mathbb{P}}^*$, we will say a tree is a $p$-tree instead of saying it is an ${\Vec{E}}(f^p)$-tree. If $S$ is a $p$-tree and $r$ is a function with domain $S$ then we define the function $\Vec{r}$ by setting for each ${{\Vec{{\nu}}}}= {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_n \rangle}} \in S$, $\Vec{r}({{\Vec{{\nu}}}}) = r({\nu}_0) {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}r({\nu}_0, \dotsc, {\nu}_{n})$. A function $r$ is said to be a ${\ensuremath{\langle p, S \rangle}}$-function if $S$ is a $p$-tree, for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{<\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow}$, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$.
One of the measures suffices.
-----------------------------
The aim of this subsection is to prove \[GetPreDense\], which together with \[DenseHomogen\] will allow the investigation of the cardinal structure below ${\kappa}$. Note the proof of \[DenseHomogen\] depends on \[GetPreDense\]. The following lemma, which is quite technical, takes its core argument from the proof of the Prikry property for Radin forcing.
Assume $p \in {\mathbb{P}}^*$ is a condition, $S$ is a $p$-tree of height one, and $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that ${\ensuremath{\{ r({\nu}) \mid {\ensuremath{\langle {\nu}\rangle}} \in S \}}}$ is predense below $p^*$.
Define the functions $r_0$ and $r_1$, both with domain $S$, so that $r({\nu}) = r_0({\nu}) {\mathop{{}^\frown}}r_1({\nu})$ will hold for each ${\ensuremath{\langle {\nu}\rangle}} \in S$. Fix ${\xi}< \operatorname{o}(\Vec{E})$ so that $S \in E_{{\xi}}(f^p)$ will hold. We need to collect the information from the sets $T^{r_0({\nu})}$ and $T^{r_1({\nu})}$ into one set $T^*$. The information from the sets $T^{r_1({\nu})}$’s is collected by setting $R = \operatorname*{\triangle}_{{\ensuremath{\langle {\nu}\rangle}} \in S} T^{r_1({\nu})}$. By \[DiagonalIsBig\] $R \in {\Vec{E}}(f^p)$.
The information from the sets $T^{(r_0({\nu}))}$’s is collected into the set $T^*$ as follows. The set $T^*$ will be the union of the three sets $T^0, T^1$, and $T^2$, which we construct now. The construction of $T^0$ is easy. Set $T^{0} = T^{j_{E_{\xi}}(r_0)(\operatorname{mc}_{\xi}(f^p))}$. It is obvious $T^0 \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(f^p)$.
The constructin of $T^1$ is slightly more involved than the construction of $T^0$. Set $T^{1\prime} = {\ensuremath{\{ {\ensuremath{\langle {\nu}\rangle}} \in S \mid T^0_{{\ensuremath{\langle {\nu}\rangle}} \downarrow}= T^{r_0({\nu})} \}}}$. From the construction of $T^0$ it is clear $T^{1\prime} \in E_{{\xi}}(f^p)$. For each ${\mu}\in T^0$ set $X({\mu}) = {\ensuremath{\{ {\ensuremath{\langle {\nu}\rangle}} \in S \mid {\mu}< {\nu},\ {\mu}\downarrow {\nu}\in T^{r_0({\nu})} \}}}$. From the construction of $T^0$ we get $X({\mu}) \in E_{\xi}(f^p)$. Set $T^1 = {\ensuremath{\{ {\nu}\in T^{1\prime} \mid \forall {\mu}\in T^0\ ({\mu}< {\nu}\implies {\nu}\in X({\mu})) \}}}$. We show $T^1 \in E_{\xi}(f^p)$. Thus we need to show $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(T^1)$. Since $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(T^{1\prime})$ it is enough to show that if ${\mu}\in j_{E_{\xi}}(T^0)$ and ${\mu}< \operatorname{mc}_{\xi}(f^p)$ then $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(X)({\mu})$. So fix ${\mu}\in j_{E_{\xi}}(T^0)$ such that ${\mu}< \operatorname{mc}_{\xi}(f^p)$. Then ${\lvert{\mu}\rvert} < {\kappa}$, $\operatorname{dom}{\mu}\subseteq j''_{E_{\xi}}(\operatorname{dom}f^p)$, and $\sup \operatorname{ran}{\mathring{{\mu}}}< {\kappa}$. Necessarily there is ${\mu}^* \in T^0$ such that ${\mu}= j_{E_{\xi}}({\mu}^*)$. Hence $j_{E_{\xi}}(X)({\mu}) = j_{E_{\xi}}(X({\mu}^*)) \ni \operatorname{mc}_{\xi}(f^p)$, by which we are done.
We construct now the set $T^2$. For each ${\mu}\in R$ set $Y({\mu}) = {\ensuremath{\{ {\nu}\downarrow{\mu}\in R_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} \mid {\nu}\in T^1,\
R _{
{\ensuremath{\langle {\mu}\rangle}}\downarrow
{\ensuremath{\langle {\nu}\downarrow{\mu}\rangle}}\downarrow}\in {\accentset{\mid}{{\nu}}}(\operatorname{dom}{\nu})
\}}}$. Now let $T^2 = {\ensuremath{\{ {\mu}\in R \mid \exists {\tau}< \operatorname{o}({\mu})\ Y({\mu}) \in {\accentset{\mid}{{\mu}}}_{\tau}(\operatorname{dom}{\mu}) \}}}$. We show $T^2 \in E_{\zeta}(f^p)$ for each ${\zeta}> {\xi}$. We need to show for each ${\zeta}> {\xi}$, $\operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(T^2)$. Fix ${\zeta}> {\xi}$. We show $\operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(T^2)$. It is enough to show there is ${\tau}< {\zeta}$ such that $j_{E_{\zeta}}(Y)(\operatorname{mc}_{\zeta}(f^p)) \in E_{\tau}(f^p)$. We claim ${\xi}$ can serve as the needed ${\tau}< {\zeta}$. Thus it is enough to show $j_{E_{\zeta}}(Y)(\operatorname{mc}_{\zeta}(f^p)) \in E_{\xi}(f^p)$. Hence we need to show $$\begin{gathered}
{\ensuremath{\{
{\nu}\downarrow \operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(R)_
{{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow}
\mid
{\nu}\in j_{E_{\zeta}}(T^1),\\
\ j_{E_{\zeta}}(R)_{
{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow
{\ensuremath{\langle {\nu}\downarrow\operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow
}
\in {\accentset{\mid}{{\nu}}}(\operatorname{dom}{\nu})
\}}} \in E_{\xi}(f^p).\end{gathered}$$ Note $R^* = j_{E_{\zeta}}(R)_{{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow}
\in {\Vec{E}}{\mathrel{\restriction}}{\zeta}(f^p)$, and if ${\nu}\in j_{E_{\zeta}}(T^1)$ and ${\nu}< \operatorname{mc}_{\zeta}(f^p)$, then there is ${\nu}^* \in T^1$ such that ${\nu}= j_{E_{\zeta}}({\nu}^*)$. Moreover, ${\nu}^* = {\nu}\downarrow \operatorname{mc}_{\zeta}(f^p)$. Hence it is enough to show $$\begin{aligned}
{\ensuremath{\{
{\nu}^* \in R^*
\mid
{\nu}^* \in T^1,
\ R^*_{
{\ensuremath{\langle {\nu}^* \rangle}}\downarrow
}
\in {\accentset{\mid}{{\nu}}}^*(\operatorname{dom}{\nu}^*)
\}}} \in E_{\xi}(f^p).\end{aligned}$$ We are done since the last formula holds.
Having constructed $T^0$, $T^1$, and $T^2$ we set $p^* = {\ensuremath{\langle f^p, T^* {\cap}R \rangle}}$. We will be done by showing ${\ensuremath{\{ r({\nu}) \mid {\nu}\in S \}}}$ is predense below $p^*$. Assume $q \leq p^*$. We need to exhibit ${\nu}\in S$ so that $q {\parallel}r({\nu})$. We work as follows. Fix ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{p^*}$ such that $q \leq^* p^*_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}}}$. There are three cases to handle:
1. Assume there is $i<n$ such that ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$ and ${\mu}_i \in T^1$. The construction of $T^1$ yields ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in T^{r_0({\mu}_i)}$ and the construction of $R$ yields ${\ensuremath{\langle {\mu}_{i+1}, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{r_1({\mu}_i)}$. Hence $r_{0}({\mu}_i)_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}}} {\mathop{{}^\frown}}r_1({\mu}_i)_{{\ensuremath{\langle {\mu}_{i+1}, \dotsc, {\mu}_{n-1} \rangle}}}$ and $q$ are $\leq^*$-compatible, by which this case is done.
2. Assume ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$. By the construction of $T^1$ the set $X = {\ensuremath{\{ {\nu}\in T^1 \mid {\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}}\downarrow {\nu}\in {{{\vphantom{T}}^{<{\omega}}{T}}}^0_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \}}} \in E_{\xi}(f^p)$. Choose ${\nu}^* \in T^{q_{n^{n^q}-1}}$ such that ${\nu}= {\nu}^* {\mathrel{\restriction}}f^p \in X$. Then $q_{{\ensuremath{\langle {\nu}^* \rangle}}} \leq^*
p_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1}, {\nu}\rangle}}}$. Now we can procced as in the first case above.
3. The last case is when there is $i<n$ such that ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$ and ${\mu}_i \notin T^0 {\cup}T^1$. By the construction of $T^2$ there is ${\tau}< \operatorname{o}({\mu}_i)$ such that $Y = Y({\mu}_i) \in {\accentset{\mid}{{\mu}}}_{i{\tau}}(\operatorname{dom}{\mu}_i)$. Hence there are ${\mu}_{i{\tau}}(\operatorname{dom}{\mu}_i)$-many ${\nu}\downarrow {\mu}_i$ such that ${\nu}\in T^1$, ${\nu}\downarrow{\mu}_i \in T^*_{{\ensuremath{\langle {\mu}_i \rangle}}\downarrow}$ and $T^*_{{\ensuremath{\langle {\mu}_i \rangle}}\downarrow {\ensuremath{\langle {\nu}\rangle}}\downarrow {\mu}_i} \in
{\nu}(\operatorname{dom}{\nu})$.
Thus there is ${\sigma}^* \in T^{q_i}$ such that ${\sigma}= {\sigma}^*{\mathrel{\restriction}}\operatorname{dom}f^{p_i} \in Y$, where ${\sigma}= {\nu}\downarrow {\mu}_i$ and ${\nu}\in T^1$. Thus $q_{{\ensuremath{\langle {\sigma}^* \rangle}}} \leq^*
p_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1}, {\nu}, {\mu}_i, \dotsc, {\mu}_{n-1} \rangle}}}$ and we can proceed as in the first case above.
Assume $p \in {\mathbb{P}}$ is a condition, $S$ is a $p_{n^p-1}$-tree of height one, and $r$ is a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that $p^* {\mathrel{\restriction}}n^p-1 = p {\mathrel{\restriction}}n^p-1$ and ${\ensuremath{\{ p {\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}r({\nu}) \mid {\ensuremath{\langle {\nu}\rangle}} \in S \}}}$ is predense open below $p^*$.
Generalize the notions of $p$-tree and ${\ensuremath{\langle p, S \rangle}}$-function to arbitrary condition $p \in {\mathbb{P}}$ as follows. By recursion we say the tree $S$ is a $p$-tree if there is $n<{\omega}$ for which following hold:
1. $\operatorname{Lev}_{< n}(S)$ is a $p{\mathrel{\restriction}}n^p-1$-tree.
2. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$, $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ is a $p_{n^p-1}$-tree.
Let $p \in {\mathbb{P}}$ be an arbitrary condition. By recursion we say the function $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function if there is $n < {\omega}$ such that:
1. $S$ is a $p$-tree.
2. $\operatorname{Lev}_{< n}(S)$ is a $p {\mathrel{\restriction}}n^p-1$-tree.
3. $ r{\mathrel{\restriction}}\operatorname{Lev}_{< n}S$ is a ${\ensuremath{\langle \operatorname{Lev}_{< n}S, p{\mathrel{\restriction}}n^p-1 \rangle}}$-function.
4. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$ the function $s$ with domain $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, define by setting $s({{\Vec{{\mu}}}}) = r({{\Vec{{\nu}}}}{\mathop{{}^\frown}}{{\Vec{{\mu}}}})$, is a ${\ensuremath{\langle p_{n^p-1}, S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \rangle}}$-function.
Assume $p \in {\mathbb{P}}$ is a condition, $S$ is a $p$-tree, and $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in S \}}}$ is predense below $p^*$.
If $S$ is a $p_{n^p-1}$-tree then we are done by \[GetPreDense1\]. Thus assume there is $n<{\omega}$ such that $\operatorname{Lev}_{<n}S$ is a $p{\mathrel{\restriction}}n^p-1$-tree. Construct the strong direct extension $q({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ and the ${\ensuremath{\langle p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}, S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \rangle}}$-function $s_{{{\Vec{{\nu}}}}}$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$ as follows. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$ let $s_{{\Vec{{\nu}}}}$ be the function with domain $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ defined by setting $s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) = r({{\Vec{{\nu}}}}{\mathop{{}^\frown}}{{\Vec{{\mu}}}})$ for each ${{\Vec{{\mu}}}}\in S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$. By \[GetPreDense1\] there is a strong direct extension $q({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ such that ${\ensuremath{\{ s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}(S({{\Vec{{\nu}}}})) \}}}$ is predense below $q({{\Vec{{\nu}}}})$. Let $q \leq^{**} p_{n^p-1}$ be a strong direct extension satisfying $q \leq^{**} q({{\Vec{{\nu}}}})$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$. Hence ${\ensuremath{\{ s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}(S({{\Vec{{\nu}}}})) \}}}$ is predense below $q$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$. Hence ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) {\mathop{{}^\frown}}s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}S({{\Vec{{\nu}}}}) \}}}$ is predense below $\Vec{r}({{\Vec{{\nu}}}}) {\mathop{{}^\frown}}q$. Let $p^* \leq^{**} p$ be a strong direct extension such that $p^*_{n^p-1} = q$ and $p^* {\mathrel{\restriction}}n^p-1 \leq^{**} p{\mathrel{\restriction}}n^p-1$ is a strong direct extension constructed by recursion so as to satisfy ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S \}}}$ is predense below $p^*{\mathrel{\restriction}}n^p-1$. Necessarily ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}}$ is predense below $p^*$
Dense open sets and direct extensions
-------------------------------------
In this subsection we prove \[DenseHomogen\], which is the basic tool to be used in the next section to analyse the properties of the cardinal ${\kappa}$ and the cardinal structure below it.
An essential obstacle in the extender based Radin forcing in comparison to the plain extender forcing is that while in the later forcig notion if we have two direct extensions $q,r\leq^* p$ then $q$ and $r$ are compatible, in the former forcing notion this does not hold. This usually entails some inductions, taking place inside elementary substructures, which construct long increasing seqeunce of conditions from ${\mathbb{P}}^*_f$, which at the end will be combined into one conditions. This method breaks if the elementary substructures in question are not closed enough (which is our case if we want to handle ${\lambda}$ successor of singular). The point of \[DenseHomogenOneBlock\] is to show how we can construction a condition $p$ such that if a direct extension $q \leq^* p$ has some favorable circumstances then the condition $p$ will suffice for this circumstances. This will enable us to work more like in a plain Radin forcing. So as we just pointed out, we aim to prove \[DenseHomogenOneBlock\]. This lemma is proved by recursion with the non-recursive case being \[DenseHomogenOneBlockCase0\]. Since the notation in \[DenseHomogenOneBlock\] is kind of hairy we present the cases $k=1$ and $k = 2$ in \[DenseHomogenOneBlockCase1\] and \[DenseHomogenOneBlockCase2\], respectively.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair and $D \in N$ is a dense open set. Let $p \in {\mathbb{P}}$ be a condition such that $f^{p_{n^p-1}} = f^*$. If there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$ and a direct extension $q \leq^* p_{n^p-1}$ such that $s {\mathop{{}^\frown}}q \in D$ then there is a set $T^* \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T^* \rangle}} \leq^{**} p_{n^p-1}$ and $s {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}} \in D$.
Assume $s \leq p {\mathrel{\restriction}}n^p-1$, $q \leq^* p_{n^p-1}$, and $s {\mathop{{}^\frown}}q \in D$. Set $D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g)\ \
s {\mathop{{}^\frown}}{\ensuremath{\langle g, T \rangle}} \in D \}}}$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. Since $f^* \geq f^{q} \in D^\in$ we get $f^* \notin D^{\perp}$, thus $f^* \in D^\in$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is an extension $s \leq p{\mathrel{\restriction}}n^p-1$ and ${\xi}< \operatorname{o}({\Vec{E}})$ such that ${\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}} \in E_{\xi}(f^*)$, then there is a $p_{n^p-1}$-tree $S$ of height one, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r, such that for each ${\ensuremath{\langle {\nu}\rangle}} \in S$, $s {\mathop{{}^\frown}}r({\nu}) \in D$.
Assume $X = {\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}
\in E_{\xi}(f^*)$. Set $$\begin{gathered}
D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g),\
\text{there is a ${\ensuremath{\langle g,T \rangle}}$-tree $S$ of height one and} \\
\text{a ${\ensuremath{\langle {\ensuremath{\langle g,T \rangle}},S \rangle}}$-function $r$ such that }
\forall {\ensuremath{\langle {\nu}\rangle}} \in S\
s {\mathop{{}^\frown}}r({\nu})\in D \}}}
\end{gathered}$$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. For each ${\nu}\in X$ fix a direct extension $t({\nu}) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and a direct extension $q({\nu}) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}t({\nu}) {\mathop{{}^\frown}}q({\nu}) \in D$. Since ${\ensuremath{\langle N, f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow} \rangle}}$ is a good pair we get by the previous lemma a set $T({\nu}) \in {\Vec{E}}(f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow})$ satisfying ${\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T({\nu}) \rangle}} \leq^{**} p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ and $s {\mathop{{}^\frown}}t({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T({\nu}) \rangle}} \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$. Set $X^* = {\pi}^{-1}_{g,f^*}(X)$. By removing a measure zero set from $X^*$ we can assume for each ${\nu}\in X^*$, $g_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = f^{t({\nu}{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p_{n^p-1}$. Define the funcrion $r$ with domain $X^*$ by setting for each ${\nu}\in X^*$, $r({\nu}) = {\ensuremath{\langle g_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}, T^{(t({\nu}{\mathrel{\restriction}}\operatorname{dom}f^*))} {\cap}T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \rangle}} {\mathop{{}^\frown}}{\ensuremath{\langle g_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, {\pi}^{-1}_{g,f^*}T({\nu}) {\cap}T \rangle}}$. Note $r({\nu}) \leq^{**} {\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}}$, thus $r$ is a ${\ensuremath{\langle g,X^* \rangle}}$-function. Since $D$ is open we get for each ${\nu}\in X^*$, $s {\mathop{{}^\frown}}r({\nu}) \in D$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is $s \leq p{\mathrel{\restriction}}n^p-1$ such that ${\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in {{\vphantom{T}}^{2}{T}}^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree, then there is a $p_{n-1}$-tree $S$ of height two, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r such that for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in S$, $s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1) \in D$.
Assume $X = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in {{\vphantom{T}}^{2}{T}}^{p^*_{n^p-1}} \mid \exists s \leq p{\mathrel{\restriction}}n^p-1\ \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}\ q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree. Set $$\begin{gathered}
D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g),\
\text{there is a ${\ensuremath{\langle g,T \rangle}}$-tree $S$ of height two and} \\
\text{a ${\ensuremath{\langle {\ensuremath{\langle g,T \rangle}},S \rangle}}$-function $r$ such that }\\
\forall {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in S\
s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1)\in D \}}}
\end{gathered}$$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. For each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X$ fix a direct extension $t({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow }$, a direct extension $t_0({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow{\ensuremath{\langle {\nu}_1 \rangle}}\downarrow}$, and a direct extension $q({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}t({\nu}_0, {\nu}_1) {\mathop{{}^\frown}}t_0({\nu}_0, {\nu}_1) {\mathop{{}^\frown}}q({\nu}_0, {\nu}_1) \in D$. For each ${\nu}_0 \in \operatorname{Lev}_0(X)$ we can remove a measure zero set from $\operatorname{Suc}_X({\nu}_0)$ so that we can assume there is a direct extension $t({\nu}_0) \leq^* p^*_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}} \downarrow}$ such that $t({\nu}_0) = t({\nu}_0, {\nu}_1)$ for each ${\nu}_1 \in \operatorname{Suc}_X({\nu}_0)$. By the previous lemma there is a $p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow}$-tree $S({\nu}_0)$ of height one, and a ${\ensuremath{\langle p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow}, S({\nu}_0) \rangle}}$-function $r_{{\nu}_0}$ satisfying for each ${\nu}_1 \in S({\nu}_0)$, $s {\mathop{{}^\frown}}t({\nu}_0) {\mathop{{}^\frown}}r_{{\nu}_0}({\nu}_1) \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$, where $\operatorname{Lev}_0(X) \in E_{\xi}(f^*)$. Set $X^* = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \mid {\nu}_0 \in {\pi}^{-1}_{g,f^*}\operatorname{Lev}_0(X), \
{\nu}_1 \in {\pi}^{-1}_{g,f^*}S({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*) \}}}$. By removing a measure zero set from $\operatorname{Lev}_0(X^*)$ we can assume for each ${\nu}_0 \in X^*$, $g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow} = f^{t({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p^*_{n^p-1}$. For each ${\nu}_0 \in \operatorname{Lev}_0(X^*)$ let $r'_{{\nu}_0}$ be the function with domain $\operatorname{Suc}_{X^*}({\nu}_0)$ defined by shrinking the trees in $r_{{\nu}_0}$ so that both $r'_{{\nu}_0}({\nu}_1) \leq^{**}
r_{{\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*}({\nu}_1 {\mathrel{\restriction}}\operatorname{dom}f^*)$ and $r'_{{\nu}_0}({\nu}_1) \leq^{**} {\ensuremath{\langle g,T \rangle}}_{{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow{\ensuremath{\langle {\nu}_1 \rangle}}}$ will hold for each ${\nu}_1 \in \operatorname{Suc}_{X^*}({\nu}_0)$. Define the function $r$ with domain $X^*$ by setting for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X^*$, $r({\nu}_0) = {\ensuremath{\langle g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow},
T_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow} {\cap}{\pi}^{-1}_{g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow},f^{t({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*)}}T^{(t({\nu}_0 {\mathrel{\restriction}}\operatorname{dom}f^*))}
\rangle}}$ and $r({\nu}_0, {\nu}_1) =
r'_{{\nu}_0}
({\nu}_1)$.
Note $\Vec{r}({\nu}_0, {\nu}_1) \leq^{**} {\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}$, thus $r$ is a ${\ensuremath{\langle g,X^* \rangle}}$-function. Since $D$ is open we get $s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1) \in D$ for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X^*$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
As discussed earlier, the following lemma is the intended one, with the previous ones serving as an introduction to the technique used in the proof.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $k<{\omega}$, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is $s \leq p{\mathrel{\restriction}}n^p-1$ such that ${\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree, then there is a $p_{n^p-1}$-tree $S$ of height $k$, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
Assume $X = {\ensuremath{\{ {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}}{\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree. For each ${\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in X$ fix a direct extension $t({\mu}{\mathop{{}^\frown}}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\downarrow}$ and a direct extension $q({\mu}{\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ such that $s {\mathop{{}^\frown}}t({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}){\mathop{{}^\frown}}q({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \in D$. For each ${\mu}\in \operatorname{Lev}_0(X)$ we can remove a measure zero set from $X_{{\ensuremath{\langle {\mu}\rangle}}}$ so that we will have a direct extension $t({\mu}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\downarrow}$ such that $t({\mu}) = t({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}})$ for each ${{\Vec{{\nu}}}}\in X_{{\ensuremath{\langle {\mu}\rangle}}}$. By recursion there is a $p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow}$-tree $S({\mu})$ of height $k-1$, and a ${\ensuremath{\langle p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow}, S({\mu}) \rangle}}$-function $r_{{\mu}}$ satisfying for each ${{\Vec{{\nu}}}}\in S({\mu})$, $s {\mathop{{}^\frown}}t({\mu}) {\mathop{{}^\frown}}r_{{\mu}}({{\Vec{{\nu}}}}) \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$, where $\operatorname{Lev}_0(X) \in E_{\xi}(f^*)$. Set $X^* = {\ensuremath{\{ {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\mid {\mu}\in {\pi}^{-1}_{g,f^*}\operatorname{Lev}_0(X), \
{{\Vec{{\nu}}}}\in {\pi}^{-1}_{g,f^*}(S({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)) \}}}$. By removing a measure zero set from $\operatorname{Lev}_0(X^*)$ we can assume for each ${\mu}\in X^*$, $g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} = f^{t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p_{n^p-1}$. For each ${\mu}\in \operatorname{Lev}_0(X^*)$ let $r'_{{\nu}_0}$ be the function with domain $X^*_{{\ensuremath{\langle {\mu}\rangle}}}$ defined by shrinking the trees in $r_{{\mu}}$ so that both $\Vec{r}'_{{\mu}}({{\Vec{{\nu}}}}) \leq^{**}
\Vec{r}_{{\mu}{\mathrel{\restriction}}\operatorname{dom}f^*}({{\Vec{{\nu}}}}{\mathrel{\restriction}}X_{{\ensuremath{\langle {\mu}\rangle}} {\mathrel{\restriction}}\operatorname{dom}f^*})$ and $r'_{{\mu}}({{\Vec{{\nu}}}}) \leq^{**} {\ensuremath{\langle g,T \rangle}}_{{\ensuremath{\langle {\mu}\rangle}}\uparrow{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ will hold for each ${{\Vec{{\nu}}}}\in X^*_{{\ensuremath{\langle {\mu}\rangle}}}$. Define the function $r$ with domain $X^*$ by setting for each ${\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in X^*$, $r({\mu}) = {\ensuremath{\langle g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow},
T_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} {\cap}{\pi}^{-1}_
{g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow},f^{t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)}
}T^{(t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*))}
\rangle}}$ and $r({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) =
r'_{{\mu}}
({{\Vec{{\nu}}}})$.
Note $\Vec{r}({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \leq^{**}
{\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}}}$, thus $r$ is a ${\ensuremath{\langle g, X^* \rangle}}$-function. Since $D$ is open we get $s {\mathop{{}^\frown}}\Vec{r}({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \in D$ for each ${\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}} \in X^*$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
Assume ${\ensuremath{\langle f, T \rangle}} \in {\mathbb{P}}$ is a condition, $k < {\omega}$, and $S \subseteq {{\vphantom{T}}^{k}{T}}$ is not an ${\Vec{E}}(f)$-tree. Then there is a set $T^* \in {\Vec{E}}(f)$ such that ${\ensuremath{\langle f, T^* \rangle}} \leq^* {\ensuremath{\langle f, T \rangle}}$ and ${{\vphantom{T}}^{k}{T}}^{*} {\cap}S = \emptyset$.
By removing measure zero sets from the levels of $S$ we can find $n < k$ so that the following will hold:
1. For each $l < n$ and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{l-1} \rangle}} \in S$, $\operatorname{Suc}_S({\nu}_0, \dotsc, {\nu}_l) \in E_{\xi}(f)$ for some ${\xi}< \operatorname{o}({\Vec{E}})$.
2. For each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in S$, $\operatorname{Suc}_S({\nu}_0, \dotsc, {\nu}_{n-1}) \notin E_{\xi}(f)$ for each ${\xi}< \operatorname{o}({\Vec{E}})$.
Shrink $T$ so that ${\ensuremath{\{ {\ensuremath{\langle f,T \rangle}}_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}} \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in \operatorname{Lev}_n(S) \}}}$ is predense below ${\ensuremath{\langle f,T \rangle}}$. We are done by setting $A = \operatorname*{\triangle}{\ensuremath{\{ T \setminus \operatorname{Suc}_{S_n}({\nu}_0, \dotsc, {\nu}_{n-1}) \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in S_n \}}}$ and $T^* = T {\cap}A$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. Assume $s \leq p {\mathrel{\restriction}}n^p-1$. Then one and only one of the following holds:
1. There is a $p_{n^p-1}$-tree S, and a ${\ensuremath{\langle p_{n^p-1}, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
2. There is a set $T^* \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T^* \rangle}} \leq^{**} p_{n^p-1}$ and for each ${{\Vec{{\nu}}}}\in {{\vphantom{T}}^{<{\omega}}{T}}^*$ and $q \leq^* {\ensuremath{\langle f^*, T^* \rangle}}_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, $s {\mathop{{}^\frown}}q \notin D$.
It is about time we get rid of the conditional appearing in the former statements show we have densely many times $p_{n^p-1}$-trees and functions.
Assume $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition. Then there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$ and $k < {\omega}$ so that ${\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q\leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^p)$-tree.
Towards contradiction assume the claim fails. Then for each $s \leq p{\mathrel{\restriction}}n^p - 1$ and $k<{\omega}$ the set $S(s,k) = {\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q\leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is not an ${\Vec{E}}(f^p)$-tree. Thus there is a set $T(s, {\kappa}) \in {\Vec{E}}(f^{p_{n^p-1}})$ satisfying ${{\vphantom{T}}^{k}{T}}(s, k) {\cap}S(s,k) = \emptyset$. Set $T^* = {\bigcap}{\ensuremath{\{ T(s,k) \mid k<{\omega},\ s\leq p{\mathrel{\restriction}}n^p-1 \}}}$. Consider the condition $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^{p_{n^p-1}}, T^* \rangle}}$. By the density of the set $D$ there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$, ${{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^*$, and $q \leq^* p^*_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, such that $s {\mathop{{}^\frown}}q \in D$. Hence ${{\Vec{{\nu}}}}\in S(s, k)$. However ${{\vphantom{T^*}}^{k}{T^*}} {\cap}S(n,k) = \emptyset$, contradiction.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. Then there is a maximal antichain $A$ below $p {\mathrel{\restriction}}n^p-1$ such that for each $s \in A$ there is a $p_{n^p-1}$-tree S and a ${\ensuremath{\langle p_{n^p-1}, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
Assume $D \in N$ is a dense open set and $p \in {\mathbb{P}}$ is a condition. Then there is a direct extension $p^* \leq^* p$, a $p^*$-tree $S$, and a ${\ensuremath{\langle p^*, S \rangle}}$-function $r$ such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$, $\Vec{r}({{\Vec{{\nu}}}}) \in D$.
${\kappa}$ Properties in the Genric Extension
=============================================
The forcing notion ${\mathbb{P}}$ is of Prikry type.
Assume $p \in {\mathbb{P}}$ is a condition and ${\sigma}$ is a formula in the ${\mathbb{P}}$-forcing language.We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vert}{\sigma}$. Set $D = {\ensuremath{\{ q \leq p \mid q {\mathrel\Vert}{\sigma}\}}}$. The set $D$ is dense open, hence by \[DenseHomogen\] there is a direct extension $p^* \leq^* p$, a $p^*$-tree $S$, and a ${\ensuremath{\langle p^*, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \in D$. Set $X_0 = {\ensuremath{\{ {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \mid \Vec{r}({{\Vec{{\nu}}}}) {\mathrel\Vdash}\lnot {\sigma}\}}}$ and $X_1 = {\ensuremath{\{ {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \mid \Vec{r}({{\Vec{{\nu}}}}) {\mathrel\Vdash}{\sigma}\}}}$. Since the sets $X_0$ and $X_1$ are a disjoint partition of $\operatorname{Lev}_{\max}S$, only one of them is a measure one set. Fix $i < 2$ such that $X_i$ is a measure one set. Set $S_i = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_k \rangle}} \mid {\ensuremath{\langle {\nu}_0, \dotsc {\nu}_n \rangle}}\in X_i,\ k \leq n \}}}$. Using \[GetPreDense\] shrink the trees appearing in the condition $p^*$ so that ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S_i \}}}$ is predense below $p^*$. Thus $p^* {\mathrel\Vdash}{\sigma}_i$, where ${\sigma}_0 = {{}\text{``} \lnot{\sigma}{}\text{''}}$ and ${\sigma}_1 = {{}\text{``} {\sigma}{}\text{''}}$.
${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is a cardinal} {}\text{''}}$.
If $\operatorname{o}(\Vec{E}) = 1$ then there are no new bounded subset of ${\kappa}$ in $V^{{\mathbb{P}}}$, hence no cardinal below ${\kappa}$ is collapsed, hence ${\kappa}$ is preserved. If $\operatorname{o}(\Vec{E}) > 1$ then an unbounded number of cardinals below ${\kappa}$ is preserved, hence ${\kappa}$ is preserved.
\[BecomesSingular\] If $\operatorname{o}(\Vec{E}) < {\kappa}$ is regular then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}=\operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$.
It is immediate ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}\leq \operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$. Hence we need to show ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}\not< \operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$. Assume ${\sigma}< {\kappa}$ and $p {\mathrel\Vdash}{{}\text{``} {\sigma}< \operatorname{cf}\operatorname{o}(\Vec{E}) \text{ and }
{\utilde{f}}{\mathrel{:}}{\sigma}\to {\kappa}{}\text{''}}$. We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $p, {\utilde{f}}, {\sigma}\in N$ and $f^* \leq^* f^{p_{n^p-1}}$. Shrink $T^{p_{n^p-1}}$ so as to satisfy for each ${\nu}\in T^{p_{n^p-1}}$, ${\mathring{{\nu}}}({\kappa}) > {\sigma}$.
Factor ${\mathbb{P}}$ as follows. Set $P_0 = {\ensuremath{\{ s \leq p{\mathrel{\restriction}}n^p-1 \mid \exists q \leq p_{n^p-1}\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}\}}}$ and $P_1 = {\ensuremath{\{ q \leq p_{n^p-1} \mid \exists s \leq p {\mathrel{\restriction}}n^p-1\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}\}}}$. For each ${\xi}< {\sigma}$ work as follows. Set $D_{\xi}= {\ensuremath{\{ q \leq p_{ n^{p}-1} \mid \text{There exists a } P_0\text{-name
${\utilde{{\rho}}}$ such that } q {\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}} {}\text{''}} \}}}$. Since $D_{\xi}\in N$ is a dense open subset of ${\mathbb{P}}$ below $p_{n^p-1}$ there is a a direct extension $p^{\xi}= {\ensuremath{\langle f^*,T^{\xi}\rangle}} \leq^* p_{n^p-1}$, a $p^{\xi}$-tree $S^{\xi}$, and a ${\ensuremath{\langle p^{\xi}, S^{\xi}\rangle}}$-function $r_{\xi}$ satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\Vec{r}_{\xi}({{\Vec{{\nu}}}}) \in D_{\xi}$. Thus for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}$ there is a $P_0$-name ${\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}$ so that $\Vec{r}_{\xi}({{\Vec{{\nu}}}}){\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$. Since ${\lvertP_0\rvert}<{\kappa}$ there is ${\zeta}^{{\xi},{{\Vec{{\nu}}}}} < {\kappa}$ such that $p{\mathrel{\restriction}}n^p-1 {\mathrel\Vdash}_{P_0} {{}\text{``} {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}<{\zeta}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$.
Let $m_{\xi}$ be a function witnessing $S^{\xi}$ is a $p_{n^p-1}$-tree, i.e., $m_{\xi}{\mathrel{:}}{\ensuremath{\{ \emptyset \}}} {\cup}\operatorname{Lev}_{<\max}S \to \operatorname{o}({\Vec{E}})$ is a function satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{dom}m_{\xi}$, $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{m_{\xi}({{\Vec{{\nu}}}})}(f^{p_{n^p-1}})$. (We use the convention $\operatorname{Suc}_S({\ensuremath{\langle \rangle}}) = \operatorname{Lev}_0(S)$.) Since $\operatorname{o}(\Vec{E}) < {\kappa}$ we can remove a measure zero set from $S^{\xi}$ (and $\operatorname{dom}m_{\xi}$) and get for each ${{\Vec{{\nu}}}}_0, {{\Vec{{\nu}}}}_1 \in \operatorname{dom}m^{\xi}$, if ${\lvert{{\Vec{{\nu}}}}_0\rvert} = {\lvert{{\Vec{{\nu}}}}_1\rvert}$ then $m_{\xi}({{\Vec{{\nu}}}}_0) = m_{\xi}({{\Vec{{\nu}}}}_1)$. Thus ${\lvert\operatorname{ran}m_{\xi}\rvert} < {\omega}$. Set ${\tau}_{\xi}= \sup \operatorname{ran}m_{\xi}$. Shrink $T^{\xi}$ so that ${\ensuremath{\{ \Vec{r}_{\xi}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}\}}}$ is predense below $p^{\xi}$. Note, if ${\mu}\in \operatorname{Lev}_0 T^{{\xi}}$, ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\operatorname{o}({\mu}) > {\tau}_{\xi}$ and ${{\Vec{{\nu}}}}\not< {\mu}$, then $\Vec{r}({{\Vec{{\nu}}}}) {\perp}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}}$. Hence $p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}} {}\text{''}}$.
Set $T^* = {\bigcap}_{{\xi}< {\sigma}} T^{\xi}$ and $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. To show this set ${\tau}= \sup {\ensuremath{\{ {\tau}_{\xi}\mid {\xi}< {\sigma}\}}}$. Note ${\tau}< \operatorname{o}(\Vec{E})$. Since ${\ensuremath{\{ p^*_{{\ensuremath{\langle {\mu}\rangle}}} \mid {\mu}\in T^{*},\
\operatorname{o}({\mu}) = {\tau}\}}}$ is predense below $p^*$ it is enough to show that $p^*_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$ for each ${\mu}\in T^{*}$ such that $\operatorname{o}({\mu}) = {\tau}$. So fix ${\mu}\in T^{*}$ such that $\operatorname{o}({\mu}) = {\tau}$. Set ${\zeta}= \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {\xi}< {\sigma}, {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}, {{\Vec{{\nu}}}}<{\mu}\}}}$. Note ${\zeta}< {\kappa}$. We get for each ${\xi}< {\sigma}$, $p^*_{{\ensuremath{\langle {\mu}\rangle}}} \leq^* p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}}<{\zeta}<{\kappa}{}\text{''}}$.
\[Gitik\] If $\operatorname{o}(\Vec{E}) \in [{\kappa}, {\lambda})$ and $\operatorname{cf}(\operatorname{o}({\Vec{E}})) \geq {\kappa}$ then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}= {\omega}{}\text{''}}$.
Fix a condition $p \in {\mathbb{P}}$ such that $\operatorname{o}({\Vec{E}})+1 \subseteq \in \operatorname{dom}f^{p_{n^p-1}}$. Partition $T^{p_{n^p-1}}$ into $\operatorname{o}({\Vec{E}})$ disjoint subsets ${\ensuremath{\{ A_{\xi}\mid {\xi}< \operatorname{o}({\Vec{E}}) \}}}$ by setting for each ${\xi}< \operatorname{o}({\Vec{E}})$, $$\begin{aligned}
A_{{\xi}} = {\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid {\xi}\in \operatorname{dom}{\nu},\
\operatorname{o}({\nu}({\kappa})) = \operatorname{otp}((\operatorname{dom}{\nu}) {\cap}{\xi}) \}}}.\end{aligned}$$ Let $G$ be generic. Choose a condition $p \in G$. Let ${\ensuremath{{\ensuremath{\langle {\nu}_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ be the increasing enumeration of the set ${\ensuremath{\{ {\nu}_0, \dotsc, {\nu}_k \mid p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p_{n^p-1{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{{\kappa}} \rangle}}} \in G \}}}$. Set ${\zeta}_0 = 0$. For each $n < {\omega}$ set ${\zeta}_{n+1} = \min {\ensuremath{\{ {\xi}> {\zeta}_n \mid
{\nu}_{\xi}\in A_{sup ((\operatorname{dom}{\nu}_{{\zeta}_n}) {\cap}\operatorname{o}({\Vec{E}})) } \}}}$ . We are done since ${\kappa}= \sup_{n<{\omega}} {\zeta}_n$.
Using the same method as above we get the following claim.
If $\operatorname{o}(\Vec{E}) \in [{\kappa}, {\lambda})$ and $\operatorname{cf}(\operatorname{o}({\Vec{E}})) < {\kappa}$ then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}= \operatorname{cf}(\operatorname{o}({\Vec{E}})) {}\text{''}}$.
\[BecomesRegular\] If $\operatorname{o}(\Vec{E}) = {\lambda}$ then ${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is regular} {}\text{''}}$.
Assume ${\sigma}< {\kappa}$ and $p {\mathrel\Vdash}{{}\text{``} {\utilde{f}}{\mathrel{:}}{\sigma}\to {\kappa}{}\text{''}}$. We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $p, {\utilde{f}}, {\sigma}\in N$ and $f^* \leq^* f^{p_{n^p-1}}$. Shrink $T^{p_{n^p-1}}$ so as to satisfy for each ${\nu}\in \operatorname{Lev}_0(T^{p_{n^p-1}})$, ${\mathring{{\nu}}}({\kappa}) > {\sigma}$.
Factor ${\mathbb{P}}({\Vec{E}})$ as follows. Set $P_0 = {\ensuremath{\{ s \leq p{\mathrel{\restriction}}n^p-1 \mid \exists q \leq p_{n^p-1}\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}({\Vec{E}}) \}}}$ and $P_1 = {\ensuremath{\{ q \leq p_{n^p-1} \mid \exists s \leq p\ {\mathrel{\restriction}}n^p-1 s {\mathop{{}^\frown}}q \in {\mathbb{P}}({\Vec{E}}) \}}}$. For each ${\xi}< {\sigma}$ work as follows. Set $D_{\xi}= {\ensuremath{\{ q \leq p_{ n^{p}-1} \mid \text{There exists a } P_0\text{-name
${\utilde{{\rho}}}$ such that } q {\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}} {}\text{''}} \}}}$. Since $D_{\xi}\in N$ is a dense open subset of ${\mathbb{P}}$ below $p_{n^p-1}$ there is a a direct extension $p^{\xi}= {\ensuremath{\langle f^*,T^{\xi}\rangle}} \leq^* p_{n^p-1}$, a $p^{\xi}$-tree $S^{\xi}$, and a ${\ensuremath{\langle p^{\xi}, S^{\xi}\rangle}}$-function $r_{\xi}$ satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\Vec{r}_{\xi}({{\Vec{{\nu}}}}) \in D_{\xi}$. Thus for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}$ there is a $P_0$-name ${\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}$ so that $\Vec{r}_{\xi}({{\Vec{{\nu}}}}){\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$. Since ${\lvertP_0\rvert}<{\kappa}$ there is ${\zeta}^{{\xi},{{\Vec{{\nu}}}}} < {\kappa}$ such that $p{\mathrel{\restriction}}n^p-1 {\mathrel\Vdash}_{P_0} {{}\text{``} {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}<{\zeta}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$.
Let $m_{\xi}$ be a function witnessing $S^{\xi}$ is a $p_{n^p-1}$-tree, i.e., $m_{\xi}{\mathrel{:}}{\ensuremath{\{ \emptyset \}}} {\cup}\operatorname{Lev}_{<\max}S \to \operatorname{o}({\Vec{E}})$ is a function satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{dom}m_{\xi}$, $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{m_{\xi}({{\Vec{{\nu}}}})}(f^{p_{n^p-1}})$. (We use the convention $\operatorname{Suc}_S({\ensuremath{\langle \rangle}}) = \operatorname{Lev}_0(S)$.) Since ${\lambda}=\operatorname{o}(\Vec{E})$ is regular and ${\lvertS^{\xi}\rvert} < {\lambda}$ we get ${\tau}_{\xi}= \sup \operatorname{ran}m_{\xi}< {\lambda}$. Shrink $T^{\xi}$ so that ${\ensuremath{\{ \Vec{r}_{\xi}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}\}}}$ is predense below $p^{\xi}$. Note, if ${\mu}\in T^{{\xi}}$, ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\operatorname{o}({\mu}) > {\mathring{{\mu}}}({\tau}_{\xi})$ and ${{\Vec{{\nu}}}}\not< {\mu}$, then $\Vec{r}({{\Vec{{\nu}}}}) {\perp}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}}$. Hence $p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}} {}\text{''}}$.
Set $T^* = {\bigcap}_{{\xi}< {\sigma}} T^{\xi}$ and $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. To show this set ${\tau}= \sup {\ensuremath{\{ {\tau}_{\xi}\mid {\xi}< {\sigma}\}}}$. Note ${\tau}< \operatorname{o}(\Vec{E})={\lambda}$. Since ${\ensuremath{\{ p^*_{{\ensuremath{\langle {\mu}\rangle}}} \mid {\mu}\in T^{*},\
\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau}) \}}}$ is predense below $p^*$ it is enough to show that $p^*_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$ for each ${\mu}\in \operatorname{Lev}_0 T^{*}$ such that $\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau})$. So fix ${\mu}\in \operatorname{Lev}_0 T^{*}$ such that $\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau})$. Set ${\zeta}= \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {\xi}< {\sigma}, {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}<{\mu}\}}}$. Note ${\zeta}< {\kappa}$. We get for each ${\xi}< {\sigma}$, $p^*_{{\ensuremath{\langle {\mu}\rangle}}} \leq^* p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}}<{\zeta}<{\kappa}{}\text{''}}$.
An ordinal ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$ if for each $d \in [{\epsilon}]^{<{\lambda}}$, ${\bigcap}_{{\xi}< {\rho}}E_{\xi}(d) = {\bigcap}_{{\xi}< \operatorname{o}(\Vec{E})} E_{\xi}(d)$.
Assume ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$.
1. If $p,q \in {\mathbb{P}}$ are compatible then $j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p) \rangle}}}$ and $j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q) \rangle}}}$ are compatible.
2. For each $p \in {\mathbb{P}}$ there is a direct extension $p^* \leq^* p$ such that $j_{E_{\rho}}(p^*)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p^*) \rangle}}} {\mathrel\Vert}{{}\text{``} \operatorname{mc}_{\rho}(p^*) \in j_{E_{\rho}}({\utilde{A}}) {}\text{''}}$.
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1. Let $r \leq p,q$. By definition of the order there are extensions $p' \leq p$ and $q' \leq q$ such that $r \leq^* p',q'$. By elementarity $j_{E_{\rho}}(p')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p') \rangle}}} \leq j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p) \rangle}}}$ and $j_{E_{\rho}}(q')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q') \rangle}}} \leq j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q) \rangle}}}$. Thus we will be done by showing $j_{E_{\rho}}(p')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p') \rangle}}}$ and $j_{E_{\rho}}(q')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q') \rangle}}}$ are compatible.
So, without loss of generality assume $p$ and $q$ are $\leq^*$ compatible. By elementarity $j_{E_{\rho}}(p)$ and $j_{E_{\rho}}(q)$ are compatible. Note $$\begin{aligned}
& p_{n^p-1} = j_{E_{\rho}}(p_{n^p-1})_{
{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}\downarrow}
\intertext{and}
& q_{n^q-1} = j_{E_{\rho}}(q_{n^q-1})_{
{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}\downarrow}.
\end{aligned}$$ Then $$\begin{aligned}
& j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}} =
p {\mathop{{}^\frown}}{\ensuremath{\langle
j_{E_{\rho}}(p_{n^p-1})_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}\uparrow} \rangle}}
\intertext{and}
& j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}} =
q {\mathop{{}^\frown}}{\ensuremath{\langle
j_{E_{\rho}}(q_{n^q-1})_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}\uparrow} \rangle}}.
\end{aligned}$$ We are done.
2. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $f^* \leq^* f^{p_{n^p-1}}$ and $p, {\utilde{A}} \in N$. Set $T = {\pi}^{-1}_{f^*, f^{p_{n^p-1}}}T^p$. Fix ${\nu}\in T$ and consider the condition $p {\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*,T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}}$. By the Prikry property there is $s \leq^* p {\mathrel{\restriction}}n^p-1$, $r \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and $q \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}r {\mathop{{}^\frown}}q {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. Since the set ${\ensuremath{\{ t \leq p \mid t {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}} \}}}$ is dense open below $p$ and belongs to $N$, we get by \[BasicReflection\] that there is a set $T_1({\nu}) \in \Vec{E}(f^*)$ such that $s {\mathop{{}^\frown}}r {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$.
Thus for each ${\nu}\in T$ there is $s({\nu}) \leq^* p {\mathrel{\restriction}}n^p-1$, $r({\nu}) \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and $T_1({\nu}) \in \Vec{E}(f^*)$ such that $s({\nu}) {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. We can find a set $T_{={\rho}} \in E_{\rho}(f^*)$ and $s \leq^* p {\mathrel{\restriction}}n^p-1$ such that for each ${\nu}\in T_{={\rho}}$, $s({\nu}) = s$. Thus for each ${\nu}\in T_{={\rho}}$, $s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. Then by removing a measure set from $T_{={\rho}}$ we can have either $$\begin{aligned}
& \forall {\nu}\in T_{={\rho}}\
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vdash}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}
\intertext{or}
& \forall {\nu}\in T_{={\rho}}\
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vdash}{{}\text{``} {\nu}\notin {\utilde{A}} {}\text{''}}.\end{aligned}$$ Let $g = f^* {\cup}f^{j(r)(\operatorname{mc}_{\rho}(f^*))}$. Set $T^* = {\pi}^{-1}_{g,f^*}T^{j_{E_{\rho}(r)}(\operatorname{mc}_{\rho}(f^*))} {\cap}{\pi}^{-1}_{g,f^*}\operatorname*{\triangle}_{{\nu}\in T_{={\rho}}} T_1({\nu})$. Setting $p^* = s {\mathop{{}^\frown}}{\ensuremath{\langle g,T^* \rangle}}$ we get for each ${\nu}\in T^*_{={\rho}}$, $p^*_{{\ensuremath{\langle {\nu}\rangle}}} \leq^*
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}}$. Thus by removing a measure zero set from $T^*_{={\rho}}$ we get either $$\begin{aligned}
& \forall {\nu}\in T^*_{={\rho}}\
p^*_{{\ensuremath{\langle {\nu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}
\intertext{or}
& \forall {\nu}\in T^*_{={\rho}}\
p^*_{{\ensuremath{\langle {\nu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\nu}\notin {\utilde{A}} {}\text{''}}.\end{aligned}$$ Going to the ultrapower we get $j_{E_{\rho}}(p^*)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(g) \rangle}}} {\mathrel\Vert}{{}\text{``} \operatorname{mc}_{\rho}(g) \in j_{E_{\rho}}({\utilde{A}}) {}\text{''}}$.
Assume ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$. Then $ {\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is measurable} {}\text{''}}$.
If $G \subseteq {\mathbb{P}}$ is generic then it is a simple matter to check that $$\begin{aligned}
U = {\ensuremath{\{ {\utilde{A}}[G] \mid p \in G,\
j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}} {\mathrel\Vdash}{{}\text{``} \operatorname{mc}_{{\rho}}(p_{n^p-1})\in j_{E_{\rho}}({\utilde{A}}) {}\text{''}} \}}}\end{aligned}$$ is the witnessing ultrafilter.
\[BecomesMeasurable\] If $\operatorname{o}(\Vec{E}) = {\lambda}^{++}$ then ${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is measurable} {}\text{''}}$.
By the previous corollary it is enough to exhibit ${\rho}< {\lambda}^{++}$ which is a repeat point of $\Vec{E}$. Fix $d \in [{\epsilon}]^{<{\lambda}}$ and consider the sequence ${\ensuremath{{\ensuremath{\langle {\bigcap}_{{\xi}' < {\xi}} E_{{\xi}'}(d) \mid {\xi}< {\lambda}^{++} \rangle}}}}$. This is a $\subseteq$-decreasing sequence of filters on $\operatorname{OB}(d)$. Since there are ${\lambda}^+$ filters on $\operatorname{OB}(d)$ there is ${\rho}_d < {\lambda}^{++}$ such that ${\bigcap}_{{\xi}< {\rho}_d} E_{{\xi}}(d) =
{\bigcap}_{{\xi}< {\lambda}^{++}} E_{{\xi}}(d)$. Set ${\rho}= \sup {\ensuremath{\{ {\rho}_d \mid d \in [{\epsilon}]^{<{\lambda}} \}}}$. Then ${\rho}$ is a repeat point of ${\Vec{E}}$.
[10]{}
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[^1]: Most of this work was done somewhat after [@Merimovich2011c] was completed. Lacking an application, which admittedly was lacking also in [@Merimovich2011c], it was mainly distributed among interested parties. Gitik, observing the utility of this forcing to some HOD constructions (see [@GitikMerimovichPreprint]), has urged us to bring the work into publishable state.
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abstract: 'Angle-resolved transport measurements revealed that planar defects dominate flux pinning in the investigated Co-doped BaFe$_2$As$_2$ thin film. For any given field and temperature, the critical current depends only on the angle between the crystallographic $c$-axis and the applied magnetic field but not on the angle between the current and the field. The critical current is therefore limited only by the in-plane component of the Lorentz force but independent of the out-of-plane component, which is entirely balanced by the pinning force exerted by the planar defects. This one-dimensional pinning behavior shows similarities and differences to intrinsic pinning in layered superconductors.'
author:
- 'V. Mishev'
- 'W. Seeböck'
- 'M. Eisterer'
- 'K. Iida'
- 'F. Kurth'
- 'J. Hänisch'
- 'E. Reich'
- 'B. Holzapfel'
title: 'One-dimensional pinning behavior in Co-doped BaFe$_2$As$_2$ thin films'
---
Flux pinning is of primary importance for power applications of superconducting materials. The different families of iron-based superconductors offer a rich variety of vortex pinning properties because of their differences in anisotropy and defect structure. Among them, the BaFe$_2$As$_2$ compound (Ba-122) is most promising for applications. Its anisotropy is small and flux pinning can be extremely strong.[@Tar12; @Fan12] In addition, the limitation of the currents by grain boundaries seems less severe than in the other Fe-based compounds. Furthermore, coated conductors[@Iid11] and untextured bulk materials [@Wei12] with high critical currents were successfully produced. The material allows for an extraordinarily high density of pinning centers without significant degradation of the transition temperature [@Tar12; @Fan12] resulting in high critical current densities, [$j_\text{c}$]{}, with a weak dependence on the applied magnetic field. The introduction of $c$-axis correlated pinning centers [@Tar12; @Lee10] leads to a maximum in the angular dependence of the critical current when the field is oriented parallel to the defects and to a reduction in current anisotropy. Here, we report on the opposite limit of a rather clean film with strong pinning only by a few planar defects oriented parallel to the FeAs planes. We performed angle-resolved measurements of the critical current density in a two-axes goniometer to carefully explore the resulting vortex-defect interactions. Similarities to the layered cuprate superconductors[@Tac89] were found, although Ba-122 is not expected to form pancake vortices.
Cobalt-doped Ba-122 films were prepared by pulsed laser deposition using a KrF excimer laser. A 15nm thick iron buffer layer was deposited onto a (001) MgO substrate before the 80nm thick $10\%$ Co-doped Ba-122 layer was grown.[@Iid10b] The transition temperature of this film is 24.7K (onset of the resistive transition at zero field).
Direct transport measurements were carried out in a helium gas flow cryostat equipped with a $6$T split coil magnet. The critical current on a small bridge was evaluated by means of the four-probe method using a criterion of 1$\mathrm{\mu}$V/cm. A two-axes goniometer enabled rotations of the sample about two axes which are orthogonal to each other ($z$- and $\bf{e_1}$-axis, Fig. \[fig:setup\]). This set-up allows anisotropy measurements of [$j_\text{c}$]{} under variable Lorentz force (VLF) in addition to the well established measurements under maximum Lorentz force (MLF) resulting in several additional measurement modes such as in-plane scans. In this case, the applied field $H_a$ is always orthogonal to the $c$-axis of the sample and only the orientation of the field with respect to the transport current and, hence, the Lorentz force changes.
The origin of the angular coordinates $\theta$ and $\varphi$ is chosen such that $H_a$ aligns with the $c$-axis at $\theta = \varphi = 0^\circ$. The measurements included MLF scans ($\theta = 0^\circ$, rotation about $\varphi$), VLF scans ($\varphi =$ const., rotation about $\theta$) and in-plane scans ($\theta = 90^\circ$, rotation about $\varphi$). The temperature range comprised seven different temperatures ranging between $20$K and $5$K.
With the azimuthal and polar angle, $\varphi$ and $\theta$, one can easily express a parametrization of the unit vectors corresponding to the three crystallographic directions. Note that the $xyz$-coordinate system as well as the magnetic field vector $\bm{B}$ are fixed within the cryostat. According to Fig. \[fig:setup\], the unit vectors are then given by $$\begin{aligned}
{\bf e_1} =
\left(\!\!\begin{array}{c}
-\sin\varphi \\
\phantom{-}\cos\varphi \\
0 \end{array}\right),\;
{\bf e_2} =
\left(\begin{array}{c}
\cos\varphi \sin\theta \\
\sin\varphi \sin\theta \\
\cos\theta \end{array}\right),\;
{\bf e_3} =
\left(\begin{array}{c}
\cos\varphi \cos\theta \\
\sin\varphi \cos\theta \\
-\sin\theta \end{array}\right).
\label{eq:unit_vectors}\end{aligned}$$
![\[fig:setup\] Geometrical representation of the experimental set-up and coordinate system used for the parametrization of the sample orientation with respect to the magnetic field $\bm{B}$.](config.eps){width="7cm"}
Since the current through the sample is supposed to flow along the $\bf{e_2}$-axis, the resulting Lorentz force (here and in the following the Lorentz force has the physical dimension of a force per length) becomes $${\bm F_\text{L}} = I_c B
\left(\begin{array}{c}
0 \\
\cos\theta \\
- \sin\varphi \sin\theta \end{array}\right) ,
\label{eq:LF}$$ where $I_\text{c}$ and $B$ are the absolute values of the critical current and the magnetic field, respectively. Hence, the projection of ${\bm F_\text{L}}$ onto the $ab$-plane of the sample is $${\bm F^\|_\text{L}} = -I_\text{c} B \cos\alpha
\left(\!\!\begin{array}{c}
-\sin\varphi \\
\phantom{-} \cos\varphi \\
0
\end{array}\right) ,
\label{eq:inplaneLFalpha}$$ where $\cos\alpha\!{\mathrel{\mathop:}=}\!\cos\varphi \cos\theta$ and $\alpha$ denotes the angle between the magnetic field and the crystallographic $c$-axis of the sample. The absolute value of the in-plane component of the Lorentz force is, therefore, only a function of $\alpha$: $$F^{\|}_\text{L} = I_\text{c} B \cos\alpha \,.
\label{eq:absFalpha}$$
![Angle-resolved measurements of $j_c(\varphi,\theta=0)$ at $T=10$K for various fields. The $j_c$-curves show a pronounced $ab$-peak due to planar defects.[]{data-label="fig:jc_phi_10K"}](mlf.eps)
![\[fig:TEM\_image\] TEM bright field image of another Ba-122 film prepared under the same conditions as the film investigated in this letter. Planar defects parallel to the $ab$-planes are clearly visible.](tem.eps)
Fig. \[fig:jc\_phi\_10K\] shows the [$j_\text{c}$]{}anisotropy in standard MLF configuration at 10K for various fields. No $c$-axis peak is observed, and the critical current exhibits a sharp peak when the field is oriented parallel to the $ab$-planes. Since the $c$-axis coherence length in Ba-122 ($\xi_c\approx 1.2$nm[@Yam09]) is similar to the distance between the FeAs layers (smaller than the lattice parameter $c \approx 1.3$nm[@Rot08]), the condensation energy is not expected to be suppressed significantly between the layers and intrinsic pinning can be ruled out as a reason for the $ab$-peak. Instead, planar defects seem responsible, which were revealed by Bright field transmission electron microscopy (TEM) in an equally prepared, although slightly thicker film (Fig. \[fig:TEM\_image\]).
To investigate the critical current behavior under variable Lorentz force, the angle $\theta$ was varied at several fixed angles $\varphi$. Fig. \[fig:VLF\_1T\_15K\] shows typical results at 15K and 1T. Note the fact that all curves converge at $\theta=90^\circ$, independent of the angle $\varphi$.
![Variable Lorentz force (VLF) scans $j_\text{c}(\varphi,\theta)$ at $T=15$K and $B=1$T.[]{data-label="fig:VLF_1T_15K"}](vlf.eps)
A special case of the VLF configuration occurs for $\theta=90^\circ$ and variable $\varphi$, where the applied magnetic field and the current remain in the same plane, i.e. parallel to the crystallographic $ab$-planes. The resulting Lorentz force varies between zero and its maximum and is always perpendicular to the $ab$-planes and, in the present case, also to the planar defects. Typical so called in-plane scans are shown in Fig. \[fig:in\_plane\_15K\] for various fields at 15K. At low fields, hardly any variation in [$j_\text{c}$]{} is detectable – apart from the scatter of the data –, which means that the out-of-plane component of the Lorentz force does not have any influence, while depinning is triggered by the in-plane component resulting from the self-field of the current. Since [$j_\text{c}$]{} does not change with angle, the self-field and the in-plane component of the Lorentz force remain constant, as required for this explanation. Applying higher fields results in a significant variation with $\varphi$ which cannot be considered as a consequence of the changing Lorentz force because the minimima and maxima of [$j_\text{c}$]{} do not coincide with those of the Lorentz force at 0 and $90^\circ$, respectively. We ascribe the variation in [$j_\text{c}$]{} to geometric imperfections. At $6$T [$j_\text{c}$]{} changes by about $\pm 15\%$ which corresponds to a tilt angle between the field and the $ab$-plane of about $1.5^\circ$ only, since the $ab$-peak is very sharp under these conditions (c.f. Fig. \[fig:jc\_phi\_10K\]). A small tilt of the $ab$-planes with respect to the sample holder together with a slight misalignment of the rotation axis would explain the observed behavior.
![In-plane scans of $j_\text{c}(\varphi,\theta=90^\circ)$ at $T=15$K for various fields. The out-of-plane component of the Lorentz force becomes zero at $0^\circ$ and $180^\circ$.[]{data-label="fig:in_plane_15K"}](inplane.eps)
![\[fig:master\_15\_10\] $j_\text{c}(\alpha)$ at $T=10$K, $B = 1$T (a) and at $T=15$K, $B = 3$T (b). Any combination of $\varphi$ and $\theta$ can be scaled to one single angle $\alpha$, which denotes the angle between the crystallographic $c$-axis and the magnetic field.](master.eps)
The independence of the critical current on the out-of-plane component of the Lorentz force is not only observed in the in-plane scans ($\theta=90^\circ$), but all data of $j_\text{c}(\varphi,\theta\!=\!const)$ and $j_\text{c}(\theta,\varphi\!=\!const)$ collapse at each field and temperature when plotted as a function of $\alpha$. Since $\alpha$ denotes the angle between the crystallographic $c$-axis and the magnetic field, this procedure corresponds to mapping all these data to the MLF configuration. Reducing the out-of-plane component of the Lorentz force – which is not a unique function of $\alpha$ – does not change [$j_\text{c}$]{}. Thus, the critical current is limited by the in-plane component of ${\bm F_\text{L}}$ only. Fig. \[fig:master\_15\_10\] shows exemplary results for $T=15$K, $B=3$T and $T=10$K, $B=1$T, respectively. It can be concluded that flux pinning is extremely anisotropic in these films and that pinning on the planar defects by far exceeds any random or $c$-axis correlated contributions. However, the planar defects can balance only one component of the Lorentz force, while the other one limits [$j_\text{c}$]{} leading to a one-dimensional pinning behavior. (In reality, the projection space of the Lorentz force onto the $ab$-planes is two-dimensional, but the rotational symmetry reduces its formal dimensionality.)
The situation resembles the intrinsic pinning of pancake vortices in layered superconductors, which would also result in the observed scaling with $\alpha$. However, our data do not follow the angular dependence predicted by the Tachiki-Takahashi model.[@Tac89] Instead, at low fields (e.g. up to about 1T at 10K), where the vortex-vortex interaction is small, data obtained at different magnetic fields coalesce when plotted directly as a function of the out-of-plane component of the magnetic field (i.e. $B\cos\alpha$), as expected from the current limiting mechanism. This difference to the Tachiki-Takahashi behavior is not unexpected since the ratio between coherence length and interlayer spacing is incompatible with the formation of pancake vortices, on which that model is based. At higher fields data obtained at different fields cannot be scaled by the Tachiki-Takahashi approach nor by $B\cos\alpha$, which is a consequence of the vortex-vortex interactions. This highlights the advantage of the $\alpha$ scaling (data obtained at VLF) for the determination of pinning dimensionality compared to scaling approaches for changing magnetic field because it is free of any assumptions of the vortex properties (changing vortex density).
In conclusion, a careful analysis of the [$j_\text{c}$]{} anisotropy revealed a one-dimensional pinning behavior in Co-doped Ba-122 films containing correlated planar defects. Only the component of the Lorentz force parallel to the defects is responsible for the critical current density in the films, and for any given field and temperature the angular dependence of [$j_\text{c}$]{} is a unique function of the angle between the magnetic field and the crystallographic $c$-axis only.
We acknowledge Bernhard Berger for his contributions to the measurements and Prof. H. W. Weber for fruitful discussions. This work was supported by the Austrian Science Fund (FWF): P22837-N20 and by the European-Japanese collaborative project SUPER-IRON (No. 283204).
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